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Sun, Stars and Planets J C Pickering 2014 Part 1: The Sun: its structure and energy generation Lecture 1. Basic properties of the Sun – an introduction

1.1 Setting the scene… 1.2 Definition of a star 1.3 Properties of the Sun 1.4 How are these properties quantified? 1.1 Setting the scene Earth- Sun distance = 1 AU (Astronomical Unit), or 8 minutes for light to reach us Distances: (in light years) Nearest star Proxima Centauri 4.2 Sirius 8.6 Orion Nebula 1500 Galactic Centre 28,000 Galactic Diameter 100,000 Andromeda galaxy 2,300,000 (biggest galaxy in our Local Group of 30 galaxies 6 million lys across) Virgo Cluster 50,000,000 (2000 galaxies, nearest other galactic cluster, and centre of our Local Supercluster – diameter 100,000,000) Most Distant Observed Object ~13,000,000,000 Number of stars and galaxies… Galaxies contain anything from between 100,000 up to 3,000,000,000,000 (3 thousand billion) stars! Our Galaxy contains 100 billion stars. Number of galaxies estimated in observable Universe – 200 - 300 billion… So, it is vital we understand the physics of stars! How do we go about describing the internal structure of stars and their evolution? Evolution of stars is very slow; for a typical ordinary star like our Sun a small change in the Sun’s properties would make the Earth uninhabitable for us, but we have been on Earth for hundreds of thousands of years. Geologists believe the Earth’s crust has been solid for several thousand million years, and that the Sun’s luminosity cannot have changed significantly during this time span. So how can we make progress in our understanding of stars and their evolution… ? What are the fundamental things we know about the Sun? and what is the Sun? 1.2 Definition of a star A star is a self-gravitating mass of gas that radiates energy. 1.3 Properties of a star we need to know about

Before we can talk about evolution and internal structure we need to know the basic properties of a star: it has a Mass , which means there is a Pressure , and a gas pressure means that we want to know about the temperature of the gas. The star radiates heat which can be quantified by Luminosity . Mass pressure temperature heat luminosity The Sun – is our closest star and has global properties: mass M

= 1.99 x 1030 kg (one solar Mass) radius R

= 6.96 x 108 m (solar radius)

lum inosity L

= 3.83 x 1026

W (solar luminosity)

Sun-Earth mean distance = 1 Astronomical Unit (AU) = 1.50 x 1011

m

2

1.4 How are these quantities determined? 1.4.1 Distance: 1.4.2 Radius of the Sun: Measure the angular size of the Sun, and knowing the Earth-Sun distance ⇒ radius

Using the small angle approximation, in radians, θ/2 = r/d where θ = angular size of the Sun, r = Sun’s radius, d = Earth-Sun distance 1.4.3 Mass of the Sun: Knowing the orbital motions of planets and their distance ⇒ G M

to high precision. [G = the gravitational constant, Ms mass of star, mp mass of planet, d star-planet separation, ω angular velocity, P orbital period] 1.4.4 Luminosity and flux: A star’s Luminosity, L, is the power (W) emitted by the entire surface of a star. At a distance d from the star, its luminosity is found by measuring flux density F (Wm-2):

At distance d from a star, its luminosity is spread over a sphere of area 4 π d2. We assume that the star radiates uniformly in all directions. So at any point on the sphere the flux density is given by: inverse-square law: F = L / (4ππππd2 ) ( d = 1 A.U. )

For the Sun we can measure flux (energy incident from the Sun per unit time per unit area) at the Sun-Earth distance, known as the solar “constant” = 1368 W m-2 and from this we can calculate solar luminosity.

Kepler’s 3rd law P2 µ D

3

P is planet’s orbital period D is the planet’s orbital semimajor axis) - gives relative scale of the solar system but not the absolute scale; then use accurate measurement of distance to a planet to set the absolute scale e.g. radar-ranging to Venus. (Earlier methods of distance measurement: transit observations; Greek astronomy)

F

d

3

Also, we can compare the solar flux at the Earth with the solar flux at other planets: for example to estimate the surface temperature of other planets. F

1

1.4.5 Surface temperature and radiation: Reminder: Black-body Radiation. A Black body: - is in thermodynamic equilibrium: photons and matter have the same temperature. Reach this through collisions/interactions, mean free path (mfp) is short, ie are opaque. - The light within the source is more likely to interact with the material of the source than to

escape, it will only escape after considerable interaction with material within the source. So a common feature of BB sources is that they are opaque.

At a given T, any body in thermodynamic equilibrium will show a black-body (≈ continuum) spectrum. For a star, the BB spectrum is a good approximation up to a point only – where the stellar atmosphere’s mfp increases, a line spectrum is formed (and this can tell us about the star’s composition, etc). Many astronomical sources produce continuous spectra with reasonably good approximation to BB form. Planck’s law : Wien’s law: Stefan’s law:

F2

d1

d2

TO DO: calculate the solar luminosity using the solar constant 1368 Wm-2.

TO DO: what would be the “solar constant” as measured on: (i) Mercury (ii) Pluto ?

4

Radiation from the surface of stars can be approximated to that of black-body radiation:

For black body, flux F per unit area of emitter F = σσσσ T4 (Stefan’s law) where σ = Stefan-Boltzmann constant

The Effective temperature is defined such that a black body of temperature Teff with the same radius as the star would radiate the same total amount of energy. Define effective temperature of surface of star by: L = 4 π π π π R2 σ (Teff )4 where R is the radius of the star.

1.4.6 Chemical composition of the Sun: We believe this to be similar to typical composition in the universe. The composition of a sample of material at any depth within the Sun can be defined using 3 simple parameters: hydrogen mass fraction X = mass of hydrogen in sample/mass of sample, and similarly helium mass fraction Y, and metallicity Z = mass of other elements in sample/mass of sample,

Hydrogen ~73% by mass X Helium ~25% Y Heavier elements ~2% Z (O, C, N, Ne, Fe, … in order of abundance) How do we know this? : Observational data: solar spectrum, meteorites 1.4.7 Age of the Sun: Only known indirectly: radioactive dating of rocks; computed from evolutionary models of the Sun. ~ 4.6 x 109 years LECTURE KEY POINTS:

• Definition of a star • Sun’s global properties: mass, radius, luminosity • Astronomical unit AU, solar units • Luminosity, inverse square law, flux density • Definition of effective temperature; Stefan’s law • Sun’s chemical composition and age

Learning outcomes include : • be able to give definitions of a star, astronomical unit, effective temperature • be able to explain what is meant by luminosity, flux, and a black body source. • have an awareness of how the Sun’s properties are measured, more details to follow later in the course as indicated during the lecture. • know Stefan’s law, and the definition of effective temperature such that you are able to write down the equations and use them To Do: - Problem Sheet 1 - Question 1 - Check you understand what a black body is - calculate the solar luminosity using the measured solar constant - Quick calculations: solar flux at Mercury and Pluto - Calculate the effective temperature of the Sun

TO DO: For the Sun, using values of L and R from 1.3 calculate the effective temperature of the Sun L = 4 π R2 σ (Teff)4 Teff = K

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Sun Stars and Planets J C Pickering 2014 Part 1 continued: The Sun: its structure and energy generation

Lecture 2: Modelling the Stellar interior Aims: derive equations governing the physical properties of a star and examine their validity

2.1 Hydrostatic equilibrium 2.2 Equation of mass continuity 2.3 How good an approximation is hydrostatic equilibrium? 2.4 How long would it take a star to collapse if pressure forces were negligible? 2.5 Mean density of the Sun 2.6 Very simple estimate of the mean pressure of the Sun 2.7 Estimate of minimum central pressure of the Sun 2.1 Hydrostatic equilibrium A star is held together by the force of gravitation, the attraction exerted on each part of the star by all other parts. If this force was the only important one then the star would rapidly shrink – but this attractive gravitational force is resisted by the pressure of the stellar material in the same way that the kinetic energy of the molecules, or equivalently the pressure of the Earth’s atmosphere, prevents the atmosphere from collapsing to the surface of the Earth. These two forces, gravitational attraction and thermal pressure, play key roles in determining stellar structure.

2.1.1. Assumptions

(a) assume stars are spherical and symmetrical about their centres

(b) The stellar properties change so slowly with time - neglect the rate of change with time of these properties.

With these assumptions a star’s structure is governed by a set of equations in which all the physical quantities depend only on the distance from the centre of the star. 2.1.2 Balance between pressure and gravitational forces: Volume of the elemental cylinder of matter is δA δr,

Mass of the infinitesimal element m = ρ(r) δA δr where ρ (r) = density of stellar material at r. 1) The gravitational force acting on the elemental mass at r in a spherical body whose density depends only on distance from the centre is the same as if all the mass interior to the element were concentrated at the centre of the body and all the remainder of the body were neglected.

Force pulling cylinder towards centre =

=

Where G is the Newtonian gravitational constant, M(r) is the mass contained within the sphere of radius r.

Consider the forces acting on a small cylinder of matter (infinitesimal element) in a spherical star, gravitational and pressure forces. The lower face of cylindrical element of matter is distance r from the centre, and upper face is distance r + δr, both faces have equal area δA.

(2) Pressure force: the forces due to the pressures balance exactly except for the forcesstellar material on the inner and outer face of the elemental cylinder at r and r +

Net pressure force = p(r+δ

For equilibrium, the forces must balance:

Equation of hydrostatic equilibrium (also known as hydrostatic support):

Applicable in stellar interiors, planetary atmospheres, etc. 2.2 Equation of mass continuity

The quantities M, ρ and r are not independent as themass M(r) contained within a sphere of radius r is determinedby the density of the material at all points within radius r.We can obtain a relation between

Stellar mass M(r) = 4 π 0∫r

Consider the mass of a spherical shell between radii r and

Mass of thin shell = 4 π r2 ρ δr

The mass of the shell can be written =

Hence So we have two differential equations but we have three functions M(r), p(r), Clearly we need a further relation between them if we are to determine all the parameters. We also need an equation for temperature (see later lecture). 2.3 How good an approximation is hydrostatic equilibrium?In deriving equation [2.1], the equation of hydrostatic equilibriumacting on any element of material in a star are exactly in balance. If example if the star was expanding or contracting, then there would be a net force on an elmatter equal to the product of its mass and acceleration (F=ma). If forces are not in balance, a fluid element would accelerate:

ρ a= - ρ g +

= λ

the forces due to the pressures balance exactly except for the forceson the inner and outer face of the elemental cylinder at r and r +

δr) δA - p(r) δA

the forces must balance:

Equation of hydrostatic equilibrium (also known as hydrostatic support):

Applicable in stellar interiors, planetary atmospheres, etc.

2.2 Equation of mass continuity

and r are not independent as the contained within a sphere of radius r is determined

by the density of the material at all points within radius r. We can obtain a relation between M, ρ and r

r 2 ρ(r) δr

spherical shell between radii r and r+δr.

provided δr is small.

of the shell can be written = M (r + δr) – M(r) =

δr for a thin shell

[2.2]

ave two differential equations [2.1] and [2.2] describing the structure of the stellar(r), p(r), ρ (r).

need a further relation between them if we are to determine all the parameters. We also need an equation for temperature (see later lecture).

2.3 How good an approximation is hydrostatic equilibrium? equation of hydrostatic equilibrium, we assumed that the forces

acting on any element of material in a star are exactly in balance. If these are example if the star was expanding or contracting, then there would be a net force on an elmatter equal to the product of its mass and acceleration (F=ma).

fluid element would accelerate:

λ ρ g say

2

the forces due to the pressures balance exactly except for the forces of the on the inner and outer face of the elemental cylinder at r and r + δr

giving:

[2.1]

for a thin shell

ibing the structure of the stellar interior –

need a further relation between them if we are to determine all the parameters. We

, we assumed that the forces these are not in balance, for

example if the star was expanding or contracting, then there would be a net force on an element of

From rest, in time t the fluid element

So, for the sun, if we consider a 10%

time 10

sec.

Solar eclipse records show that there has been

therefore λ < 10-14

Since geological evidence concerning the ages of radioactive elements in the Earth’s crust and of fossils suggests that the properties of the Sun have not changed significantly for at least 10(3 x 1016 seconds), the present λ

So, hydrostatic equilibrium is a very good approximation. 2.4 How long would a star take to collapse if pressure forces were negligible (dynamical timescale) ?

and

2

~

2.5 Mean density of the Sun Simply the mass per unit volume: What is this density similar to? 2.6 Very simp le estimate of the Sun’s mean

Using Make approximation for mean value, mass = Mdensity [2.5], approximating

To do: calculate the dynamical timescale for the Sun.

To do: calculate a value for the mean pressure

From rest, in time t the fluid element travels distance

thus

10% decrease in radius, s = R /10 = 7 x 10

7

show that there has been no appreciable radius change in 10

Since geological evidence concerning the ages of radioactive elements in the Earth’s crust and of fossils suggests that the properties of the Sun have not changed significantly for at least 10

λ < 10-27.

So, hydrostatic equilibrium is a very good approximation.

2.4 How long would a star take to collapse if pressure forces were negligible (dynamical

Putting s=R, the stellar radius, gives

2.4

per unit volume:

le estimate of the Sun’s mean pressure

value, mass = M/2 and take radius R/2 and

2 2

2 2.6

calculate the dynamical timescale for the Sun.

calculate a value for the mean pressure for the Sun using [2.6]

3

thus

[2.3]

7 m, and g=300 ms

-1,

change in 103 yrs ~ 10

10 sec.

Since geological evidence concerning the ages of radioactive elements in the Earth’s crust and of fossils suggests that the properties of the Sun have not changed significantly for at least 109 years

2.4 How long would a star take to collapse if pressure forces were negligible (dynamical

, the stellar radius, gives

[2.5]

/2 and take mean

4

2.7 Estimate of the Sun’s minimum central pressure Use the equations of hydrostatic equilibrium [2.1] and mass continuity [2.2] to estimate a minimum value of the central P of a star of known mass and radius: ⁄

⁄ ≡ ()

()=

− () ()

4 ()=

−()

4

()

()=

−()

4

−

= − = 4

Where c= centre of star, s surface, MS total mass, PC central pressure, PS surface pressure.

But in a star r < rS and so

>

So we can estimate the minimum value of the integral:

4

> 4

=

8

Rearranging, and taking PS << PC gives:

> +

8 >

8 [2.7]

For the sun MS = M

and rS = R

so PC

> 4.5 x 1013 N m-2 . This result requires no knowledge of the chemical composition or physical state of the solar material. For stars other than the Sun, we can write:

>

8

⨀

[2.8]

LECTURE KEY POINTS: • Derivation of equation of hydrostatic equilibrium – including assumptions: spherical symmetry, static - some assessment of their validity • Equation for dM/dr mass continuity • Dynamical timescale t dyn • Mean density of Sun • Estimates of mean pressure of Sun, and minimum central pressure

To Do: calculate the dynamical timescale for the Sun, and the mean pressure. Problem sheet 1 Question 2

Learning outcomes include : • be able to derive the equations of hydrostatic equilibrium and mass continuity, giving assumptions and reasoning, and a comment on their validity. • be able to derive, calculate and explain what is meant by the dynamical timescale, • be able to estimate the mean density and minimum central pressure of the Sun

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Sun Stars Planets 3. J C Pickering 2014

Part I continued : The Sun: its structure and energy generation

Lecture 3. Modelling the Sun’s interior: pressure, density, temperature 3.1 Local thermodynamic equilibrium, and stellar plasma 3.2 Pressure in solar interior, and the equation of state 3.3 Mean molecular weight 3.4 Estimate of central temperature of Sun 3.5 The virial theorem 3.6 Contraction of a star 3.1 Local thermodynamic equilibrium, and stellar plasma The interiors of stars are in thermodynamic equilibrium to high degree of accuracy: well-defined temperature T(r), pressure p(r), etc 3.1.1 What do we mean by thermodynamic equilibrium?

• If a physical system is isolated and left alone for a sufficiently long time it settles down into a state of thermodynamic equilibrium

• In thermodynamic equilibrium the overall properties of a physical system do not vary from point to point and do not change with time • But individual particles of the system are in motion and do have changing properties, eg electrons being removed from or attached to atoms • But there is a statistically steady state in which any process and its inverse occur equally frequently • Because the properties are not varying from point to point when in thermodynamic equilibrium, all parts have the same temperature • Key point – in thermodynamic equilibrium all physical properties of the system (eg P, internal energy, specific heat) can be calculated in terms of its density, temperature and chemical composition alone.

3.1.2 Stellar plasma: an ideal gas and local thermodynamic equilibrium, radiation The stellar material is an ionised gas or plasma, and is assumed to be an ideal gas. Because of the high T in stars all but the most tightly bound electrons are separated from the atoms. This allows a greater compression of stellar material without deviation from the perfect gas law because a nuclear dimension is 10-15 m compared with a typical atomic dimension of 10-10 m, and so the plasma has a higher density than you might intuitively expect is possible for an ideal gas. Plasma also differs from an ordinary gas because the forces between electrons and ions have a much longer range than the forces between neutral atoms.

In contrast to many space plasmas that are low density, most of the plasma in a star is very dense (as mentioned above), i.e., it has a short mean free path and there are many collisions and interactions between the electrons, ions and photons. As the timescale between collisions is much shorter than the changes in pressure, temperature and composition, one might expect the plasma to be in thermodynamic equilibrium. As temperature and pressure are functions of the stellar radius, we do not have global, but local thermodynamic equilibrium (LTE) . This is a huge simplification and one of the reasons that we can easily define the temperature in these regions: In LTE the electrons, photons and ions all have the same temperature and this is equivalent to the

kinetic temperature of the gas (in temperatures are not the same!).

• In the deep interior of a star it is

• Near the surface of a star there start to bethere is a net outflow of energy. However if there are enough collisions then we have a kinetic temperature and a state of local thermodynamic equilibrium.

Unlike in typical laboratory conditions, in a stellar interior the radequilibrium with matter – and in thermodynamic equilibrium the intensity of radiation is given by the Planck function:

3.2 Pressure in the solar interior 3.2.1 Radiation Pressure Just as the particles in a gas exert a pressure which can be calculated from the kinetic theory of gases, by considering collisions of particles with an imaginary surface in the gas, the Planck distribution exert a pressure known as

(where a is the radiation constant= 7.55 x 3.2.2 Gas pressure Pressure contribution from ions and electrons. Particles in the gas exert P calculattheory of gases, and we will see that:

( where R is the gas constant = 8.26 x

3.2.3 Pressure in the Solar interior

But for a star like the Sun it has been

Sun.

ure of the gas (in contrast, in many space plasmas, the electron and ion the same!).

of a star it is very close to being in thermodynamic equilibrium

of a star there start to be departures from thermodynamic equilibrium and there is a net outflow of energy. However if there are enough collisions then we have a kinetic temperature and a state of local thermodynamic equilibrium.

Unlike in typical laboratory conditions, in a stellar interior the radiation is in thermodynamic and in thermodynamic equilibrium the intensity of radiation is given by the

3.2 Pressure in the solar interior

Just as the particles in a gas exert a pressure which can be calculated from the kinetic theory of gases, by considering collisions of particles with an imaginary surface in the gas, the Planck distribution exert a pressure known as radiation pressure.

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is the radiation constant= 7.55 x 10-16 J m-3 K-4)

Pressure contribution from ions and electrons. Particles in the gas exert P calculatgases, and we will see that:

is the gas constant = 8.26 x 103 J K-1 kg-1 and µ is the mean molecular weight

Pressure in the Solar interior In general P P P

for a star like the Sun it has been found that and so we can neglect

2

many space plasmas, the electron and ion

very close to being in thermodynamic equilibrium

ermodynamic equilibrium and there is a net outflow of energy. However if there are enough collisions then we have a kinetic

iation is in thermodynamic and in thermodynamic equilibrium the intensity of radiation is given by the

Just as the particles in a gas exert a pressure which can be calculated from the kinetic theory of gases, by considering collisions of particles with an imaginary surface in the gas, the photons in a

Pressure contribution from ions and electrons. Particles in the gas exert P calculated from kinetic

mean molecular weight )

and so we can neglect Prad for the

3

3.2.4 Equation of state Rather than using the usual expression for the ideal gas equation, P = n kB T, it is customary to express the number density n in terms of the mass density and introduce the mean molecular weight, µ. The mean molecular weight is the mean mass of the gas particles in units of the hydrogen mass, mH. The mean particle mass is thus µmH.

The mass density of the gas, ρ, can be expressed in terms of its number density n and mean particle mass:

= ⟹ =

The ideal gas equation then becomes:

= =

= ℛ

Where specific gas constant for hydrogen ℛ =

≃ 8300 3.3 Mean Molecular Weight µ Now we need an expression for the mean molecular weight as a function of ρ, T and chemical composition. Calculating µ for completely general values of ρ and T is very complicated because to find n (number density of each species), the fractional ionization of all the elements has to be computed. But this can be simplified as follows: Recall that in order to find µ we will need to calculate the total number density n for all the species in the gas. Let us define X, Y and Z as the mass fractions for hydrogen, helium and other elements heavier than He (astronomers conventionally call these the ‘metals’). X is thus the ratio of the mass of the gas that is in the form of hydrogen to the total mass of the gas, or ()/ . The total mass of hydrogen is the number of hydrogen atoms (or, for the fully ionised gas, the total number of protons, N(H+)) multiplied by the mass of a hydrogen atom, mH. We can neglect the electron masses here, and we also do not have to distinguish between proton, hydrogen or atomic mass. We thus have:

= H mass fraction = (H)

≃

(H)

=

(H)

= He mass fraction = (He)

≃

(He)

=

(He)

4

= mass fraction for metals = (metal)

Here ρ is the mass density and n denotes number density. As we have absorbed all the metals in Z, we have X + Y + Z = 1. We now need to consider how each element contributes to the mass and the number density.

4

When ionised, each hydrogen atom contributes an electron and a proton to the number density; it contributes mH to the mass density. Each He atom contributes two electrons and its nucleus (an α particle, made up of 2 neutrons and 2 protons), i.e., 3 particles, to the number density and adds 4 mH to the mass density. Finally, consider the species heavier than helium. For a species with electron number l, a fully ionised ion will contribute l electrons along with one nucleus. The mass due to this species is approximately 2 l mH (this assumes that the number of neutrons and protons is equal in a nucleus, which is not a bad assumption). The total number density is thus given by:

= 2 + 3 + ( + 1)metal

We can now express the number densities for the different species in terms of their mass densities and mass fractions, for example according to = /, so

= 2

+ 3

4 + + 1

2

≃ 2 + 3

4 +

1

2

X + Y + Z = 1. We can now simplify, and say Z≈0, and so:

=

= 1

2 + 34

= 4

5 + 3

TO Do: Problem sheet 1. Add your more accurate calculation of µ, where Z≠0, to your notes here. For the Sun, the surface values for X and Y are about 74% and 24%, so µ ≈ 0.6. In the solar core, the value of Y increases to more than 60% and µ ≈ 0.8.

…We can now use the equation of state as giving an expression for P…

3.4 Estimate of Sun’s central temperature Take

= ℛ

and simplify for rough estimate, using ≈ 1, ≈ , =

recall from lecture 2: ≃ ⨀

⨀ [2.6] and =

⨀

43 ⨀

3 [2.5]

so

⨀

⨀ =

ℛ

⨀

⨀

⟹ = ⨀

ℛ ⨀

Note: how does TC vary with stellar mass? Since =

, =

, if we assume that the

mean ρ ≈ constant, if M increases by factor of 2, radius increases by 21/3 ≈1.26, ie mass increases more rapidly than R. So M/R is larger for stars of larger mass, and hence Tc is larger for greater M.

Our calculation has made no assumptions about the method of energy release, but is not a bad estimate. So, with modest densities and high temperature we can see that the stellar material is gaseous and ionised. With the high central pressure (as found in lecture 2) it might initially be surprising that the stellar material is an ionised gas, or plasma, but as it is ionised there can be much greater compression without deviation from the perfect gas law as the nuclear dimension is 10-15 m compared to typical atomic dimension of 10-10 m.

To do: calculate T c Compare this with the value from a more accurate solar model = 1.56 x 107 K.

3.5 The virial theorem A further consequence of the equations: can be found by integrating the equations over the entire volume of the star: …

Start with: Integrate LHS by parts: and since

we have First term : negligible. If the star were surrounded by vacuum, its surface pressure p(R) would be zero and first term = 0. In fact, surface pressure will not be zero, but will be many orders of magnitude smaller than the central the other two terms that it may be neglected.

Third term: gravitational potential energy forming the star from its component parts dispersed to infinity.So we have:

for a perfect gas, the internal energy per unit mass

Where Utot is the total thermal energy, heat at constant V, and since a fully ionized gas is a monoatomic gas, Putting it all together gives the virial theorem

Thus the negative gravitational energy is just equal to twice the thermal energy.

equations: and

can be found by integrating the equations over the entire volume of the star: …

Multiply by 4πr3 and integrate over the interior of the star:

: negligible. If the star were surrounded by vacuum, its surface pressure be zero and first term = 0. In fact, surface pressure will not be zero, but will be many

orders of magnitude smaller than the central P, or mean P, so this term is so small compared to terms that it may be neglected.

nal potential energy Ω of the star (negative). It is the energy released in forming the star from its component parts dispersed to infinity.

3 Ω 0

r a perfect gas, the internal energy per unit mass u is given by

3 3 1

3 1tot 2 tot

is the total thermal energy, γ = ratio of the specific heat at constant P to the specific heat at constant V, and since a fully ionized gas is a monoatomic gas, γ = 5/3.

the virial theorem:

Thus the negative gravitational energy is just equal to twice the thermal energy.

5

can be found by integrating the equations over the entire volume of the star: …

and integrate over the interior of the star:

: negligible. If the star were surrounded by vacuum, its surface pressure be zero and first term = 0. In fact, surface pressure will not be zero, but will be many

so small compared to

of the star (negative). It is the energy released in

ratio of the specific heat at constant P to the specific 5/3.

Thus the negative gravitational energy is just equal to twice the thermal energy.

3.6 Contraction of a star (aside) An important result of using The total energy of a star can be defined by E = U + energy. If the star radiates into space, E must decrease. Combining 2U + Ω = 0 and E = U – 2U = so E = – U = Thus, the total energy of the star is negative and is equal to half the gravitational energy, or minus the thermal energy. So, a decrease in E leads to a decrease in star composed of a perfect gas, with no hidden enerradiates energy. (Ω is always –ve for a self gravitating body and becomes more negative as a body of fixed mass contracts.) This may seem rather paradoxical, such a star finds it difficult cool down; any attempt to lose energy causes the star to contract and to release energy at a rate that not only supplies the energy loss from the surface, but also heats up the material of the star. We had assumed a fully ionised gas, with

Contrast the behaviour of a star with that of a cooling ember surroundings, radiates into the surroundings. As it radiates, it cools, eventually coming to thermodynamic equilibrium with the surroundings classical self gravitating mass of gas, a star, that radiates into cooler surroundings (universe) becomes hotter and hotter, and increases the disparity between its temperature and the surrounding temperature. However the laws of thermodynamics are not broken requires heat to flow from hot (star) to cold (universe), and for T to become uniform when the system has reached thermodynamic equilibrium, and of course the star has thermodynamic equilibrium with the universe! Lecture 3: Key points • Gas pressure and radiation pressure• Mean molecular weight • Estimate of central temperature of• Solar interior is gaseous and • Virial theorem, and contraction of a star After this lecture you should:

- Be able to explain what is meant by local thermodynamic equilibrium- Understand that the Planck function describes the radiation of a star- Understand that the Sun is composed of a plasma, and understand why this may be

approximated as an ideal gas- Understand what is meant by- Be able to estimate the mean molecular weight- Know what the equation of state is, and be able to use it where necessary- Be able to estimate the central temperature of the Sun- Virial theorem: be able to derive this and use it as demonstrated in notes and lecture

To Do: Calculate TC for the Sun Problem sheet 1, Q3 and Q4

Contraction of a star (aside)

is as follows:

The total energy of a star can be defined by E = U + Ω provided there are no other sources of energy. If the star radiates into space, E must decrease.

= 0 and E = U + Ω, gives:

2U = -U = Ω / 2

U = Ω / 2

Thus, the total energy of the star is negative and is equal to half the gravitational energy, or minus the thermal energy. So, a decrease in E leads to a decrease in Ω, but to an increase in U. Thus a star composed of a perfect gas, with no hidden energy supplies, contracts and heats up as it

ve for a self gravitating body and becomes more negative as a This may seem rather paradoxical, such a star finds it difficult

se energy causes the star to contract and to release energy at a rate that not only supplies the energy loss from the surface, but also heats up the material of the star. We had assumed a fully ionised gas, with γγγγ = 5/3, but this is also true as long as

Contrast the behaviour of a star with that of a cooling ember – the ember, hotter than its surroundings, radiates into the surroundings. As it radiates, it cools, eventually coming to thermodynamic equilibrium with the surroundings (same temperature). This is not so for a classical self gravitating mass of gas, a star, that radiates into cooler surroundings (universe) becomes hotter and hotter, and increases the disparity between its temperature and the

However the laws of thermodynamics are not broken requires heat to flow from hot (star) to cold (universe), and for T to become uniform when the system has reached thermodynamic equilibrium, and of course the star has not yetthermodynamic equilibrium with the universe!

Gas pressure and radiation pressure , thermodynamic equilibrium

central temperature of the Sun Solar interior is gaseous and essentially fully ionized Virial theorem, and contraction of a star

xplain what is meant by local thermodynamic equilibrium;

Understand that the Planck function describes the radiation of a star; un is composed of a plasma, and understand why this may be

approximated as an ideal gas; Understand what is meant by gas and radiation pressure, know the expressions for these

stimate the mean molecular weight; of state is, and be able to use it where necessary

e able to estimate the central temperature of the Sun; be able to derive this and use it as demonstrated in notes and lecture

6

Ω provided there are no other sources of

Thus, the total energy of the star is negative and is equal to half the gravitational energy, or minus but to an increase in U. Thus a

gy supplies, contracts and heats up as it ve for a self gravitating body and becomes more negative as a This may seem rather paradoxical, such a star finds it difficult to

se energy causes the star to contract and to release energy at a rate that not only supplies the energy loss from the surface, but also heats up the material of the star.

as long as γγγγ > 4/3.

the ember, hotter than its surroundings, radiates into the surroundings. As it radiates, it cools, eventually coming to

(same temperature). This is not so for a classical self gravitating mass of gas, a star, that radiates into cooler surroundings (universe) – it becomes hotter and hotter, and increases the disparity between its temperature and the

However the laws of thermodynamics are not broken – the zeroth law requires heat to flow from hot (star) to cold (universe), and for T to become uniform when the

not yet reached

un is composed of a plasma, and understand why this may be

know the expressions for these;

of state is, and be able to use it where necessary;

be able to derive this and use it as demonstrated in notes and lecture;

1

Sun Stars Planets 4. J C Pickering 2014

Part 1 : The Sun: its structure and energy generation, continued.

Lecture 4. Energy production in the Sun 4.1 Estimate of the minimum mean temperature of the Sun 4.2 How does the Sun shine? 4.3 The source of the Sun’s energy 4.4 Nuclear fusion 4.5 Stability 4.6 Solar neutrino problem 4.1 Estimate of the minimum mean temperature of the Sun using the virial theorem. In lecture 3 we derived the virial theorem: 2 Utot + Ω = 0 and used this to explain the phenomenon of a star contracting and heating up as it radiates heat. Here we will use the virial theorem to estimate the minimum mean temperature of the Sun. The virial theorem can be written as:

3 + Ω = 0 (1)

But we know that

Ω = −

But r < rS (surface radius) everywhere, so 1/r > 1/rS is true. Therefore:

−Ω >

=

2 (2)

If we have an ideal gas = ℜ

, so

3 = 3

ℜ =

3 ℜ

(3)

Where ₸ is the mean temperature, defined by

=

0

Combining (1) (2) and (3):

3 ℜ

>

2

> 6 ℜ

We assume Sun is composed of ionised H, so µ = ½, take MS = M

, rS = R

, and so

₸

> 2 x 106 K

2

4.2 How does the Sun shine ?

4.2.1 Could the Sun’s energy source be gravitational energy?

-----No, since… Total available gravitational energy = G M

2 / R

This could sustain the Sun’s present luminosity for time = gravitational energy/luminosity = (G M

2 / R ) / L

~ 107 yrs

4.2.2 Is the Sun were shining by cooling down? If it were then by the virial theorem,

the thermal time is half of the time that the Sun could shine through gravitational energy (above), since thermal energy ~ ½ gravitational energy. Neither can explain how Sun has shone for > 109 yrs This means that, if the Sun’s radiation were supplied by either contraction (gravitational) or cooling it would have changed substantially in the last 10 million years, but geologists estimate it can hardly have altered in a time a hundred times longer > 109 yrs The thermal timescale (also known as the Kelvin-Helmholtz timescale) is:

So, there has to be another source for the Sun’s radiant energy. 4.3 The source of the Sun’s energy So, if the sun radiates energy at rate = 4 x 1026 W, using E = m c2 the Sun is losing mass at a rate of 4 x 109 kg s-1. From geologists, we know L

has not changed significantly over last few thousand

million years, so over this time mass loss ≈ 2 x 10 - 4 M

.

So if neither gravitational energy nor thermal energy can account for the sun’s energy, then the source of the Sun’s energy must be released by the conversion of matter from one form to another, and be capable of releasing at least 2 x 10-4 of the rest mass energy of the Sun. This rules out chemical reactions such as combustion of coal, gas and oil which only release up to 5 x 10-10 of the rest mass energy. The only way known in which quantities of energy as large as this can be released, by the change of matter from one form to another, is through nuclear reactions .

- either fission reactions of heavy nuclei, like in an atomic bomb and nuclear reactions which can release 5 x 10 -4 of the rest mass energy

- or fusion reactions of light nuclei that occur in the hydrogen bomb and can release almost 1 % of the rest mass energy.

It is believed that nuclear fusion reactions are the source of the energy radiated during most phases of a star’s evolution.

3

4.4 Nuclear fusion Hydrogen Helium 4 1H ⇒ 4He

Mass: 4 mH 3.97 mH

E = m c2 ∴ energy production = (0.03 mH) c2 i.e. fraction 0.007 of mass converted to energy This could power the Sun for tnuc ~ 0.007 M

c2 / L

~ 1011 yr

Note:

tdyn << tK-H << tnuc

Main energy source in Sun: Proton-proton reaction

Branch I 85%

Branch II 15%

Branch III <1% [In main-sequence stars more massive than the Sun, the main energy source is also H burning but by the CNO cycle . Requires higher temperature.]

Nuclear fusion is a process in which nuclei of relatively low mass are fused together to form nuclei of somewhat greater mass. Fusion is brought about by a sequence of nuclear reactions in which colliding nuclei combine and fragment, to produce new nuclei together with other particles. No energy is actually created in these reactions: energy is liberated from the reactants and is redistributed amongst the products so that some of it replaces the energy radiated by the Sun, thus maintaining the high core T and sustaining the nuclear reaction rates.

The most important series of nuclear reactions occurring in main sequence stars are those converting hydrogen into helium – this is termed hydrogen burning . There are several routes by which H can be converted to He, and they must obey conservation laws: conservation of charge and energy, and conservation of baryon number. There are two basic reaction chains for conversion of H to He: the Proton-Proton (PP) chain and the carbon-nitrogen cycle in which nuclei of C and N are used as catalysts in the conversion of H to He. The particular temperature in the Sun’s core dictates that the PP I chain dominates . But in stars with progressively higher T than the Sun, the other 2 chains PP II and PP III become important. For stars with even higher central temperatures the CNO cycle dominates.

4

The net effect for each of the PP chains is production of a helium nucleus from four protons.

PP I overall 4 11H → 42He + 2e+ + 2νe + 2γ e+ = positron νe = neutrino γ = gamma ray

+

Step [3] 32He + 32He → 42He + 11H + 11H

[The Appendix of this handout gives a schematic diagram of the PP I, II, and III reactions and the CNO cycle.] Stellar central temperature and relative rate of energy release for PP and CNO reactions.

The overall PPI 4 11H → 42He + 2e+ + 2νe + 2γ produces neutrinos. The theories of thermonuclear reactions can be tested by calculating the rate of neutrinos expected at Earth, and checking these with experiments. The reaction rates are very sensitive to temperature, so this will allow an estimate of the core temperature.

To Do: Estimate the rate of solar neutrinos passing through the top of your head: [2005 Q1]

The overall nuclear PPI reaction chain is: 4 11H → 42He + 2e+ + 2νe + 2γ Αssuming the neutrino (νe) energy to be negligible, calculate approximately the energy liberated from just one completion of this PPI chain. Assuming this PPI chain were the only kind of nuclear reaction taking place, derive an expression for and estimate the number of solar neutrinos passing through the top of your head per second. [Take helium nucleus mass = 3.97 mH, proton mass mH =1.673 x 10-27 kg. 1 AU = 1.5 x 1011m. Solar luminosity=3.83x1026 W]

Step [1] 11H + 11H → 21H + e+ + νe

Step [2] 21H + 11H → 32He + γ

X 2

∝

∝

Reaction rates = rate of energy generation per unit volume of gas. n = number density of reactant nuclei T = temperature

Stellar properties are very sensitive to T. L very dependent on T and hence M.

5

We know that stars on the main sequence have a stable “hydrogen burning” lifetime that is over 3 billion years. The thermonuclear energy source is however not infinite, and gravity is relentless. Every star must confront the paradox by which it gets hotter and hotter while losing more and more energy to the cold dark universe. This confrontation and its ultimate resolution is the central plot to the life stories of the stars. 4.5 Stability The hydrogen bomb works on the same principle as the thermonuclear reactions which power the sun. Why does the Sun “burn” stably for billions of years, why does it not just explode like a bomb? It has a built in “safety valve”, and “burns” by “controlled thermonuclear fusion”. Make notes in lecture. 4.6 Solar neutrino problem “Missing” neutrinos due to neutrino oscillations… TO DO: Additional reading: - read and make notes! John N. Bahcall, How the Sun Shines available at: http://www.nobelprize.org/nobel_prizes/themes/physics/fusion/

Although EM radiation has a hard time escaping from the Sun’s core, the neutrinos produced there have no such difficulty. Observing these solar neutrinos gives a direct test of our solar models and solar nuclear reactions. Although huge numbers are produced (100 billion pass through your thumbnail every second) their low interaction rate makes them hard to detect when they reach and pass through the Earth. The higher the energy of the neutrino the greater the probability of detection (generally probability of detection µ (neutrino energy)2 ). The neutrino emitted by β decay of 85B (PP III chain) is much more energetic than the other neutrinos emitted by H burning, and the number of reactions going through PP III at the estimated solar central T is µ T18. Because of this high dependence on T, if these neutrinos could be detected, the central T of the Sun could be determined very accurately and it would be compared with the central T predicted by theoretical calculations.

How to measure the neutrinos? Davis and Bahcall, experimentalist and astronomer, proposed an experiment :– The only way to detect neutrinos, is to cause a stable atomic nucleus to be transformed into an unstable nucleus by capturing a neutrino and then to observe the β decay of the unstable nucleus. Neutrinos must be captured in a place where no other particles could produce the same unstable nucleus – eg in a deep mine shielded from cosmic rays. The process studied was: 37 Cl + νe → 37

18Ar + e -

37 Ar → 37Cl + e + + νe

Chlorine was in perchloroethylene (cleaning fluid) C2Cl4 400,000 Litres in a tank deep in a mine. Argon was removed before it decayed. At the end of a typical 80 day run, the tank was emptied, and the contents were analysed, and the 37

18Ar nuclei were counted – no mean feat, as there were usually only about 50 37

18Ar nuclei among the 1031 nuclei in the tank! The result however, was only about 1/3 of the neutrinos expected! This result was also confirmed by a Japanese group.

This became known as the solar neutrino problem. Initially people thought the solar models might be flawed in some important aspect – and attempts were made to construct models giving lower central T – but this was never achieved. Helioseismology experiments started to show that solar models were valid.

6

Physicists know of 3 kinds of neutrino: Electron neutrino νe , Muon neutrino νµ ,Tauon neutrino ντ. Neutrinos created in the Sun’s core are electron neutrinos, and this is the only kind that can be detected by the C2Cl4 tank experiment. There was a suggestion that a particular kind of interaction, between the electron neutrino leaving the core and the solar material through which they pass, causes the neutrinos to change type – so that only 1/3 of the neutrinos leaving the Sun are still electron neutrinos.

Experiments were constructed, Sudbury Neutrino Observatory, Canada, which can detect all 3 kinds of neutrino – big tank of heavy water. The flux of all 3 types of neutrino can be measured – confirming the hypothesis that neutrinos can change their type. Lecture 4: SUMMARY outcomes and TO DO:

Summary: • estimate of minimum mean temperature of the Sun • Insufficiency of gravitational and thermal energy to power Sun • Nuclear power – fusion : be able to describe the H burning chains of nuclear reactions and discuss with respect to the Sun and hotter stars. • p-p reaction chain as source of solar energy – be able to calculate resultant energy • Solar neutrino problem : be able to discuss the solar neutrino problem, and how investigated, and current explanation • be able to discuss the contraction and heating up of a star, and stability of nuclear fusion

TO DO: - • Calculate the number of neutrinos from the Sun passing through your head per second

• Read web site article on the solar neutrino problem – add to notes in this handout on this topic

• Problem sheet 1, Q 5 and Q6

7

APPENDIX.

8

Copy Right The Open University

1

Sun Stars Planets 5. J C Pickering 2014 Section 1: The Sun: its structure and energy generation, continued. Lecture 5: Energy production and heat transport 5.1 Luminosity, the energy generation equation 5.2 Heat transport 5.3 Radiative heat transport 5.4 Opacity 5.5 Photon energy during diffusion process 5.6 Heat transport: conduction 5.7 Convective heat transport 5.1 Luminosity Derive an equation relating the rate of energy release and the rate of energy transport. Assume: star is spherically symmetrical and energy is transported in a radial direction. Suppose energy flows across a sphere of radius r, and at a rate L(r) (in units W). Define luminosity L(r) = energy per unit time crossing sphere of radius r, centred on centre of Sun, Let ε = energy per unit time produced per unit mass at any given location in Sun (W kg-1). Consider the energy release in a spherical shell. L (r + r) exceeds L(r) by the energy released in the shell. Equate the difference between energy crossing a sphere of radius r + r, and a sphere of radius r, to the energy released in the spherical shell: ADD NOTES DURING LECTURE Where ε is function of ρ, T and chemical abundances.

2

Only in the inner 25% of the Sun are temperatures high enough for significant energy production. We have neglected the change with time of the stellar properties. We can neglect the time dependence if tnuc >> t th , that is if the energy sources being used are capable of supplying the star’s radiation for a long time compared to the thermal time. We have also not considered the possibility that some of the energy released in the shell is used to heat up or change the volume of the shell. We now have one further equation for the structure of a star, but only by introducing 2 more unknown quantities: and L – so we will still need several more equations… 5.2 Heat transport in the Sun Consider the way in which energy is transported outwards in a star: 3 ways of transporting heat: radiation, conduction, convection Radiation and conduction both depend on the collision of energetic particles with less energetic particles resulting in an exchange of energy. In the case of radiation the energy is carried by photons. 5.3 Radiative heat transport (heat transport by photons) If photons could escape freely from centre of Sun, they would take R

/ c = 2 seconds to reach the

surface. But we know it must take at least more like the Kelvin-Helmholtz time (lecture 4) The energy released at the centre of the Sun slowly diffuses outwards. We have seen that the total thermal energy of the Sun could supply its rate of radiation for about 3 x 107 years – so this gives us an estimate of how long it takes for a photon to diffuse from the solar centre to the surface. Photons escape by random walk, scattered many times by ionized matter in the solar interior. 5.3.1 Random walk With a central T 107 K photons associated with black-body radiation have wavelength in the X ray range. But light from the stellar surface is typically in the visible spectral range – so the photon energy is about 104 times smaller than the average energy per photon in the stellar core. The source of this degradation of photon energy is the coupling between radiation and matter. Photons diffuse through most stellar matter, a process in which a given photon travels, on average, a mean free path l before being scattered or absorbed and re-emitted in a random direction. The description of this process is similar to the statistical problem of a random walk, where N random choices for the next step (here – re-emission in arbitrary direction) result in net displacement N1/2 l from starting point.

3

Steps: all steps of equal length l for simplicity; directions random

Total net displacement after N steps r = r1 + r2 + … + rN And root mean square radial displacement after N scatterings: < rN

2 >1/2 = N1/2 l So, the net distance travelled in N steps |r| = N1/2 l The number of steps to travel from the star’s centre to the star’s surface is N = ( R

/ l ) 2

Time taken for one step = l / c On average in the Sun, l ≈ 10 -3 m = 1 mm (l is the mean free path – m.f.p.) Hence the total time for photon to escape (the diffusion time) is:

[5.2]

Note - N ~ 10 24 - It takes 1012 times longer to escape than if photons had free flight - The mean free path is the distance particles travel between collisions – if m.f.p.is large then particles can get from a point where the T is high to one where it is significantly lower before colliding and transferring energy, and a large transport of energy results.

O

r3

r2 r1

r4

4

5.3.2 Heat transport by radiation Derive an expression for dT/dr: Assume: l , the mean free path, is much much less than the scale on which temperature T and density vary, and that energy transport by photons is a diffusive process. Radiative energy density in photons u = a T 4

Net flux (in +r direction) across surface r is (1/6) v u(r-l) - (1/6) v u(r+l) = - (1/3) v l du

dr So

We have v = c and u = a T 4 so:

and

Define opacity κ by Also, F = L / 4π r 2

So finally, [5.3] Opacity к depends on density, temperature, chemical abundances.

Flux (1/6) v u(r+l) u(r+l)

u (r–l)

(1/6) v u(r–l)

Only 1/6 of photons travelling in any one principal direction.

r–l

r+l

r

5

5.4 Opacity The flow of energy by conduction and radiation is essentially similar in nature, and the rate at which energy flows by these processes is determined by the opacity . The radiative transport equation (derived in 5.3) relates the rate of energy transport to the temperature gradient and the opacity. The opacity of stellar material is a measure of the resistance of the material to the passage of radiation.

[The probability that a photon will be absorbed in travelling distance δx = к ρ δx . So if l= δx the probability of absorption is 1, and so 1 = к ρ l.] The calculation of stellar opacity is a very complicated process as all atoms and ions must be considered. Looking at sources of opacity all the microscopic processes contributing to the absorption of radiation at different frequencies need to be considered. There are 4 basic types of process involved: 5.4.1 Bound-bound absorption An e- is moved from one bound orbit in an atom or ion to an orbit of higher energy with the absorption of a photon 5.4.2 Bound-free absorption (Most important in Sun) An e- in a bound state around a nucleus is moved into a free hyperbolic orbit by the absorption of a photon. 5.4.3 Free-free absorption An e- initially in a free state absorbs a photon and moves to a state of higher energy. 5.4.4 Electron scattering (Most important at higher temperatures) It is possible for a photon to be scattered by an e- or an atom. Although this process doesn’t lead to true absorption of radiation it does slow down the rate at which energy escapes from a star because it continually changes the direction of the photons.

6

5.5 Photon energy Photons are generated in the Sun’s core through thermonuclear reactions as rays, typical wavelength 10 -12 m, at high temperature ~15 x 10 6 K. But photons at the Sun’s surface have typical wavelength 10-6 m and the temperature is 5800 K. How does the average energy of photons degrade from high energy from the PP chain reactions, to the relative low energy of visible light ? It is a 2 stage process:

(i) rays from PP (proton-proton thermonuclear reactions) chains undergo multiple scattering with electrons and ions in the star’s core. Each scattering redistributes the energy between photon and particles. So, some of the energy of a very high energy photon is transferred to the surrounding plasma. Eventually the distribution of photon energies changes from that of PP ray to a blackbody spectrum characterised by the temperature of the core. We can say thermalisation takes place.

(ii) electrons and ions in the solar material interact with photons (scattering or absorption and re-emission) as they make their way out of the Sun by diffusion (random walk). At each point in the star the spectrum is characteristic of a black body source at the temperature of that point. Overall, photons have energies distributed according to the characteristic black body curve dependent on the temperature at the point in the star – and we see that an initially relatively small number of high energy photons produced in the core of the star, eventually becomes at the star’s surface a large number of lower energy photons. 5.6 Heat transport by Conduction Is heat transport by particles (ions and electrons) as opposed to photons in radiative heat transport. Is heat transport by particles important in the Sun? Consider the flux of energy:

For a gas: Energy density ug = 3/2 n kB T (monoatomic gas) and Pgas = n kB T, so

For radiation: Energy density urad for radiation urad = a T4 , and Prad = 1/3 a T4 , so Therefore

(1) (2) (3)

104 << 1 << 1

Find n and T: If T = ₸ (from 4.1) then T≈ 2 x 106 K

(

)

Where we have taken μ≈1/2 as we are assuming fully ionised H. So:

( )

So (1)

.

However, this is more than offset by (2) and (3) being <<1, so that overall conduction is unimportant in the solar interior. But this is not the case in dense stars, like white dwarfs, where conduction is very important.

7

5.7 Convective heat transport (heat transport by bulk motion of material) We have seen that the energy generated in the core by nuclear reactions must escape from the star. If the outer parts of the star are in radiative equilibrium, a certain T gradient must be set up to maintain the requisite energy flow L (see equation of radiative heat transport [5.3]). If the opacity к becomes high, or if L is high, then a steep T gradient must be set up to maintain energy flow – a steep T gradient is unstable in a star just as it is unstable in the Earth’s atmosphere. A star will be unstable against convection if the actual T gradient > T gradient if an element of mass rose adiabatically. Under what conditions does convection occur? Schwarzschild stability criterion Energy transport by conduction and radiation occurs whenever there is a temperature gradient maintained in a body. However, convection, which occurs only in liquids and gases, only occurs when the temperature gradient exceeds some critical value. We must ask whether a given run of T(r), p(r), ρ(r) is stable. Consider an element of fluid displaced upwards, slowly so it remains in pressure balance but quickly enough that no heat is exchanged with its surroundings (adiabatic). Change is adiabatic means And so: Pressure Density If fluid element denser than new surroundings, sinks back: stratification stable If fluid element lighter than new surroundings, keeps rising: stratification unstable Unstable if: [5.4] This is the Schwarzschild stability criterion. If unstable this means bulk motion (convection) will transport heat

Stable if

i.e.

i.e.

Original

New position (surroundings)

New position (fluid element)

ρf P f ρ1 P1

ρo P o

Fluid element

surroundings

δr

8

Note: Mixing length Phenomenological picture: fluid elements rise distance lmix and then merge with surroundings ( lmix is called the mixing length): It is assumed that convective elements of a characteristic size rise or fall through a distance, comparable with their size before exchanging their excess heat with their surroundings. Estimate lmix - pressure scale height (~R/10 in solar interior) Find an expression for dT/dr for convective heat transport Convection is very efficient at transporting heat. Stratification has only to be marginally unstable to transport the solar energy output. So in convectively unstable regions: is an excellent approximation Equation of state

, hence P T so: ln = ln P - ln T + const

Differentiating:

Substituting in:

Gives [5.5] in convectively unstable regions Convection will thus occur when the magnitude of dT/dr is greater than the adiabatic temperature gradient. This will be the case in cooler regions with high opacity and the hot core. Key Points Equation for dL/dr – be able to derive this Heat transport by radiation – diffusive process – be able to discuss Photons escape by random walk – very slow – be able to estimate diffusion time, and discuss Equation for dT/dr if radiation is heat-transport mechanism – be able to derive this Sources of opacity – be able to list, and explain these Photon energy – understand and be able to explain Conduction is not important inside the Sun: be able to estimate importance of conduction Schwarzschild criterion for convective stability: be able to derive this and explain Convection very efficient at transporting heat: be able to explain why Equation for dT/dr if convection is the mechanism of heat-transport: be able to derive this Heat transport will be by: convection if stratification unstable or radiation if stratification stable To do: Problem Sheet 1 – finish questions 1-6

In

1

Sun, Stars, Planets J C Pickering 2014 Lecture 6: Overview of the Sun’s Structure 6.1 Summary of equations of stellar structure 6.2 The solar interior 6.3 Overview of the structure of the Sun: interior and atmosphere 6.4 The atmosphere of the Sun 6.5 Sunspots and magnetic activity Lectures 1-5 have completed the derivations of all the equations of stellar structure: 6.1 Summary of equations of stellar structure equation of hydrostatic equilibrium equation of mass continuity equations of heat transport radiative and convective Energy generation equation

Also need an equation of state (ideal gas equation)

[where the following symbols are: T temperature, P pressure, density, r radius, L luminosity, m mass, energy per second generated per unit mass, mean molecular weight, opacity, G gravitational constant, c speed of light, a radiation constant, gas constant] Appropriate boundary conditions for the differential equations need to be considered: Boundary conditions:

m = 0, L= 0 at r = 0

m = M

at r = R

p = T = 0 at r = R

(zero boundary conditions: can do better, but this is a good enough approximation for simple use of equations)

2

Also we have expressions for P (Pressure), (energy per unit time produced per unit mass at any given location in Sun (W kg-1)), and opacity :

( )

( )

( ) Given the chemical composition of a star, we now have 7 equations with 7 unknowns: P, T, m, L, and as functions of r.

Calculations of the structure of a star now involve obtaining expressions for P, and and then the solution of the 4 differential equations. Comment on boundary conditions: (i) at centre m = 0, L = 0 when r = 0 (ii) at surface: use approximations – if the stars were truly isolated bodies then we would expect and P = 0 at the surface. But, there is no sharp edge to a star: the density of the Sun near to the visible surface is estimated to be 10-4 kg m -3 which is much less than the Sun’s mean density (see Lecture notes section 2.5).

Also, surface temperature is much less than mean temperature, for sun surface T = 5780 K, compared to typical mean T ~ 2 x 10 6 K (see lecture notes section 4.1).

So it is possible that the solution of the equations of stellar structure throughout the interior of a star will not be seriously affected if the true surface boundary conditions were replaced by the assumption = 0, T = 0 at r = R

6.2 The solar interior 6.2.1 Composition. Variation of the mass fractions X, Y and Z with the fractional radius r / R

for

the Sun: add diagram during lecture Throughout most of its interior the Sun is by mass ~ 73% hydrogen, 25% helium and ~ 2% of everything else. But in the central 30% of its radius the percentage of hydrogen is increasingly depleted (and He increased), with about half the hydrogen initially present in the core of the Sun having been converted into He.

3

6.2.2 Variation of pressure, density and temperature with fractional radius in the Sun

The rise of T, P and with increasing depth is as expected. It is not until a fractional radius of about 0.5 that the density is equal to that of water on Earth (1000 kg m -3). Even at the centre of the Sun, where the T ~ 15.6 x 106 K, the density is predicted to be only fourteen times that of lead, though the pressure is more than 1010 times that of the Earth’s atmosphere at sea-level.

Lecture 7

Hydrogen abundance

Luminosity

Energy generation rate

X

0.7

0

r / R

Fractional radius

0 0.5 1.0

r / R

0 0.5 1.0

r / R

0 0.5 1.0

2

0

4

ε(1

0-3

J s-

1 kg-

1 )L

(102

6W

)

radiative

Lecture 7

Temperature

Density

PressureT (1

06K

)15

0

r / R

(Fractional Radius)

0 0.5 1.0

convective

r / R

0 0.5 1.0

r / R

0 0.5 1.0

2

0

150

p (1

016

Nm

-2)

ρ(1

03kg

m-3

)

Structure of Sun according to aStandard solar model

0

6.2.3 Variation of luminosity L and energy generation rate with fractional radius in the Sun

4

6.3 Overview of the Structure of the Sun: interior and atmosphere 6.4 The atmosphere of the Sun The surface of the Sun is not really a surface at all, but a thin, semi-transparent shell of gaseous material: 6.4.1 The Photosphere: The light that we see from the sun comes from the photosphere (= sphere of light), a layer of atmosphere about 500 km thick (very thin compared to Sun ~1.4 million kms across!). In physical terms the photosphere is more like an atmospheric layer and is the closest thing to a surface. The air you are breathing now is over 1000 times denser than the material of the photosphere! The photosphere emits light as it is hot: 9000 K at its lower boundary, and ~4500 K at the top. Most photospheric light comes from region 5800 – 6000 K, and this represents the surface temperature of the sun. What wavelength does this correspond to? If we take the emitted light as being that of a blackbody at 5800K,

Wien’s law: maxT = 2.8979 x 10-3 m K gives max ~ 500 nm – visible light (We will discuss the photosphere in greater depth later in the course…)

Photosphere Chromosphere Transition Region Corona Solar wind

5

6.4.2. The Chromosphere During a total eclipse of the Sun, the moon blocks the light from the photosphere for a few minutes (it is an amazing coincidence that, although the Sun and Moon differ greatly in size and distance from the Earth, their diameters and distances are just right to allow such a precise blockage to occur). The solar “atmosphere” can then be seen – mostly white, but close to the photosphere is a narrow region with pink/red tinge, the chromosphere…

The gases of the Sun extend far beyond the photosphere, which may be considered the lowest level of the solar atmosphere. The region immediately above the photosphere is called the chromosphere. The chromosphere is 2000-3000 km thick. It glows faintly relative to the photosphere and can only be seen easily in a total solar eclipse. The faint glow of the chromosphere is due to an emission spectrum from hot, low density gases emitting at discrete wavelengths. In the chromosphere energy continues to be transported by radiation. Hydrogen atoms absorb energy from the photosphere and most of the energy is then emitted as red light (because of strong Balmer H-alpha emission). This colour is the origin of its name (chromos meaning “colour''). The chromosphere is most easily viewed by filtering out all other wavelengths of light from the Sun and only letting the red light from the chromosphere through. (The discovery of helium was from emission lines seen in the chromosphere during an eclipse in 1868. Helium was found on the Earth in 1895.)

6.4.3. Transition region Above the chromosphere is a very thin layer of the Sun's atmosphere about 100 km thick over which the temperature rises drastically from 20,000 degrees Kelvin in the upper chromosphere to over 2 million degrees Kelvin in the corona. This region is called the transition region. The large temperature rise was unexpected these outermost layers of the Sun which are far from the heat producing core. The complicated structure of the Sun's magnetic field provides clues to the dramatic increase in temperature over such a small change in radius. 6.4.4. The Corona The outermost layer of the Sun’s atmosphere is called the corona. It gets its name from the crown like appearance evident during a total solar eclipse. The corona stretches far out into space and, in fact, particles from the corona reach the Earth's orbit. The corona is very extensive, very tenuous and has an extremely high temperature (~106 K). It does however appear faint compared to the photosphere, and therefore can only be seen from Earth during a total solar eclipse or by using a coronagraph telescope which simulates an eclipse by covering the bright solar disk. What is the source of heating in the Corona ? The Corona cannot be heated simply by energy radiated by the photosphere or by heat conducted through the chromosphere; 2nd law of thermodynamics says no transfer of energy from a cooler body to a hotter one is possible in this situation by either of these methods… But, the energy must come from the lower regions of the Sun somehow – how? The behaviour of the magnetic field in the lower parts of the Corona plays an important role in heating the coronal gas… The shape of the corona is mostly determined by the magnetic field of the Sun.

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Charged particles can stream away from Sun along open field lines - fast solar wind several hundred km/s. The free electrons in the corona move along magnetic field lines and form many different structures including helmet streamers which can be seen as long, spiked cones in solar eclipse images. Sometimes the magnetic field emerges from the lower regions and loops back down into the Sun. These magnetic structures can be seen extending up into the corona. As particles follow the path created by the magnetic field they form dynamic loops and arches that are most readily visible with special telescopes. These structures are known as solar prominences. As the density is so low, the corona is a poor emitter compared to the photosphere. The Corona is not a black-body because it has such a very low density and is not opaque. Although not a black-body source of radiation, we can still estimate the wavelength at which the Corona predominantly radiates: Photon energy kT where k = Boltzmann const.

Temperature of corona 2 x 106 K E kT 1.38 x 10-23 x 2 x 106 = 2.76 x 10-17 J

E = h = hc/ so = hc/E = 6.63 x 10-34 x 3 x 108 / (2.76 x 10-17 )

= 7.21 x 10-9 m 7 nm (X ray)

Some phenomena arise when the magnetic fields undergo very rapid changes:

Solar flares: rapid energetic outbursts of EM radiation. Duration 100-1000s, and may release up to 1025 J in radiation in this time! And the material may have a temperature of over 107 K! Incidence of flares follows solar activity cycle. Radiation emitted by flares is dominant in X ray and EUV.

Coronal mass ejection (CME): is a violent outflow of coronal material. The mass of ejected material is typically 5 x 1012 – 5 x 1013 kg, and frequency of CMEs is typically 1 per day, or 3 per day at solar maximum of the Sun’s activity cycle. It appears that the magnetic field is moving and the plasma is forced to move with the field. Origin of these CME events is probably in a rapid reconfiguration of the magnetic field in the lower parts of the solar corona – this allows energy stored in the magnetic field to be released suddenly, causing a violent outflow of coronal material.

Magnetic reconnection occurs: – which converts energy stored in magnetic field into kinetic energy of particles. In the case of solar flares and CMEs the reconnection event occurs high in a coronal loop – particles are accelerated down to the foot points of the loops to give solar flares. Much research into this area is still going on…

Closed field lines - helmet streamers

Open field lines

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The incredibly hot temperature of the corona, however, requires a permanent heating mechanism, or the plasma of the corona would cool down rapidly. There are many mechanisms which could heat some gas above the surface of the Sun, but none of those mechanisms could account for the large rate of heating necessary to heat the corona. This phenomenon remained a mystery for more than 50 years. However, although not all details are clear there is a likely solution to this mystery:

Using data from instruments onboard the SOlar and Heliospheric Observatory (SOHO) and from the Transition Region And Coronal Explorer (TRACE), solar physicists have identified small patches of magnetic field covering the entire surface of the Sun. Contrary to the bright, large magnetic field loops which are linked to the "active regions" during periods of solar maxima, these patches seem to appear and disappear randomly in time scales on the order of 40 hours. This “magnetic carpet” is probably a source of the corona's heat. It is now believed that the heating of the corona is linked to the interaction of the magnetic field lines radiating out of the small patches mentioned above. Because the laws of electromagnetism prohibit the intersection of two magnetic field lines, every time magnetic field lines come close to crossing they are "rearranged," and this magnetic reconnection continuously heats the solar corona. It's a fairly inefficient source of energy, but the vast number of these small magnetic patches on the Sun’s surface makes the process a viable solution to the 50 year old problem of what heats the solar corona.

6.4.5. Solar Wind Particles from the corona also stream out along the magnetic field lines of the Sun that extend into interstellar space. This "solar wind" transports particles through space at 400 kilometers per second. (At the Earth the number density of protons in the Solar wind 7 x 10 6 m-3 ; very low). When the solar wind reaches the Earth the magnetic field of the Earth will sometimes trap these electrons and protons and pull them into the Earth's atmosphere. Atoms in the Earth's atmosphere interact with these high energy particles by accepting energy from them and then releasing that energy in the form of coloured light – aurora.

6.5 Sunspots and magnetic activity Sunspots are indicators of solar activity. Measurements of magnetic fields reveal that all sunspots are characterized by magnetic field strengths that are much higher than elsewhere in the photosphere. Possibly the intense magnetic field suppresses convection, which reduces the rate of energy transport to the photosphere resulting in the relatively cool regions that characterise a sun spot (hence they appear darker than their surroundings). Sunspots tend to occur in groups or pairs with opposite polarities. Magnetic field lines in a sunspot Add diagram during lecture

Magnetic field lines and a pair of sunspots

The field lines emanate from one sunspot and form a loop which arches above the photosphere and returns to enter the Sun at the other member of the pair.

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Why does the sun have a magnetic field? We believe that the Sun’s magnetic fields are generated by electrical currents acting as a magnetic dynamo inside the sun. Magnetic fields can be formed when electrically conducting fluids flow according to certain patterns, and we think such electrical currents are generated by the flow of hot ionised gases in the Sun’s convection zone.

The Sun’s magnetic dynamo has a 22 year cycle. During the first half of this cycle, the Sun’s magnetic north pole is in its northern hemisphere and the magnetic south pole is in the Sun’s southern hemisphere. At the time of the peak in sunspot cycle (solar maximum) the magnetic poles flip, exchanging places, and the magnetic north is now situated in the Sun’s southern hemisphere. This “flip” happens every 11 years at solar maximum.

Lecture 6: Summary • Summary of eqs. of stellar structure; boundary conditions –be able to discuss, and derive these equations (see previous lectures) • Solar atmosphere structure: photosphere, chromosphere, transition region, corona, and solar wind – be able to discuss, describe, comment on. • Coronal heating problem – be able to discuss • Importance of magnetic fields in the corona - be able to describe & discuss related phenomena • Solar magnetic activity cycle , 11-year sunspot cycle, 22 year magnetic dynamo cycle – be able to comment on TO DO: look through the first 6 lectures of the course which have covered the derivations of the stellar structure equations. Make sure you understand this material, and attempt past exam questions of relevance.

The sun has an 11 year cycle of sunspot activity, which can be seen in the varying number of sunspots observed. The Approximately 11 year solar

sunspot cycle.

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Sun, Stars and Planets 7. J C Pickering 2014 Part 1 continued: The Sun: its structure and energy generation Lecture 7: The solar spectrum and stellar spectra 7.1 Thermal radiation spectrum 7.2 Absorption lines 7.3 Non thermal radiation 7.4 Common uses of stellar spectrum analysis 7.5 Spectral Classification of stars 7.6 Estimating luminosity and luminosity Classification of Stars To Do: re-read sections covered in previous lectures of particular relevance: 3.1 Thermodynamic equilibrium 5.5 Photon Energy 7.1 Thermal radiation spectrum 7.1.1 Black body radiation We see a thermal radiation spectrum from the Sun’s photosphere (the Sun’s visible bright surface). In the photosphere the photons are in thermodynamic equilibrium with matter and so have a black-body spectrum with intensity distribution B or Bλ : Black body [Reminder]: There are 2 key features of sources that produce black-body spectra: i) The energy that is emitted as light has its origin in the internal, or thermal, energy of the material making up the source – such sources are thermal sources of radiation. ii) In addition to being a thermal source of light, the condition is also needed that the light within the source is more likely to interact with the material of the source than to escape, it will only escape after considerable interaction with material within the source. So a common feature of BB sources is that they are opaque. Many astronomical sources produce continuous spectra with reasonably good approximation to BB form. The spectral distribution of this intensity is dependent solely on the temperature of the material T, and is given by the Planck function: Planck Function:

B has dimensions of: energy / unit area / unit time / unit frequency / unit solid angle B has dimensions of: energy/unit area/ unit time/ unit wavelength/unit solid angle

( )

(

⁄ )

( )

( ⁄

)

h = 6.6 x10-34 Js-1

k =1.4 x10-23 JK-1

c= speed of light, λ wavelength, υ frequency, T temperature

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7.1.2 Wien’s displacement law: maxT = 2.8979 x 10-3 m K gives maximum of Planck distribution. So provided an object is a black-body source, by studying its spectrum one can determine its temperature. Temperatures of stars can be found using this. For solar surface temperature (T=5800K), Bλ(T) peaks in the visible range of the spectrum, at a wavelength of about 500nm, in yellow part of the visible spectrum: violet 400nm (4000 Å) red 700nm (7000Å) (Å is Angstroms) Cooler stars - redder (longer wavelength) radiation Hotter stars - for hotter stars than the Sun the peak of the Planck distribution of flux density moves to shorter wavelengths and these stars will appear white, or bluer than the Sun. So a star’s colour will give an indication of its surface temperature (more on this in lecture 9) Question: A star has luminosity 5 times the solar luminosity, and from its spectrum its effective temperature is 30,000K. What is the stellar radius? At what wavelength does the star’s radiation peak? [ The solar max= 500nm and Teff = 5800K ] 7.2 Absorption lines 7.2.1 Formation of absorption lines in stellar spectra

Planck distribution for black body objects at a range of temperatures

Notice that: The area under the Planck curve increases dramatically as T increases. This is not surprising as for a blackbody flux (W/m2) per unit area of emitter is given by Stefan’s law:

Also, since where R is stellar radius, luminosity:

So as temperature of the star increases the luminosity increases dramatically.

Dark lines are seen in the continuous blackbody spectrum. These lines are formed in the cooler overlying material in the Sun’s atmosphere.

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7.2.2 Energy levels and transitions A photon can be absorbed if its energy matches the difference between two energy levels of an atom. An electron is excited to a higher lying energy level or atom is ionized: Photon of energy h is absorbed: ---------------------------------- En’ h = ΔE ------------------------------------ En ΔE = hc/, =hc/ΔE, h = Planck Constant (Js), = wavelength, ħ =h/2, me(e- mass)= 9.11 x 10-31kg o = electric permittivity of free space. For hydrogen: Or En= -h c R / n2 (J) = -13.6/ n2 (eV) where the Rydberg constant, R = m e4 /(8 o

2 h3 c), and 1eV=1.6 x 10-19 J.

In fact, about 25,000 lines have been identified in the Sun’s visible spectral region, most originating in the photosphere, which is responsible for nearly all the light between the lines. Each element, each atom and ion has its own unique spectrum – ranging from simple hydrogen spectrum, to the complex spectrum of iron with 1000s of transitions. 7.2.3 Occurrence and strength of lines depends on: i) amount of element present (relative abundance) ii) probability that electron is in appropriate energy level (depends on T; the higher the T the more likely that energy levels corresponding to higher energies will be occupied) iii) probability that photon of given frequency will be absorbed So, solar absorption lines give information on: Chemical abundance and temperature, and hence density and pressure. The number of lines observed depends on the complexity of an atom or ion’s energy level structure. The energy level structure of hydrogen is very simple, Helium is more complicated, and the level structure of the iron group of elements (3d transition elements) is complex with hundreds of energy levels and so these have thousands of lines in their spectra. In fact astronomers use vast databases of atomic and molecular data in order to identify transitions seen in stellar spectra, and to interpret what these tell them about a star. The data bases contain both laboratory observed experimental spectra for each atom and ion, as well as calculated values – data on millions of lines!

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7.2.4 The spectrum of the Sun (see Slides file for lecture 7, on course web site) An image of a spectrum can be represented graphically in terms of flux m-2 nm-1 against and the absorption lines have different relative depths, or strength (corresponding to different levels of blackening of photographic plate or nowadays, detection by detector.) The spectrum of the Sun follows the curve for a blackbody at 5800K, with absorption and emission lines also.

[Interesting web site: allows you to plot a segment of solar spectrum in detail at: http://bass2000.obspm.fr/solar_spect.php - select “tools – solar spectrum”] 7.3 Also non-thermal radiation is observed 7.3.1 Bremstrahlung: (free-free transitions – see opacity, Lecture 5) Transitions are possible that involve only the continuum of the H atoms. These are called free-free transitions or Bremsstrahlung (braking radiation) – and the corresponding absorption is free-free opacity. (In stellar interiors, where H is completely ionized free-free absorption is one of the most important contributors to opacity.) 7.3.2 Synchrotron radiation: electrons spiral around B-lines Synchrotron radiation is seen in the X-ray and radio emission from hot coronal plasma and from hot plasma in flares. Any charged particle moving in a magnetic field is forced to move in a spiral around magnetic field lines. The spiraling motion of electrons in a magnetic field gives rise to the emission of electromagnetic radiation. 7.4 Some common uses of stellar spectral analysis a) classification of spectral type of star (7.5 below) b) classification of luminosity class (7.6 below) c) measurement of photospheric chemical abundances d) measurement of radial velocity of star from Doppler shift of centre of spectral line e) measurement of stellar rotation from additional broadening of spectral line by the rotational velocity f) measurement of the mass inflow or outflow from asymmetries in the line profile g) measurement of photospheric magnetic fields by the Zeeman effect 7.5 Spectral Classification of stars In order to be able to systematically discuss and understand the behaviour of stars we need a system of classification of stars. We can use the temperature and colour, as well as the main spectral characteristics of the star for such as system:

Solar spectrum overview. On this scale the emission and absorption lines cannot be seen. Much higher resolution would be needed to see the spectral line features.

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We have looked at the solar spectrum, and how absorption lines are seen in stellar spectra – these absorption lines can be used as an indicator of the star’s temperature, as they show us what species (atoms/ions or molecules) are a dominant or noticeable source of opacity, and thus roughly what the temperature is – in simple terms: cool stars will show lines from molecules and atoms, hot stars will show H and He lines relatively more strongly.

Spectral Classification The spectroscopic method of obtaining photospheric temperatures was well established by the 1920’s, led by astronomer Anne Jump Canon at Harvard University. On the basis of the strengths of their spectral lines, stellar spectra are classified by letter in a scheme called the Harvard Spectral Classification. In order of descending temperature the spectra classes are labeled O B A F G K M Each class also divided into 10 subdivisions, from 0 to 9, with 0 being the hottest. So F9 and G0 stars are very similar. so e.g. …, B8, B9, A0, A1, A2, … A9, F0, F1, … Example: the Sun is a G2 star, and Rigel is classified as a B8 star. If we can establish a star’s spectral type, we can determine its average photospheric temperature with an uncertainty of only about 5 %. [Note: OBAFGKM phrases can be used to remember order… ]

7.6 Estimating luminosity and luminosity Classification of Stars In addition to the spectral classification scheme in 7.5 which assigns a letter to a star of particular spectral characteristics and temperature, we also have a luminosity classification based upon the widths of the absorption lines in the star's spectrum. (An absorption line is not infinitely thin. It has a strength and a width – the strength is the area within the line profile.) A luminosity classification is necessary as stars may have a similar temperature and spectral characteristics, but have very different luminosity – they may be a different kind of star (will be discussed later in the course…).

Lecture 11

Betelgeuse, Antares

Molecular bands noticable

< 3500 Red M

Arcturus, Aldebaran

Metal lines dominate 3500 – 5000 Orange K

Sun,Capella Solar-like spectrum 5000 – 6000 Yellow G

Procyon Weak metal lines 6000 – 7500 Yellow-white F

Sirius, Vega Strong H lines 7500 – 11000 White A

Rigel, Spica Neutral He lines 11000–25000 Blue-white B

Mintaka He+ lines; strong UV > 25000 Blue O Examples Main characteristics Teff (K) Colour Type

The strengths of various absorption lines versus photospheric temperature (From: S.Green & M Jones, An Introduction to the Sun and Stars (Cambridge University Press))

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If one looks at spectra of stars of the same temperature, but different luminosity (where L has been determined non-spectroscopically), then at a given temperature, the more luminous the star, the narrower are its spectral absorption lines, and the stronger are the absorption lines due to certain ionised atoms. Luminosity has an effect on the line width – because of the conditions in stellar atmospheres: A large star has lower density in its outer layers than a small star of the same temperature, because the mass of these layers is spread over a larger volume. In small stars, density and pressure are higher, atoms and ions collide and interact more, causing distortions in the energy levels of the atom, and thus a wider range of wavelength for a particular photon. So, we get broader absorption lines for smaller stars (lower luminosity). This process is called pressure broadening. Also, because of more collisions and recombinations between particles in the atmosphere, there are less ions present in a smaller star, and so the spectral lines from ions are weaker for a lower luminosity star than for a larger star of the same temperature.

This correlation of luminosity and spectral features is used to determine luminosity for stars of unknown L, it is not precise but gives luminosity classes which can distinguish different groups of stars. So: Absorption lines are Pressure-sensitive:

Spectral lines get broader as the pressure increases. Big stars have lower pressure in their atmospheres compared to smaller stars.

Implications: Larger stars have narrower absorption lines. Larger stars are brighter at the same temperature.

Since larger stars are brighter at a given stellar temperature (more surface area to radiate), measuring differences in the line widths for stars of the same temperature gives an estimate of the stellar luminosity. This gives us a way to assign a relative luminosity to stars based upon their spectral line properties!

Lecture 7: Summary • Thermal radiation – form given by Planck’s function: be able to describe, and explain • know Wien’s law - know and be able to use maxT=constant. • know (be able to write down) and be able to use Stefan’s law • Absorption and emission lines at frequencies determined by energy levels of electrons in atom: be able to understand and explain origin of lines in stellar spectra and what information may be obtained about the star from stellar spectra • Balmer and Lyman series for hydrogen: be able to explain and describe these • Stellar spectral types – be able to understand and explain these • Luminosity and line widths – understand and be able to discuss the link between these • Luminosity classes – know what this is, more details later in the course. TO DO: Problem sheet 2 - Questions 1, 2, 8

Luminosity Classes: Ia: the most luminous supergiants

Ib: less luminous supergiants II: luminous giants III: normal giants IV: subgiants V: dwarfs (main-sequence stars) The full designation of a star’s spectral type also includes its luminosity class. The Sun is spectral type G2 V, Betelgeuse is a spectral type M2 Ia. ←Diagram showing approximate positions of luminosity classes I to V. (From: S.Green & M Jones, An Introduction to the Sun and Stars (Cambridge Univ. Press))

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Sun Stars Planets 8. J C Pickering 2014 Section Stars, putting the sun in context. Lecture 8: Understanding the Main Sequence 8.1 The main-sequence 8.2 Equations of stellar structure, state, and boundary conditions 8.3 Homology transformation, and relationships between parameters 8.4 Main sequence lifetime 8.1 The main-sequence. 90% of the stars lie on the main-sequence, the longest phase in their life, during which the energy generation is hydrogen burning. We can use the stellar structure equations to understand observed stellar properties.

8.2 Equations of stellar structure, equations of state and boundary conditions 8.2.1 Given here is the summary of the equations covered so far during the course – see your lecture notes for the derivations, etc Four differential equations governing stellar structure: In the stellar interior we have:

Hydrostatic equilibrium

Mass conservation

Radiative heat transport

Convective heat transport

(

)

Radiative equilibrium

[1]

[2]

[3a]

[4]

These diagrams (known as Hertzsprung-Russell diagrams, more on this in later lectures) plot luminosity and temperature of each star: in the solar neighbourhood and in the Pleiades star cluster.

[3b]

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We need expressions for P, and . If a star is in a steady state, and in a state close to thermodynamic equilibrium, then these three quantities depend on , T and C (chemical composition), and these will generally be functions of radius r. So we have a problem of basic physics, to determine the pressure, opacity and rate of energy generation of a medium for given conditions of density and temperature: Opacity = (, T, C) We could assume opacity of stellar material is given by Kramer’s opacity formula:

= constant Energy generation rates = (, T, C) For many thermonuclear burning stages, the energy-generation rate, may, within a limited T range be approximated by: where: = constant, and are usually slowing varying functions of and T, and may be taken as constants for simple calculations. For most fuels 1, and 4 for hydrogen burning, and 30 or more for carbon burning. Chemical composition this enters through parameters μ (mean molecular weight, к (opacity) and (energy generation rate). Equations of state P = P (, T, C) Assume either gas pressure or radiation pressure operate, but not both. Possible expressions for pressure:

Where is the gas constant and a the radiation constant. 8.2.2 Boundary conditions and calculations

(i) Inner boundary conditions At r =0 m(r) = 0 L(r) = 0

(ii) surface boundary conditions At r = R T=0 and P=0 (since TS << Tc and PS <<Pc) So given the chemical composition of a star, we have 7 equations for 7 unknowns P, , T, M, L, and as functions of r. Model calculations then involve: obtaining expressions for P , and and then solving the four differential equations [1] – [4] with boundary conditions. This yields P, ,T, (C), as functions of r, L, Teff

[5]

[6]

[7a]

[7b]

r O

R

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8.3 Homology transformation The equations (listed in 8.2) are subject to a homology transformation: that is, given a solution to the equations, with the quantities (P, T, L, ) stated as functions of r, for a given total mass M and given chemical composition C, then we can find a new solution for a new total mass, simply by multiplying the other physical variables by appropriate scaling factors. So since the input physics is the same, we expect that in terms of dimensionless coordinate x = r/R the internal structure of stars will be identical except for a scaling: So for two stars, say star 1 and star 2:

and so for example: Add notes

This expresses the fact that each star contains the same fraction of its mass within the same fraction x of its radius. And (add notes →):

We have the same curves for all stars with the same physics and composition. So, for example, if we know the P in one star at a particular fraction of its radius, then we would know there is a rule to scale this to determine the P in another star at another fraction of its radius. So we can write:

r O

R

Where Po , To , mo , o , and Lo functions are the same for all the stars. We thus have a set of answers, which are the same for all stars. We can find the relations, for how ρc, Pc, Tc and L scale with M and R:

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8.3.1 Central density, M and R - make notes during lecture shows how the central density scales from star to star. 8.3.2 Central P, M and R - make notes during lecture Shows how the central pressure scales from star to star 8.3.3 Central T, M and R TO DO You work through this one And similarly using equation of state for pressure: substitute in for P = Pc Po ρ = ρc ρo T = Tc To …gives Pc ρc Tc But from [8] ρc M / R3 and from [9] Pc M2/ R4 so: M2/ R4 ( M / R3 ) Tc giving: 8.3.4 Luminosity-mass and also, similarly, using the radiative heat transport equation: [where of course T, L, and ρ, are functions of r] substitute in for: ρ = ρc ρo T = Tc To

L(r) = L Lo = r / R , d /dr=1/R and = o T- giving:

[-------------------------------------------------] Same for all stars

But ρc M / R3 [8] and Tc M / R [10] so this gives, after substituting in for Tc and ρc :

ρc M / R3 [8]

Pc M2/ R4 [9]

Tc M / R [10]

L M 3+ β –α R 3α –β [11]

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For low-mass stars ( M 1 M

) : Kramers opacity = 1, = 7/2

: p-p chain 4

- so [11] becomes: L M

5.5 R -0.5

M5.5 approximately because R dependence is so weak

For higher mass stars : electron scattering = 0, = 0

: CNO cycle 17

so [11] becomes: So a reasonable compromise across the entire mass range is: [14] is an important relation – you should know this TO DO: PLEASE work through the above calculations yourselves 8.3.5 Radius- mass A similar method, as used in the derivations 8.3.1 – 8.3.4, is used here. Starting with [4] where L, ρ and are functions of r Substituting in [4] for: (r) = o ρ(r) T(r) T(r) = Tc To(x) ρ = ρc ρo =

L = L Lo ….gives

(you work this through): ( L/R ) dLo /d = [4 2 ρo o ρo To ] R2 ρc2 Tc but comparing with [4] we can say that: dLo /d = 4 2 ρo o ρo To will be true for all stars (it is one of the 4 equations describing stellar structure) in terms of our new variable , fractional radius.

L M 5.5

[12] (for low mass stars)

L M 3 [13] (for higher mass stars)

L M 4 [14]

L M 3+ β –α R 3α –β [11]

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So we can therefore state that : Because energy generation [6] (r) = o ρ(r) T(r) does vary with the mass of the stars, we consider now the two cases for low and higher mass stars : (i) low mass stars ( 4) Using equation [15] L/R R2 ρc2 Tc And substituting in for L, Tc, and ρc into [15] using:

L M 5.5 R -0.5 [12] (for low mass stars – more precise version before simplification of neglecting of R) … and ( 4) gives R3 ρc2 Tc M5.5 R -0.5 then substitute in for ρc M / R3 [8] and for Tc M / R [10] gives R3 (M2 / R6) (M4 / R4) M5.5 R -0.5 giving M 0.5 R 6.5

so: (ii) higher mass stars ( 16) You Do this: follow the same procedure as in (i) above, with 16 but use [13] L M 3 which is for higher mass stars. You should get the result: These relations [16] and [17] between R and M, allow you to revisit relations [8], [9] and [10] for low and higher mass stars to now give the dependence of central density ρc, central pressure Pc and central temperature Tc on stellar mass M.

For example: ρc M / R3 [8] would become: (i) for low mass stars, using [16], ρc M /M3/13 so approximately ρc M for low mass stars [18]

L/R R2 ρc2 Tc [15]

M R 13 or R M 1/13 [16] for low mass stars

R M 15 / 19 [17] For higher mass stars

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(ii) but for higher mass stars, using [17], ρc M / M45/19

so approximately ρc 1 / M for higher mass stars [19] These are strikingly different relations for ρc ! 8.3.6 The slope of the main-sequence

We already know L = 4 R2 Teff4 from previous lectures, so

we can write Teff

4 L/ R2

(i) low mass stars: Use [12] L M 5.5 and [16] R M 1/13 and substitute into Teff

4 L/ R2

gives: Teff4 M5.5 / M2/13 or approximately Teff

4 M5.5

but L M 5.5

so approximately, L Teff

4 for the lower main sequence [20] (ii) higher mass stars: use [13] L M 3 and [17] R M 15/19 and substitute into Teff

4 L/ R2 gives: Teff

4 M3 / M30/19 , or approximately Teff4 M 3/2

so M (Teff

4) 2/3

but L M 3 so approximately, L [ (Teff4 ) 2/3 ] 3

→ L Teff

8 for the upper main sequence [21] These relations of L and Teff for upper and lower MS are qualitatively correct, but not as steep in observation.

Log L

Log Teff

–8

–4

The H-R diagram is a plot of log L against log Teff . So we are looking for the relationship between these two physical parameters, for both low and higher mass stars.

8

We have seen that for stars of the same chemical composition, chemically homogeneous stars, the equations governing the stellar structure need only to be solved once, and the properties of stars of all masses could then be obtained. Such a set of models of stars in which the dependence of the physical quantities on fractional mass mo and x is independent of the total mass of the star is known as a homologous sequence of stellar models. 8.3.7 Departures from homology scaling: Any departures from homology scaling are due to:

(i) Radiation pressure (important in high-mass stars) (ii) Convection in (a) the core and/or (b) the outer part of the star

8.4 Lifetime of a star on the main sequence Amount of fuel available during hydrogen burning phase But

So

Time t for star to exhaust fuel in its H burning phase is, since [14],

Massive stars are quickest to exhaust their hydrogen fuel. LECTURE 8: SUMMARY – learning outcomes Can qualitatively understand form of MS On Main Sequence (MS), L increases with M More massive stars have shorter main sequence lifetimes Homology scaling, a useful tool: be able to quickly show useful relationships between parameters starting from equations of stellar structure, and equations of state

TO DO: Problem Sheet 2 Q 7.

Main sequence lifetime [22]

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Sun, Stars, Planets 9. JC Pickering 2014 Lecture 9: The Sun in context, stellar magnitudes and measurement of distance 9.1 Basic parameter ranges for stars 9.2 Stellar magnitudes 9.3 Stellar colours and temperatures 9.4 Measuring stellar distances 9.5 Proper motion 9.6 Summary of methods of distance determination 9.1 Basic stellar parameter ranges: (mass M, luminosity L, radius R, effective temperature Teff) 0.1 M

< M < 50 M

0.01 R

< R < 1000 R

10 - 4 L

< L < 106 L

2000 K < Teff < 100 000 K

How are these measured? We have already described briefly how M

R

L

and Teff

are determined. We can now turn to the determination of these parameters for more distant objects. We can use theoretical models (for example as discussed in lecture 8) then in order to understand the values we find for a large number of stars. 9.2 Stellar magnitude How bright are the stars? When we observe stars to determine their brightness, we are actually measuring the amount of light from the star reaching us – its apparent brightness. To study and compare stars and their properties we have to know their intrinsic brightness. By intrinsic brightness we are talking about the total amount of power a star radiates into space over all wavelengths. This is known as luminosity, (measured in Watts). Hipparchus, the Greek astronomer, catalogued and classified stars visible to the naked eye according to their visual brightness - into six magnitudes: the brightest being first magnitude and the faintest 6th magnitude. A 1st magnitude star is approx 100 times brighter than a 6th magnitude star. The human perception of brightness as measured by the eye is logarithmic in nature (equal steps of perceived brightness correspond to equal ratios of flux density). So the magnitude scale is a logarithmic scale. The modern quantification of brightness was then designed to agree with the old Greek measures. The modern system: quantifies magnitudes as: 5 steps of magnitude = factor of 100 in Flux. So for example:

a 10th magnitude star is 100 times fainter than a 5th magnitude star. a 20th magnitude star is 10,000 times fainter than a 10th magnitude star.

If you look up tables of stellar properties, commonly you would find values given for the stars’ apparent or visual magnitudes rather than values of their luminosity L. 9.2.1. Definition of stellar magnitude

is the apparent magnitude (definition) is the flux of light (energy per unit time per unit area) received from star at the observing point. The value of the constant K is chosen to match to the scale of Hipparchus. Note: the larger the value of (i.e. brighter star), the smaller the magnitude The apparent magnitude scale can be applied to any object in the sky: Sun: -26.73 Full moon: -12.7 Venus at its brightest: -4.1

2

Brightest star: -1.46 Naked eye dimmest stars (urban): +3 Naked eye, faintest visible, dark skies: +6 Faintest objects detectable by HST ≈ +31 Faintest objects detectable through the largest telescopes (eg ELT) ≈ +36 The Sun and the dimmest objects detectable through telescopes are 62 magnitudes apart, this corresponds to an actual difference in brightness of over 6 x 10

24.

The difference in magnitudes of two stars can be expressed:

( )

(

)

Where

where d = distance of star to the Earth.

Values of flux density range from 10 -8 W m -2 for Sirius, the brightest star in the sky, to 10-20 W m -2 for the faintest detectable objects, ranges from –1.5 to 36. So apparent magnitude depends on the star’s luminosity L and its distance d from the observer. is what can be observed To compare intrinsic properties of stars, we need to remove dependence on distance. So we want to define absolute magnitude M of a star to be what its apparent magnitude would be if it were at some chosen standard distance away. What distance? 9.2.2 Parsec: unit of distance Astronomers measure distances in units of a parsec. 1 parsec (pc) is defined as the distance at which a length of 1 A.U. would subtend an angle of one arc-second. Add diagram By definition (using small angle approximation):

Aside:

1 pc = 206265 AU = 3.09 x 1016 m. (Just for reference, 1 light-year = 9.5 x 1015 m) 1 AU = 1.5 x 1011 m = average distance from Earth to Sun.

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9.2.3. Definition of absolute magnitude The absolute magnitude M of a star is the apparent magnitude the star would have if it were placed at a standard distance of 10 pc away. Thus we can calculate the absolute magnitude M of a star as follows: Where is the flux of the star observed at Earth with the star at its actual distance , and is the flux that would be received if the star were at a standard distance =10pc from Earth. We know

and L is an intrinsic property of the star. So

(

)

(

)

(

)

( ) Where d=distance to star in parsecs. Most stars have absolute visual magnitudes within the range -6 < Mv < 16 Sun has absolute visual magnitude Mv = 4.8, and it is a very average star. 9.2.4 U B V passbands So absolute magnitudes of stars can be used as a comparison of the stars’ brightness. But to determine the magnitude we need to measure the stars’ flux. In practice the flux density is measured using detectors and filters which are sensitive to different wavelength ranges. We can measure magnitudes in different wavelength bands. The most well known of these is the “UBV “ system. Three standard filters are: U “Ultraviolet” B “ blue” V “ visual” passband Each of U, B, V filters about 100 nm wide, centred on following wavelengths: λU ~ 365 nm λB ~ 440 nm λV ~ 550 nm So e.g. we talk about: apparent visible magnitude mV And absolute visible magnitude MV These magnitudes measured in the different bands can be used to assign a “colour” to the star and give an indicator of T.

But d2 = 10pc

Example: what is the luminosity of a star whose distance is 60pc and with an apparent magnitude of 3.61 ?

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9.2.5 Bolometric magnitude The total apparent or absolute magnitude of the star, taken over all wavelengths, is usually called the bolometric magnitude. The difference between the bolometric magnitude of a star and its magnitude in, for example the V band, is called the bolometric correction, BC. Theorists are interested in the total energy output of star. The magnitude corresponding to total energy output is called the bolometric magnitude: mbol , Mbol. [Where, as before, Mbol and mbol - M denotes “absolute”, and m denotes “apparent” bolometric magnitudes.]

A star’s energy output is approximately black-body in distribution, so its effective temperature can be calculated from measuring two of mU, mV, mB. Then from say mV, the bolometric magnitude mbol is calculated by adding the bolometric correction BC: mbol = mV + BC BC values are tabulated for each type of star (OBAFGKM and Luminosity class), and are generally negative, since there is more energy in the whole spectrum than in a limited part of it. Exercise: Mbol = 0 for a main-sequence star with L = 3 x 1028 W. Show that Mbol

= -2.5 log ( L /(3 x 1028))

and calculate Mbol

and mbol

9.3 Stellar colours and temperatures Recall that stars radiate approximately as black bodies, so their intensity is given by the Planck function: Measuring a star's brightness in (say) U, B, V bands gives a measure of its effective (surface) temperature. Define colour indices: U – B = mU – mB and B – V = mB – mV Teff 40 000 K 9 900 K 4 900 K U – B -1.15 0.0 0.47 B – V -0.35 0.0 0.89 Cooler stars are redder ; hotter stars are bluer U-B, B-V positive U-B, B-V negative We can therefore determine a star’s temperature using colour indexes, known as the photometric method of temperature determination. (Note: this is independent of distance.) However, a more accurate measurement of photospheric temperature can be obtained for individual stars by the spectroscopic method (explained in lecture 7) based on analysis of the spectral absorption lines in the observed stellar spectra.

T

V B U log B

log

Note: x axis is frequency in this plot, unlike the Planck distribution plots shown in lecture 7, which are against wavelength.

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9.4 Measuring stellar distances: Method of trigonometric parallax Distance is measured directly geometrically by this method. It is important to know the distances in order to: be able to estimate the luminosity of an object; to find masses of objects from their orbital motions (using Kepler’s 3rd law); and for estimating physical sizes of objects. Nearby stars have small periodic apparent motions (with respect to more distant background stars) due to Earth’s motion in its orbit around the Sun. This is not a “true” motion and is called Stellar Parallax.

Naturally as we observe the parallax of more distant stars the parallax decreases with increasing distance to the star. As the nearest stars are still far from Earth, the largest measured parallaxes are of the order of less than an arcsecond. Our nearest star, alpha Centauri, has a parallax angle of 0.76 arcsec. Parallaxes are measured by both photography and digital imaging, and preferably from space to avoid blurring and problems observing through the atmosphere of the Earth. d=1/p Where: p = parallax angle in arcseconds , d = distance from Earth to star in parsecs [ Reminder: Parallax Second= Parsec (pc) Fundamental unit of distance : "A star with a parallax of 1 arcsecond has a distance of 1 Parsec." ] 1 parsec (pc) is equivalent to: 206,265 AU, 3.26 Light Years, 3.086x1013 km ] eg: a star has a parallax of 0.02 arcsec – what is its distance? [Ans:50pc] If p = 1 arc-sec, d = 1 parsec and If p = N arc-secs, d = (1/N) parsecs Limitations: if the stars are too far away, the parallax is too small for accurate measurement. Smallest measurable parallax from the ground is ~ 0.01 arcsec, so this method of trigonometric parallaxes is limited: although it is a good method to distances up to 100 pc, there are not many stars that are this close. However, satellite measurements (Hipparcos) measure parallaxes to accuracy of 0.001 arcsec. Hipparcos measured parallaxes for about 120,000 stars, out as far as 1000pc for brighter stars. Gaia space mission will measure parallax accurate to a few arcsecs !

9.5 Proper Motion Stars not fixed in space: they move relative to the Sun. The motions however are very tiny seen from the Earth.

Two components of the motion: Motion along line of sight does not change star’s position on sky. Motion perpendicular to it does change star’s position in the sky (by tiny amount). Such a change in position of a star is called its proper motion, so called because it is intrinsic to the star, and not a result of the motion of the observer or a moving reference point. Proper motion is usually expresses in seconds of arc per year (3600 arc sec = 1 degree).

μ

d

In the diagram the line of sight to the star in June differs from that in December, when the Earth is on the other side of its orbit around the Sun. The nearby star, as seen from Earth, appears to sweep through the angle shown. Half of this angle, is the parallax, p. d=1/p

d

6

We can find the traverse speed of the star using the proper motion from vt = d sin μ (μ = proper motion, d = distance to star, which could be obtained by measuring its parallax) small angle, sin μ ≈ μ Example: Barnard’s star has the largest known proper motion: 10.3 arc-sec per year and also has parallax 0.55 arc-sec (hence lies at 1.8 pc from Earth). The radial velocity (vr), the component of the star’s motion relative to us (toward or directly away from us) can be obtained by a method relying on the Doppler effect. This gives a change in the wavelength of a spectral line emitted by this object ( ). Knowing this wavelength shift (by measuring it in a stellar spectrum), and comparing it to the known wavelength of the transition line, (measured in the laboratory) the rest wavelength() , and using:

( )

gives us the radial velocity . [ redshift: increase in wavelength, object moving away from us, blueshift: decrease in wavelength, object moving towards us]. Note: since, in general, the proper motion of a star would be expected to decrease with distance from Earth, the proper motion of a star can be used as a rough guide to the star’s distance. The radial velocity and transverse velocity specify the overall motion of the star through space with respect to Earth. True Space velocity of star √

These overall motions of the stars are not random, they are in part related to the large-scale motions in our Galaxy, and by the groupings of many stars in clusters. In fact, whether a star belongs to a cluster can often be decided by comparing its motion through space with that of the other cluster members.

9.6 Brief summary: Some important distance measurement methods: 1] Parallax - method possible for nearer stars 2] Proper motion – greater for nearer stars in general, measure many stars’ proper motion and we get a rough guide to distance 3] Variable stars – Period-Luminosity relation, measure period, calculate L, then measure apparent magnitude (flux), and so get absolute magnitude – and then get distance (not covered as part of this lecture course, but it is a very important method) 4] Colour and spectral type: gives Teff, but L=4R2 T4

eff, assume stars of same type have approximately the same size, so use R of similar spectral type star whose radius is known, then with T and R, get L. Measure apparent magnitude with detectors from Earth, knowing L, then calculate absolute magnitude – which then gives you the distance. LECTURE 9: SUMMARY and outcomes Ranges of stellar parameters M, R, L, Teff – be aware of, and how the sun compares Definition of apparent and absolute magnitude – be able to define, know and use Apparent magnitude– relation between apparent magnitude, distance, luminosity, be able to

write down relevant equations and use. Definition of parsec – be able to define a parsec U B V photometry – be able to explain what this is Bolometric magnitude, bolometric correction – be able to define and explain Colour indices – be able to explain, and describe Stellar temperature measurement from colour indices – be able to explain and discuss Measuring stellar distances – by trigonometric parallax – be able to explain and calculate

parallax or distance Proper motion – be able to explain & describe this, and comment on use as distance estimate TO DO Questions now do-able: Problem sheet 2, Q 4,5,6,8,9, Problem sheet 3: Q 1,2,7

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Sun Stars Planets 10. J C Pickering 2014

Stars - Putting the Sun in context Lecture 10: The Hertzsprung-Russell Diagram

10.1 Observers’ H-R diagram 10.2 Theorists’ H-R diagram 10.3 H-R diagrams: 100 brightest stars, 100 nearest stars 10.4 Distribution of stars on the H-R diagram 10.5 Mass and the H-R diagram 10.6 H-R diagram and clues to stellar evolution 10.7 Mass and rate of stellar evolution – clusters: open and globular To understand more about stars and obtain insight into their evolution we need to look at the overall distribution of stellar properties rather than individual properties of individual stars.

We can make use of Hertzsprung-Russell Diagrams to systemize our observations of stars. Suitable properties for comparing stars are Teff, L and R – however we don’t need all three – because stars emit like black bodies, Teff L and R are related (L = 4 π R2 σ Teff

4) - so if we know two of Teff , L and R, then we can get the third property. 10.1 Observers’ H-R diagram: Colour-magnitude diagram - add during lecture Note: magnitude is always plotted increasing downwards (larger magnitude=lower luminosity) Absolute magnitude is related to luminosity, (see lecture 9) B-V colour index, spectral type and effective temperature are related (see lectures 7 and 9)

So possible HR diagrams include: Colour-magnitude, and log L – log Teff, etc 10.2 Theorists’ H-R diagram: log L – log T eff

Since L = 4 π R2 σ Teff4 , R= [L/(4 π σ Teff

4) 1/2 so for each point on the HR diagram there is a unique stellar radius. So for stars of the same radius : d log L = 4 and stars of the same radius lie on lines of slope –4 in the H-R diagram. d log T

[ NB: temperature always plotted increasing to the left]

Annotate during lecture

2

10.3 HR diagrams – brightest and nearest stars

10.3.1 HR diagram of the 100 brightest stars

10.3.2 H-R diagram of the nearest 100 stars 10.4 Distribution of stars on the H-R diagram Assume: • any particular star is luminous for only a finite time

• there are distinct stages between star’s birth and death – each stage characterised by some range of Teff and L; thus the star moves around the H-R diagram as it evolves

• stars we see today are not all at the same stage of evolution

So: • if a large population of stars is observed today, then the longer a particular evolutionary stage lasts, the greater will be the number of stars that we observe in that stage. Only very few stars will be observed going through a short lived stage.

• the concentrations of points on the H-R diagram are those regions where stars spend a comparatively large fraction of their lives. On this basis a star must spend most of its life on the main sequence, because this is where 90% of the stars lie. Based on the above

-8

Luminosity Class I

Luminosity Class II

Luminosity class V

2

16

Mv

A0 M5

4

B0 G0 M0

Mv

absolute

magnitude

Annotate during lecture

3

assumptions we can say that the red giant, supergiant and white dwarf regions are where we might expect some stars to be for a significant time.

• the concentration of stars on the H-R diagram depends not only on how quickly a star passes through a region, but also on what fraction of stars pass through the region at all.

• some regions of the H-R diagram might appear to have no stars, or scarcely any, simply because they correspond to stages in a stellar lifetime when stars tend to be shrouded in cooler material and are therefore not observable directly.

Hipparcos

The Hipparcos satellite carried out precise measurements of all stars down to approximately magnitude +9 of: position, parallax, proper motion and photometry (colour index etc) to unprecedented accuracy. http://www.rssd.esa.int/Hipparcos An H-R diagram of the stars measured using Hipparcos is shown below:

4

10.5 Mass and the H-R diagram So we have our observational data: absolute magnitude and colour index, which gives us L against Teff – and hence R, but one further property of a star is enormously important – mass.

•••• Stellar masses are measured from 0.08 M

to about 50 M

. The Sun is a very average star. •••• The lower the mass, the greater the number of stars – huge stars are rare. If we have a first look at stellar evolution: (see diagram above) – looking at the different masses on an H-R diagram – notice:

(i) supergiants tend to have greater masses than red giants which have a greater mass than white dwarfs.

(ii) Within the class of supergiants, red giants or white dwarfs, there is no correlation of M with L or photospheric Teff – masses appear “jumbled”.

(iii) Along the main sequence (MS) stars, mass does correlate to L, and hence Teff. As mass increases, L and Teff increase. In fact we see a 500 times increase in mass giving a 1010 increase in L along the MS.

(iv) In the lower MS, masses are comparable with red giants, in the upper part of the MS the masses are comparable with supergiants.

Do stars change their mass during their evolution?

We now have information on masses of stars on the H-R diagram, but before we use our L, Teff and M information to look at stellar evolution we need to know whether a star’s mass varies during its lifetime. From observational evidence:

• one observes MS stars, red giants and super giants losing mass in terms of stellar winds. But this mass loss is very much less than the star’s initial mass.

• In images of planetary nebulae we can see impressive mass loss, with shells of material having been flung off by a central star. Some stars end their live by shedding a planetary nebula or more violently a supernova – great mass loss…

• So for most of the life of a star, severe mass loss occurs only when a planetary nebula is shed, and the stellar remnant becomes a white dwarf, or when a massive star ends its life as a supernova.

10.6 Using H-R diagram features to devise a model for some of the stages of stellar evolution At this stage, with the features we see in the H-R diagrams, and knowledge of stellar masses, and mass loss, we can deduce the following…

1] Main sequence phase : during the MS phase a star does not change L or photospheric Teff very much (otherwise it would move along the MS and this doesn’t fit with the range of masses seen).

(From: S.Green & M Jones, An Introduction to the Sun and Stars (Cambridge University Press)

Diagram: Stellar mass and the H-R diagram Masses are given in multiples of M

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However stars do drift very slightly above the MS during the MS phase, hence it appears as a band in the H-R diagram rather than a narrow line.

2] After the MS phase – less massive stars become red giants , and more massive stars become supergiants . This is consistent with stellar masses seen in these regions in the HR diagram. Also consistent with the rarity of supergiants – there are very few very massive stars.

3] red giants evolve to a point where they shed planetary nebulae and the stellar remnant evolves to become a white dwarf

4] super giants become star destroying type II supernovae .

10.7 Do stars of different masses evolve at different rates? To answer this question we can look at star cluster HR diagrams. Make 2 assumptions:

1) That star clusters occur because the stars in them form at about the same time. 2) The compositions of stars in a cluster are similar as they formed from the same cloud of gas.

These two points are vital – if the stars form at the same time and have the same composition then the stars in the cluster only differ in their mass, they are a homologous group. So, looking at an HR diagram of stars from a single cluster will show if stars with different masses evolve at different rates.

10.7.1 Open Clusters (galactic clusters) These are comparatively young systems (age 100 million to a few billion years),with 102–105 stars, egs: the Pleiades (M45) and NGC 188.

Example HR diagrams for open clusters: In the H-R diagram of stars in the Pleiades cluster notice: • Almost all the stars are on the main sequence. This cluster is not old enough for many stars to have reached the end of their MS phase.

• The most luminous stars visible on diagram appear to be moving away from the MS. • The upper end of the MS, where the most massive stars are expected to be, is unpopulated.

In the H-R diagram of NGC 188, notice: • The absence from the MS of all but the low-mass stars. • The presence of considerable numbers of stars between the main sequence and the red giant region, which could represent the higher masses missing from the main sequence. This suggests the more massive a star, the sooner it leaves the main sequence, and that most stars that have left the MS go on to become red giants.

• The most massive stars have left the MS, and must therefore have shorter MS lifetimes. • The point at which this depopulation of the MS occurs is called the Main sequence turn-off, and can be used as an indicator of the ages of clusters.

• Supergiants are absent in this cluster, which could be because massive MS stars, which are their precursors, are rare. Also, if massive stars evolve rapidly, then any supergiants could have become Type II supernovae, and have thus vanished from the H-R diagram.

Open cluster NGC 188 The Pleiades open cluster

6

• The absence of white dwarfs is presumably because they are too faint to detect. • This is an older cluster than the Pleiades, because in the Pleiades the MS is populated to higher stellar masses than the MS in NGC 188. This occurs because the more massive the star the sooner it leaves the MS. In NGC 188 there has been enough time for all but the low mass stars to leave the MS, whereas the Pleiades is too young (100 million years) for this to have happened.

10.7.2 Globular clusters are far older than open clusters, billions of years old, typically 106 stars.

10.7.3 Estimating cluster age from MS Turn-off point At the Turn-off point the star leaves the main sequence as it ends H burning in its core.

Cluster age =

Total energy released on MS()

Luminosity

But = mass H "burnt" × Nuclear energy released/Kg ()

= × Where = fraction of mass depleted of H before the star leaves the MS ≈ 20% = mass fraction of hydrogen

But ∝ and so ∝ / and we can write = ⨀

⨀/

∴ = =

⨀ ⨀

/

= ⨀⨀

⨀

/

⨀ can be read from the HR diagram at the MS turn-off point, ≈ 20%, estimated (eg 0.6),

can be calculated (eg ≈ 6.575 x 1014 J/kg). Putting these into the above expression allows the age of the cluster to be calculated. LECTURE 10: SUMMARY •••• Hertzsprung-Russell diagram : be able to understand, sketch, explain, discuss •••• Main sequence, Red giants, White dwarfs: introduction to these stars •••• Typical HR diagrams: be able to discuss and describe, understand various regions, and what we can deduce about stellar evolution. Nearby stars – Brightest stars – be able to describe, explain and comment on the difference in the HR diagrams for these •••• Open and Globular clusters - be able to comment, explain and describe the HR diagrams. •••• Estimate of cluster age using MS Turn off point - be able to estimate using HR diagram

In the H-R diagram of globular cluster M13 the more massive stars have evolved to become red giants, and more details of the further evolution will be given in the next lectures. The MS Turn-off point can be used to estimate the cluster’s age, in this case 14 billion years!

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Sun Stars Planets Lecture 11 J C Pickering 2014 Life of a star: part 1. The main sequence and post - main sequence evolution 11.1 Main sequence life 11.2 Life beyond the main sequence: post main sequence evolution Consider the three phases of stellar evolution: - The first phase follows the formation of a star from a cloud of interstellar material through its gravitational contraction and subsequent heating to the main sequence, defined as the point at which hydrogen burning begins. - The second phase follows the development of a star on through the main sequence (MS): a star spends most of its lifetime on the MS so this is an important, though least spectacular place. - The final phase, post main sequence life, has more rapid evolution that begins when H burning has been exhausted in the stellar core, through to the star’s death. 11.1 Main sequence life 11.1.1. The zero age main sequence ZAMS The curve defined by values of L and Teff corresponding to static stars that are homogeneous and have just commenced hydrogen burning forms the zero age main sequence (ZAMS). The star is essentially static, supplying energy losses by nuclear burning. Equations of stellar structure hold well (see earlier lectures). As we saw for the Sun, nuclear processes take place in the core. Nuclear reaction rates vary with T, so moving out from the centre of a star, the boundary defining the limit of nuclear reactions is sharp. The size of the core varies with stellar mass, as do the reactions and the mechanisms of energy transport to the outer layers of a star. On the main sequence: 11.1.2. Nuclear reactions and energy transport: Lower main sequence stars ( < 1.5 M

) Core H burning. Radiative transport from core, further out there is a convective envelope. Upper main sequence stars ( > 1.5 M

) Core T high enough for CNO H burning to take place, and energy release in the core is sufficiently concentrated to trigger convective instability – and the centre of the star is convective. Diagram: Structure of 1 M

MS star 5 M

MS star

2

11.1.3 Why do we have a main sequence? Consider L and T and why they remain constant for so long: (i) stability A stable star is in hydrostatic equilibrium: the inward force of gravity on each layer of a star is balanced by the net outward pressure forces. - If there is a small change what happens? – if a star’s radius decreases a little at constant T, then (a) the density of gas in each layer of the star would rise as the volume has decreased. As P = k T/m for each layer of the star, the P will also rise, and (b) the gravitational force will also rise a small amount if the radius is decreased slightly – but this will be accompanied by a rise in T due to conversion of the gravitational potential energy into kinetic energy of the gas particles, and hence the P also rises. (c) because of these P rises the net outward force due to the variation of P with depth will also rise and there will no longer be a balance with the inward gravitational force. The star will thus expand back to its original dimensions. A star on the MS is very stable against any dimensional changes. (ii) The Russell-Vogt theorem states “The equilibrium structure of an ordinary star is determined uniquely by its mass and chemical composition”. A certain mass of stellar material of fixed composition can only reach one stable configuration. This stable configuration corresponds to one point on the H-R diagram. A star of different mass occupies a different point on the H-R diagram. We have a MS because the stars on it are stable, with similar chemical composition but with different masses. ( Stars away from the MS have a different chemical composition.) 11.1.4 Main sequence lifetimes We have already seen in previous lectures that the more massive the star the shorter its MS lifetime.

Lifetime on MS:

So for a star double the mass of another, its MS lifetime is 1/8 as long. Note: to calibrate with lifetime scale we could use the Sun, with 1010 years and M= 1 M

So massive stars, eg M = 15 M

are predicted to have relatively short MS lifetimes, even as brief as 10 million years. So, many of the massive upper MS stars currently observed must have formed fairly recently on the astronomical timescale. [NB age of Sun 4.5 x 109 years] 11.1.5 range of stellar masses There are far more low mass stars than high mass stars. Are we seeing all the low mass stars? The less massive a star is, the less luminous it is, and the harder it is to observe – the true number of low mass stars is easy to underestimate. Lower limit of the mass of a star: The lower the mass of a star, the lower is the core T. At lower T, energy generation from nuclear processes is reduced and becomes insignificant. Objects, more massive than planets, but < 0.08 M

, which is not sufficient mass to run nuclear reactions at a rate high enough to match the surface radiation rate, are called brown dwarfs. These brown dwarfs have a low T and don’t emit much visible radiation, they radiate in the IR (infrared). Brown dwarfs can be distinguished from gas giant planets as: they form from the ISM (interstellar medium) and so have a similar starting composition as other stars; whilst planets form by accretion of dust in nebulae surrounding protostars.

3

Upper limit of the mass of a star: - limited initially obviously by the mass of the original contracting cloud (not a severe limit as typical cloud masses are several thousand M

s). - and also, in the most massive stars there is an important process which is not important for lower mass stars: radiation pressure, for photons within a black-body source, is given by: Prad = 1/3 aT4 (a=radiation density constant) whereas gas pressure Pgas= n k T (n=number density of particles) and so Prad / Pgas = aT3 / 3 nk Note: For the Sun, radiation pressure is negligible compared to the gas pressure. But as T increases with higher mass stars, the effect of Prad will increase rapidly compared to the effect of Pgas . The stability of lower/intermediate mass MS stars results from the balance of gravitational force

and force due to gas pressure. For most massive stars stability needs a balance between gravitational force and the force due to

radiation P. Modelling has shown that radiation P increases so rapidly with T that a very massive star would

easily be blown apart by the radiation P. Upper limit to mass of a star is 100 M

but its lifetime would of course be very short. 11.2 Life beyond the Main sequence There are two main principles important for understanding stellar evolution

1. a battle between pressure and gravity 2. fusion of heavier elements takes place at successively higher temperatures

H-R diagram showing the evolutionary track of a moderate/low mass star – eg the Sun

Add labels A – H during lecture

4

Recapping, for the main sequence (MS): [A] Main sequence the distribution of stars on the HR diagram shows that most stars lie on the MS stellar models show that the stability of MS stars is the result of a balance between inward force of

gravity and outward force of gas pressure (or radiation P for higher M stars) outward pressure is sustained by the energy provided from fusion of H to He the rate of energy production and lifetime of a star on the MS is highly dependent on the T and

hence the M of the star. as H burning continues, H is converted to He in the star’s core, so its composition is changing a

little. the mean molecular weight, increases and thus central Pc drops. As P drops the core contracts a little, so central density c increases, and hence Tc increases to restore the P balance, resulting in a slight increase in L during a star’s MS life. This explains the finite width of the MS on the H-R diagram, with stars of the same mass but different ages, and hence different compositions, being in slightly different locations.

The end of the star’s MS lifetime is marked by exhaustion of hydrogen in the star’s core. Structure of a: (a) MS star and (b) star leaving the MS because of core H exhaustion. 11.2.1 Low/moderate mass star post main sequence evolution – eg Sun [B] A critical point will come when hydrogen in the star’s core is all gone. Nuclear reactions in the core stop and the star leaves the main sequence. The core slowly starts to contract as it is no longer releasing energy at a sufficient rate to generate

a pressure gradient sufficient to support the surrounding layers. Because of the contraction of the core, gravitational potential energy is converted into thermal

energy and hence the T will begin to rise. [C] Eventually the shell of unprocessed hydrogen surrounding the original core will be heated

sufficiently for hydrogen burning to start in a clearly defined shell – shell hydrogen burning. The star is now ascending the red giant branch (RGB) with energy production from

hydrogen shell burning. Because of the shell source of H burning thermonuclear reactions (no reactions yet in core of He),

the envelope of the star expands as the core continues to contract, – during this process the L remains nearly constant.

L = 4 R2 Teff4 so surface T (Teff) actually drops (as L constant and R is increasing).

Because of dropping surface T, the star moves horizontally to the right on the HR diagram. Eventually L increases as convection carries energy to the star’s surface and the star continues

ascending the RGB.

H – burning core

Add in lecture

5

For a star of 1 M

: the core has been compressed to 1/50 th of its original size, the core T has risen from 15 x 106 K 100 x 106 K, radius has increased from 1 R

10 R

and surface T has dropped to 3500 K. The star now shines orange-red: it is a red giant:

- add - Structure of a red giant star

[D] Once the core T has risen sufficiently, at this point helium burning starts in the star’s core.

3 4He 12C “Triple-alpha thermonuclear reaction” How does He burning start? In the red giant star core contraction had continued, driving the envelope out rapidly. In low mass stars electron degeneracy sets in before core He ignition is reached. On He ignition, the energy release raises the core T. When He ignition does occur, because matter in the core is degenerate, there is no significant

change in P; so the increased core temperature Tc leads to even further energy release (nuclear fusion at a faster rate), and a thermal runaway develops – known as a helium flash – and evolution is dynamic.

The huge energy release rate at the peak of the He flash (~ 1011 L

) lasts very briefly and the large energy release rapidly lifts the electron degeneracy and allows the core to expand

(because of the high Tc) as this happens the core T drop means that He burning ceases so the envelope contracts slightly. Most of the increased luminosity of the core goes into expansions so star’s luminosity decreases. Finally the core re-adjusts its structure so that He burning can occur under non degenerate

conditions. The star is now on the horizontal branch of the HR diagram

[E] The star is now in the region of the HR diagram called the horizontal branch (HB) On the HB the star is now in a state of core He burning and shell H burning. Structure of a horizontal branch giant star. Enlarged view of the core -ADD

6

Eventually the He in the core of the HB star is exhausted (has burnt C and O). As before, the core contracts, and so core P and T increase until He ignites in a shell just outside

the core, with H burning in the next shell outside of that. The star is in a double shell burning stage. The mass of the now inert core, consisting of C and O, continues to increase as the He shell

produces more C and O, and the core continues to contract further raising the core T. As the T continues to rise, the energy generation of the two shell sources goes even faster, and

the increasing L distends the outer envelope of the star. The star moves up to the asymptotic giant branch.

HR diagram showing evolutionary track of a solar mass star from MS to AGB.

[F] On the asymptotic giant branch (AGB) – the asymptotic giant star is in a state of double shell burning (with an inert core, and H and He burning shells). Add diagram below during lecture: Structure of an asymptotic giant star. Enlarged view of the core

HR diagram of a globular cluster showing the: Main Sequence (MS), Red giant Branch (RGB), Horizontal Branch (HB) and Asymptotic giant branch (AGB)

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Eventually: more contraction of the core causes free electrons there to become degenerate. As the shell sources burning H and He generate higher luminosities the star may become a large

red giant or supergiant, depending on its mass, with high energy expenditure, and so is not able to live much longer. This is not a stable situation!

The He-burning shell is thin causing a thermal runaway: - as He shell burning gets underway, the shell (because it is so thin and insubstantial), cannot lift the material above it – and so cannot expand. So the temperature rise resulting from He burning is not moderated. With this temperature increase, the He burning rate increases yet more, and so increasing the T even further. - This leads to rapid energy release, a thermal pulse, (which can last a few 100 years), which is approximately periodic with intervals of 104 – 105 years, and considerable expansion. After the thermal pulse the T has dropped and so the cycle of thermal runaway then repeats itself. This thermal runaway, and thermal pulse means that the star is variable and undergoes mass loss from its outer envelopes. The ejected material may form a circumstellar shell. High mass loss is characteristic of the

very late stages of evolution of a giant. These extended gaseous and dust envelopes were historically called planetary nebulae.

The central hot star is still contracting, and the envelope continues to expand and is thrown off as a planetary nebula

[G] what is now left behind after mass loss as a planetary nebula is the naked core of the star – a white dwarf. In a white dwarf degeneracy pressure and gravity are in balance. [H] The only source of energy remaining for the white dwarf is thermal energy, and the star gradually cools getting dimmer and dimmer.

H-R diagram with evolutionary track for a star from red giant, through planetary nebula stage, and cooling to white dwarf stage (plotted are dots showing central stars of planetary nebulae, and open circles cooling white dwarfs).

8

11.2.2 high mass star post main sequence evolution (M >10 M

) what we already know: Massive stars have a shorter lifetime on the MS than less massive stars Due to their higher mass they have higher central T, so H burning reactions proceed faster Because of the faster nuclear reactions, energy is released more quickly and so the stars are

brighter – more luminous For a short while these stars are hot and bright on the MS – O and B type stars. Evolution for high mass stars: [A] hot and bright O and B type stars on the Main sequence. H burning through CNO cycle Towards the end of the MS life, H becomes depleted in the core – end of MS life [B] H depleted in core – H burning in core stops Same process as for lower mass stars now happens: core contracts and so core T rises. [C] Red giant – H burning shell H shell burning stars. The core becomes choked with “ash” = nuclei that are a products of the

nuclear reaction. As nuclear reactions diminish (moves to right in HR diagram), there is no longer the pressure

gradient from the escaping radiation to balance the gravitational force, and so the core contracts under gravity

Contracting core raises the density and T of the core, so this “ash” (Helium) begins to burn [D] He burning begins in core. The ignition of He begins without a helium flash (as M > 2.25 M

) With He burning, the T in the core rises, increasing the rate of energy release (nuclear reaction

rate depends on T). (star moves to left in HR diagram) If stellar material is fairly opaque, energy finds it hard to escape. So the T rises, and then the gas

expands and cools, the drop in T leading to a drop in the rate of nuclear reaction Star is stable – lies on the horizontal branch and is a He burning supergiant the colour of the star depends on its surface T – which depends on its mass.

Structure of a He burning supergiant

9

As the He in the core diminishes noticeably, the He burning nuclear reaction wanes. This means that the P gradient due to escaping energy diminishes, so the core of the star once again contracts under gravity and so the core T goes up again. When the core T reaches 3 x 108 K carbon burning can start in the core.

12C + 4He 16O + Outside the core (C burning) there are shells burning HeC and HHe.

The supergiant star goes through this pattern of steps a number of times as successively heavier elements become scarce in the core. But each new fuel releases less energy than the previous one on burning. There is a process of diminishing returns… This cycle continues and the star evolves by successively using up one round of fuel in the core,

undergoing core contraction to raise the T sufficiently to ignite the previous ash into a new fuel. Finally iron is produced in the core – and the story reaches a final climax Iron is the ultimate slag heap of the universe – no further nuclear extraction of energy is possible

by burning of heavier elements at this stage. Once the core is iron, the core has no alternative but to contract because of the halt in nuclear

reactions. The core contracts and heats up catastrophically – no new source of energy can be found to balance the gravitational force. Every channel of heat loss only results in the star contracting even further…

Very high temperatures are reached, several billion K. Intense temperatures lead to iron photo-disintegration, taking the core’s heat energy to overcome the particles’ binding energy.

Now there is no balance for the gravitational force – catastrophic core collapse results, and it falls inwards freely under its own self gravity.

The rapidly rising density squeezes the free electrons into the protons to form a huge hot mass of neutrons.

A supernova explosion results leaving either a neutron star or a black hole 11.2.3 Diminishing returns – nuclear reactions in core Each new thermonuclear reaction is less efficient than previous one at releasing energy So the reactions need to proceed at a higher rate to yield radiation to balance gravity Faster reaction rates mean more neutrinos are produced – these carry away a growing proportion of the energy generated

Schematic diagram of structure of a supergiant star, last day! H-rich outer envelope limits are not shown.

Schematic of supergiant star structure: C burning stage, with He shell burning and H shell burning

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The star goes through its life cycle at an ever increasing pace – squandering its reserves at a faster and faster rate:

Duration: Burning reaction stage: H 7 x 106 years

He 5 x 105 (~10% of MS lifetime) C 600 years Ne 1 year O 6 months Si 1 day !

HR diagram showing evolutionary tracks for low mass (1 M

) and high mass (5M

) stars.

LECTURE 11: SUMMARY •Hertzsprung-Russell diagram: evolutionary tracks of low mass and high mass stars: understand, and be able to sketch, explain, and compare and contrast • Post Main sequence evolution: know different stages, be able to describe and explain what happening in star in terms of nuclear reactions, physical changes and processes – and position at each stage of evolution on H-R diagram. Be able to sketch star’s structure at each evolutionary stage.

Predicted paths of stars on the H-R diagram as they evolve off the MS to the red (or supergiant phase): Note- detail for 1 and 2 M

tracks is not included (see earlier diagram). A onset of hydrogen core fusion – start of MS life B dashed line B marks end of H core fusion – end of MS life – and start of H shell fusion C hydrogen shell fusion continues D helium core fusion starts E helium core fusion continues F helium shell fusion starts

All extra diagrams in this handout are taken from “Sun and Stars”, Green and Jones.

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Sun Stars Planets 12. J C Pickering 2014 Lecture 12: The Life of a star, part 2: STAR FORMATION 12.1 Where do stars form? 12.2 Formation of a protostar. 12.3 H-R diagram, Hayashi tracks, evolution onto the main sequence (MS) 12.4 Observations of protostars, bipolar outflows, and protoplanetary disks Consider the three phases of stellar evolution: • The first phase follows the formation of a star from a cloud of interstellar material through its gravitational contraction and subsequent heating to the main sequence (MS), defined as the point at which hydrogen burning begins. • The mid phase follows the development of a star on through the main sequence: a star spends most of its lifetime on the MS so this is an important, though least spectacular place • The final phase of stellar evolution, has more rapid evolution that begins when H burning has been exhausted in the stellar core, through to the star’s death. This lecture covers star birth. (Lecture 11 covered main sequence and post MS phases.) Stars are born from more widely dispersed gas which condenses because of the gravitational attraction of the gas on itself. 12.1 Where would we expect to find star birth? Interstellar space is not empty – it contains dust and gas; the Interstellar Medium (ISM). Observational evidence seems to point to the dense clouds as being interstellar nurseries:

we observe young star clusters which seem to be surrounded by the remnants of the original cloud from which they are formed (dust and gas – eg Orion nebula)

some dense clouds have a large number of compact infra red (IR) light sources – this is due to either light being absorbed by dust and re-emitted at the temperature of the dust (in the IR: from Wien’s law), or the source is cool and emits only in the IR.

The IR radiation comes not from the star itself but from the cocoon of dust still surrounding it. 12.2. Formation of a protostar 12.2.1 Contraction of a dense cloud: (i) spontaneous and (ii) triggered cloud collapse (i) spontaneous collapse – Jean’s instability

All atoms, molecules and particles in a cloud are attracted to each other by gravitational forces. Each particle is affected by the gravitational attraction of the combined mass of all the others.

However observations show many clouds appear to be in a state of equilibrium – ie not contracting. So each gas particle is in continuous motion (average translational KE = 3/2 kT). This motion produces a gas pressure that provides an outward force to counteract the tendency of the gas to contract. So there is a balance of gas pressure against gravitational force: if the force due to gravity is greater, then gravitational contraction occurs.

For a uniform spherical cloud “Jean’s mass”, MJ , is the mass above which the gravitational force will overcome the opposing pressure due to particle motion:

(

)

(

)

Where n = particle number density, m=mass of “average” gas particle, T=gas temperature So at lower T, lower mass clouds will contract more easily than at higher T.

2

(ii) collapse triggered externally Trigger mechanisms are probably needed to cause a cloud to change from an equilibrium state to one in which contraction has been initiated. A slight increase in density is required to trigger Jean’s collapse. Possible triggers include: a shock front: from a supernova has a compressed gas region just inside the expanding shell. regions where several O and B stars form (OB associations). These, forming close together,

produce large amounts of visible and UV light causing a shock in the material in the surrounding ISM (due to force of radiation pressure).

Spiral density waves in a galaxy maintain a galaxy’s spiral structure and compress all material they pass. Star formation regions are more concentrated in spiral arms.

Close approach or collision of another cloud or star: produces a local gravitational disturbance which could trigger gravitational contraction.

Star formation can be triggered throughout an entire galaxy (called a starburst galaxy) by the interaction with another nearby galaxy.

12.2.2 Fragmentation Stars are often found in clusters – they appear to form from fragments of a cloud, each collapsing on its own. They form open clusters of typically a few hundred stars. 12.2.3 From fragment to protostar A star is formed in a region of large radius and low temperature in the HR diagram. If the collapse is quasi-static according to the virial theorem (see earlier lecture), half the energy of collapse goes to heating the material, the other half is radiated away. If the collapse is not quasi-static, some of the energy of the collapse goes into rotational energy of the whole star (this is what happens in practice). Initially - the gas is transparent.

- Collapse occurs at nearly uniform T – most of change in gravitational PE goes into the moment of inertia of the star – collapse is rapid.

Then - as contraction occurs, R (fragment radius) decreases so gravitational PE decreases. Energy is conserved, so gravitational PE converts to KE of the molecules - the KE increases, and is converted by mutual collisions of molecules into thermal energy of the gas, and T goes up. The excited molecules emit radio waves, microwaves and IR radiation, which escape the cloud as it is initially transparent, so the T rise remains minimal (only 10-20 K). As contraction progresses further: - the number density of molecules increases, so it is harder for the radiation to escape – it becomes trapped by the surrounding layers. The gas becomes opaque to radiation; so now some of the gravitational PE goes into heating, and the fragment’s internal T rises more rapidly. This heating reduces the rate of collapse- a state of quasi-static equilibrium is approached. The interior of the star may be hot enough for H burning to begin, at which point the collapse is halted and the star adjusts itself to equilibrium on the zero age main sequence. Note: for this H burning to begin there must be enough gravitational PE available to raise the T to the hydrogen burning ignition point. For a gas cloud of very low mass, this will not happen as the core T will be insufficient for H burning, and in this case the collapse is halted by electron degeneracy pressure. The lowest mass star than can burn H is 0.08 M

. Objects of lower mass than this (13-80 MJ, Jupiter masses), substellar objects, brown dwarfs, have core deuterium burning if they are sufficiently massive for core T to exceed 106 K. Brown dwarfs may shine for a hundred million years at most before their deuterium supply is burned out.

3

12.3 HR diagram and evolution onto the Main sequence On an H-R diagram a fragment of dense ISM cloud falls in the low T region. The track of the contracting cloud fragment across the H-R diagram is far from certain, because the process takes place behind gas and dust, screening the fragment from view and takes place over a short timescale - so we are less likely to observe stars in this phase. We therefore rely on models, that predict: after a few thousand years of gravitational contraction, surface T rises to 2000 – 3000 K. the contracting fragment is still quite large and so L is high (L = 4 R2 T4

eff), approximately 10- 100 times its eventual L as a star on the MS. the fragment eventually becomes a protostar. The H-R diagram below shows the predicted tracks for protostars of various masses as they evolve towards the MS region – called Hayashi tracks.

Notice: the timescale for this early stage of evolution – the more massive the protostar, the quicker it reaches the MS. A 15 M

protostar takes 105 years to reach the MS, this is less than 1% of the time for a 1 M

protostar to reach the MS. Virtually all protostars that then become MS stars take less than 108 years to pass through the protostar phase, this is very quick compared to other phases of their life. The HR diagram track shapes depend on changes in internal structure and the way energy is transported through protostars as they collapse. Recall L = 4 R2 T4

eff , and notice: intermediate/low mass protostars – an early drop in L is due to the effect of the increase in surface T being more than off-set by the effect of the decrease in R > 2 M

higher mass prototstars – effects of increasing surface T and decreasing surface area just about balance, so L changes little as T increases (tracks are ≈ horizontal on H-R diagram). shortly before joining the MS, tracks generally show a drop in L as the effect of the contraction of the protostar tends to dominate over T effects.

Diagram: theoretical tracks (Hayashi tracks) of protostars of various masses across the H-R diagram as they evolve towards the MS. (Times for the protostars to reach different stages of their evolution are shown.). [Diagram from Sun and Stars, Green and Jones]

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12.4. Observations of protostars: Observation is difficult: the number of stars observed in a given phase of evolution should be roughly proportion to the time an individual star spends in that phase. When evolution is fast, as in this phase of evolution, then the statistics are against our finding any stars in that place. However, with radio and IR telescopes we are learning much more about star formation: for example, many protostars show a phenomenon called bipolar outflow.

As the surrounding envelope of dust around the protostar disperses, the accretion process stops, the central globe of gas is no longer a protostar – it becomes a pre-main sequence star. Eventually when the core has contracted further, the core T will be sufficient for H burning and it become a main sequence star.

12.5 Why do the planets orbit the Sun close to the ecliptic plane? Why are there bi-polar mass outflows observed in protostars ?

12.5.1 conservation of angular momentum, accretion disk formation Typically a real cloud from which a star would form would have a small amount of net angular momentum (for example due to galactic rotational shear, or other bulk motions or interactions). This angular momentum is conserved as the cloud collapses. Consider angular momentum L, particle of mass m, at distance from cloud centre r, with angular velocity , with centripetal acceleration ac and gravitational acceleration ag.

Since angular momentum l = m r2 ω, and l and m are constant:

But as r decreases during cloud collapse, ω must increase. What does this imply for a collapsing cloud?

The radial acceleration of a particle of mass m at radius r has 2 components:

(1) Gravity directed towards the collapsing cloud centre:

where M(r)is the mass contained within sphere of radius r.

As the cloud fragment contracts it spins faster and flattens into a circumstellar disc, or torus, with the protostar at the centre. Material in a disk rotating around the star moves radially and is gradually accreted. At the same time mass loss from the star is probably channelled in a bipolar outflow by the star’s magnetic field and the disk. Mass loss is a strong stellar wind - radiation pressure contributes to propelling the stellar wind (pressure exerted by photons on any object that absorbs or reflects them).

Diagram: Gas flowing away from protostar at high speed (50 km s-1 ) in opposite directions. Phase lasts ≈ 104 years.

5

(2) Centripetal term directed away from the axis of rotation

With the radial component thus being

Net radial acceleration

Note: a(r) for a rotating cloud is less than for a non-rotating cloud.

At the pole but at the equator

In the equatorial plane, accelerations are both radial (and assume the mass inside the contracting mass point is constant):

We have

and , but from earlier we know,

, so

As the contraction proceeds, and r decreases:

Initially r large, ω small: increases faster than

r decreases further, and eventually = , and at this point the gas doesn’t contract any closer

to the centre. (At points with even lower r, > ).

So contraction continues only in regions outside of the equatorial disk, but not in the plane of the disk. It is the balance between the centripetal acceleration and gravitational acceleration that causes a collapsing cloud core to form an accretion disk: ac is acting perpendicular to the axis of rotation of the gas cloud, and ag is acting radially. Flow of mass parallel to the axis of rotation is unimpeded by rotation, whilst flow towards the centre near the equatorial plane is prevented by the rotation.

← Annotate diagram during lecture

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12.5.2 Why are forming stars seen to have bipolar jets and outflows? Whilst the physics of precisely what is going on is still not clear, these outflows can in part be understood by considering the conservation of angular momentum. We have just seen that because of conservation of angular momentum: r 2 1/, so for a collapsing cloud, with initial radius ri

and angular velocity and i, and final stellar radius rf and angular velocity f

if ri 1015 m say, then (

)

The resulting star might be expected to have a rotation period of under a second if angular momentum were strictly conserved as the cloud contracts, and an equatorial rotation velocity greater than the speed of light… The Sun however has a rotation period of about a month… So a protostar must be losing angular momentum somehow. It does this by throwing matter outward – we believe this because of observations of jets of particles outward along a forming star’s axis of rotation. So the star is formed by accretion of matter from an accretion disk, but it is simultaneously throwing matter outwards at high speed along its axis of rotation. As this mass is thrown out, the rotation decreases, and then mass can once more flow inwards in the accretion disk to the forming star. The flow outwards at the star’s poles prevents accretion from the pole direction, so accretion is primarily from matter in the accretion disk. This accretion disk, and the matter remaining after the star has formed, is involved in planet formation, and hence accounts for the plane of the orbits of the planets, the prograde rotation and revolution of planets.

LECTURE 12: SUMMARY of stellar formation Protostars and evolution onto the Main sequence: be able to describe the different stages of evolution, physical processes, changes. Be able to sketch diagram of protostar with bipolar outflow. ISM – be able to comment on the ISM briefly with reference to star formation. Description of cloud collapse, and triggers: be able to list and comment on triggers, and describe processes in cloud collapse. HR diagram, Hayashi tracks for protostars: be able to sketch and comment on for 1 M

star. Protostars and bipolar outflow: be able to sketch and describe

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Sun, Stars, Planets 13. J C Pickering 2014 Lecture 13: Part I: Binary stars: measuring stellar masses Part II : The Sun in the Galaxy 13.1 Binary stars 13.2 Binary star orbits 13.3 Measuring stellar masses using binary star observations 13.4 The Sun in the Galaxy 13.1 Binary stars and measuring stellar masses Approximately 2/3 of stars are in binary systems. (Also even triple systems.) This is however not very surprising given that young stars are found in clusters. A “binary star” is in fact actually two stars – but they are so far away that they appear as one star in the sky – their angular separations are so small that the binary star usually cannot be resolved. Binary star systems provide one of the few ways of measuring stellar masses. The basis of the method of measuring the mass of a star is to observe how it moves when a force is applied to it: Newton’s 2nd law of motion F=ma. We must observe the star being accelerated. Any star is accelerated by any other mass in the Universe through gravitational attraction, but the effect is only large enough to measure if the other mass is large and relatively close to the star – a binary system. 13.2 Binary orbits The stars of a binary orbit a common centre of mass, COM, on elliptical orbits. Annotate during lecture: Define the angle i to be the angle between the line of sight to the binary system and the normal of the orbital plane of the binary system: Annotate during lecture:

2

We will assume that the stars have circular orbits about the centre of mass of the binary system (Annotate the diagram below during lecture:)

( ) ( )

These results are also true for elliptical orbits (radius of orbit r can be replaced by semi-major axis a). Notice: as M1 or M2 increase, P decreases, and as r (or a) increases, P increases. 13.3 Measuring the stellar masses Stellar masses for the two components of a binary system can be measured in three cases:

(i) visual and astrometric binaries (ii) spectroscopic binaries (iii) eclipsing binaries

(i) Visual binaries: here we can see both stars separately as 2 distinct points of light. We can measure each star’s orbit on the sky.

We can measure the ratio of the orbit sizes and use [13.3]

The angular separation of the stars, α, is measured. Distance, d, from the Earth to the binary star system is found from the measured parallax. Then: a = sinα /d Using [13.4]

and knowing the stars’ separation, a , and measuring the period P, gives M (=M1 + M2 ). ...and thus, having the ratio of the stellar masses and their sum we can find M1 and M2 individually.

Balancing the gravitational and centrifugal forces: For Star 1:

For Star 2:

[13.1]

[13.2]

[13.3]

[13.4]

α

d

a

3

(ii) Spectroscopic binaries: these are known only from the periodic variation of the Doppler shift of the spectral lines observed.It is not possible to resolve these as two objects when seen visually alone.

We know the velocity components along the line of sight (measured from the stellar spectra) but we do not know r. Annotate diagram during lecture Therefore [13.3] can be written:

And so we can also write:

(

)

We want to use [13.4] and express mass in terms of what can be measured, v1 or v2 . To do this we need to replace r with v. Start by multiplying [13.4] by (sin i)3 :

( )

( )

( )

(

)

( )

( )

“mass function” And we can also get a similar expression to [13.7] containing v2 and M1 . If one of the two stars is very faint (a “single-lined binary”), we can measure P and v1 , and derive only the mass function from these. For a “double-lined binary”, we can measure P, v1 and v2 , allowing us to get the stellar mass provided that we know inclination i. For an unknown inclination we therefore find a minimum mass by setting sin i =1.

Period P is measured from the spectroscopic observations. Projected velocities v1 and v2 are found, where:

[13.5]

[13.6]

[13.7]

4

If the orbit of the stars is seen nearly edge-on, we have … (iii) Eclipsing binaries: a bound pair of stars is deduced from periodic changes of the total light from the system arising from eclipses of one star by the other - variations in light curves due to eclipses i 90o

Get an accurate measure of m1 and m2 (Also the lengths of the eclipses can be used to estimate the radii of the stars,) [An analysis of the light curve can often yield an estimate of the inclination of the orbit of the system relative to the line of sight.] Further comments about binary systems Stars’ radii tend to increase with age (e.g. red giants!) If one star of binary system expands, it may reach stage where its outermost material is more attracted to the companion star. Mass transfer from one star to the other follows. Also can transfer angular momentum. These are called interacting binaries. This interaction may affect the evolution of both stars. Examples include: e.g. Algol (β Per) – first eclipsing binary discovered (P=2.87 days) high-mass MS star and a low-mass red giant e.g. dwarf novae brighten by 2-5 mag for a few days each month (probably an interacting binary with WD accreting matter onto a disk) e.g. binary pulsar neutron star rotating hundreds of times a second, spun up by mass

Brightness

Time

Example: Light curve of an eclipsing binary.

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13.4 The Sun in the Galaxy Galaxy classifcation The best known and most often used general scheme, in which galaxies are grouped according to their appearance was devised by Edwin Hubble. It splits galaxies into ellipticals, spirals (normal and barred), and irregulars, and is represented by the “tuning-fork” diagram below. Elliptical galaxies are graded from E0 (spherical) to E7 (very elongated) in terms of increasing eccentricity). Spirals have a planar disk – and are divided into those without (S) and with (SB) central bar. Normal spirals range from Sa (arms tightly wound around the nucleus) to Sc (arms widely spread from the nucleus), and, similarly, barred spirals from SBa (arms tightly wound) to SBc (arms widely spaced). Irregulars are designated Ir. To this original scheme, another category, S0, was added to describe lenticular systems with a nucleus surrounded by a disk-like structure that lacks spiral arms. Hubble’s tuning-fork diagram of galaxy morphologies

Comments on galaxy types Spirals have obvious rotation – and gas and young stars can be seen In ellipticals – stellar motions appear to be mainly random – so no rotation. There is little gas or dust, and little current star formation Origin of different galaxy types – may depend on initial conditions, whether there was significant rotation or not. The type will also depend on the history of the galaxy – it may have experienced collisions and mergers.

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Components of our Galaxy Schematic diagram of our Galaxy – ADD DURING LECTURE

(i) thin disk of stars, gas and dust (ii) central bulge (iii) halo of old stars. Globular clusters are found in a roughly spherical halo, distances

to these can be measured using variable stars. The Sun is 8.5 kpc from the Galactic centre in a spiral arm. Our Galaxy rotates with a period of 2 x 108 years – so the Sun’s velocity around the Galactic centre is 220 km s-1 . Of course the stars also have random motions (for example due to gravitational interactions.) Our neighbours…and beyond Our Galaxy has satellite galaxies: well known are the Large Magellanic Cloud (LMC) and Small Magellanic Cloud (SMC). The Galaxy is part of the Local Group of galaxies (includes M31). Galaxies tend to be grouped into clusters and superclusters. Galaxies are found on large scale to be moving away from one another: Hubble expansion of the universe. The recession velocity: v = H d Where H ~ 70 km s-1 Mpc-1 On very large scales we find galaxies tend to be confined to 2-D structures in space (a bit like walls of a bubble-foam.) LECTURE 13: SUMMARY Derivation of relations between binary masses, separations and orbital period – be able to do this from basics Using binaries to estimate stellar masses – be able to calculate Visual binaries – get masses if distance known – be able to calculate Spectroscopic binaries – be able to calculate msin3i (inclination i) Eclipsing binaries: most useful – understand and be able to use light curve information Mass transfer between binary companions leads to interesting evolution –be aware of this Galaxy types: elliptical, spiral, irregular – be aware of the different types, especially with respect of star formation Main components of our spiral Galaxy: bulge, disk - be able to sketch the components of our Galaxy, and comment on Sun’s place in Galaxy; Galactic rotation - be able to comment on this TO DO: questions 4 and 5, problem sheet 3.

Imperial College London

Physics Department

Sun, Stars and Planets

An introductory course in astrophysics

Dr Juliet Pickering, Dr David Clements

2013-14

Comments and corrections to d.clements@imperial.ac.uk

Dave Clements Office 1011 Blackett

Lecture notes may be found on Blackboard (http://blackboard.ic.ac.uk)

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Contents

1 The Planets Half of the Course 1

2 Planets: Textbooks 3

3 Outline Syllabus 5

3.0.1 The Examination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

4 An Overview of the Solar System 7

4.1 Aims of this Lecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4.2 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4.3 Overall Inventory of the Solar System . . . . . . . . . . . . . . . . . . . . . . 7

4.4 Mercury . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4.5 Venus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4.6 Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4.6.1 The Moon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4.7 Mars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4.7.1 Phobos and Deimos: The Moons of Mars . . . . . . . . . . . . . . . . 10

4.8 The Asteroid Belt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4.9 Jupiter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4.9.1 The Moons of Jupiter . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.10 Saturn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.10.1 The Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.10.2 The Moons of Saturn . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.11 Uranus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.12 Neptune . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.13 Pluto, Trans-Neptunian Objects (TNOs) and the Kuiper Belt . . . . . . . . . 13

4.14 Comets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.15 The Oort Cloud . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.16 Formation of the Solar System . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.17 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5 Planetary Orbits: Kepler’s Laws 17

5.1 Historical Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5.2 Kepler’s Three Laws of Planetary Motion . . . . . . . . . . . . . . . . . . . . 18

5.3 Derivation of Kepler’s Three Laws . . . . . . . . . . . . . . . . . . . . . . . . 19

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Contents

6 Terrestrial Planets: Heating, Cooling Processes and Interiors 23

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

6.2 The Unquiet Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

6.3 Primordial Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

6.4 The Structure of the Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

6.5 Long Duration Heat Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

6.6 The decay of long term heating sources . . . . . . . . . . . . . . . . . . . . . 27

6.7 Heat Loss from Planets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

6.8 Cooling Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

6.9 Volcanism and Tectonics on Other Terrestrial Planets . . . . . . . . . . . . . 30

7 Terrestrial Planet Surfaces and Temperatures 33

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

7.2 Major Factors in Shaping Planetary Surfaces . . . . . . . . . . . . . . . . . . 33

7.3 Impact Cratering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

7.4 Volcanism and Tectonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

7.5 Erosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

7.6 The Surface Temperatures of Planets . . . . . . . . . . . . . . . . . . . . . . . 36

7.7 The Greenhouse Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

8 Terrestrial Planet Atmospheres 39

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

8.2 Why do we have an atmosphere at all? . . . . . . . . . . . . . . . . . . . . . . 39

8.3 Atmospheric Density and Pressure . . . . . . . . . . . . . . . . . . . . . . . . 39

8.4 Temperature Variations with Height . . . . . . . . . . . . . . . . . . . . . . . 40

8.5 Thermal Escape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

8.6 Current Atmospheric Composition . . . . . . . . . . . . . . . . . . . . . . . . 43

8.7 Origin of Atmospheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

9 Gas Giants: Structure and Atmospheres 47

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

9.2 Basic Properties of Gas Giants . . . . . . . . . . . . . . . . . . . . . . . . . . 47

9.3 The Internal Structure of Jupiter and Saturn . . . . . . . . . . . . . . . . . . 48

9.4 Excess Heat in Jupiter and Saturn . . . . . . . . . . . . . . . . . . . . . . . . 49

9.5 The Internal Structure of Uranus and Neptune . . . . . . . . . . . . . . . . . 50

9.6 Gas Giant Atmospheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

9.7 Ring Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

9.7.1 Derivation of the Roche Limit . . . . . . . . . . . . . . . . . . . . . . . 53

10 Moons: Formation and Properties 55

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

10.2 Orbits and Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

10.3 Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

10.3.1 The Moon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

10.4 Tidal Forces and Tidal Heating . . . . . . . . . . . . . . . . . . . . . . . . . . 57

10.5 Tidal Locking, Libration and Circularisation . . . . . . . . . . . . . . . . . . . 58

10.6 Orbital Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

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Contents

11 Small Bodies: Comets, Asteroids and the Outer Solar System 6111.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6111.2 Asteroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6111.3 Kuiper Belt and Trans-Neptunian Objects . . . . . . . . . . . . . . . . . . . . 6411.4 Comets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

12 Detecting Exoplanets 6912.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6912.2 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6912.3 What is a Planet Anyway? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7212.4 Direct Detection: How Hard Can it Be? . . . . . . . . . . . . . . . . . . . . . 7212.5 Reflex Motion and Doppler Measurements . . . . . . . . . . . . . . . . . . . . 7312.6 Planetary Transit Searches . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7512.7 Other ways to detect planets . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

12.7.1 Pulsar Planets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7712.7.2 Gravitational Lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

13 The Exoplanet Population 7913.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7913.2 The Current State of Planet Searches . . . . . . . . . . . . . . . . . . . . . . 7913.3 Selection Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8013.4 Exoplanet Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8013.5 Exoplanet Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8213.6 Exoplanet Orbits: Hot Jupiters and Planetary Migration . . . . . . . . . . . . 8313.7 Host Star Metalicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8513.8 Exoplanets: A young Science . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

14 Astrobiology: Life on Other Planets 8714.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8714.2 Life on Earth: History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8714.3 Lessons from the History of Life on Earth . . . . . . . . . . . . . . . . . . . . 8814.4 Life Elsewhere in the Solar System . . . . . . . . . . . . . . . . . . . . . . . . 91

14.4.1 Mars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9114.4.2 Europa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9214.4.3 Enceladus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

14.5 Life Outside the Solar System . . . . . . . . . . . . . . . . . . . . . . . . . . . 9214.5.1 Host Star . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9414.5.2 Gas Giant Moons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

14.6 The Galactic Habitable Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . 9414.7 How to Find Life on Other Planets . . . . . . . . . . . . . . . . . . . . . . . . 95

15 The Search for Extraterrestrial Intelligence 9715.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9715.2 The Drake Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9715.3 The Fermi Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9815.4 SETI and CETI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9915.5 The Future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

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Contents

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Chapter 1

The Planets Half of the Course

The second half of the Suns Stars and Planets lecture course concerns Planets, both withinour solar system and those outside it that are now being discovered by both ground and spacebased satellites. This section of the course will comprise 12 lectures, one guest lecture (TBD)and three problems sheets. As with the first part of the course, your tutors will not coverthe course, so you will need to find solutions for the problem sheets on Blackboard. Lecturenotes, problem sheets, as well as their solutions, and other materials as necessary, will alsobe available on Blackboard.

Dr Clements’ office is 1011 Blackett, and his office hours for this course are 12pm-1pm onTuesdays.

If necessary you can also contact him via email at: d.clements@imperial.ac.uk

He can be found online on Twitter as @davecl42 and on Wordpress blogs as davecl.wordpress.com.Some of the items on these sites may occasionally be of interest to those doing this course.

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Chapter 2

Planets: Textbooks

There is no single textbook for this part of the course, but much useful material can be foundin the following books:

• An Introduction to the Solar System, edited by David A. Rothery, Neil McBride andIain Gilmour, published by Cambridge University Press

ISBN: 978 1 107 60092 8

As with Introduction to the Sun and Stars, this is a large, colourful and very wellillustrated introductory text to the Solar System side of this course. It has lots of facts,descriptions, diagrams and figures, but is rather light on mathematics. It covers basicideas well, but without the rigour that is usual for any Imperial College course.

• Exploring the Solar System, by Peter Bond, published by Wiley-Blackwell

ISBN: 978 1 4051 3499 6

This is another well presented and illustrated introductory text much like the Rotherybook above. It covers a number of topics somewhat more deeply, and in addition includeschapters on the Sun and on Explanets. It is also rather lightweight on mathematics.

• Planets & Planetary Systems, by Stephen Eales, published by Wiley-Blackwell

ISBN: 978 0 470 01693 0

This is a shorter text book than many of the others listed, and lacks many of the colourfulillustrations. However, it makes up for this in taking a much more mathematical pointof view of the subject material. It also includes sections on exoplanets and on life inthe universe. It is thus a very useful textbook for this part of the course. If you onlyget one textbook, this should probably be it.

• An Introduction to Astrobiology, edited by David A. Rothery, Iain Giilmour and MarkSephton, published by Cambridge University Press

ISBN: 978 1 107 60093 5

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Astrobiology is a relatively young subject that brings together astronomy, physics, bi-ology and much else besides. This book provides an excellent introduction to thesedisparate fields and their contributions to astrobiology. As such, it goes rather furtherthan this course will in many areas, but will provide a lot of extra material for thosewho are interested in this new and growing field.

• Transiting Exoplanets, by Carole Haswell, published by Cambridge University Press

ISBN: 978 0 521 13938 0

This book provides in depth coverage of the increasingly important study of transitingexoplanets, as well as good coverage of the overall field of exoplanet searches. Thecoverage includes mathematical discussion of a wide range of topics concerned withtransiting exoplanets, including their atmospheres, the structure of giant planets and theoverall exoplanet population. This textbook goes well beyond the material this coursewill cover on exoplanets, but it provides a lot of interesting and rigorous material onexoplanets in general, and the application of transit methods to the search for exoplanetsand exo-planetary systems.

There are, of course, many other textbooks, popular books and well illustrated coffee tablebooks available that cover these areas as well, and the library is well stocked with such texts.

In addition to academic texts there are a number of popular non-fiction and fiction bookswhich deal with the planets, moons and other objects found in the solar system. These canprovide a more intuitive feel for what the landscapes of the solar system might be like, whetherdealing with the giant volcanic landscape of Olympus Mons on Mars, or the icy surfaces ofthe moons of Saturn. Of the many such books available I would recommend Kim StanleyRobinson’s Mars series (Red Mars, Green Mars and Blue Mars) for their coverage of thelandscapes, geography and geology of Mars, Paul McAuley’s The Quiet War and Gardensof the Sun for their coverage of the moons of the outer planets, and Alastair Reynold’s BlueRemembered Earth for a tour of the solar system. Kim Stanley Robinson’s recent 2312 alsoadds some interesting coverage of Mercury and Venus.

Chapter 3

Outline Syllabus

The lectures dealing with the planets in our own and other solar systems will be divided upas follows:

1. An Overview of the Solar System and its formation

2. The orbits of planets and Kepler’s Laws

3. Terrestrial Planets: Heating, Cooling and Interiors

4. Terrestrial Planets: Surfaces and Surface Temperatures

5. Terrestrial Planet Atmospheres

6. Gas Giants: Structure and Atmospheres

7. Moons: Formation and Properties

8. Small Bodies: Comets, asteroids and the Outer Solar System

9. Exoplanets: Detection

10. Exoplanets: Properties and Characterisation

11. Astrobiology: Life on Other Planets

12. The Search for Extraterrestrial Intelligence

3.0.1 The Examination

In principle, all the material in these lecture notes, and the lectures, is examinable. Inpractice, less than that is easily examined. At the end of each chapter I will highlight thingsto remember, which are the central issues in each chapter that may appear in the examination.

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This course contains a lot of information. You will need not only to recall that for theexamination, but also be able to use it. An exam question might ask you to compare andcontrast the properties of two different types of planetary body covered in separate parts ofthe course. To achieve a high mark you will need not only to recall the basic facts but usethem to draw such contrasts and make conclusions.

Chapter 4

An Overview of the Solar System

4.1 Aims of this Lecture

Much of this half of the course will deal with the objects in our own Solar System. This firstsection is meant to provide an overview of those objects, planets and otherwise, where theyare, what their key properties are and how they differ, and what the key physical drivers werebehind their formation. There will also be some historical discussion about how we arrivedat our present knowledge of the Solar System, both theoretical and observational.

When looking at the properties of extrasolar planets and planetary systems it is also usefulto see how these compare and contrast with the local example of the Solar System. A broadidea of what our solar system contains is thus a necessary first step in our study of planets.

4.2 Units

In many circumstances, astronomers do not use standard SI units since the numbers involvedare, literally, astronomically large. We thus use, in addition to SI, units that might be called‘astronomer’s units’ which are based on scalings from known objects or places. We may thustalk about numbers of solar masses or solar luminosities, scaling to the mass and luminosityof the Sun. Such quantities are denoted with or Sun eg. M or LSun. Similarly we alsoscale to the Earth using ⊕ or E and Jupiter using Jup.

We also have the special unit of distance called the Astronomical Unit, or AU. This is thedistance between the Earth and the Sun, which is 149.6×106km.

4.3 Overall Inventory of the Solar System

The Solar System consists of the following ingredients:

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• The Sun - a star with a surface temperature of ∼5780K, mass of 2×1030kg., and radius7× 108km, with a rotational period of ∼ 27 days and magnetic activity. The first partof this course will have told you much more about the Sun.

• 8 planets. Four of these are terrestrial planets, like the Earth, four are gas giants likeJupiter. Many of these planets have moons.

• Asteroids

• Kuiper-Belt objects and Trans-Neptunian objects

• All of the above are in the same orbital plane, known as the ecliptic, and are on mostlycircular, prograde (ie. in the same direction as the Sun) orbits. A small number ofKBOs and comets are exceptions to this.

• The Oort cloud - which is roughly spherical, contains about 1011 nascent comets, andpossibly extends out to 10000 AU.

The age of the Solar System is roughly equal to the age of the Sun and the age of the Earth,which are found to be ∼ 4.6× 109 years.

We shall now look at each of these objects in turn.

4.4 Mercury

Mercury is the closest planet to the Sun. It has no atmosphere and images reveal that itssurface is heavily cratered. Surface temperatures range from 740K in the powerful glare ofthe Sun, to 80K on the far side of Mercury from the Sun. It is clearly a very harsh place.The NASA satellite Messenger is currently in orbit around Mercury, and the ESA missionBepiColombo will be launched in 2015, due to arrive in 2022.

Mercury has a weak magnetic field about 1.1% as strong as Earth’s. This has implications,as we shall see later, about the internal structure of the planet.

The heavily cratered surface implies that the surface is very old. Some regions are lesscratered, suggesting that they have been resurfaced at some point in the distant past bygeological activity. Other surface features suggestive of tectonic activity exist (eg. faults),but there are no indications of recent geological activity.

4.5 Venus

Venus is the next planet as we travel outwards from the Sun. Unlike Mercury, its surfacefeatures cannot easily be studied since it has a thick, opaque atmosphere, mostly made upof carbon dioxide. Clouds can be seen in the atmosphere - they are made from tiny droplets

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of sulphuric acid. Venus is actually a more hostile environment than Mercury. Apart fromthe sulphuric acid rain. the atmospheric pressure is 100 times that of Earth, and the surfacetemperature is 670K, hot enough to melt lead. Carl Sagan frequently described Venus asHell.

Despite the challenges of these conditions, several Russian probes have managed to land onthe surface of Venus and beam back images during their brief lifetimes, while the NASAMagellan satellite used radar to see through the obscuring atmosphere and map the surface.These missions have revealed a surface with few impact craters and, instead, signs of lavaplanes and volcanoes.

Venus lacks a magnetic field, making it different from the Earth and Mercury, and implyingthat its internal structure may be rather different to the Earth, which, given that they havesimilar mass, is surprising. The young surface, with an absence of cratering, and the absenceof plate tectonics suggest the interesting possibility that Venus goes through periodic totalresurfacing events, where the entire surface is covered by layers of lava. Counting impactcraters suggests that the most recent resurfacing event could have occurred 300-500 Myr ago.

4.6 Earth

The Earth is the planet we are most familiar with. In the context of this survey its mostimportant aspects are that it has an atmosphere that is ∼80% Nitrogen and 20% Oxygenand has a surface temperature of 288K. The presence of oxygen in the atmosphere is uniquein the Solar System and is something we will discuss later on.

Why do you think oxygen is so unusual as an atmospheric constituent?

Earth has few visible impact craters, indicating that the surface is young. It has a strongmagnetic field, and active volcanoes and tectonic plates. These combined, as we shall see,provide information on the internal structure of the planet. Water is common, with oceanscovering 70% of the surface.

4.6.1 The Moon

The Earth also has an unusually large moon, the Moon, which has no atmosphere. It showsmany impact craters but also signs of historic lava flows, the mare or seas. The Moon is ina synchronous orbit with the Earth, so that the same face of the Moon always points to theEarth. The Moon does not have a significant global magnetic field. The Earth-Moon systemis thought to have been formed through a huge impact between the proto-Earth and a Marssized body about 100-150Myr after the formation of the Solar System.

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4.7 Mars

Mars is smaller than the Earth or Venus, but larger than Mercury. It has a very thin atmo-sphere, with a pressure only about 0.6% of Earth’s and mostly made up of carbon dioxide.The mean surface temperature is 233K. The surface of Mars has a distinct orangey-red colourthanks to the colour of the rocks and dust on its surface. There are many huge geologicalfeatures on Mars, including the largest volcano in the Solar System, Olympus Mons, whichis 24km high, and a huge canyon system, Valis Marineris, that extends 400km across thesurface.

Despite the thin atmosphere Mars has strong weather systems with seasons, and dust stormsthat can last for weeks and that can cover a significant fraction of the planet’s surface.

The current central question about Mars is whether it was once hospitable for life, and whetherlife ever formed there. Mounting evidence for the existence of substantial quantities of waterice adds credence to these ideas, and the new generation of Mars rovers and orbiting satellitesare gathering large volumes of data about the role of water on the surface of Mars in itsdistant past. New results from the Curiosity rover will continue to emerge during the courseof these lectures.

4.7.1 Phobos and Deimos: The Moons of Mars

Mars also has two small moons, Phobos and Deimos. They are likely asteroids that have beencaptured by Mars’s gravitational field.

4.8 The Asteroid Belt

The asteroid belt lies between the orbits of Mars and Jupiter and is made up of a largenumber of rocky and metallic bodies ranging in size from Ceres, with a diameter of 950 km,downwards, with many, many more small bodies than large. Those that have been studiedin detail have plentiful impact craters. The NASA Dawn mission is currently in the asteroidbelt, studying various asteroids in detail, so we will soon learn much more about these objects.

4.9 Jupiter

Jupiter is the largest planet in the Solar System and the first ‘gas giant’. It is composedmostly of hydrogen (90%) and helium(10%) with traces of methane, ammonia and watervapour. The features we see on Jupiter and not those of a solid ‘surface’ but are in factever-changing cloudscapes that lie at the top of its deep atmosphere. The different colours ofthe cloud bands represent detailed differences in content, chemistry and depth. An exampleof one of these weather systems is the Great Red Spot, a storm system larger than the Earth,

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that has persisted for several hundred years. The temperature of the cloud tops that we seeis ∼120K, but the temperature and pressure will rise as you go deeper into the atmosphere.At a depth of 10000km, the temperature should be ∼6000K with a pressure 106 times thaton the surface of the Earth. Jupiter also has a large and powerful magnetic field, 20000 timesstronger than that of Earth.

4.9.1 The Moons of Jupiter

Jupiter has a large number of moons, dominated by the four large ‘Galilean’ satellites, so-called because they were first observed by Galileo, called Io, Europa, Ganymede and Callisto,in order outward from Jupiter. Io is the most volcanically active body in the Solar System,with a large number of active volcanoes and a surface covered by sulphur deposits from theeruptions. Because of this, it’s surface visibly resembles a giant pizza. The heating necessaryto maintain this level of volcanic activity comes from ‘tidal heating’, something we will discusslater. Io is largely made up of rocky material, not dissimilar from the Earth. Europa, thenext moon outwards from Jupiter, is very different from Io, with a surface that is made upof ice. Despite this, Europa is a largely rocky body, but the icy layer is expected to be about100km deep. The icy surface shows few impact craters, and instead appears to be made upof broken ice packs and fractured plains. This suggests that liquid water may at times reachthe surface through ‘cryovolcanoes’. Water vapour escaping form these may have recentlybeen detected by the Hubble Space Telescope. This in turn suggests the possibility that asubsurface ocean of liquid water might lie beneath the surface, kept liquid through similartidal heating processes to those on Io, but operating at lower temperatures. The remainingGalilean moons, Ganymede and Callisto are broadly similar, made predominantly of ice andwith heavily cratered, old surfaces. Ganymede is the largest moon in the Solar System, witha diameter larger than that of Mercury, but, since it is largely made of ice rather than rock,it has a substantially smaller mass.

All four of Jupiter’s Galilean moons lie within its magnetosphere, and are thus bombardedby charged particles. Io is especially strongly affected, and suffers from an especially harshradiation environment as a result. Sulphur and Oxygen atoms released by Io’s volcanism areheated by the charged particles in Jupiter’s magnetosphere and escape the moon’s gravityto eventually form a ring of plasma around Jupiter. Ions streaming from this ‘plasma torus’are picked up by the magnetic field and accelerated into Jupiter’s ionosphere, producing anelectrical current of several million amps and leading to spectacular aurorae around Jupiter’spoles.

4.10 Saturn

Saturn is the second biggest gas giant in our Solar System, having a radius about 15% smallerthan that of Jupiter. Its atmosphere also has a banded appearance similar to that of Jupiter.Storm systems have been observed in Saturn’s atmosphere by the Cassini spacecraft, butnothing on the scale of Jupiter’s Great Red Spot. Saturn has such a rapid rotation speed,

12 Sun Stars and Planets 2012-13

with a day only 10.7 hours long, that there is significant atmospheric bulging at Saturn’sequator. Similar to Jupiter, Saturn also has a large magnetic field.

4.10.1 The Rings

The most distinctive feature of Saturn, of course, is the ring system. While all the gas giantshave ring systems of some kind, Saturn’s is the most visible. It is not solid, but is made upof many small icy and rocky particles, ranging in size from 1cm to a few metres. The ringsare most likely the result of the break up of a moon following a catastrophic impact. Thering particles all orbit in the equatorial plane of Saturn, creating the disk we see. The ringsthemselves are surprisingly thin, only 100m thick, but the reflectivity of the particles makesthe ring system highly visible in reflected sunlight.

The rings are structured into many smaller subrings as a result of the gravitational influenceof small ‘shepherd’ moons that are among Saturn’s 61 satellites.

4.10.2 The Moons of Saturn

Saturn’s moons are generally small, with only seven having radii greater than 200km. How-ever, the largest of these, Titan, is one of the most interesting objects in the entire SolarSystem. Titan is one of the largest moons in the Solar System, and is roughly half the sizeof the Earth. It has a thick atmosphere that is predominantly nitrogen and methane. Theremainder of the atmosphere, <1%, is made up of complex hydrocarbons. These make Ti-tan’s atmosphere opaque, but also indicate that complex hydrocarbon chemistry is takingplace. The Cassini spacecraft and the Huygens lander have examined Titan in detail, andhave revealed a surface of ice beneath the smoggy atmosphere, together with lakes and seasof liquid hydrocarbons, filled by a rain of ethane and methane.

Another moon of Saturn that has aroused considerable interest of late is Enceladus. Its surfacehas few impact craters, but is instead covered by many cracks, suggesting that it has beenresurfaced through cryovolcanic activity at some time in its past. Direct evidence for this wasfound by Michele Dougherty of Imperial College, who discovered geysers of water vapour,mixed with other compounds, being vented into space from cracks in Enceladus’ surface.

4.11 Uranus

Uranus, like Neptune, is smaller than Jupiter or Saturn, but still has a mass 15 times that ofEarth. It appears as a rather featureless blue-green planet. It is unusual in the Solar Systemin that its axis of rotation is tipped ∼98 degrees away from being ‘vertical’ to the plane of theecliptic. It is suspected that this is due to a major impact in its earlier history that knockedthe planet onto its side. One pole of Uranus thus always points towards the Sun, while theother always points away. This results in unusual atmospheric flows with one side of the

13

planet always warmer than the other. Uranus has a ring system, the second most prominentin the Solar System, but its ring particles are much darker than those found in Saturn’s rings.Like all the other gas giants, Uranus has a large magnetic field.

Uranus has at least 27 moons, but only five are larger than 200km in radius. Some of theseshow evidence for cryovolcanism in their past.

4.12 Neptune

Neptune is the last planet in our Solar System, and is the last of the four gas giants. It hasa mass about 17 times that of Earth. Its atmosphere is a distinct blue colour resulting fromthe small amount of methane it contains absorbing light at the red end of the spectrum whilethe rest is reflected. More features are observable in Neptune’s atmosphere than in Uranus’,with banding and pale clouds being detectable by both flypast missions such as Voyager andby remote observation from the Hubble Space Telescope. Giant storm systems also occur,although the ‘Great Dark Spot’ detected by Voyager 2 in 1989 had gone away by the timeHST observed the planet in 1994.

Neptune has at least 13 moons, but only three have a radius larger than 200 km. The largestof these, Triton, is an unusual object that orbits in the opposite direction to all the otherNeptunian moons. This suggests that it did not form at the same time as Neptune and therest of its moons, but was instead captured by the planet at a later date. Such a capturewould have been associated with impacts and other activity that would leave their mark onthe surface of Triton, and indeed we find that Triton has a strange divided surface, withgeyser-like plumes evident in one area, and a rough, resurfaced, geography elsewhere. Thegeysers are likely responsible for Triton’s tenuous nitrogen atmosphere.

4.13 Pluto, Trans-Neptunian Objects (TNOs) and the KuiperBelt

Beyond Neptune, there are no single dominant mass planets. Instead, there is a plethora ofsmall bodies that form a belt of objects, known as the Kuiper Belt, extending outwards fromthe orbit of Neptune. The first of these to be discovered was Pluto, long regarded as a planetin its own right, but the discovery of other, similarly sized, if not larger, TNOs starting inthe 1990s has led to a re-evaluation of Pluto’s status. The discovery of Eris, which is largerthan Pluto, tipped the balance, and Pluto was demoted to being a ‘minor planet’ by theInternational Astronomical Union in 2006.

Kuiper belt objects are left overs from the formation of the Solar System and, since theyrepresent relatively pristine material from the formation epoch, they are of great interest.Their distance from the Sun makes them difficult to study, but the NASA New Horizonsmission will fly past Pluto in July 2015, and will tell us much more about these objects.

14 Sun Stars and Planets 2012-13

4.14 Comets

Comets are small bodies from the outer Solar System whose orbits take them close to theSun. When this happens, their surface heats up, and volatiles boil off, forming the distinctivetail that, in the case of bright comets, can even be visible during daylight. Comets come intwo different types, defined by whether they are short or long period. Short period cometsare thought to come from the Kuiper Belt, while long period comets, with periods greaterthan about 100 years, come from further away. They come from the last, and most distant,part of the Solar System to be discussed here: The Oort Cloud.

The ESA Rosetta mission will rendezvous with the comet 67P/ChuryumovGerasimenko inAugust 2014, drop[ping a lander on the comet, and then following it as its orbit takes thecomet on its closest approach past the Sun. If all goes well, we will know much more aboutcomets by the time this course is given in 2015.

4.15 The Oort Cloud

The Oort Cloud is named after Jan Oort, the astronomer who first suggested its existence.Oort’s idea was that a large population of cometry bodies, that formed in the inner SolarSystem at the same time as the rest of the planets, would be thrown out of the Solar Systemby gravitational interactions with giant planets like Jupiter and Saturn. The resulting cloudof comets could include as many as 1011 objects and extend to tens of thousands of AUin distance, forming a roughly spherical cloud surrounding the Solar System. Long periodcomets, with high inclinations relative to the ecliptic, many of which have retrograde orbits,fall into the inner Solar System from the Oort Cloud.

4.16 Formation of the Solar System

The physics of star and planetary formation is a large and complex topic, the theory of whichwill be covered in the 3rd Year Astrophysics course. The core concepts, though, emerge fromseveral observational facets of our Solar System, which allow us to get some basic idea forthese processes without going into details. The first key observation is that the orbits of mostbodies in the Solar System are roughly circular, and they are all prograde ie. orbiting in thesame direction as the Sun’s rotation. This suggests that the Sun and planets all formed aspart of the collapse of a solar nebula. This broad picture was first developed by Laplace inthe late 18th century.

The starting point is a slowly rotating molecular cloud, which starts to collapse under theforce of its own gravity. As this happens, thanks to conservation of angular momentum, thecloud contracts and spins faster. The collapse then continues, perpendicular to the rotationaxis. The pre-stellar material, made up of dust and gas, settles into the rotation plane andthe collapse proceeds fastest at the centre, where the Sun will eventually condense. Away

15

from the centre, clumps of dust and gas start to coalesce as the diffuse disk material breaksup. Once these planetesimals reach 10km in size they begin to accrete material themselvesthrough runaway gravitational attraction, eventually forming the planets. At the same timethe infant Sun ignites and the resultant stellar wind clears away gas and dust that is notalready gravitationally bound into condensed objects. The eventual composition of a planetwill depend on where in the solar nebula the planet formed.

This is the broad picture of star and planet formation that astronomers work with, but thedetails are still uncertain. Observations of planets in other star systems, ie. exoplanets, areprompting rapid development in this field. For example, it seems that gas giant planets oftenmigrate to the inner regions of a star system, even though they have to form at distancesfrom their parent star comparable to those of Jupiter and Saturn.

4.17 Summary

This part of the course has provided a rapid tour of the Solar System, showing both thevariety of objects in it, and looking at some of the features common to all of them. SolarSystem science is a very rapidly moving field, with active research going on from the groundand in space. The Curiosity rover is busy on Mars, the Cassini spacecraft is continuing itssurvey of the Saturnian system, and the Dawn satellite is providing us with our first viewof the largest asteroids. Meanwhile, the New Horizons mission is getting closer to Pluto andRosetta is nearing its target comet.

Things to Remember

• The names of the planets

• Their order going out from the Sun and that they lie in the same plane - theecliptic

• The basic geography of the Solar System including the asteroid belt, the mostfamous moons/rings, the Kuiper Belt and the Oort Cloud

• The basic principles behind our model of the formation of the Solar System

• The natural consequences of this model with respect to prograde orbits, andvariation of planetary composition with distance from the Sun

16 Sun Stars and Planets 2012-13

Chapter 5

Planetary Orbits: Kepler’s Laws

5.1 Historical Introduction

The predominant view of the structure of the Solar System until the late 16th century wasestablished in Ancient Greece by Aristotle. This model of the Solar System was geocentric- the Earth was at the centre and all other bodies, including the Sun, orbited around it,against a backdrop of ‘fixed’ stars. This was not the only model of the Solar System duringthat period, and in fact Artstarchus suggested, in 200BC, that the Sun rather than the Earthwas at the centre of the Solar System, but the geocentric view prevailed.

The geocentric model explained the usual motion of the planets against the fixed stars as aresult of the planets’ motion around the Earth in their respective orbits. However, one aspectof planetary motion that was difficult to explain in the geocentric system was retrogrademotion. This is the stage in the orbit of planets like Mars and Uranus where their directionof motion in their orbit, as seen from Earth, reverses, and they appear to move backwardsagainst the frame of reference of the fixed stars. The solution to this in the geocentric modelwas add an epicycle to the planet’s orbit, and epicycle being an additional circular motion ofthe planet about its circular orbital track around the earth.

In 1543 Copernicus began a revolution in our understanding of the Solar System and the restof the Universe. In his book De revolutionibus orbium coelestium (On the Revolutions of theCelestial Spheres), published just before his death1 he describes how a heliocentric model ofthe Solar System, with all the planets including Earth orbiting the Sun in circular orbits, canexplain retrograde motion without the need for epicycles. In the heliocentric view, epicyclesarise when the Earth catches up and overtakes a planet, like Mars, that is further from the Sunand thus orbiting more slowly. Some tweaks and contrivances are still necessary to explainanomalies that arise because the actual orbits of the planets are somewhat elliptical, but thiswork represented a huge advance in our understanding.

1Legend has it that the very first copy of De Devolutionibus was placed in his hands as he lay on hisdeathbed in a stroke induced coma. He is said to have woken, glanced at the book, and then died peacefully

17

18 Sun Stars and Planets 2012-13

The publication of De Revolutionibus did not change things immediately, and did not causeany great controversies. At that stage the geo- and heliocentric models were simply that- competing ideas that had yet to be fully tested by observations. Beginning in 1576, theastronomer Tycho Brahe began observations at his observatory at Uraniborg, near Copen-hagen, that would provide the data necessary for a far more precise understanding of theorbits of the planets. He and his staff and students produced highly accurate measurementsof planet positions, without the aid of telescopes, between 1576 and 1597. He also discovereda supernova in our own galaxy which still bears his name.

Johanes Kepler worked as an assistant to Tycho at the end of Tycho’s life, when he hadmoved to Poland. On Tycho’s death in 1601, Kepler was appointed his successor as imperialmathematician to Emperor Rudolph II. While the main part of his work for the Emperor wasto provide advice of an astrological nature (astrology paid better than astronomy, even then!),Kepler continued his work analysing the detailed observations that Tycho had obtained of themotions of the planets. Over the next decades, Kepler derived three laws for planetary motionfrom Tycho’s observations, and also attempted to provide a physically-based explanation forthese laws, arguably making him the first astrophysicist (not that we would recognise hisphysics today). These results were published in 1609 in Astronomia Nova. He also wrote abook, Somnium published posthumously, discussing the possibility of astronomy from anotherplanet, which has been described as the first work of science fiction.

At the same time, Galileo was working with the first optical telescopes in Italy. Among manyother things, in 1610 he discovered the four large moons of Jupiter, known to this day asthe Galilean moons, and found that they orbited Jupiter in just the same way as the planetsorbitted the Sun in the heliocentric model.

Between Kepler and Galileo, the stage was set for a revolution in our understanding of theSolar System.

5.2 Kepler’s Three Laws of Planetary Motion

Kepler derived three laws of planetary motion from Tycho’s data:

1. Planets follow an elliptical orbit with the Sun at one focus

2. The line joining the planet and the Sun sweeps out area at a constant rate

3. The square of the time a planet takes to go round the Sun, P, is proportional to thecube of the semi-major axis of its orbit ie.

P 2 ∝ a3

Kepler’s explanation for these laws involved Platonic solids and the harmony of the celestialspheres. It wasn’t until Newton turned his attention to planetary orbits that we arrived atsomething we would recognise today as a full physical explanation of Kepler’s laws, usingNewton’s laws of motion and gravitation.

19

θ

Origin

Sun

Planet in orbitr

r

Figure 5.1

5.3 Derivation of Kepler’s Three Laws

The natural coordinate system for studying the orbits of the planets is a plane polar system,with the origin at the centre of mass of the massive object, in this case the Sun, around whichthe planets are orbiting.

Basis vectors in a plane polar system are r, along the direction joining, in this case, a planetto the Sun’s centre of mass, and θ, along the tangent to this direction. Expressing these incartesian coordinates, in terms of the angle θ we find that:

r =

(cosθsinθ

), θ =

(−sinθcosθ

)(5.1)

From this you can show that:

d

dtr = θθ,

d

dtθ = −θr (5.2)

Exercise Use the definition of plane polar coordinates in terms of cartesian coordinates givenin equation 5.1 to show that the definition of the differentials in equation 5.2 are correct.

Given these definitions, and using the general result for vector differentiation that:

d

du(φa) = φ

da

du+dφ

dua (5.3)

we can obtain the velocity and acceleration:

r = rr (5.4)

r = rr + rθθ (5.5)

r =(r − rθ2

)r +

(2rθ + rθ

)θ (5.6)

The gravitational force of the Sun on an orbiting planet will be given by Newton’s inversesquare law of gravity. Combining this with Newton’s first law of motion F = ma we then get:

20 Sun Stars and Planets 2012-13

rθ∆t

r

Figure 5.2: Area traced out by an orbiting body in time ∆t.

mr = f(r)r (5.7)

f(r) =GMm

r2(5.8)

where f(r) is the force due gravity, M is the mass of the Sun, and m is the mass of the planet.Using the formula for r given in equation 5.6, and examining the r and θ directions separatelywe find:

r − rθ2 = −GMr2

(5.9)

Since there is no force in the θ direction, as gravity only acts along r, the vector joining theplanet to the Sun, the term in θ must vanish:

2rθ + rθ = 0 (5.10)

This last term can be written more usefully, and more compactly as:

1

r

(d

dtr2θ

)= 0 (5.11)

Exercise: Use the chain rule for differentiation on equation 5.11 to show that this is equalto equation 5.10.

Equation 5.11 is equivalent to saying that r2θ is a constant. r2θ is the specific angularmomentum, ie. the angular momentum per unit mass, and we will define this as equal to theconstant h.

r2θ = h (5.12)

What does this last result mean in terms of Kepler’s Laws? Figure 5.2 shows how area istraced out by a body as it orbits around the Sun. The area of a triangle is simply

1

2base× height =

1

2r(rθ∆t

)=

1

2h∆t (5.13)

21

But we know that h, the specific angular momentum, is constant, so the area covered in equaltimes ∆t, is also constant. Kepler’s Second Law is proven.

Before going further in the mathematics, let us also look at the combination of equations 5.9and 5.11.

Circular motion will have a constant distance form the Sun, r, which we will set to the valuea, and a constant rate of change of θ, ie. θ = ω, the angular velocity. This is constant soequation 5.11 is satisfied. In equation 5.9 we then have r = 0 which means:

rθ2 = aω2 =GM

a2(5.14)

Both sides of this equation are constant, so as long as they are equal we have solved both 5.9and 5.11, indicating that circular motion is an acceptable solution for the orbit of a planet.

Furthermore, since the time taken for a planet to orbit the Sun, p = 2πω by definition, we thus

have:

aω2 = a

(2π

p

)2

=GM

a2=⇒ a3

p2=GM

4π2= const (5.15)

Which is Kepler’s Third Law proven, at least in the case of circular orbits.

For a more general solution to 5.9 and 5.11 we define a new variable u such that r = 1/u anddifferentiate to obtain u and u.

r = − 1

u2u = −r2du

dθ

dθ

dt= −r2θ du

dθ= −hdu

dθ(5.16)

and

r = −h ddt

(du

dθ

)= −hdθ

dt

d

dθ

(du

dθ

)= −hθd

2u

dθ2(5.17)

Substituting this into equation 5.9 gives:

hθd2u

dθ2+r2θ

rθ =

GM

r2θθ (5.18)

Given that r2θ = h this can be rewritten:

hθd2u

dθ2+ huθ =

GM

hθ (5.19)

Dividing through by h and θ then yields:

d2u

dθ2+ u =

GM

h2(5.20)

This is the equation of simple harmonic motion with constant forcing, the general solution towhich is:

u = Acos(θ − θ0) +GM

h2(5.21)

22 Sun Stars and Planets 2012-13

We can choose a coordinate system where θ0 = 0 and rewrite this as:

1

r=

1

r0(1 + ecosθ) (5.22)

where r0 = h2/(GM) and where we can set e ≥ 0 without loss of generality. There are thenfour cases, depending on the value of e.

• For e = 0 we simply get a circle, with r = r0 at all times.

• For 0 < e < 1 we have an ellipse with eccentricity e - this is in fact the situation for theorbits of all the planets, proving Kepler’s first law.

• For e = 1 we get a parabola.

• For e > 1 we get a hyperbola.

The first two of these options are bound orbits, so is what we see for the planets. The lasttwo options are unbound orbits, and are what is seen for objects that achieve escape velocity.In the case of the Solar System, the Pioneer and Voyager satellites have managed this, andwill travel forever between the stars of our Galaxy.

Things to Remember

• Kepler’s 3 laws as stated in section 5.2

• How to explain all three, and the consequences with respect to circular, ellipti-cal, parabolic and hyperbolic orbits

• Be able to show the 3rd law in the context of a circular orbit

Chapter 6

Terrestrial Planets: Heating,Cooling Processes and Interiors

6.1 Introduction

The Earth is a terrestrial planet, along with Mercury, Venus and Mars. There are common-alities between them, but also substantial differences. In this chapter we will look at theinternal structure of terrestrial planets and the factors that drive that structure. This willprovide some key insights into how and why the four terrestrial planets differ. Along the waywe’ll also uncover some of the forces that have shaped the Earth and its geography over its4.5 billion year history.

6.2 The Unquiet Earth

In London, it is easy to think of the Earth as fixed and unchanging, but we know that thisisn’t in fact the case. Earthquakes and volcanic eruptions are just two reminders that ourplanet is a dynamic system, even if much of that dynamism operates on timescales far longerthan that of a human life.

The Earth beneath us is in fact a lot more dynamic even than that, as can be seen whenmaterial from deeper beneath the surface bursts out during a volcanic eruption. Where didthe energy for that heat come from and how has this driven the large scale structure of theEarth and the surface features we see today?

23

24 Sun Stars and Planets 2012-13

6.3 Primordial Heating

The formation of the Earth was a violent process, with large impactors peppering the formingplanet and with smaller bodies accreting at a high rate. The kinetic energy of these impactorsis largely turned to heat during the collision, and the accretion of smaller bodies also leadsto heating of the young earth. In general, the potential energy of all the mass that falls ontoa planet during its formation, which is converted to heat, is given by:

PE ∼ GM2

R(6.1)

where M is the mass of the planet, G has its usual value, and R is the radius of the planet.

The immediate consequence of all this energy being deposited into the young planet as it isforming, is to make much of the material molten, and to keep it that way for many thousandsof years.

6.4 The Structure of the Earth

The material that made up the forming Earth includes substances with a range of densities. Ifa mixture of liquids with different densities is allowed to settle, the densest material will endup at the bottom, in this case at the centre of the Earth. The constituents of the young Earthcan be determined by looking at the constituents of meteorites. These include Al, Si, Ti, Fe,Ni, Mg, Ca, with Fe and Ni being quite abundant. The densest of these materials are Fe andNi, so these constituents separated out, and fell towards the core of the Earth, leading to theformation of a solid, largely iron core, surrounded by a liquid nickel-iron outer core. Abovethis is the mantle, made of a material called peridotite which includes minerals containingMg, Ca, Fe, Al, Si, Na, O, Cr, but which is essentially 40-60% SiO2. The minerals that makeup peridotite include feldspar, olivine, pyroxene, spinel, garnet and others. Above the mantlelies the crust, made of basalt and ∼75% SiO2. On top of this crust are the sedimentary rocksproduced by erosion processes which make up most of the landscapes that we can see on thesurface (see Fig. 6.1).

The separation of the Earth into core, mantle and crust is based on the results of seismology- essentially looking at how the speed of sound changes as seismic waves travel through theEarth. Figure 6.2 shows how the physical properties that affect the transmission of seismicwaves change with depth, and their effect on the passage of seismic waves.

An alternative way of thinking about the structure of the Earth is based not on the con-stituents of the material but on its physical state. This leads to a different classification thatwe will find useful later on. In this approach the core is the same, but the mantle is then di-vided not into the upper and lower mantle, which is based on composition, but into the regionwhere the rock is molten or under sufficient pressure that it can flow — the asthenosphere —and the region where the rock is rigid — the lithosphere — where flow is not possible. Theupper parts of the mantle and the crust make up the lithosphere. This is also shown in Fig.6.1.

25

Figure 6.1: Structure of the Earth from Fig. 2.1 of Rothery, McBride & Gilmour.

26 Sun Stars and Planets 2012-13

Handout: Structure and Atmospheres of Planets

To Section 3.3.4: Interiors of the Earth and terrestrial planets

Description of the Earth’s structure Seismic / compositional:

• core (Fe/Ni rich)

• mantle (40 to 60% SiO2)

• crust (~ 75% SiO2) State of material:

• lithosphere: rigid outer shell

• asthenosphere: convecting part of the mantle

from USGS http://pubs.usgs.gov/gip/dynamic/inside.html

Interior structure of the Earth, taken from Karttunen et al Fundamental Astronomy

Left: Comparison of the composition of the terrestrial planets and the Moon (McBride & Gilmour) Below: relative sizes & core masses (in %) taken from Karttunen et al

Figure 6.2: Internal structure of the Earth and how this changes the results of seismology eg. speedof seismic waves vs. depth. Taken from Karttunen et al. Fundamental Astronomy.

27

Isotope Half-life (109) yrs Present Rate of Heat Generation (10−12W kg−1)235U 0.71 0.04238U 4.5 0.96232Th 13.9 1.0440K 1.3 2.8

Table 6.1: Half-lives of the most important radiogenic heat sources in the Earth’s crust and mantletoday.

6.5 Long Duration Heat Sources

As well as the initial heat input from the formation of the planet, there are two long durationsources of heat that help to keep the Earth’s interior hot. The most important of these isradiogenic heating from long lived unstable isotopes. The radiation given off by their decay isabsorbed by their surroundings, leading to an increase in temperature. Table 6.1 summariesthe most important long lived isotopes in the Earth for long term radiogenic heating. TheEarth’s age is 4.5 billion years, which is comparable to the half lives of these isotopes.

Tidal heating is the other potential source of long term heating. Tidal heating comes fromthe effects of a nearby orbiting massive body - in the case of the Earth, the Moon producestidal effects. The most noticeable are the tides in the Earth’s oceans, but there is also a ∼1mmaximum rise and fall of the Earth’s rocky surface due to the Moon. This deformation ofthe Earth imparts energy which appears as heating. It is thought that thsi heating is largelydeposited in the crust and mantle. The amount of energy imparted to the Earth from theMoon by this tidal interaction is small, nearly two orders of magnitude less than the energyinput from radiogenic heating, but tidal heating is very important for other bodies in theSolar System, as we will see later.

6.6 The decay of long term heating sources

While radiogenic heating and tidal heating persist today, they are not an infinite resource.With time the radioactive species responsible for heating will decay away. The tidal heatingrate will also fall as angular momentum is transferred from the rotation of the Earth to theorbit of the Moon, and as the Moon moves away from the Earth. Ultimately, therefore, theinteriors of all terrestrial planets cool.

6.7 Heat Loss from Planets

The Earth cools by radiating heat away into space. Volcanic eruptions are the most obviousexample of this, but there are many other, less dramatic ways in which the heat from theupper layers of the asthenosphere travels through the lithosphere. The amount of heat that theEarth, or any other planet, can thus expel is determined by its surface area. In contrast, the

28 Sun Stars and Planets 2012-13

Comparison between the different

terrestrial planets

Relative planet sizes and core

volumes

McBride & Gilmour Introduction to the Solar System

planet and terrestrial moon structure

core as % of volume

12%

16%

9%

42%

4%

Figure 6.3: Comparison of the interiors of the 4 terrestrial planets and the Moon. Note that thesmaller the object, the thicker the crust. The Moon, for example, has a crust that is 1000km deep.From McBride & Gilmour.

amount of heat, and, indeed, the amount of active radiogenic heating underway, is dependenton the volume of the Earth, or other object.

Heat loss rate ∝ Surface area

Volume=

4πR2

4/3πR3∝ 1

R(6.2)

Thus smaller planets cool more rapidly than larger planets. The long term result of coolingis that the lithosphere thickens and the asthenosphere becomes thinner. Mars, for example,has a much thicker crust than the Earth, and Mercury has an even thicker crust.

6.8 Cooling Processes

How does the heat travel from the core of the Earth to the surface?

29

Figure 6.4: Convection transferring heat from the core to the surface. Fig 2.14 of Rothery, McBride& Gilmour.

There are four key processes that allow the Earth to cool:

• Conduction

This is the most familiar process, whereby heat transfers from a hotter to a cooler regionthrough thermal conduction. In the lithosphere, where rocks are rigid and cannot flow,this is the main method of heat transference.

• Convection

In the asthenosphere, where material is able to flow, convection operates, and is themost efficient way that heat is transferred. Hotter material expands, and is thus lessdense, so rises, while cooler material contracts, becomes more dense, and falls. Thecooler material then warms up and the process continues. Large scale convection cellsexist in the asthenosphere, where this process can operate. Solid state convention, inwhich rocks flow by a few cm/year, drives this process. See Fig 6.4 for a diagram ofhow this operates.

• Eruption/Advection

The lithosphere is too rigid to allow convection, so the last stage of the process of heattransference from the core to the surface takes place when molten rock, or magma,spreads over the surface and cools, or as it is injected into the lithosphere and coolsbeneath the surface, and the heat is conducted away to the surrounding crust.

30 Sun Stars and Planets 2012-13

Figure 6.5: The key features of plate tectonics: sea floor spreading, continental plates, subductionzones, and arcs of volcanoes around the edges of continental plates. From USGS

• Plate techtonics

The surface of the Earth is made up of a series of plates that essentially float on thesurface of the asthenosphere. Some of these plates are thicker, and form the continents,while others are thinner, and form the floor of the oceans. The plates move relativeto each other, with new material being produced by hot magma emerging from theasthenosphere at mid-ocean ridges, leading to sea floor spreading, and with old, cold,material sinking into the asthenosphere at the edges of continents in subduction zones.

6.9 Volcanism and Tectonics on Other Terrestrial Planets

Given that the cooling rate of a planet is set by its surface area to volume ratio, we wouldexpect smaller planets, and similar objects like the Moon, to have thicker lithospheres and thusbe less tectonically active. The smallest terrestrial planet in our Solar System is Mercury.Studies show that its surface is old, as evidenced by heavily cratering, but that there areregions where some resurfacing has taken place. Our best estimate is that this resurfacingtook place roughly a billion (109) years ago (1 Gyr). The Moon, while not a planet, sharesmany of the properties of a terrestrial planet, so is another useful check of our ideas aboutplanetary volcanism. Like Mercury, there are regions of the Moon that are old and heavilycratered, but others, the maria, or seas, that are younger and appear to have been resurfaced

31

by more recent lava flows. These are also old, about 1 Gyr old, and are thought to be relatedto impact events that punched holes through thinner portions of the Moon’s lithosphere,allowing lava to flow over the surface.

Venus, with a similar size and mass to Earth, would be expected to have a similar lithospherethickness and thus similar tectonic activity. However, observations, conducted using the radarmapping instruments of the Magellan spacecraft, have found no evidence for tectonic activity.The surface of Venus, though, is young, showing none of the extensive cratering that is seenon the older surfaces of the Moon or Mercury. If there are no tectonic plates on Venus, howdoes its interior cool?

One idea is that the lithosphere of Venus acts like the lid on a huge pressure cooker. Instead ofthe continuous leaking of internal heat that we see on Earth, the idea is that Venus occasionallyblows its top, with the entire planet being resurfaced through periodic volcanic catastrophes.The surface of Venus appears to be between 700Myr and 500Myr old, which would set thedate of the most recent catastrophic resurfacing. Volcanoes are seen on Venus, often showinga strange, flat topped appearance suggesting slow growth. Historical lava flows up to 2000kmhave been found, but there is no evidence of ongoing volcanic activity. The volcanoes of Venusare presumably awaiting the next epoch of catastrophic volcanism.

Mars is intermediate in mass between Mercury and the Earth, so might be expected to havean intermediate level of tectonic activity. There is indeed some evidence of tectonics on Mars,with significant differences between the northern and southern hemispheres. The southernhighlands are similar to the thick crust of Earth’s continents, while the northern lowlandsare similar to the thinner crust of the oceans. However, the lithosphere of Mars is now muchthicker than that of Earth, so any ancient tectonic activity is likely to have stopped longago. Age estimations using cratering statistics suggest that the southern highlands are older,at about 4.5 Gyr, while the northern lowlands and the Tharsis Bulge, home to Mars’ giantvolcanoes, are younger at 3.7Gyr.

Mars has the largest volcanoes in the Solar System, including the giant Olympus Mons. Theseare all found in the Tharsis Bulge region, which appears to be similar to the ‘hot spots’ foundin several locations on the Earth. These hot spots seem to be the result of upwellings in theasthenosphere at certain positions in the mantle. These mantle plumes bring heat into thelithosphere in a way that is largely separate from tectonic activity. The Hawaiian Islandson Earth are a result of a hot spot located in the middle of the Pacific Ocean plate, wellaway from any region of sea floor spreading. Since the Pacific plate is moving, the volcanicislands, that grow around the location of the hot spot, are gradually dragged away from thehot spot, leading to the production of a chain of volcanic islands and, further away, a series ofsub-surface sea mounts. On Mars, there is no tectonic activity, so the large ‘shield’ volcanoesthat grow from them just continue growing. This is why Olympus Mons is the largest volcanoin the Solar System.

32 Sun Stars and Planets 2012-13

Things to Remember

• The structure of the Earth (Fig 6.1) including the names, constituents and propertiesof different layers

• Heat sources for terrestrial planets, long and short duration including primordialheating

• Heat loss processes, including conduction, convection, advection/eruption & platetechtonics

• Dependence of heat sources and heat losses on the size of body

• Consequences of this for the internal structure and surface volcanism on otherplanets

Chapter 7

Terrestrial Planet Surfaces andTemperatures

7.1 Introduction

In the previous chapter we found that the surface properties of planets are far from typical oftheir interiors. The vast majority of the Earth’s volume is made up of molten rock, flowing,albeit slowly, in giant convection currents. The continental plates that make up the surface ofour planet are just low density material floating on this ocean of rock. However, the surfaceof the Earth is something that we are intimately concerned with, and the surfaces of otherplanets are, by and large, all that we can see of them. Understanding the processes responsiblefor the planetary surfaces that we see, and that determine the basic properties of planetaryenvironments, including surface temperature, are thus a key ingredient to understanding thenature of planets in our own and other solar systems.

7.2 Major Factors in Shaping Planetary Surfaces

While the four terrestrial planets are all very different in appearance, they are all shaped bysimilar physical process. The importance of these different processes, though, varies betweenplanets, and this will apply just as much in other solar systems as it does in our own. Thereare four central processes that determine the surface geography of planets:

• Impact cratering

The majority of impact events occurred during the earliest stages of the Solar System,but impacts continue to happen today, albeit at a much lower rate. Examples on Earthinclude Meteor Crater in Arizona, a crater 550 feet deep and about 1 mile across thatwas produced by an impact about 50000 years ago, and the Tunguska event, likelyan airbursting small meteor, that levelled 2150 sq. km of forrest in Siberia in 1908. In

33

34 Sun Stars and Planets 2012-13

Planet Cratering Tectonics Volcanism Erosion Comments

Mercury Heavily cratered, No (some in past) No No Most geologically inactivevery old surface (∼4.5 Gyr) terrestrial planet

Venus Few craters, surface No tectonics seen Past volcanoes seen Yes Catastrophiconly ∼500Myr old resurfacing possibility

Earth Few craters, geologically Yes Yes Yes Interior not yet solidifiedactive, erosion effects

Mars Heavily cratered in parts No current tectonics Past giant volcanoes seen Yes Probably almost solidified

Table 7.1: Summary of role of shaping processes in terrestrial planets

February 2013, a somewhat smaller meteor passed over the Russian city of Chelyabinsk.The shockwave as it passed through the atmosphere causing moderate damage over alarge area and injuring roughly 1500 people. With an estimated mass of 12000-13000tonnes and a size of 20m, the Chelyabinsk meteor is probably the largest natural objectto enter the Earth’s atmosphere since Tunguska.

• Volcanism

As discussed in the previous lecture, volcanoes are where the heat contained withina planet can escape to the surface and, over geological timescales, allow the planet’sinterior to cool. The effects of lava flows can be seen in many places in the Solar System,and there are giant volcanoes on Mars.

• Tectonics

The surfaces of some terrestrial planets are made up of tectonic plates that float on topof the hot, molten, interior. The interactions of these plates, and their movements onthe surface, are a driving force for shaping the surface of planets.

• Erosion

In the presence of a fluid, whether gas or liquid, surface features are eroded and modifiedover time.

The importance, or otherwise, of each of these factors varies form planet to planet. Thoseplanets that have cooled rapidly, for example, so that their lithospheres are thick, are less likelyto experience tectonic or volcanic activity, while those with no atmosphere will not experienceextensive erosion. The importance of these factors for each of the terrestrial planets is listedin Table 7.1.

7.3 Impact Cratering

Cratering is ubiquitous throughout the Solar System, caused by the impact of small bodieswith larger objects. Impacts can be thought of as the process of accretion of planetarymaterial that continuing to this day, though at a much lower rate than during the epoch ofplanet formation. Younger surfaces on planets and moons have fewer craters, and this can beused to date the surfaces that are seen. Where large scale resurfacing has occurred, through

35

volcanic activity, for example, signs of cratering are erased. Erosion, also, can erase the signsof cratering given sufficient time.

During an impact, rocks are heated and subjected to very high pressures. Rocks melt andfracture as a result. Material will be ejected from the impact crater, in both solid andmolten form, and a cavity is excavated leading to the familiar shape. A sufficiently powerfulimpact will expel large quantities of material, leading to large area effects. Ejected rockscan sometimes be expelled from the atmosphere and even given sufficient kinetic energy toachieve escape velocity. In fact some meteorites found on Earth were ejected from Mars bypast impacts.

Impacts on water, which will be common in the case of Earth since it is 70% covered by water,would produce massive tsunamis.

A large enough impact will produce significant environmental damage. The extinction of thedinosaurs has been linked to the Chixulub crater, found beneath the Yucatan Penunsula inMexico. Still bigger impacts can change the nature of the objects involved. The Moon, forexample, is thought to have been formed as a result of an impact between the young Earthand a body roughly the size of Mars.

The impact rate in the Solar System has been in decline for at least the past ∼4 Gyr, butthere are suggestions, based on crater counting studies on the Moon, that there was a briefincrease in the impact rate about 3.8 - 4 Gyr ago. This phase in the development of theSolar System has been termed the Late Heavy Bombardment, and may be linked to broaderaspects of the evolution of the Solar System.

7.4 Volcanism and Tectonics

The physical background to volcanism and plate tectonics were discussed in the previouschapter. Both can have a considerable effect on shaping planetary surfaces. Evidence forhistorical large scale resurfacing events involving volcanic activity can be seen on the Moonand Mercury, but these most ended 3 Gyr ago. Volcanoes are clearly present on Mars,including the massive Olympus Mons. The surface of Venus appears to be geologically young,less than 0.5 Gyr, and volcanoes have been seen on its surface in radar mapping observationsby the Magellan satellite, but the lack of tectonic activity on Venus has led to the idea thatits surface is periodically subject to catastrophic volcanism, where it is completely resurfacedfrom time to time.

On Earth, there is evidence for large scale resurfacing during events known as flood basalts.Examples of these include the Deccan Traps in India, about 65 million years old, the ColumbiaRiver flood basalts, about 15 million years old, and the Siberian Traps, about 248 millionyears old. These flood basalts are made up of tens to hundreds of separate lava flows stackedon top of each other, reaching thicknesses of 1 to 3 km and covering thousands of squarekilometres of the surface. They are the result of the production of lava volumes up to 2million cubic kilometres in size that erupted over timescales of 1 to 5 million years. This

36 Sun Stars and Planets 2012-13

Planet Albedo Mean Surface Temperature (K) Surface Atmospheric Pressure (bar)

Venus 0.77 733 92Earth 0.3 288 1.0Mars 0.25 223 6× 10−3

Mercury 0.10 443 10−15

Table 7.2: Albedos of terrestrial planets. From Rothery McBride & Gilmour

represents an annual eruption rate over twenty times greater than those observed for presentday hot spots such as Hawaii. Many of the historical flood basalts are associated with massextinctions, and the volume of lava, ash and gas they can produce is certainly enough to causemajor environmental effects.

Plate tectonics are responsible for the shape and distribution of continents and oceans acrossthe surface of the Earth, driven by the convection currents in the upper parts of the mantle.There is evidence for historical tectonic activity on Mars. As with volcanism, tectonic activityis expected to decline with time as a planet cools, and the lithosphere extends to greaterdepths. Smaller planets, such as Mars or Mercury, will cool much faster than the Earth,leading to the cessation of tectonic activity, and the geologically quiescent state we see todayon these planets.

7.5 Erosion

Where fluids are able to flow on a planet’s surface, erosion can take place. The flowingfluids may be gas or liquid, leading to two different types of erosion: fluvial erosion whereliquids are involved - this can be seen as water erosion on Earth and, possibly, on Mars; andaeolian erosion where the flowing fluid is the gas of an atmosphere - this can be seen in dryenvironments on Earth, on the surface of Mars and, to some extent, on Venus. These differenterosive processes leave different signatures on the environment, allowing us to get some ideaof the presence, or absence, of water on the surface of Mars in the past. Both fluvial andaeolian erosion leads to the formation of stratified sedimentary rocks, like sandstone.

7.6 The Surface Temperatures of Planets

The presence, or absence, of liquid water on the surface of a planet is of great interest inthe context of exobiology - the search for life on other planets. In the case of Mars we areinterested in whether water flowed on its surface in the past, and in the case of planets aroundother stars - exoplanets - we are interested in determining whether life, or the conditions forlife, might persist today.

The temperature of a planet can be estimated, under certain assumptions, quite simply bylooking at the energy balance of incoming to outgoing radiation.

37

The energy input to a planet is the radiation received from its parent star, minus the fractionof that radiation that is reflected away. The reflection fraction is given by the planet’s albedo,a, where a = 1 means total reflection and a = 0 means total absorption. The total energyreceived by a planet can then be calculated as follows:

Flux density in Wm−2 received = F =L

4πd2(7.1)

where L is the total luminosity of the star and d is the distance from the star to the planet.The planet intercepts a total of:

πR2F = πR2 L

4πd2(7.2)

where R is the radius of the planet. Some of this energy is reflected by the planet’s albedoso the total power absorbed by the planet is then:

Total Power Received = PR = πR2 L

4πd2(1− a) (7.3)

If we then assume that the planet radiates this heat away as a perfect black body we canfind its no-atmosphere temperature TNA. The reason why we have to specify that this isa no-atmosphere temperature will become apparent shortly. The total power emitted by ablack body at temperature TNA and radius R, PE is:

PE = 4πR2.σT 4NA (7.4)

from the Stefan-Boltzman equation, and where σ is the Stefan-Boltzman constant, σ =5.670373× 10−8Wm−2K−4. By setting the power received equal to the power radiated awaywe can determine the temperature at which these balance, and find TNA.

4πR2.σT 4NA = πR2 L

4πd2(1− a) (7.5)

⇒ TNA =

(L (1− a)

16 π σ d2

)1/4

(7.6)

Note that this is independent of R, the radius of the planet.

How well does this equation work?

Given the albedo values for the various planets found in table 7.2 we can calculate the no-atmosphere temperatures of a variety of planets in our Solar System.

Exercise: Calculate TNA for the planets listed in Table 7.2.

When you do this, you will find that the TNA values for Mars and Mercury are a good matchto the results of the energy balance equation, but the Earth and Venus are anything but, withVenus having a surface temperature 500K higher than that estimated by TNA. Why is this?

38 Sun Stars and Planets 2012-13

7.7 The Greenhouse Effect

The answer to this is that Venus and the Earth both have significant atmospheres, and thatthose atmospheres contain gases that allow more heat to be retained than is assumed by theno-atmosphere approximation. Solar radiation peaks in the optical part of the electromagneticspectrum. The atmosphere is (largely) transparent at these wavelengths, so the light ofthe Sun passes straight through, allowing us to see, and allowing its radiation to heat theplanet. The re-emitted thermal radiation, however, peaks at longer wavelengths, in the mid-infrared (you can determine this using the Wien Displacement law). The atmosphere is notas transparent at these wavelengths as in the optical, thanks to the presence of CO2, methaneand other so-called greenhouse gases. These gases in the atmosphere absorb some of the mid-IR radiation from the surface, warm up a little, and reradiate it, again as thermal emission,in all directions. A reduced fraction of the radiation emitted from the surface thus reachesspace, and the planet therfore retains more of the energy received form the Sun than it wouldwithout an atmosphere. The with-atmosphere temperatures, as observed for Venus and theEarth, are thus higher than the calculated no-atmosphere temperatures, TNA.

The overall climate system on the Earth is of course more complex than this simple analysissuggests, with the presence of clouds leading to local increases in albedo, the detailed contentof the atmosphere changing the albedo further, modifying the degree of the greenhouse effect,and other factors such as the condensation of water vapour into rain providing other inputsof heat. The details of these and other factors are studied in the Atmospheric Physics course.One thing, though, is clear - if the fraction of greenhouse gases, such as CO2 and methane,in the atmosphere increases, there will be more mid-IR absorption, and greater retention ofheat.

Things to Remember

• The major factors shaping planetary surfaces: impacts, volcanism, tectonics and erosion

• The dependence of impact rate on time

• The presence or absence of volcanism on other Solar System bodies and the rea-sons for this

• How to calculate the no-atmosphere surface temperature of planets

• How the Greenhouse Effect can change these no-atmosphere temperatures

Chapter 8

Terrestrial Planet Atmospheres

8.1 Introduction

In the last chapter we saw how important the presence, or absence, of an atmosphere is for thesurface temperatures of terrestrial planets. Atmospheres are also important for many otherprocesses, including erosion and, of course, anything biological. In this chapter we will lookat the origin of terrestrial planet atmospheres in the Solar System, how atmospheres escapefrom a planet, how atmospheres are structured, and what they contain.

8.2 Why do we have an atmosphere at all?

Venus and the Earth have significant atmospheres. Mars has only a thin atmosphere, whileMercury, the Moon and smaller rocky bodies in the asteroid belt and elsewhere usually havelittle or no atmosphere at all. What makes Venus and the Earth so different? The answer tothis is that Venus and the Earth are more massive bodies than the others, and the atmosphereis held onto their surface by the effects of gravity. We can examine the effects of gravity on thestructure of the atmosphere of a planet using the principle of hydrostatic equilibrium, wherebythe downward force due to gravity on each part of the atmosphere, must be balanced by thepressure gradient in the atmosphere.

8.3 Atmospheric Density and Pressure

If the pressure gradient is to balance the force of gravity then:

dP

dz= − ρg (8.1)

39

40 Sun Stars and Planets 2012-13

where P is the atmospheric pressure, z is height above the surface, ρ is the atmosphericdensity and g is the gravitational acceleration. Pressure and density are connected by the gaslaw:

PV = NkBT (8.2)

where P is pressure, V volume, kB is Boltzmann’s constant, and T is temperature. We canrewrite this in terms of the density of a gas of mean molecular weight 〈µA〉 and the atomicmass unit mamu as follows:

P 〈µA〉mamu

ρ= kT (8.3)

Since V = Mρ and the mass M of a given mole of material, M = N 〈µA〉mamu. Combining

equations 8.1 and 8.3 we get:

dP

dz= − P 〈µA〉mamu g

kT(8.4)

To be able to solve this equation we need to make two simplifying assumptions. The first isthat g doesn’t vary with height. This is a good assumption since the atmospheres of terrestrialplanets are thin compared with the sizes of the planets. The second is that T doesn’t varywith height. This isn’t as good, but is adequate since pressure changes more rapidly withheight than temperature. Equation 8.4 can then be solved by separation of variables, and weget:

P = P0e−〈µA〉mamu gz

kT (8.5)

and there is a similar exponential fall off for density as a function of height. The quantity(kT/ 〈µA〉mamu g) is known as the scale height and gives height at which pressure and densityfall to 1/e times their surface values. The scale height is actually quite small on Earth, withpressure falling to about 60% of the sea level value at the summit of Mauna Kea, a heightof 4200m, and to just 33% of the sea level value at the height of Everest, about 6000m.This is why mountaineers on Everest often develop serious breathing and respiration relatedproblems. This also shows that our approximation that g doesn’t vary with height is a goodone.

8.4 Temperature Variations with Height

The atmosphere is largely transparent to solar radiation, so does not absorb energy from theSun. Instead, solar radiation is absorbed by the surface of the Earth, which heats up, andthis heats the atmosphere around it. The temperature of the Earth’s atmosphere, and thatof other terrestrial planets, thus decreases with height. The behaviour of temperature withheight divides the atmosphere into different regions:

• Troposphere

This is the lowest layer of the atmosphere, in contact with the surface of the planet.Temperature drops rapidly with height in the troposphere

41

Atmospheric structure

Troposphere: T drops dramatically with height

Mesosphere: T drops, but more gently

Thermosphere: T rises; large day-night variations

Stratosphere: T rises due to atmospheric absorption (O3mostly); unique to Earth

Venus Earth Mars

temperature [K] temperature [K] temperature [K]

Altitu

de

[km

]

Figure 8.1: Temperature vs. height for Venus, Earth and Mars, showing the separate regions of theatmosphere. Only Earth has a stratosphere as only Earth has Ozone in its atmosphere to absorb UVradiation. From McBride & Gilmour.

• Stratosphere

Only the Earth has a stratosphere, characterised by temperature rising with height. Thisis a result of ozone molecules absorbing ultra-violet radiation from the Sun, leading toan injection of energy, and thus heat, in this layer. Other planets, such as Mars orVenus, do not have any ozone in their atmospheres, and thus lack a stratosphere

• Mesosphere

Temperature falls with height in the mesosphere, though at a slower rate than in thetroposphere

• Thermosphere

Temperature rises with height in the thermosphere as a result of several energy in-jection processes, including the absorption of extreme-UV photons from the Sun, andinteractions with the Solar Wind.

8.5 Thermal Escape

If gravity keeps an atmosphere around a planet, how do low mass planets like Mercury andMars, lose their atmospheres?

42 Sun Stars and Planets 2012-13

Figure 8.2: The shape of the Maxwellian velocity distribution for three different temperatures. Notethat it has a tail to high velocities. These high velocity particles are those most likely to escape froma planetary atmosphere. From Tanner McCarron and Weston McCarron.

For an atom or molecule of a gas to be able to escape from a planet, it must have a velocitygreater than the escape velocity of the planet. Escape velocity is such that the kinetic energyof the particle is equal to the change in potential energy required to climb out of the planet’spotential well ie. ∆PE = KE. This is given by:

1

2mv2esc =

GMm

R⇒ vesc =

√2GM

R(8.6)

where M is the mass of the planet and R is its radius. Note that escape velocity is independentof the mass of the object trying to escape the planet’s gravity well.

The velocity distribution of particles in a gas at a temperature T is given by the Maxwelldistribution:

P (v)dv =4√π

( m

2kT

) 32v2e−

mv2

2kT dv (8.7)

where m is the mass of the particle, T the temperature, v the velocity and P (v) is theprobability of that velocity. This distribution has a high velocity tail, shown in Fig. 8.2.These are the particles most likely to escape from a planetary atmosphere.

The most probable and rms velocities in a Maxwellian are:

vp =

√2kT

m; vrms =

√3kT

m(8.8)

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Escape velocity and thermal velocity

For planet to keep itsatmosphere:

thermal velocity has tobe lower (by ~ factor 6)than the escapevelocity

All gas giants can sustainH/He atmospheres;

Earth and Triton can keepwater in theiratmospheres

Venus and Mars canmaintain CO2 and N2 intheir atmospheres

Figure 8.3: Escape velocity plotted against surface temperature for a number of planets and otherSolar System bodies.

We can set a rough requirement for atmospheric retention, for example, that the most probablevelocity should be less than a tenth the escape velocity, ie. vp < (1/10)vesc. A more accurateformula, called the Jeans Escape Fraction, gives the rate of particles escaping by thermalevaporation:

φJ =n

vp2√π(λ+ 1)e−λ ; where λ =

(vescvp

)2

(8.9)

where n is the number density of gas particles.

One common factor in all these equations is that, at a given Temperature, lighter particleswill have a higher velocity. We would thus expect that lighter species, such as Hydrogen andHelium, will be more likely to escape, and this is in fact what we find in Figure 8.3. The gasgiants, Jupiter, Saturn, Neptune and Uranus, are all massive enough to retain hydrogen andhelium, the most abundant elements in the universe. Earth and Triton can keep water in theiratmospheres, while Mars, Venus and Titan can retain carbon dioxide. Mercury and the Mooncannot even retain carbon dioxide, and thus have absent or extremely thin atmospheres.

8.6 Current Atmospheric Composition

The current atmospheric composition of the terrestrial planets with appreciable atmosphereis shown in Fig 8.4. The atmospheres of Venus and Mars can be classed as ‘oxidised’ inthat there is no free oxygen. Instead, oxygen is bound into compounds, mostly CO2 but also

44 Sun Stars and Planets 2012-13

water, H2O and sulphur dioxide, SO2. The gas giants, as we will see in more detail later, haveatmospheres rich in hydrogen, since they are massive enough to retain this light species, andare dominated by hydrogen, helium and compounds like methane (CH4), ammonia (NH3.)water and hydrogen sulphide (H2S). They thus have atmospheres classed as ‘reducing’. Earthis the only planet to have an oxidising atmosphere, containing free oxygen (and also thecompound ozone, O3, which shields the surface from UV light and, thanks to the heatingfrom this absorption, a stratosphere).

The presence of free oxygen in the Earth’s atmosphere is rather special since Oxygen is areactive element that would usually be bound into compounds, as seen on Venus and Mars.The lack of carbon dioxide in Earth’s atmosphere, a major constituent in the atmospheres ofMars and Venus, is also rather odd. Oxygen was not always a major constituent of Earth’satmosphere. In fact, the latest estimates suggest that it was only a major constituent for thepast billion years. Oxygen, of course, is produced by photosynthesis, so the presence of lifeon the Earth is the reason that we have oxygen in the atmosphere. If life were to suddenlydisappear, the oxygen levels would gradually decrease as it combines chemically with otherelements, like carbon.

The relative absence of CO2 in the Earth’s atmosphere is another issue. This is the result ofthe Urey weathering process, whereby carbon dioxide dissolved in water reacts with silicatesin rocks, leading to the deposition of calcium carbonate (CaCO3). An example of this reactionis:

CaSiO3 + 2CO2 + H2O → Ca2+ + SiO2 + 2HCO−3Ca2+ + 2HCO−3 → CaCO3 + CO2 + H2O (8.10)

The CaCO3 produced by this reaction is dissolved in the water and eventually precipitatesout to form sedimentary rocks such as limestone and chalk. These same rocks, on Earth atleast, can also be produced by biological processes, that also serve to remove CO2 from theatmosphere.

8.7 Origin of Atmospheres

Where did planetary atmospheres come from originally?

Any primordial atmosphere is likely to have been lost, since the planets were all very hotafter formation, the solar wind would have been quite powerful in the early Sun, and thebulk of the content would have been the most common gases, hydrogen and helium, whichthe terrestrial planets are too small to retain. Instead, what we seen now are likely to besecondary atmospheres, the content of which will be controlled by the balance between gassources and gas sinks:

• Sources

45

Composition of atmospheres:

Venus Earth Mars

Venus & Mars: oxidised atmospheres

Earth: rather special! Most of CO2 fixed in rocks and in oceans. O2

consequence of photosynthesis at later stage: oxidising

atmosphere

Figure 8.4: Current composition of terrestrial planet atmospheres (Mercury essentially has no atmo-sphere). Note that both on Venus and Mars CO2 dominates.

Outgassing from planetary interiors eg. volcanoes

Evaporation and sublimation of material on the surface (eg. water, solid CO2)

Bombardment by bodies rich in volatiles

• Sinks

Condensation and chemical reactions (temporary)

Stripping by the solar wind

Impacts

Thermal escape

The key difference between Earth and Venus, which are otherwise quite similar planets, maybe that Venus never acquired a significant amount of water - forming from planetesimals thatlacked it, missing out on bombardment by material rich in water ice, or because it lost itswater through photoionisation through being closer to the Sun, with the resulting hydrogenbeing lost to thermal escape. A lack of water would have meant that the Urey weatheringprocess was never able to leach CO2 out of the atmosphere. Outgassing from the core wouldthen not have been counteracted by chemical reactions, and so the thick atmosphere couldbuild up and make Venus the unpleasant place it is today. Mars, conversely, had too littleoutgassing to counteract the increased rate of thermal escape due to its lower mass, at leastin recent times, and thus lost its atmosphere, becoming the cold, low pressure environmentcurrently being explored by the Mars rovers

46 Sun Stars and Planets 2012-13

Things to Remember

• That atmospheric pressure drops exponentially with height, and the definition ofatmospheric scale height

• The structure of Earth’s atmosphere, with troposphere, stratosphere, mesosphereand thermosphere, and the way temperature varies with height in these four regions

• The atmospheric structures of Venus and Mars, and why they differ from theEarth

• Know what escape velocity is and how to derive it

• Be able to derive a simple formula (using a constant times vp rather than theJeans Escape Fraction) to show whether a given molecular species can be retained by aplanet of given mass and temperature

• The consequences of this for the atmospheres of planets in the Solar System

• The compositions of the atmospheres of the terrestrial planets and how they re-late to other properties & the Urey weathering process

• The sources and sinks of terrestrial planet atmospheres

Chapter 9

Gas Giants: Structure andAtmospheres

9.1 Introduction

The planets Jupiter, Saturn, Uranus and Neptune are collectively known as the Gas Giants.They are the four largest planets in the Solar System and have properties that are very dif-ferent from the terrestrial planets. In this chapter we will examine their properties, includingtheir structure and atmospheres, and look at the differences between Jupiter and Saturn,which share a number of properties, and Uranus and Neptune, which are similar to eachother but subtly different from the other gas giants. We will also examine the origins of afeature that is common to all the gas giants, but which is most distinctive in Saturn - ringsystems.

9.2 Basic Properties of Gas Giants

The Gas Giants are much lower density than the terrestrial planets, and have much deeperatmospheres. Saturn, for example, has a sufficiently low density that it would float on water- if you could find a big enough bucket. This means that they cannot be dominated by thekind of rocky and metallic material that makes up most of the structure of the terrestrialplanets. Instead, the bulk of their mass comes from light elements, and they have muchhigher Hydrogen and Helium abundancies. Consideration of the mass and temperature of thegas giants in the context of the thermal escape of gases from an atmosphere (see Fig. 8.3)shows that Hydrogen and Helium can be retained by them.

Since the gas giants are effectively large balls of gas, it is difficult to define a ‘surface’ forthem, or to probe very far beneath the cloud layers that we see from outside, to determinewhat their internal structure might be. The ‘surface’ issue is solved by arbitrarily definingthe surface of these planets to be where their atmospheric pressure equals that of Earth i.e.

47

48 Sun Stars and Planets 2012-13

the radius of these planets is defined as the radius at which P = 1 bar. What we know oftheir internal structure has largely been determined by examinations of their effect on thepassage of spacecraft that fly past or orbit them, measurements of their magnetic fields, andthe combination of these results with models of their deep interior. These results indicatethat the gas giants have rocky and icy cores that are at high temperature and pressure - inthis case we use the term ice to refer to volatile materials such as H2O, CH4 and NH3 ratherthan something that is actually frozen. Observations and flypasts also reveal that the gasgiants are all somewhat flattened in shape - they are prolate, bulging a little at their equatorsas a result of their high rotation speeds. All of the gas giants have magnetic fields.

The gas giant planets formed further out in the Solar System than the terrestrial planets,and the formation process took longer, especially for the outermost planets. The increaseddistance from the Sun allowed them to accrete more Hydrogen and Helium, leading to theirgreater mass. Models of the early Solar System suggest that the gas giants formed somewhatcloser to the Sun than we see them now, and then proceeded to migrate outwards, clearingthe Solar System of debris as they did so. The asteroid belt and the Kuiper belt are all thatthey left behind.

Examination of the temperatures of the Gas Giants compared to the energy they receivefrom the Sun reveals that all except Uranus are emitting excess heat ie. they are warmerthan the energy they receive from the Sun would suggest. The origin of this excess heat willbe discussed below.

9.3 The Internal Structure of Jupiter and Saturn

The internal structure of the two largest gas giants is shown in Fig 9.1. Their cores arethought to be a mixture of rock and ices (ie. volatiles), surrounded by a layer of ices. Thismaterial is at temperatures and pressures of up to ∼16000K and 50 Mbar (ie. 50 million Earthatmospheres) in Jupiter, and ∼10000K and 18 Mbar in Saturn. The rocky/icy cores in bothplanets have masses of about 10 Earth masses in total. There may be further differentiationin the cores, leading to a metallic iron centre, as in Earth, but we do not have sufficient datato be sure of this.

Outside the rocky/icy cores is a region made up of helium and metallic hydrogen. The latteris a form of hydrogen that only arises under intense pressure, where the hydrogen nuclei arepressed together so hard that their electrons become delocalised from their parent nuclei,and form a fermi gas of free electrons that can flow throughout the volume of the metallichydrogen. This material is conductive and liquid at the temperatures and pressures prevalentin the cores of these planets. You can think of this material as being somewhat like mercuryat room temperature and pressure.

Further out form the centre, the pressure subsides to values below 2Mbar, where hydrogenreturns to its more familiar molecular form as H2. This is a gradual process so there isno sharp boundary between metallic and molecular Hydrogen layers. Further out still, asimilar smooth transition occurs between liquid and gas. While hydrogen and helium are the

49

Figure 9.1: The internal structure of Jupiter and Saturn (from Rothery, McBride & Gilmour).

dominant materials in these outer laters, they are also mixed with other icy material andsome small amount of rocky material.

9.4 Excess Heat in Jupiter and Saturn

Jupiter and Saturn are both warmer than the energy they receive from the Sun would suggest.This is known as having excess heat. The origin for this excess is probably different for eachplanet.

• Jupiter

There are three likely explanations for excess heat in the case of Jupiter. Firstly, asthe largest planet in the Solar System, it may still be radiating away the residual heatfrom its formation - the cooling rate for this primordial heating, as for all heat loss inplanets, goes as the inverse of the radius. Secondly, there is the possibility that Jupiteris still slowly contracting, converting potential energy to thermal energy as it does so.Finally, and uniquely for Jupiter in the Solar System, there is the possibility that theremay be a low rate of deuterium fusion in the hottest densest regions.

• Saturn

Saturn is too small to have significant residual heat from its formation. Instead, the bestidea for how it generates its excess heat is that it comes form the separation of Helium

50 Sun Stars and Planets 2012-13

Figure 9.2: The internal structure of Uranus and Neptune (from Rothery, McBride & Gilmour).

from Hydrogen in the metallic Hydrogen layer. The Helium then rains downwards,releasing potential energy as it does so. This process cannot work in Jupiter as themetallic hydrogen layer there is hotter, allowing helium to be dissolved in it, and stirredby convection, keeping the materials well mixed.

9.5 The Internal Structure of Uranus and Neptune

The internal structure of the smaller two gas giants, Uranus and Neptune, is shown in Fig.9.2. There are some similarities between them and Saturn and Jupiter, but also some strikingdifferences. The first difference is that Uranus and Neptune have less Hydrogen and Heliumthan the larger gas giants. This leads to the second key difference, which is that they are toosmall to be able to produce the conditions necessary for the formation of metallic hydrogen.The two outer gas giants have more volatiles in them than hydrogen or helium - roughly 20%of their mass comes from these gases while they account for about 90% of the mass of Jupiterand Saturn. Some people therefore classify them as ‘ice giants’, rather than the more genericgas giants.

Apart from the absence of metallic hydrogen and the reduced amount of H and He, theirstructure is broadly similar to that of the other two gas giants, with a rocky and icy core,and inner region of icy material, and then an outer region of Hydrogen and Helium.

51

Figure 9.3: The atmospheric structure of Jupiter and Saturn (from Rothery, McBride & Gilmour).

Neptune is found to produce excess heat. This is likely a result of continuing differentiation inits internal structure, with denser material falling towards the centre and releasing potentialenergy as heat. Uranus, in contrast, releases no excess heat. This is a puzzle since the twoplanets are very similar. Where one has an internal heat source, one would thus expect theother to have one as well. The only significant difference between the two is that the orbitalaxis of Uranus is pointed towards the Sun. This leads to a very different distribution of heatwithin its atmosphere and could, in principle, lead to the disruption of the convection flowsthat would otherwise allow internally generated heat to reach the surface and be radiatedaway. Further research is needed to determine whether this explanation is correct.

9.6 Gas Giant Atmospheres

When we observe a gas giant planet, we are looking at their atmospheres not their surfaces.These have many colours and structures in them, arising from the various processes, chemical,physical and meteorological that drive them. The temperature profile of the atmospheres showsimilarities to some aspects of the terrestrial planets, with a troposphere, a convective layer,where the temperature falls with height, and then a thermosphere where the temperaturerises with height. Jupiter and Saturn (see Fig. 9.3) have highly reflective clouds in theirtopmost layers, with the constituents of lower layers not yet fully identified. The colouring in

52 Sun Stars and Planets 2012-13

Figure 9.4: The atmospheric structure of Uranus and Neptune (from Rothery, McBride & Gilmour).

the atmospheres is not yet fully understood, but is likely due to trace elements, including, inthe case of Jupiter, sulphur.

A similar division between troposphere and thermosphere is seen in the atmospheres of Uranusand Neptune, but the temperature increase in the atmosphere of Uranus with height in thethermosphere is very slow, indicating that there is little or no convection and that energytransport is very inefficient. This relates to the issue of there being no excess heat detectedin Uranus, as discussed above. Methane in the upper layers of both these planets give themtheir bluish colour.

All the gas giants have banded structures in their atmospheres which are related to variationsin wind speeds, with different bands traveling at different speeds, and with turbulence occur-ring at the interfaces between different bands. The differing colours are related to differentmaterials, with dark bands, called belts, coming from rising material and light bands, calledzones, from sinking material. This can be explained in two ways - either the planets can beseen as a series of coaxial rotating cylinders, or as a number of convection cells.

Long duration weather systems also appear within this banded structure, the most obviousof which is the Great Red Spot on Jupiter, a 14000 x 26000 km storm that has been ragingfor at least 170 years. Smaller storms have been seen on the other gas giant planets, but noneas persistent as this.

53

M m m

2 R

d

r

Figure 9.5: Diagram showing two self gravitating bodies of mass m close to a larger body of massM.

9.7 Ring Systems

All of the gas giant planets have ring systems. Saturn has the most obvious, partly because itsring material has a high ice content and is thus highly reflective, but rings have been detectedfor all of the others. The rings are composed of orbiting debris, with particle sizes rangingfrom 1cm to 5m. Ring systems have a wide range of structures. In Saturn we see gaps anddivisions in the rings due to small moons clearing their orbit, moons that shepherd material,and to the effects of orbital resonance (see later). Formation scenarios for rings include theidea of satellites shattered by an impact, that they are made up of material left over from theformation of the solar system that never coalesced to form a planet, and the suggestion thatthey are the remains of a moon that migrated towards the plant and was then disrupted.

Central to all these ideas is the result that the rings of all the gas giants are within theirRoche Limit. This is the radius around a planet within which an object that would otherwisebe held together by its self-gravity, will be torn apart by tidal forces.

9.7.1 Derivation of the Roche Limit

We have two bodies of mass m a distance r apart, lying a distance d from a body of largermass M and radius R, where M m and d r.

The tidal force from the body of mass M that is working to pull the mass m bodies apart isthe difference in gravitational attraction on them. Thus:

Ft = ∆F (d, d+ r) =GMm

d2− GMm

(d+ r)2(9.1)

Multiplying out the term on the right we get:

Ft = GMm

((d+ r)2 − d2d2(d+ r)2

)= GMm

(d2 + 2dr + r2 − d2d2(d2 + 2dr + r2)

)(9.2)

54 Sun Stars and Planets 2012-13

we know that r is small compared to d, so we can eliminate terms in r2 and, in the denomi-nator, terms in r as well, giving:

Ft = 2GMmr

d3(9.3)

The Roche Limit is then defined as the distance at which this tidal force from the large massM is balanced by the gravitational attraction between the two smaller masses m. Thus:

Gmm

r2= 2

GMm

d3r ⇒ dR =

(2M

m

)1/3

.r (9.4)

The Roche Limit is usually expressed in terms of densities, with ρP for the planet and ρs forthe satellite, and using R for the radius of the planet and rs for the radius of the satellite.Looking at things this way we find that:

ρP =M

43πR

3P

; ρs =m

43πr

3s

(9.5)

Substituting this into equation 9.4 we get:

dR = 21/3

(R3pρP

r3sρs

)1/3

rs ⇒ dR = 21/3(ρpρs

)1/3

R (9.6)

A more detailed calculation by Edouard Roche in 1848 leads to the actual value of RocheLimit:

dR = 2.456

(ρpρs

)1/3

R (9.7)

Things to Remember

• The internal structures of the gas giants and the reasons for differences between them

• The atmospheric structures of gas giant planets and their constituents

• The properties of gas giant ring systems

• The simple derivation of the Roche Limit that leads to equation 9.6

• How to apply the Roche Limit to moons and ring systems

Chapter 10

Moons: Formation and Properties

10.1 Introduction

Nearly all of the planets in the Solar System have moons, with some of these moons, such asTitan or Ganymede, being larger than the planet Mercury. Even minor planets, such as Plutoand Eris, have their own moons. Eris has one known moon, called Dysnomia, while Pluto hasfive detected moons - Charon, Hydra, Nix and the recently discovered, and recently namedStyx and Kerberos. The presence of these moons around Pluto, and the likelihood that thereare other, smaller, and thus harder to detect, companions, is causing difficulties in plans forthe NASA New Horizons flyby mission to Pluto.

The number and range of properties of moons around planets is summarised in Table 10.1.The history and formation of moons can provide extra information about the formation ofthe solar system and about the planets around which they orbit.

10.2 Orbits and Masses

The mass of a planet can be determined if you know the details of the orbit of a moon aroundit. If we make the simplifying assumption that the orbit of a moon around a planet is circular,and that the planet’s mass is much greater than that of the moon, then we can use equation5.14 to derive the planet’s mass. All we need to know is the distance between the planet andthe moon, and the moon’s orbital period. These can all be determined observationally. Thus:

aω2 =GM

a2⇒M =

ω2a3

G(10.1)

For the Earth-Moon system the parameters are: a = 386 × 106m while the orbital periodof the Moon is 27.3 days, which converts to ω = 2π/2.36 × 106s. Put these numbers intoequation 10.1 and we get the mass of the Earth = 6× 1024 kg.

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Planet Moons Comments

Mercury 0Venus 0Earth 1 The Moon, 1700km radiusMars 2 Phobos and Deimos, both irregular and small

(22 and 15km diameter respectively)Jupiter 8 + >50 captured Ganymede (2600 km), Callisto (2400 km)

Io (1800 km), Europa (1600 km)Saturn 23 +>30 captured Titan (2600 km) earthlike atmosphere!

Rhea (800 km), Iapetus (700 km)Enceladus (250 km; water)7 large + other Trojans, co-orbiting moons, rings...

Uranus 27+ Titania (800 km), Oberon (800 km)Neptune 13+ Triton (1400 km)

Table 10.1: Summary of the number of moons around each planet in the Solar System, with commentson some of their properties.

Exercise: Calculate the mass of Mars and Jupiter given that Phobos has an orbital diameterof 9.4× 103km and an orbital period of 0.32 days, and that Callisto has an orbital diameterof 1883× 103 km and an orbital period of 16.7 days.

10.3 Formation

One of the reasons why the moons of the Solar System have such a wide range of properties isthat they have formed in a variety of ways. The three principle ways that moons are formedare:

• Condensation

In a process similar to that which formed the plants around the Sun, smaller bodies arethought to be able to form in orbit around a larger planet. Moons formed through thisprocess will be prograde ie. orbiting the planet in the same direction that the planetrotates, and, since they formed in relatively dense material around a forming planet,from a proto-satellite condensation disk, they will be relatively higher mass objects.The four Galilean moons of Jupiter, Io, Europa, Ganymede and Callisto, are likely tohave formed through condensation.

• Capture

Gravitational interactions between planets and smaller, free floating, bodies can leadto the smaller bodies becoming gravitationally bound to planets. Such captured moonsmay have retrograde orbits. Examples of these include the moons of Mars, Phobos andDeimos. Larger retrograde moons, such as Triton, are likely to be protoplanetary coresthat were captured by a larger planet during the later stages of planet formation.

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• Collision and fragmentation

Small prograde moons, especially those close to a planet, are likely to have been formedby collisions between moons that were once larger. These larger bodies are then splitinto smaller, separate moons. This process takes place predominantly near to a planetsince the orbital velocities will be higher and thus the collisions more energetic, leadingto increased chances of fragmentation during a collision. Saturn’s moon Hyperion, forexample, is likely to have formed this way.

Smaller moons in general, whether formed through fragmentation or capture, are likelyto have irregular shapes since their gravitational fields are too weak to produce a spher-ical surface.

10.3.1 The Moon

Our own Moon is somewhat of an exception among this range of formation methods, since itappears to have formed as the result of a giant impact between the young Earth and a Marssize body roughly 50-100 Myrs after the formation of the proto-Earth. Denser material in theimpactor would have remained with the young Earth, while the Moon subsequently formedfrom the lighter ejecta that resulted from the collision. This explains why the Moon has asmaller nickel-iron core, relative to its size, than other terrestrial-type bodies. The energy ofthis collision would have re-melted the surfaces of both bodies.

10.4 Tidal Forces and Tidal Heating

We have already looked at one aspect of tidal forces when we examined the Roche limit, buttidal effects also apply that are less dramatic than the production of ring systems, throughobjects being broken apart. Where bodies orbit each other there will be a difference in forcesfrom one side of the object to the other. As discussed above (equation 9.3), the tidal forceon a body of mass m a distance r from the centre of mass due to its orbit a distance d froma larger body of mass M is:

Ft = 2GMmr

d3(10.2)

If some part of the object is liquid, then this fluid will flow in response to the tidal force andyou get what we see in the seas of Earth - tides.

Exercise: Using the orbital parameters of the Earth, Sun and Moon, compare the tidalforces on a kg of water on the surface of the Earth due to the Moon and due to the Sun. Youwill find they are of comparable magnitude. This is what gives rise to ‘spring tides’ whenthe Moon is full or new, and thus aligned with the Sun. You might also want to comparethe magnitude of the tidal forces on Earth with the tidal forces experienced by Io as it orbitsaround Jupiter.

These forces will also result in the deformation of the moon or planet subject to the tidalforces. This leads to heating in just the same way that repeatedly squashing a tennis ball

58 Sun Stars and Planets 2012-13

produces heating. In the case of the Earth and Moon there isn’t very much tidal heating, butin other systems around massive planets, like Jupiter, there can be a considerable amount ofheat generated. This is what powers the volcanoes of Io, leads to the water geysers foundon Saturn’s moon Enceladus, and which may maintain a liquid ocean beneath the surfaceof Europa. Indeed water geysers may have recently been found on Europa, similar to thosealready found on Enceladus.

10.5 Tidal Locking, Libration and Circularisation

A rotating moon (or planet) will, in general, tend to pull the bulge produced by tidal forcesaway from perfect alignment with the centre of mass of the body around which it is orbiting,producing a force that acts to bring the tidal bulge back into alignment. Over time, this forcewill act to synchronise the orbital period and rotational period of the objects. The end pointof this effect, known as tidal locking, can be seen in the Earth-Moon system, where the Moonalways points the same face at the Earth. A one-to-one ratio in this kind of tidal lockingis not always the end point. Mercury, for example, has 1.5 rotation periods for each orbitalperiod.

Our discussion of tidal effects to this point has assumed that the orbits are circular, but thisis not, in general, the case. Instead, most orbits are elliptical to some extent, with the orbitalspeed varying in accordance with Kepler’s Laws. The result of this is that the face of themoon that points to the planet at the closest (pericentre) and furthest (apocentre) points ofthe orbit, wobbles from side to side during the rest of the orbit. (see Fig 10.1). Over time,the forces that result from this libration will tend to circularise the orbit.

10.6 Orbital Resonances

One object orbiting around another in isolation from everything else is a simple physical sys-tem, where the orbital parameters can be calculated analytically. However, in the real world,there are always other bodies involved which can add complexity to the orbital mechanics. Inthe case of a planet with many moons, the orbit of one moon can be affected by contributionsfrom other moons. In the Solar System more broadly, as we will see in the next lecture,planets like Jupiter or the Earth can influence the orbits of other objects around the Sun.

In many circumstances, the gravitational interactions between orbiting bodies will occur atrandom intervals and will average out over time. However, if an orbital configuration re-peats regularly and with a period that is a small integer number of orbits, then the smallperturbations from these interactions will not average out. This is a process known as orbitalresonance and it occurs quite often in complex systems of orbiting bodies. Such resonancesare described in terms of the number of orbits of the inner body to the number of orbits ofthe outer body (or bodies, for more complex interactions) eg. the Galilean moon Io is in a4:2:1 resonance with the moons Europa and Ganymede. So Io orbits Jupiter 4 times for every2 orbits that Europa makes and for every one orbit that Ganymede makes.

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Libration and circularisation

On elliptical orbit, object moves faster near pericentre

But rotation is constant, so point on object appears to wander back and forth over a rotation (see red dot shown here)

Example: Moon (shown here) only roughly faces Earth, can see more than 50% of surface as the spin lags and leads the orbital motion

Non-balancing forces will act to circularise the orbit

Exception: Io… it is in 4:2:1 resonance with Europa and Ganymede tidal heating volcanism

http://www.astrosurf.com/cidadao/moon_obs_04.htm

and http://antwrp.gsfc.nasa.gov/apod/ap070902.html Figure 10.1: How the face of a moon on an elliptical orbit changes during the course of that orbit,producing the effect known as libration. Taken from www.astrosurf.com

There are two possible effects from such an orbital resonance:

• The body (or bodies) are locked into its orbit, and cannot, for example, move outwardsas a response to tidal forces. This effect happens typically when the orbits of the objectsconcerned never approach each other very closely, and are called stable resonances.

This is the situation for Io in its orbit around Jupiter. In isolation, the tidal effects thatsqueeze Io as it orbits around Jupiter would have led it to move outwards, in the sameway that the Earth’s Moon has moved away over millions of years. Orbital angularmomentum is thus transferred to the Moon from the Earth, and the Moon climbs outof the Earth’s gravity well. This cannot happen to Io because of the orbital resonanceit is in with Ganymede and Europa. The energy that would otherwise move Io upthe gravity well from Jupiter instead goes into tidal heating of Io’s interior, leadingto the rampant volcanism that we can see on its surface. Europa, too, is involvedwith this orbital resonance and, like Io, is also locked into its orbit. It is further fromJupiter, so there is less tidal heating as a result, but this is still enough to melt someof Europa’s icy interior, leading to a layer of liquid water beneath its surface, andproducing cryovolcanism. Saturn’s moon Enceladus is in a 2:1 orbital resonance withthe moon Dione. Tidal heating here is likely to be responsible for the cryovolcanismthat produces the geysers seen on this moon.

• The body gets accelerated or decelerated in its orbit until it is no longer in resonance.In effect this means that the orbital configuration affected is cleared of objects subjectto the resonance. These are called unstable resonances.

Observation of the rings of Saturn show a variety of gaps. The most obvious of these,observable from Earth and discovered originally in 1675 by Giovanni Cassini and named

60 Sun Stars and Planets 2012-13

after him, is the Cassini Division. This gap in the rings is produced by a 2:1 orbitalresonance between any particles in this ring and the moon Mimas.

Orbital resonances and other effects in fact make the structure of Saturn’s rings verycomplex, with a wide variety of different gaps, sub-rings, and sub-structures such asspokes, braids and shepherd moons.

Orbital resonances are not restricted to the orbits of moons around planets, but also applyin the broader Solar System. They produce the Kirkwood Gaps seen in the distribution oforbits in the asteroid belt, and, in the outer Solar System, lead to the class of objects calledPlutinos (see the next chapter for more details).

Orbital interactions and resonances can also lead to quite bizarre non-circular and non-elliptical orbits. The Saturnian moons Janus and Epimetheus, for example, have ‘bean’shaped orbits around Saturn. This kind of thing is not restricted to the outer Solar System.A near Earth asteroid, Cruithne, lies in a 1:1 orbital resonance between the Earth and theSun, leading to a strange ‘bean’ shaped orbit that reaches outwards to the orbit of Mars andinwards beyond the orbit of Venus.

Things to Remember

• Use of the orbital period of an orbiting body to calculate the mass of its parent planet

• The names and general properties of the most famous moons in the Solar Sys-tem (the Moon, Mars’ moons, the Galilean moons of juliter, Titan, Enceladus, Triton)

• The main formation mechanisms for moons

• Tidal forces, tidal heating, tidal locking and libration & circularisation

• Orbital resonances in moon systems and how this can lead to tidal heating, es-pecially in the example of Io

Chapter 11

Small Bodies: Comets, Asteroidsand the Outer Solar System

11.1 Introduction

A full analysis of the orbital dynamics of the Solar System shows that the regions betweenmost of the planets lack stable orbits because of gravitational resonance effects. The onlyregions in the Solar System where this is not the case are between Mars and Jupiter, andoutside the orbit of Neptune. Unsurprisingly, we find plentiful small objects orbiting in theseregions, forming the Asteroid Belt, between Mars and Jupiter, and the Kuiper Belt, beyondthe orbit of Neptune. Further out, there is also the Oort cloud, where proto-comets kickedout of the young Solar System, live. There are also other, shorter lived, populations ofobjects, such as the Centaurs, and comets, which occasionally visit the inner Solar System.All together, these objects form the small bodies and minor planets of the Solar System.

While they are not a significant constituent of the Solar System by mass, small bodies havenot been through the reprocessing involved in planet formation that the rocks and gases inlarger bodies have endured. The small bodies can thus provide us with clues about whatmaterial in the early Solar System might have been like. Asteroids also occasionally collidewith planets, including the Earth, so keeping an eye on them is not only useful scientifically,it may provide early warning for major disasters.

11.2 Asteroids

There are approximately 100,000 asteroids currently known in the main asteroid belt betweenMars and Jupiter, and about 30,000 of these have well determined orbits. The total massof these asteroids only amounts to about 0.001 Earth masses. The largest asteroid in themain belt is Ceres, with a diameter of 900km. The vast majority are much smaller than this,

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62 Sun Stars and Planets 2012-13

Kirkwood gaps

Plot of the number of asteroids vs period Kirkwood gaps (discovered 1857)

These correspond to orbital

resonances with Jupiter.

Most prominent at

2:1, 3:1, 5:2, 7:3

[Remember

# orbits (inner body) : # orbits (outer body)]

but additional gaps, not explained

by obvious orbital resonances.

New studies (e.g., Minton &

Malhotra 2009, Nature 457,

1109) suggest it is a result of

orbital resonances + planetary

migration.

Nature (2009) 457

Figure 11.1: The distribution of main belt asteroids as a function of their orbital radius. The gaps inthe distribution are known as the Kirkwood Gaps. Also noted are the orbital resonances with Jupiter,which coincide with most, but not all, of the Kirkwood Gaps. From Minton & Malhotra, Nature(2009), 457, 1109.

with their size distribution mirroring the size distribution of impact craters on objects likethe Moon and Mercury. This isn’t surprising since the impacts were produced by asteroids.

While most asteroids lie between Mars and Jupiter, some exist in the inner Solar System.Near Earth Asteroids are bodies that come close to the Earth. About 7000 of these arecurrently known. Potentially Hazardous Asteroids (PHAs) are those that come very close toEarth and might collide with it at some point in the future. About 1000 of these are currentlyknown.

Most asteroids are too small to have gone through any surface differentiation themselves.Rather then being solid bodies, like Earth or Mercury, they are thought to be ‘rubble piles’made up of lots of separate sub-fragments bound together by mutual gravity. The differentfragments can move relative to one another, leading to a somewhat ‘molten’ appearance, withfiner, dusty regolith material settling to lower points in the local gravitational potential, andlarger fragments moving upwards in a manner similar to the motion of brazil nuts in museliwhen it is shaken. Data taken by the Hyabusa spacecraft on its mission to the near-Earthasteroid Itokawa are consistent with this idea.

The distribution of Main belt asteroids as a function of their semi-major axis (in AU) isshown in Fig. 11.1. As you can see, the distribution is not uniform, but is characterised by

63

several distinct gaps. These were discovered in 1857 and are still known as Kirkwood Gaps inhonour of the astronomer who found them. Most of the Kirkwood Gaps are easily explainedby orbital resonances with Jupiter. The 2:1, 3:1, 5:2 and 7:3 resonances are clearly seen.However, there are other gaps that do not coincide with resonances with the current orbit ofJupiter.

The explanation for this, as proposed by Minton and Malhotra (Nature (2009), 457, 1109,M&M) is that the giant planets went through a period of migration in the early Solar System.M&M’s model suggests that Jupiter moved inwards by about 0.2 AU, while Saturn, Uranusand Neptune moved outwards by 0.8, 3 and 7 AU respectively. This migration is the resultof the exchange of angular momentum between planetesimals, left over from the main phaseof planet formation, and the giant planets. They tested this idea by modelling the effects ofthe known planetary orbits on an initially uniform distribution of main belt asteroids overa period of 4 Gyr and comparing this to the same simulation, but with the added assumedperiod of planetary orbit evolution. These simulations were then compared with the observedasteroid distribution. The simulation with the planet orbit evolution was by far the better fit.The planetary orbit evolution would also give rise to the observed late heavy bombardmentof the inner Solar System, and fits with other results.

Asteroids come in a range of classes, largely determined by their reflectance spectrum whichallows an estimation of the material on their surface. Classes include:

• C class: surface dominated by carbon (carbonaceous), with reflectance ∼5%. These arethe most common type, representing 40% at 2AU and 80% at 4AU.

• S class: surfaces dominated by silicates (stony material), with reflectance ∼16%, andwith a distinct spectral absorption signature at ∼1µm. These are the second mostcommon class.

• M class: these asteroids are almost entirely metal, containing Ni and Fe. They are rarer,but have reflectance ∼15%.

• D-class: these asteroids are very dark, with reflectance only ∼3%. They are increasinglycommon at greater distances from the Sun. Their surfaces may include organic material.

There are also other classes including E and P. The overall numbers of different classes,especially the low reflectance ones, are difficult to judge since different classes are detectedwith differing efficiencies, so the statistics are dominated by selection effects.

The size distribution of asteroids is a power law, with roughly equal amounts of mass in eachlogarithmic mass bin, so there are many small asteroids, but only a few very large ones.

One dynamically interesting subclass of asteroid are the Trojan asteroids. They share thesame orbit as Jupiter, but lie 60 degrees ahead and 60 degrees behind the planet’s orbitalposition. These points are the so-called L4 and L5 Lagrange points, where the gravitationaland centrifugal forces of two orbiting masses cancel out for a third, smaller, orbiting body.The L4 and L5 points are stable saddle points in the gravitational potential, so that objectsthat arrive there will stay there. The Trojan asteroids might have arrived at Jupiter’s L4 andL5 points during the period in which Jupiter migrated closer to the Sun.

64 Sun Stars and Planets 2012-13

Figure 11.2: The orbits of a selection of Kuiper Belt Objects compared to the orbits of Jupiter,Saturn, Uranus, Neptune and Pluto (J, S, U, N, P). From Rothery, McBride & Gilmour.

11.3 Kuiper Belt and Trans-Neptunian Objects

The migration of the giant planets, as described above, has largely cleared small bodies fromthe orbit of Jupiter to the orbit of Neptune. Beyond the orbit of Neptune, though, smallbodies can persist undisturbed. As early as the 1950s, Gerard Kuiper proposed the existenceof a belt of small bodies beyond the orbit of Neptune. Pluto, and its largest moon Charon,were already known at this point, discovered in 1930, but it wasn’t until 1992 that anyfurther trans-neptunion objects (TNOs) were discovered. We now know of at least 70,000such objects, with diameters >100km, forming what has been called the Kuiper belt, whichlies 30-50 AUs or more from the Sun (see Fig. 11.2). The total mass of objects in the KuiperBelt has been estimated to be ∼0.1 Earth mass, meaning that the Kuiper Belt actuallyincludes more mass than the ‘main’ asteroid belt between Mars and Jupiter.

Kuiper Belt objects (KBOs) are thought to be the left overs from earlier stages of the for-mation of the Solar System, made up of material with a high fraction of ices and volatiles.Reflectance spectra of KBOs have a wide range of properties, which may be the result of longterm changes in their surface properties resulting from exposure to UV light from the Sun,but also resulting from more abrupt surface changes coming from impacts between KBOs.Our knowledge of the outer solar system is still very incomplete. The NASA New Horizonsmission will help with this when it encounters Pluto in 2015 and then moves on to studyother KBOs in 2016-2020.

65

Figure 11.3: The orbital eccentricity plotted against semi-major axis for KBOs. From Rothery,McBride & Gilmour.

While we await the results from New Horizons, and other detailed studies using groundand space based telescopes, the classifications of KBOs are largely based on their orbitalproperties, and in particular on their orbital eccentricity and semi-major axis. When theseare plotted together you can see three separate groups of orbital characteristics which leadsto the division of KBOs into separate classes. The majority are classified as Classical KBOs,and have low eccentricity orbits with radii of ∼44 AU. The second largest group have a rangeof eccentricities and all lie at a radius close to 39.4 AU. Pluto is one of these objects, leadingthem to be termed Plutinos. If the orbital period of the Plutions is compared with that ofNeptune, you find that Neptune orbits the Sun three times for every two orbits that a Plutinomakes: they are in a 3:2 orbital resonance with Neptune, keeping them in this orbital position.Pluto is in many ways indistinguishable from the other Plutinos, a result which eventuallyled to the reclassification of Pluto as a dwarf planet. The final group of KBOs have higheccentricities and large semi-major axis. They are classified as Scattered Disk Objects andare likely the source of short period comets.

An additional class of small solar system body that is likely associated with KBOs are the so-called Centaurs. These are objects whose orbits cross the orbits of one or more major planet.Such orbits will not be long lasting because they will eventually encounter the gravitationalfield of a major planet and be captured or scattered into a different orbit. They may evenhit one of the giant planets. An example of this was the impact of Comet Showmakey-Levy9 with Jupiter in 1994. The discovery of Centaurs predates that of KBOs other than Plutoby a number of years. The first Centaur, called Chiron, was found in 1977. The fact that itwas in an orbit that was not stable in the long term hinted at the existence of a larger bodyof similar objects in more stable orbits that would be able to feed Centaurs into the Solar

66 Sun Stars and Planets 2012-13

System. The reservoir for the Centaurs, and for short period comets, is the Kuiper Belt,which was discovered 15 years later.

11.4 Comets

The final class of small body in the Solar System that we will discuss are perhaps the mostspectacular: comets.

Comets are objects on highly eccentric orbits that come from the outer to the inner parts ofthe Solar System. They are objects rich in ices and volatiles which are released when theyheat up in the inner parts of the system, giving rise to the spectacular tails (one of dust oneof ionised material, which interact differently with the solar wind). They have very shortlives, only a few 104 years before mass loss and dynamical interactions with planets lead totheir destruction. Their internal structure is thought to resemble a ‘dirty snowball’, wheredust and rock is mixed with ices, including water ice and other frozen volatiles. The dirtysmowball forms the comet’s nucleus, which is small (10-20km) and very porous, When theynear the Sun they heat up and volatiles boil off, leading to the familiar shape of these objects.

There are two classes of comet based on their orbital period - short period (<200 years) andlong period. The short period comets, like Halley, have low orbital inclination and are usuallyprograde. These comets are thought to originate in the Kuiper Belt. Long period comets havemuch higher orbital inclination and are as often retrograde as prograde. They are thoughtto come from the Oort cloud, a spherical reservoir of comets believed to lie at much greaterdistances from the Sun than the Kuiper Belt, out to as far as 50000 AU, about a quarter ofthe way to the nearest star. So far, no definitive detection of an object in the Oort Cloud hasbeen made. Instead, its existence is currently inferred from the presence of the long periodcomets, in much the same way that the existence of the Kuiper Belt was once inferred fromthe discovery of Centaurs. Sedna, a TNO with a very eccentric orbit with an aphelion of∼1000 AU is our best current candidate for a member of the Oort cloud.

Towards the end of 2014 the Rosetta mission will rendezvous with the periodic comet 67P/ChuryumovGerasimenko,and send a lander down to the surface in November 2014. Over the following year, the space-craft will follow the comet as it falls into the inner Solar System. Depending on the resultsof this mission, by the time this lecture course is given next year the section on comets mightbe very different.

67

Things to Remember

• The basic parameters of the asteroid belt and the classes & constituents of asteroids

• The Kirkwood gaps and their origin in orbital resonances. The Trojan aster-oids.

• Properties of Kuiper Belt objects. The Centaurs.

• The internal structure and origin of comets, including the orbital properties oflong & short period comets

• How these relate to the Kuiper Belt and Oort Cloud

68 Sun Stars and Planets 2012-13

Chapter 12

Detecting Exoplanets

12.1 Introduction

We have spent the last 8 chapters, and 8 lectures, looking at the properties of objects in ourown Solar System. Not so long ago, that would be where things stopped, since we knew ofno other planetary bodies in the Universe. Whole theories of planet and planetary systemformation were developed on the basis of the 8 planets and many minor bodies of our ownSolar System, but there was lack of other places where these models could be tested.

Over the last 15 years, though, there has been a revolution in our understanding of planetarysystems, resulting from a series of technological breakthroughs that have allowed planetsin other solar systems - exoplanets - to be discovered in ever greater numbers. Specificobservatories on the ground and in space are completely dedicated to planet searches, andover a thousand planets in other systems are now know 1iscoveries continue at a fast pace -last years lecture notes said ’nearly a thousand’. More planet discoveries are announced everyday, so some of the raw numbers in these notes will already be out of date. You can keeptrack of the latest results through dedicated websites such as exoplanet.eu.

12.2 Units

For the rest of this course we will be looking at objects far away from the Solar System,and will thus have to use astronomer’s units. Many of these are based on scalings to knownobjects eg. the mass of the Sun (M), the astronomical unit, the Solar luminosity (L), butthere are other more specialised units that we will use.

The first, and most important, of these is the parsec. This is a distance of 3.26 light years,3.08×1016m, or 2.1 ×105 AU. The parsec is derived from the distance at which an objectmust be for it to have a proper motion on the sky of 1 arcsecond when the Earth moves by 1

1D

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70 Sun Stars and Planets 2012-13

Figure 12.1: The number of exoplanets discovered each year. Plot produced using the tools anddatabase available at exoplanet.eu

71

Figure 12.2: Diagram showing how the parsec is derived

AU. Essentially this means that a parsec is the length of the adjacent side of a triangle whichhas an angle of 1 arcsecond and an opposite length of 1 AU. See Fig. 12.2 for a diagram.

The other thing that we will be using that could be termed an astronomer’s unit are magni-tudes. These will be used to express the brightness of stars and changes in the flux receivedfrom them. Magnitudes where introduced to you in the first (Stars) part of this course, butas a reminder, the difference in magnitude between two objects is given by:

m1 −m2 = −2.5 log

(F1

F2

)(12.1)

where m1 and m2 are the magnitudes of the two objects, and F1 and F2 are the fluxes of thetwo objects. This just deals with how to compare two objects. To place magnitudes on anabsolute scale, one also has to define the flux that corresponds to zero magnitude. There aretwo ways of doing this in astronomy. The traditional way has been to define the magnitudeof the star Vega (a bright A-type star) to be identically zero for all observations. Theseare called Vega magnitudes. The second, is to define a specific flux density, of 3631 Jy (aunit which will be explained shortly) to be zero magnitude for all observations. These arecalled AB magnitudes. Since the star Vega has (roughly)2ega is a good, single temperature,black body at optical wavelengths, which led to its adoption as a standard star. However,it was later found that it is surrounded by a dust disk, left over from its formation, whichcontributes significantly in the mid- and near-IR. This causes some confusion when Vega isused as a standard star at these wavelengths, though these considerations are not importanthere. a black body spectrum, its flux varies with wavelength, so the flux corresponding to

2V

72 Sun Stars and Planets 2012-13

zero magnitudes in the vega system varies with wavelength. This is not the case for ABmagnitudes.

The Jansky (Jy) is another astronomer’s unit, but is more closely related to standard SI units.

1Jy = 10−26Wm−2Hz−1 (12.2)

12.3 What is a Planet Anyway?

Within our own Solar System, there is a formal definition of a planet. It is a body orbitingthe Sun, whose gravity is strong enough to make it spherical, and which has cleared itsneighbourhood of smaller bodies. The latter part of this definition, agreed by the InternationalAstronomical Union (IAU) in 2005, is what demoted Pluto to minor planet status.

Outside our Solar System the definition of what is a planet is less clear. The IAU definitionstates that to qualify as an exoplanet a body must be orbiting a star and have a mass below thethreshold at which thermonuclear fusion of deuterium can take place. This sets the maximummass for a planet at ∼ 13 MJ . No consideration is given to how these bodies formed, andthe minimum mass should match the minimum mass to qualify as a planet in our own SolarSystem.

Things that are not considered exoplanets in this scheme include objects above the deuteriumburning mass limit, which are defined as brown dwarfs (these are essentially failed stars) andfree floating bodies that are low enough mass to qualify as an exoplanet, but which do notorbit around a star. These are termed sub-brown dwarfs, and recent results suggest that theymay be more numerous than stars in our galaxy.

A physical definition of a planet based on formation history and/or composition, which mightbe a more scientific approach, is still lacking. The discovery of objects like Cha 110913-773444,which appears to be a sub-brown-dwarf (ie. with mass <13 MJ) but which has a dust diskin which planets (moons?) might be forming, just makes the issue more complicated.

12.4 Direct Detection: How Hard Can it Be?

In the current era of space telescopes and large, 8-10m, telescopes on the ground, one mightthink that directly detecting an exoplanet orbiting around another star would be easy. Un-fortunately, this is far from true, mainly because of the huge contrast between the light thatcomes from the star and that which is reflected from the planet, and because of the smallangular separation between any planet and its parent star.

Consider a planet with an albedo of 1, visible only because of light reflected from its parentstar, radius Rp, orbiting a distance d from a star of luminosity L∗. The stellar flux receivedby the planet will be:

F =L∗

4πd2(12.3)

73

The planet’s luminosity, Lp then comes from the total power it intercepts, assuming that ithas an albedo of 1:

πR2p ⇒ Lp = FπR2

p =L∗4

(Rpd

)2

(12.4)

If we put in numbers appropriate for Jupiter into this equation - Rp ∼ 7 × 107m and d ∼ 5AU - and calculate its relative luminosity to the Sun, we find:

LJL

=L4

(RJd

)2

× 1

L=

1

4

(7× 107

5× 1.5× 1011

)2

⇒ LJL∼ 2× 10−9 (12.5)

So, to directly detect a planet like Jupiter orbiting another star you will have to remove thelight of that star to an accuracy of about 1 part in a billion - the star outshines the planet bythat much. This is very difficult to achieve. The situation is somewhat better in the infrared,where the Black Body spectrum of the hot star is declining, but where that of the coolerplanet is peaking, but this still requires better than 1 part in a million exclusion of stellarlight. There are ways that this can be achieved, using techniques such as choronography andnulling interferometry, but this all means that direct detection is not an efficient way to searchfor planets. However, it can be, and has been, used to follow up planets that have alreadybeen detected by indirect methods, so as to better characterise the objects. For example,the first direct spectrum of an exoplanet was obtained in 2010 by Bowler et al using such anapproach.

If we cannot search for exoplanets directly, how can they be found?

Fortunately there are a range of indirect methods that look for the effects of any planets thatmay be present on their parent star.

12.5 Reflex Motion and Doppler Measurements

A family of detection methods are based on studying the dynamical effects of an orbitingplanet on the parent star.

Just as with binary stars, a star and planet actually orbit around a common centre of mass,but with the planet mass much smaller than the stellar mass. Viewing this in the centre ofmass frame it looks like Fig. 12.3. From the stars part of the course we know that for binarystars:

M1

M2=r2r1

and M1 +M2 =4π2

G

r3

P 2(12.6)

where P is the orbital period. So, for a planet of mass mp orbiting a star of much larger massMs:

mp

Ms=asap

and Ms =4π2

G

a3pP 2

(12.7)

where as and ap are the distances from the star and planet to the common centre of massrespectively.

74 Sun Stars and Planets 2012-13

x

a_p

a_s

Figure 12.3: A star orbits around the common centre of mass with radius a s, while the planet orbitsaround it with a radius a p.

The first result from this analysis, as can be seen from Fig. 12.3 is that the position of astar being orbited by a planet will appear to wobble around on the sky. While the planetcannot be directly seen, its existence can be inferred if we can measure the star’s regulardisplacement as. In fact the angular displacement β = as/d, where d is the distance to thestar from the observer, is what is usually measured. From the above we can find that

β =asd

butmp

Ms=asap⇒ β =

mpapMsd

(12.8)

So you get a larger, and thus more detectable, angular displacement β for large mass planetsin wide orbits around low mass stars. For our own Solar System viewed form a distance of10 pc, you would see a displacement lower than 0.4 milliarcsec per year from the effect ofJupiter orbiting the Sun. This is a very small angular shift, corresponding to the width of afinger at 5000 km, so it is not a particularly viable method of planet detection.

EXERCISE: Calculate the displacement you get from the Earth’s orbit around the Sun,when viewed from 10 pc.

However, an alternative method based on this same idea comes from looking at the motionof the star along the line of sight, rather than in the plane of the sky. What can be measuredhere are changes in the velocity of the star in the line of sight, which can be done by lookingat Doppler shifts from spectral lines.

Recalling the results for binary stars:

M32 sin3(i)

M2= v31

P

2πG(12.9)

75

where i is the inclination angle between the orbital plane and the line of sight, and P is theorbital period.

For planets M2 = mp, M = Ms +mp = Ms and v1 = vs, so:

mp sin(i) =

(M2sP

2πG

)1/3

vs (12.10)

Without knowing the inclination angle i, this method only allows you to calculate the mini-mum mass for a planet.

The largest values of vs come for large mass planets orbiting low mass stars with short orbitalperiods.

If you were observing along the plane of the ecliptic, the velocity shifts from the Sun-Jupitersystem amount to 12 m/s, and from the Sun-Earth system amount to 0.1 m/s.Velocity shiftsas low as 0.5 m/s have been measured, and the first clear detections of exoplanets aroundmain sequence stars were obtained using this method.

EXERCISE: Calculate the velocity shift you would get if Jupiter was orbiting the Sun withan orbital period of 88 days (ie. in the orbit of Mercury) rather than its actual orbital periodof 11.9 years.

12.6 Planetary Transit Searches

If a planetary system’s orbital plane lies along our line of sight, planets will from time to timepass in front of their star, absorbing some of the light from the star that would otherwisereach us. This kind of thing can be seen in our own Solar System where Venus or Mercurycan be seen to pass in front of the Sun. The last transit of Venus was in June 2012. Planetarytransits will cause a small, but potentially measurable, dip in the brightness of a distant starobserved from Earth (see Fig. 12.4).

What flux decrease will a planetary transit produce?

If the uneclipsed flux of the star is Fs, the eclipsed flux Ft, the flux of the planet is Fp, theradius of the star is Rs and of the planet is Rp then:

Ft = Fs −(RpRs

)2

Fs + Fp (12.11)

Since Fp << Fs we get:

Ft = Fs

(1−

(RpRs

)2)

and the flux decrease ∆F = Fs − Tt =

(RpRs

)2

Fs (12.12)

76 Sun Stars and Planets 2012-13

Transit MethodTransit Method

…measure change in flux as planet crosses stellar disk; accessible parameters:

* orbital period

* planetary radius RP/RS

* planetary mass (need MS)

* inclination of orbit

flux decrease ≈ (RP/RS)2

Sensitivity/bias:

• easier to detect larger planetary radii and

• small semi-major axes

→ short periods

→ `hot Jupiters’

∆F

Figure 12.4: A diagram showing schematically what happens to the light received form a star as aplanet transits along our line of sight.

So:

∆F

F=

(RpRs

)2

(12.13)

For the Solar System, Jupiter would cause a 1% drop in the light seen from the Sun, whichis large enough to be measurable from the ground, while the Earth would produce a 0.01%drop, which can be measured from space.

If you know that the planet is transiting then doppler measurements can determine theplanet’s mass. The star’s radius can be determined from the duration of the transit, leadingto the radius of the planet. That, combined with the mass, allows the density to be calculated,which is the first step towards understanding what the planet is made of.

For a transit to be detected the planet’s orbital plane must be quite closely aligned with theline of sight to the star. Assuming a random orientation of orbital inclinations for planetarysystems, and considering the diameter of the Sun, it can be shown that there is a chance ofabout 1 in 200 for the transit of an Earth-like planet around a Sun-like star to be visible.Such transits would happen only once a year for the Earth, and you would need to observeat least two such transits to be sure that it was detected, and to measure the orbital period.The Kepler satellite was thus designed to monitor a total of 105 stars for a period of 3 to 5years in search of, among other things, Earth like planets. If they are common, it should beable to detect several such systems. Sadly, a technical failure on the satellite has led to thepossible end of its planet hunting mission after only about 3 years, but efforts are under wayto restore its planet hunting capabilities.

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12.7 Other ways to detect planets

While the transit and doppler methods are responsible for the detection of most of the exo-planets that we know, there are a couple of other methods that have proven useful.

12.7.1 Pulsar Planets

The first of these is the timing of pulsars, which led to the detection of the very first exoplanets.Pulsars are rotating neutron stars, remnants of supernova explosions, which emit beamsof electromagnetic radiation from their magnetic poles. These beams act like lighthouses,producing regular pulses which can be timed to picosecond accuracies. Regular deviations inthese pulses, produced by the same centre-of-mass shifts seen above for the transit method,can be measured to accuracies better than 1m/s in velocity and 1000 km in distance. Thishas allowed a small number of planets, with masses down to 0.0004 Earth masses, to bediscovered. None of them are likely to be particularly nice places to live, though, since thesesystems have survived a supernova and are now bathed in hard radiation from the pulsar.

12.7.2 Gravitational Lensing

Gravitational lensing is the process by which light is bent and focussed as it passes close to alarge mass. Stars and planets are both large enough to produce a measurable magnificationof the light of a background star if they pass close enough to our line of sight. Large scalemonitoring projects like OGLE, originally intended to search for Baryonic Dark Matter, canin principle detect the lensing amplification produced by a planet orbiting a star responsiblefor lensing, and there are a small number of cases where this has been found. The advantageof this approach is that it is sensitive to essentially all possible planetary masses, but thedisadvantage is that the lensing signal is not repeatable, so one can never be absolutelycertain what has produced it, or determine the full characteristics of any planetary systemthe lensing has revealed.

78 Sun Stars and Planets 2012-13

Things to Remember

• The definitions of parsec, AU, magnitude and Jy

• The definition of a planet, and how to distinguish it from a brown dwarf and aminor planet

• The problems of direct detection of exoplanets; derivation of luminosity ratios

• The derivation of reflex motion

• The derivation of the change in the flux from a star due to a planetary tran-sit

• The use of the above methods in detecting planets and other planet detectionmethods

Chapter 13

The Exoplanet Population

13.1 Introduction

At the time of writing, over a thousand planets are now known outside our own Solar System,including at least 129 in multi-planet systems. More are being discovered all the time thanksto ongoing survey programmes such as the Kepler satellite and SuperWASP survey, lookingfor planetary transits, and the HARPS radial velocity survey. We have reached the pointwhere we can draw conclusions about some aspects of the exoplanet population. However,the methods of exoplanet detection all have limitations, so our view of the population as awhole is necessarily incomplete, and biased by what are known as ‘selection effects’. In thischapter we will look at what is known about the exoplanet population, try to deconvolve someof the selection effects, and draw some conclusions about the overall population of planets inour galaxy.

13.2 The Current State of Planet Searches

New planet discoveries are announced all the time, so any attempt to describe the currentstate of planet searches is doomed to become rapidly out of date. However, the broad pictureis that we now (Feb 2014) have discovered over 1000 planets. Most of these were discoveredusing the radial velocity technique, but the transit method is rapidly producing new candidateand confirmed planets thanks to the Kepler mission. About 15% of discovered planets arefound in multiple planet systems. The parent stars of the planets discovered so far are largelyF, G and K type main sequence stars. This does not, however, mean that other stellar typesdo not have stars, since the majority of planet searches have been targeted at F, G and K typestars. This is because these stars are sufficiently long lived that life might have developedon planets around them, they are relatively bright, compared to the more common low massM stars, and they are well suited to the radial velocity method, since they have many welldefined spectral lines, and have stable stellar atmospheres. The nature of planets aroundother stellar spectral types is thus largely unconstrained.

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80 Sun Stars and Planets 2012-13

On the basis of current results we can say that at least 20-50% of F, G and K-type stars haveat least one giant planet, comparable in mass to Jupiter, in an orbit whose semi-major axisis <20 AU.

13.3 Selection Effects

The issue of host stellar type is the first in a series of ‘selection effects’ that constrain andbias what we are able to say about the exoplanet population. Selection effects arise in awide variety of sciences, especially observational ones like astrophysics, where one does nothave full control over what you find. Selection effects are often quite subtle and can requirecareful consideration, but they can also be quite obvious once the observational problemis understood. The preponderance of F, G and K-type stars in the radial velocity methodsearches is a case in point. Another is the sensitivity, or lack of it, of various methods tovarious types of planets. The radial velocity method, for example, is not sensitive enoughto discover an Earth mass planet in an Earth-like orbit around any other stars. The radialvelocity changes that the Earth produces on the Sun have an amplitude of ∼ 0.1 m/s. Themost accurate radial velocity measurements so far achieved are five times bigger than this, at0.5 m/s, and that was only possible after a long monitoring programme and heroic efforts toexclude the effects of other motions on the surface of the star being observed (Alpha Centauri- Udry et al., 2012, Nature, 491, 207).

While some selection effects will exclude some classes of planet from what we can detect, otherselection effects will lead to other classes of planet being much easier to detect. High massplanets that are in orbits very close to their parent stars produce the largest radial velocityshifts. Such ‘Hot Jupiters’ are thus very well represented in the current results of exoplanetsearches.

These and other selection effects must be carefully considered when using the current set ofknown exoplanets to derive conclusions about the overall population of exoplanets. Never-theless, this is what we are about to do.

13.4 Exoplanet Masses

Figure 13.1 shows a histogram of currently known planet masses, measured in terms of Jupitermasses. As can be seen, the vast majority of currently known exoplanets have masses thatwould class them as gas giants if they were in our own Solar System, with masses > 0.05MJ , the mass of Uranus. In fact, most of these masses are not actual masses but are lowerlimits to the mass of planets detected using the radial velocity method, and thus are infact measurements of M sin i where i is the angle of inclination of the orbit to the line ofsight. To go from such a radial velocity minimum mass to an actual mass measurement, adetermination of the inclination angle is needed. If a transit observation is available thenwe known the the inclination angle i is high. An example of this is the planet HD209458b,where a transit observation and a radial velocity shift is seen. Estimates for the inclination

81

Figure 13.1: Histogram of planet masses. The x-axis shows log(planet mass/Jupiter mass). Earth’smass is 0.003 times that of Jupiter. As can be seen, nearly all currently known planets have masses ionthe range of gas giants, with hardly any planets known that have masses comparable to that of Earth.This is because current detection techniques make it very difficult to detect and confirm a planet withmass comparable to that of Earth. Generated using data from exoplanet.eu

82 Sun Stars and Planets 2012-13

angle can also be obtained by other observations. In the case of Epsilon Eridani, a dust ringaround the star is observed. This appears elliptical in the observations, but such rings areexpected to be circular. From the dust ring ellipticity an inclination angle of 46 degrees canbe estimated. The orbital plane of the planets in this system should match that of the dustring, so we can then estimate the true mass of the planet Epsilon Eridani b. From radialvelocity measurements we have M sin i = 0.86MJ , thus:

M sin i = 0.86MJ ; i = 46 ⇒M =0.86MJ

sin 46= 1.2MJ (13.1)

The fact that most of the planets known have masses in the gas giant range is not surprising.Our current detection techniques are largely insensitive to lower mass planets. We thus wouldnot expect to have found many, or in fact any, Earth mass planets in our studies to date. Thefew Earth and lower mass planets that appear in Fig. 13.1 are the result of pulsar timingor gravitational lensing detections which are not subject to the selection biases in favour oflarge mass planets that apply to the radial velocity and, to some extent, transit methods thatare responsible for the majority of planet detections. Nevertheless, a statistical analysis ofKepler planet candidates, including many that are potentially Earth-mass but which cannotbe unconfirmed by radial velocity measurements (Petigura et al., 2013), has concluded that22% of Sun-like stars harbour planets in orbits such that liquid water is possible on theirsurfaces (a region known as the habitable zone - see next chapter). The nearest star withsuch a planet could be as close as 12 light years away.

13.5 Exoplanet Composition

Detailed analysis of the composition of an exoplanet is not something we can yet achieve.However, simply being able to measure the density of an exoplanet would be a big steptowards understanding what it might be made of, especially bearing in mind the range ofdensities of planets in our own Solar System, with the terrestrial planets being much denserthan the gas giants. Masses and planetary radii are available for about a few hundred planetsso far. The vast majority of these turn out to have low densities, comparable to those ofour own gas giants. There are a handful of exoplanets that have higher densities, though,and these can be considered candidate terrestrial planets. In most of these cases their massestimates are currently rather uncertain, so there are large uncertainties on their deriveddensities.

The best way to determine the composition of an exoplanet, or at least its atmosphere, isto obtain spectroscopy, but this is an even more difficult job than direct detection of thecontinuum light of a planet. However, for transiting exoplanets, a number of tricks arepossible. Time resolved spectroscopy allows us to look at the effect of the exoplanet on thelight of the star as it passes through the planet’s atmosphere. Comparison of stellar spectrabefore, during and after the transit allow the size and some aspects of the planet’s compositionto be measured. This was first achieved on the planet HD209458b, a gas giant orbiting 0.045AU from its parent star. Absorption in the Lyman α line coming from hydrogen in theplanet’s atmosphere was found to cover 15% of the stellar disk, rather than the 1.5% covered

83

Figure 13.2: Mass and orbital semi-major axis for non-pulsar planets. Many more gas giants closeto their parent star are found than expected. From Rothery, Gilmour & Sephton.

by the opaque core of the planet, implying that the atmosphere of this planet is very extended,resulting from the fact that it is being heated to high temperatures by the star. Since theseinitial observations, similar spectroscopic transit studies have revealed water vapour, carbondioxide and methane in HD209458b’s atmosphere, as well as hydrogen. These are all thingsyou would expect to find in a gas giant’s atmosphere.

13.6 Exoplanet Orbits: Hot Jupiters and Planetary Migration

One of the big surprises when exoplanets started to be discovered was that there are a largenumber of ‘hot Jupiters’ - gas giant planets that orbit very close to their parent stars (seFig. 13.2). These are in fact the easiest objects for both radial velocity and transit studiesto detect, but there was no expectation at all, before their detection, that such things wouldexist. The reason for this is that gas giants are expected to form much further out in theirsolar systems since the young star will, on first ignition, heat and boil off all the volatiles inthe inner regions of the protoplanetary disk. This is why we see terrestrial planets close toour own Sun and gas giants further out.

Hot Jupiters must therefore migrate inwards, from their formation location, to where they

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Figure 13.3: Orbital eccentricities vs. semi-major axis for exoplanets compare to those of Jupiterand Saturn. From Rothery, Gilmour & Sephton.

are seen by our exoplanet observations. The best current idea for how this occurs is thatthere is an interaction between the forming gas giant and the protoplanetary disk during theprocess of formation that causes it to move inwards. The infall cannot proceed too far orthe gas giant will end up in the star, so some other process has to terminate the migration,possibly as a result of the young star boiling away the protoplanetary disk. As a gas giantmoves inwards in its system, smaller terrestrial planets will be scattered out of their systemsor dumped into their stars.

If this is common, how did our own solar system and planet stay as they are? There is someevidence from our asteroid belt that Jupiter moved inwards by about 0.2 AU, but then thismotion stopped. It turns out that interactions between gas giants when there is more thanone in a system can slow or halt any inward migration. Perhaps our existence on Earth is aresult of such an interaction between Jupiter and Saturn.

The orbital eccentricities of exoplanets are often much larger than those seen in our own SolarSystem (see Fig. 13.3). This may come about through orbital resonances between two gasgiants, or through close encounters between gas giants, which would result in one gas giantbeing expelled from the system and the other acquiring a high eccentricity orbit, passing closeto its parent star.

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13.7 Host Star Metalicity

One other result that has emerged from studies of exoplanets and their host stars is that itappears that planets are more likely to be found orbiting stars with higher metallicities - ie.that contain more enriched material. The origin of this effect is currently unclear, and it maybe that this is actually the result of a subtle selection effect and not a genuine signal. If real,two possible explanations are:

• That an inherently more metal rich star will have more metals in its protoplanetarydisk, possibly enhancing the condensation of dust into planetessimals and increasingthe likelihood of planet formation.

• Alternatively, it might be that inner, rocky terrestrial planets often have their orbitsdisrupted and end up falling into their parent star, enriching its atmosphere

13.8 Exoplanets: A young Science

The study of exoplanets is still very young. We have seen many surprises so far, includingthe discovery of hot jupiters and more broadly that solar systems cover a much wider rangeof properties, such as orbital eccentricities, than was once expected. It seems possible thatour own Solar System is rather more stable gravitationally than many of the other systemsuncovered so far. Whether we are lucky in living in a solar system where the young Earthcould survive or not is unclear. However, we still cannot detect terrestrial plants in othersystems, and the observations we do have are potentially subject to a wide range of selectioneffects and biases. There is much work to be done in this field in exploring the properties ofexoplanets and how they are related. You can do some of this yourselves with exoplanet.eu,which collects data on all exoplanets as they are discovered and provides tools for analysingtheir properties.

Things to Remember

• The current state of exoplanet searches

• The results of selection effects in exoplanet searches

• The calculation & observed distribution of exoplanet masses

• The determination of exoplanet composition

• Exoplanet orbits, hot jupiters and planetary migration

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Chapter 14

Astrobiology: Life on Other Planets

14.1 Introduction

Not so long ago, the quest for life elsewhere in the universe could be regarded as speculationthat would remain impossible to test. A range of discoveries of the last 20 years, however, havedrawn this topic into the scientific mainstream, and there are now many people working in thegeneral area of astrobiology. This includes astronomers and physicists, but also biologists andgeologists, since studies of the history and diversity of life on Earth can inform our searchesfor life elsewhere.

14.2 Life on Earth: History

The Earth formed roughly 4.5 Gyr ago, and the late heavy bombardment ended about 4Gyago. The earliest clear signs of life on Earth are structures called stromatolites, which arebuilt up by the action of a thin later of photosynthesising blue green algae. The oldest knownstromatolites are 3.46 Gyr old, and were discovered in Australia. Evidence for life arisingeven earlier than this is provided by carbon isotope ratios in Earth’s oldest sediments. Thesesuggest that autotrophic organisms that fixed atmospheric carbon were well established 3.8Gyr ago, though it has also been suggested that simple chemical processes might be mimickingthis signature of life. If life was genuinely well established 3.8 Gyr ago that is only a veryshort time, compared to the age of the Earth, after the planet became inhabitable at all,following the late heavy bombardment.

How life arose is a subject of great debate. One well established scenario suggests thatlife started with simple, self-sustaining chemical reactions which gradually increased in com-plexity. These chemical reactions likely took place in places where there was a rich mix ofchemicals and plentiful available energy. Hydrothermal vents in the deep ocean are one pos-sible site for the first emergence of these processes. The self sustaining chemical networksrequire catalysts to operate. These may originally have been mineral catalysts, such as the

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iron oxides available in hydrothermal vents, but other organic materials, proteins and RNA(Ribonucleic acid), are also capable of such catalysis. In addition to catalysis, RNA is alsocapable of self-reproduction, which would have given it such an advantage over the otherprocesses operating at the time, that it likely took over and the earliest biology on Earth wasbased on RNA. Later, DNA (Deoxyribonucleic acid) came to dominate since, as long as theproteins necessary for its reproduction are around, since it is more stable and less subject toreproduction errors.

The early history of life on Earth may thus move from mineral catalysed chemistry, to asimple RNA-world, which then suffered a genetic takeover as the more stable and efficientDNA came to dominate.

While photosynthesis is the dominant energy generation mechanism on Earth today, this isdependent on the availability of sunlight. Deep ocean hydrothermal vents, while possessing arich chemistry, are well away from sunlight, and thus feed themselves not through photosyn-thesis but through chemosynthesis, deriving energy from chemical processes rather than fromlight. These organisms would have lived in what we consider extreme environments, and wecan see their descendants today, a class of single cell organism known as archaea, in similarlyextreme environments like hydrothermal vents and hot springs.

As life developed and spread, photosynthesis started, and began producing oxygen as a byproduct. This was not a significant constituent of the atmosphere until very recently ingeological terms. It was only about 500 Myr ago that oxygen levels approached those of today(see Fig 14.1) and were high enough to allow the ozone layer to form. Up until this point mostof life on Earth was anaerobic - ie. operated in the absence of oxygen. In fact, oxygen is toxicto anaerobic life, so the first mass extinction we know about was the result of photosyntheticorganisms polluting the Earth with the deadly poison that is oxygen. Anaerobic life is, ofcourse, still with us in the oxygen free slime at the bottom of oceans and in stagnant poolsof water. Many of the waste products of anaerobic life are in fact toxic to us, which is whypond gas and the results of other anaerobic processes smell so bad.

For most of the history of life on Earth, life was made up of single celled organisms. Multi-cellular life, like us, only emerged about 1 Gyr ago, first as multicellular algae, then as thefirst attempts at more complex multicellular life, the still poorly understood ediacarans. Itwas not until about 550 Myr ago that what we would regard as modern multicellular lifeemerged during the Cambrian explosion, a sudden flowering of great diversity in forms of life,the results of which we can still see today.

14.3 Lessons from the History of Life on Earth

What lessons can be drawn for the search for life elsewhere from this rapid overview of thehistory of life on Earth?

Firstly, we can look at what appear to be the essentials for life:

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Figure 14.1: The oxygen content of the Earth’s atmosphere over time. The Berker-Marshall Point isthe stage at which there is enough oxygen in the atmosphere for the ozone layer to form. From Paumannet al., Biochimica et Biophysica Acta (BBA) - Bioenergetics, Volume 1707, Issues 23, AprilMay 2005,Pages 231253

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Figure 14.2: Key dates in the history of life on Earth. From Rothery, Gilmour & Sephton.

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• A supply of energy of some kind (photosynthesis dominates currently, chemosynthesis,in the absence of light, probably dominated the early stages of life on Earth).

• The presence of liquid water. This is necessary to allow chemical reactions to take placeat all.

All other things that we might think are essential, such as DNA or the presence of oxygen,are likely to be beneficial to specific types of life, but not to the existence of life in general.

Secondly, the world as we know it today is in fact a relatively recent occurrence. For much ofthe history of life on Earth there were only unicellular species existing in a largely anaerobicenvironment.

14.4 Life Elsewhere in the Solar System

Using the lessons gained from examining the history of life on Earth, what can we say aboutthe potential for life elsewhere in our own Solar System?

14.4.1 Mars

The place in the Solar System most likely to have once had an environment fairly similarto the Earth is the planet Mars. While it is currently a cold, dry place with a very thinatmosphere, there is now a growing body of evidence that suggests that liquid water onceflowed on the surface of Mars during a warm wet phase as recently as 3 Gyr ago. Otherobservations suggest that small amounts of water may have flowed on the surface much morerecently.

If liquid water existed, or exists today, on Mars, is there any evidence for life? As yet, thereis nothing unambiguous - if there was you would have heard about it - but there are someinteresting hints. The presence of methane in the atmosphere of Mars suggests that thereis something on the planet producing this gas. It is one of the byproducts of anaerobiclife on Earth, but it can also be produced by geological processes. Examination of isotopeabundances in any methane detected by the Curiosity Rover should be able to determine theorigin of this gas, since biological processes operate differently for different isotopes. So far,however, Curiosity has found no methane on Mars, despite its detection by orbiting satelliteselsewhere on the planet. Meanwhile, possible evidence for historical life on Mars may haveemerged in meteorites from Mars that have landed on Earth. In 1996 the discovery of martianmicrofossils was claimed in the meteorite ALH84001, which originated on Mars. This resultis far from agreed, but the possibility of finding fossil martian lifeforms, whether in martianmeteorites or in situ on Mars using rovers such as Curiosity, is one way in which the presenceof ancient life on Mars could be confirmed.

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14.4.2 Europa

As discussed under the section on the Solar System’s moons, there is evidence for a liquidocean beneath the icy surface of the Galilean moon Europa. The conditions in such a sub-surface ocean are very uncertain, but it is possible that the tidal heating of the moon by itsorbit around Jupiter, could lead to the presence of hydrothermal vents in this ocean, similarto those thought to have been the cradle for life on Earth. Similar processes within Europacould lead to the same kind of primitive life that emerged on the young Earth. Future mis-sions to the Jovian moons such as ESA’s JUICE project will be looking for signs of this oceanand any biological processes that might be taking place within it. The recent discovery of awater plume on Europa, similar to that found en Enceladus, means that we may be able toget an idea of subsurface conditions on this moon through observations from JUICE or evenfrom observatories closer to home. There may also be a similar subsurface ocean on the moonGanymede as well, though this would be buried under an even thicker layer of ice since it issubject to less tidal heating than Europa.

14.4.3 Enceladus

The one other place in the solar system where there is clear evidence for the presence of liquidwater is Enceladus, the moon of Saturn, where jets of water vapour emerge from cracks inparts of its surface. Enceladus, like Europa, is tidally heated, so here too there may be asubsurface ocean and hydrothermal vents that could host biological systems. There is alsosome evidence of a subsurface water layer in Saturn’s moon Titan.

14.5 Life Outside the Solar System

Having looked at possible homes for life in our own Solar system it is now time to look forit elsewhere. The requirements for a habitable planet outside our own Solar system will bebroadly similar to what we have found locally, with the presence of liquid water being ofparamount importance. For an Earthlike planet to be capable of supporting life the followingconditions would have to hold:

• Large enough mass so that the atmosphere can provide sufficient pressure for water tobe a liquid on its surface. Atmospheric pressure is given by:

P = mcGMp

R2(14.1)

where P is the pressure, mc is the mass of a column through the atmosphere, Mp is themass of the planet and R is the radius of the planet.

• The planet must be large enough to have geological activity so that volatiles can beincorporated into the crust, as seen in the carbon cycle on Earth. It should be notedthat Venus is an interesting exception to this rule, since it is similar in mass to Earth,

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Handout: Life as we don’t know it

... one of the ‘big’ questions: is there life elsewhere than on planet Earth? While much of thisused to be pure speculation, the discovery of other planetary systems along with the discoveryof water in planetary atmospheres has put some of the speculations on a much firmer footing.

Life outside the solar system

Let us look at some of the preconditions (as we currently understand them!) in turn. Note thatthey mostly assume that life will be carbon-based!

1. Habitable planets: planet masses

(a) Large enough mass so that atmosphere provides enough pressure for water to beliquid. Pressure P = mcGMp/R

2 (mc: column mass, Mp: mass of planet, R: radiusof planet).

(b) Large enough mass to have geological activity (so that volatiles can be incorporatedinto crust ! see carbon cycle on Earth).

(c) Large enough to retain atmosphere, vexc =q

2GMp

R, typically need Mp 0.5M.

(d) Small enough not to have accreted extended gas-giant mantle; Mp 10M.

We thus want planetary masses between 0.5 and 10 Earth masses; these are sometimescalled ‘Earth-mass planets’.

2. Planet positions: the habitable zoneFor carbon-based life, (and carbon cycle) need liquid water, and thus temperatures be-tween 273 and 373 K. For naive calculation of habitable zone, see PS 4 where we founda distance of 0.6 to 1.1 AU for the habitable zone.

from http://www.geosc.psu.edu/ kasting/PersonalPage/Kasting.htm

Figure 14.3: The location of the Continuously Habitable Zone for a range of stellar types comparedto the position of the planets in our own Solar System. From www.geosc.psu.edu.

but lacks a carbon cycle, so while this is a necessary condition it is not sufficient forthere to be an active geological cycle.

• The planet must be large enough to retain an atmosphere: remembering the thermalescape of atoms from atmospheres discussed in Section 8.5. This implies that the planetmust have a mass ≥ 0.5 M⊕.

• The planet must be small enough not to have accreted an extended hydrogen richatmosphere, and to have become a gas giant. This implies a mass ≤ 10 M⊕

For liquid water to exist the planet must also be at an appropriate temperature, between273 and 373K. These surface temperature limits define what is called the Habitable Zonefor planets in any given system. A simple calculation of the width of the Sun’s habitablezone, using the considerations discussed in section 7.6, would estimate it to lie from 0.6to 1.1 AU in distance form the Sun. Lower mass stars will have smaller habitable zonescloser to them, while higher mass stars will have them further out.

The power output of stars changes over the course of geological time. For life to havetime to evolve, we are actually interested in a narrower region where liquid water canpersist for the entire history of the planet. For our own Solar System this extends from

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0.95 to 1.15 AU. Figure 14.3 shows where the continuously habitable zone lies for arange of stellar types.

14.5.1 Host Star

The host star has influence beyond just keeping a planet’s temperature at the right level forliquid water to exist. High mass spectral types, such as O, B and A stars, evolve too quicklyfor life to have time to evolve - recall that stellar life time decreases as M−3 on the mainsequence. High mass stars also have high surface temperatures and would thus emit copiousamounts of UV light which might be harmful to life.

Very low mass stars such as M stars have their CHZ closer than the tidal locking radius. Thiswould mean that one side of the planet has permanent day and the other permanent night.This might be a problem, with the two sides being respectively very hot and very cold, so thecold side could act as a trap, freezing out the atmosphere over time. Recent modelling work,though, suggests that a sufficiently dense atmosphere can circulate heat from one side of atidally locked planet to the other, avoiding this problem.

Many stars lie in binary systems which may lead to instability in planetary orbits. Orbitsmay also be affected by other bodies in the same solar system as we have seen with the inwardevolution of hot jupiters in Chapter 13. Conversely it may be that a gas giant in a stableorbit further from the star than a terrestrial planet, as is the case for Jupiter in the SolarSystem, might limit the number of impacts in the inner system.

14.5.2 Gas Giant Moons

The considerations given above apply to life on the surface of a terrestrial planet. As discussedin section 14.4, life might also exist beneath the icy surfaces of gas giant moons, like Europaand Enceladus, in our own Solar System. Gas giant moons elsewhere might also be capable ofharbouring life in this way, or, if they are warm enough and have a dense enough atmospherefor the presence of surface water, they might also harbour life on their surface.

14.6 The Galactic Habitable Zone

The large scale geography of our galaxy influences where it is most likely to find life bearingplanets. The formation of terrestrial planets requires high-metallicity stars. These are mostlikely to be found in the thin disk or bulge of the galaxy. The outer regions of the disk havelow metallicity so would have fewer terrestrial planets, while regions closer to the centre ofthe galaxy would suffer from two disadvantages: firstly the stars are closer together, leadingto overcrowding and the possible gravitational disruption of a stellar system by a passingstar; secondly, a given star will be more likely to be close to an energetic event, such as a

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supernova, which could wipe out life in nearby star systems. The core of our galaxy is thusnot thought to be hospitable for life.

The orbit of a star around the galaxy is also important. The Sun’s orbit is nearly circular, soit is less likely to stray into crowded regions in the core. The Sun’s orbit also avoids crossingthe spiral arms of the galaxy, which are also regions of high stellar density and thus hazardousto life.

14.7 How to Find Life on Other Planets

We have discussed the considerations for an extrasolar planet to be able to support life, butwould we be able to detect such life if it were present?

We have already seen how exoplanet studies are beginning to be able to determine variousparameters for the atmospheres of hot jupiters. Our observational capabilities are improv-ing and there are now plans for instruments that will eventually be able to take spectra ofterrestrial exoplanet atmospheres. There are a number of ‘biomarkers’ that could appear inthese spectra if life is present. Chief among these is ozone, which has a prominent absorptionfeature in the infrared at about 10µm. This would be a clear sign of the presence of lifesince oxygen is a highly reactive molecule, which, unless constantly replenished, would soonbe locked up in other compounds like CO2. The only process we are aware of that can keepoxygen levels high enough for an ozone absorption layer to exist is photosynthesis in plants.Other possible biomarkers include methane and spectral features associated with chlorophyllin plants.

However, as we have seen, the ozone layer in the Earth’s atmosphere is relatively recentin geological terms, and chlorophyll, while common on Earth, will not necessarily be themolecule of choice for photosynthesis on other planets. A more general signature of life willbe signs of any chemistry which is out of equilibrium - the abundance of oxygen in the Earth’satmosphere is an example of this - since the action of biological processes are the only waywe know that can maintain such a disequilibrium over time. Quite what we might find in theatmospheres of biologically active exoplanets remains to be seen.

Things to Remember

• The history of life on Earth

• The requirements for life and the likely sites elsewhere in the Solar System

• The requirements for and potential sites of life around other stars

• The galactic habitable zone

• How life might be found on exoplanets

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Chapter 15

The Search for ExtraterrestrialIntelligence

15.1 Introduction

Assuming that life does exist on other planets, the next great question is whether intelligentlife exists elsewhere. We do not yet have any evidence for extraterrestrial intelligence (ETI),but absence of evidence is not evidence of absence. There are many issues surrounding thesearch for extraterrestrial intelligence - how it should be conducted, whether we should try tomake contact ourselves, what to do if we ever do find evidence for it - but few hard and fastresults. The lecture for this section of the course will thus largely take the form of a discussionabout how we can guesstimate the number of extraterrestrial intelligences in the galaxy, ofwhat uncertainties there are in such a prediction, and about broader issues concerning thesearch and possible discovery of ETI.

The two key results in this area are the Drake Equation and the Fermi Paradox, which willbe described here in turn.

15.2 The Drake Equation

The Drake equation (devised by Frank Drake in 1961) encompasses the terms needed to pre-dict the number of intelligent civilisations in the galaxy that at any given time are interestedin communication with other civilisations. At the time the equation was devised, there werefew constraints on any of the terms, but an extra 50 years of astrophysics has begun to tiesome of them down.

N = R× fp × nE × fL × fi × fc × Lc (15.1)

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where:

N = the number of technological civilisations in the galaxy that are interested in communi-cationR = the average rate of star formation in the galaxy (in stars per year)fp = the fraction of those stars with planetary systemsnE = the average number of habitable planets in each systemfL = the fraction of those habitable planets on which life developsfi = the fraction of those planets on which intelligent life developsfc = the fraction of intelligent species interested and able to communicate with other speciesLc = the lifetime of a communicating civilisation

At this point the only terms in the Drake Equation where we have accurate values are Rand fp, which have values of roughly 8 to 20 for R and about 0.5 for fp. nE is a term that weshould have good constraints on fairly soon, from long term transit studies using instrumentslike Kepler. Our best guess at it so far is that it is likely to be close to 1, and the bestcurrent results suggest a value of 0.2 (which, for our purposes here, is very close to 1). fL is,essentially, the goal of the whole field of exobiology, but that study is currently in its infancyand any guess that can be made at this stage will be highly uncertain.

That leaves fi, fc & Lc, which are not easily determined, and which are controlled by factorsthat are biological and sociological.

What values do you think are reasonable for fi, fc & Lc, and what are your justifications forthese estimations?

15.3 The Fermi Paradox

The next important consideration in this field is known as the Fermi Paradox. This arisesfrom a question that the famous physicist Enrico Fermi ask in an informal discussion in 1950.Fermi made the observation that, given the great age of the universe (about 13.5 Gyr) and thevery large number of stars in the galaxy (about 1011), then we should be able to see evidenceof intelligent life through interstellar probes or spacecraft, unless intelligent life capable ofinterstellar travel and/or communication is very rare. His key question was ‘Where are they?’since it can be shown quite easily (I have run a 3rd year UG project on this) that a civilisationcapable of interstellar travel and colonisation can spread throughout the galaxy in what is,in geological and cosmological terms, a relatively short time - 1-100 Myr.

This issue is also known as the ‘great silence’ since it applies just as much to communicationas it does to physical contact with alien intelligences.

What possible resolutions can you come up with for the Fermi paradox, and what is yourjustification for these solutions?

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15.4 SETI and CETI

The Search for Extraterrestrial Intelligence (SETI) and attempts at Communication withExtraterrestrial Intelligence (CETI - confusingly pronounced the same way) are terms forvarious attempts to observationally test the idea that extraterrestrial intelligences exist andmight be contacted by us. Much of the work has focussed on radio observations looking fornarrow band, artificial signals coming from Sun-like stars, but there are many other possibleways in which extraterrestrials might communicate with each other.

Can you think of alternatives to radio waves for interstellar communication?

What kind of problems exist when trying to detect a narrow band radio signal at an unknownfrequency?

Within CETI, a small number of attempts have been made to transmit powerful radio signalstowards certain locations in the galaxy. The most significant of these was the use of theAreceibo interplanetary radar to send a message coded as an image towards the globularcluster M31. The message will take 25000 years to reach the cluster and any reply wouldtake 25000 years to come back, so we don’t expect a snappy conversation. Meanwhile, normalradio and TV broadcasts from Earth are propagating through the nearby galaxy. While cableand satellite TV mean that less power has been expended on such transmissions over the lastfew decades, powerful transmissions of previous years are still heading out into space. One ofthe most powerful and furthest travelling signals so far is that of the 1936 Olympics in Berlin,presided over by Adolf Hitler.

Given the experience on Earth of what happens when two cultures with very different levelsof technological development interact, is it a good idea to be advertising our presence on thegalactic stage?

15.5 The Future

Developments in radio astronomy currently underway will allow us to have far greater sensi-tivity to narrow band signals in the next decade. The Square Kilometer Array project (SKA)in particular will be able to detect airport radars 50 to 60 light years away, while more pow-erful early-warning type radars could be detected at even greater ranges. At the same time,large ground based telescopes such as the E-ELT, and space-based projects such as SPICA,JWST and, eventually, Darwin, will be able to detect terrestrial planets and search for signsof life in their atmospheric spectra.

In the next few decades we will thus be able to not only start filling in some more of theastrophysical and astrobiological terms in the Drake Equation, but might also be able toconduct the first studies in SETI that could detect nearby civilisations like our own.

SETI, and CETI, might soon stop being the preserve of scientific speculation, and becomeactual observational and practical studies in their own right.

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Things to Remember

• The Drake Equation

• The Fermi Paradox

• The current status and potential for SETI observations