Introduction P 1

PART II ETP — ASTROPHYSICS

22 Lectures Prof. S.F. Gull & Prof. A.N. Lasenby

• 16 lectures this Term. First, an introduction (2 lectures) to astrophysics

and the content of the visible universe. Prof. Lasenby will then deal

with the large-scale structure and evolution of the universe and give an

introduction to General Relativity (8 lectures).

• I will then describe the astrophysics of (more or less) normal matter

from scales of stars to clusters of galaxies, using ordinary physics and

Newtonian gravitation theory. There are 6 lectures nest Term, the last

few of which will concentrate on the qualitative understanding of

astrophysical “test cases”, such as supernovae and their remnants,

radio galaxies and quasars, and the amazing detective story told by

meteorites.

• HANDOUT — Syllabus; books; essential astronomical facts and

jargon; orders of magnitudes; how we measure distances, velocities,

masses; basic information about astrophysical objects we will meet;

states of condensed matter. Please report any errors or typos.

• NOTES — Provisional hardcopy available in advance. Definitive copies

of overheads available on web.

• SUMMARY SHEETS — 1 page summary of each lecture.

• EXAMPLES — 4 in all, 3 sheets this Term — 2 examples per lecture.

• WORKED EXAMPLES — Will be available on the web later.

• WEB PAGE — For feedback, additional pictures, movies etc.

http://www.mrao.cam.ac.uk/∼steve/astrophysics/

Introduction P 2

THE OBSERVABLE UNIVERSE

• Astrophysics is the extension of laboratory physics to large-scale

structures in the universe.

• “Large” means bigger than the Earth (radius 6400 km) — the nearest

external object is our Moon (distance 400, 000 km radius 1738 km).

• The universe seems to organise itself preferentially into stars, which

are objects of size 109 m and 1030 kg.

• The nearest star is the Sun at distance 1.5 × 1011 m (8 light min).

• Sun is a fairly typical star: radius 7× 108 m; mass (M�) 2× 1030 kg.

• Next nearest stars are 30, 000 times further away at about 5 light

years 5 × 1016 m).

• Stars organise themselves into various scales, but there is another

preferential scale: galaxies (1021 m, 1042 kg).

• Other larger scales: clusters, superclusters.

• Universe originated from hot, dense state 15 billion years ago.

• Finite observable universe 2 × 1026 m containing 1012 galaxies.

• And that’s just the stuff we can see. . .

Introduction P 3

ASTROPHYSICS AND ASTRONOMY

• Universe contains wide range of exotic phenomena: stars; star

formation; supernovae; galaxies; radio galaxies; quasars; clusters;

cosmic microwave background.

• Laws of physics governing the behaviour of astrophysical objects is

exactly the same as here on Earth (we keep an open mind on this, but

would need a lot of convincing otherwise).

• Extreme conditions found in the cosmic laboratory can provide tests of

our understanding of physics.

• Astronomy is an observational science: we have to make do with the

objects Nature provides, and we can only see them from one viewpoint.

• Astrophysical timescales can be very long (e.g. dynamics of radio jets

in galaxies: 107 years). Can often only see a snapshot of a long

evolutionary process; have to infer evolution from many examples.

• Inevitable selection effects: we can only see objects that emit radiation;

we have limited dynamic range of instruments; we can only see rare,

very luminous phenomena to the greatest distances; some objects

emit anisotropically – interstellar masars, pulsars, relativistic jets.

Introduction P 4

PHYSICS NEEDED FOR ASTROPHYSICS

• We will need almost everything you have been taught – and a bit more.

• Gravity The glue of the universe — makes all objects tend to attract

and collapse. Needed throughout the course: particle orbits in globular

clusters, galaxies and clusters; Hydrostatic equilibrium of stars and

cluster gas; star formation; evolution and death of stars; formation of

galaxies and other structures from the expanding universe; dark matter.

• Fluid dynamics and plasma physics Many astrophysical

phenomena involve jets, turbulence, shocks, explosions. Hot gas in

clusters of galaxies.

• Nuclear and Statistical physics Needed to understand the

physics of normal and degenerate stars; end points of evolution of

stars; supernovae and synthesis of post-Fe elements; nucleosynthesis

in the early universe.

• Radiation mechanisms and radiative transfer This is how

we see astrophysical objects; also important for energy transport in

stars; cooling and determining the evolution of objects.

• Exotic physics General relativity and Black holes; dark matter;

dark energy; the epoch of inflation.

Introduction P 5

STUDYING ASTROPHYSICS

• Overall aim We will try to develop an intuitive view of the various

astrophysical phenomena we observe, and be able to apply simple

physical models to explain them quantitatively.

• Realistic goal I’d like you enjoy astrophysics, to be able to do the

problems on the question sheet and to do well in the Exam.

• Astronomical context We have to have some appreciation of the

observational data: wavebands used; resolution achieved relative to

scale of the object; nature of radiation processes; spectral and velocity

information (if any). The is the necessary astronomical legwork. You

can’t progress in astrophysical research without building telescopes,

doing surveys and studying sources in gruesome detail.

• Theoretical treatment This is very difficult. Theories we can

compute with are approximate (no one knows how to do the 2-body

problem in GR. Even numerical N -body Newtonian gravitational work

and 3-d fluid dynamics are fraught with difficulties, though can give

insight. Analytical treatments are usually worse, but can still be very

useful. We use approximate theory and imperfect simulations to try to

educate our physical intuition. We iterate round the loop of theory,

observation and simulation and eventually hope to progress.

• Method This course will consist of generally applicable theoretical

ideas, a necessary minimum of astronomical facts, illustrated by

examples of astrophysical test cases which will be described in more

detail.

Introduction P 6

ASTRONOMICAL FACTS AND JARGON

• Angles are measured in degrees, arc minutes and arc seconds:

π/180 radians = 1 degree = 60 arcmin = 3600 arcsec

• 1 AU (astronomical unit) = 1.5 × 1011 m

• 1 pc (parsec) = 1 AU subtends 1 arcsec = 180 × 3600/π AU

= 3 × 1016 m

• 1 M� (Solar mass) 2 × 1030 kg

• Redshift: z ≡λobs − λrest

λrest

• m = apparent magnitude = −2.5 log10

(

Flux(ν)

StandardFlux(ν)

)

M = absolute magnitude = m − 5 log10

(

Distance

10 pc

)

m M

Sun −27 +5

Full Moon −13 +32

Sirius −1.5 +1.5

A0 star at 10 pc 0 0

White dwarf at 100 pc +20 +15

Galaxy at z = 1 +22 −22

Type Ia SN at z = 1 +24.5 −19.5

Introduction P 7

THE EXPANDING UNIVERSE AND HUBBLE’S “CONSTANT”

• In 1929 Edwin Hubble found that all galaxies seemed to be mov-

ing away from us, with velocity v proportional to distance D: v = H0D.

• He determined the constant H0 to be 500 km s−1 Mpc−1.

• In real units this is 1/H0 = 1.5 Gyr.

• This caused a huge problem, because the age of the Earth is 4.5 Gyr.

• Cosmologists were resourceful and invented lots of crazy theories to

account for this impossible observation.

• Hubble’s calibration had underestimated the luminosity of Cepheid

variables. When corrected, the problem disappeared.

• The value of H0 is still uncertain.: H0 = 75 km s−1 Mpc−1 is

currently popular.

Introduction P 8

EVOLUTION OF HUBBLE’S CONSTANT

• Published values of Hubble’s constant up to 1980.

Introduction P 9

THE BIG BANG

• Although the Hubble constant settled down so that the universe was

older than the Earth, people still didn’t like the idea that the universe

had a definite “beginning” in a hot, dense state 10–15 billion years ago.

• Fred Hoyle (one of the originators of the“Steady-State” theory) said:

“You might as well say it all started in a big bang!”

• The discovery of the relic 2.7 K cosmic microwave background

radiation and the clear evidence of evolution in the radio source counts

(radio source were much more common at z = 2 than they are today)

largely settled the matter, though there were persistent pockets of

resistance for many years afterwards.

• So now we call the dense, hot early universe the “Big Bang”.

• A version of the “Steady-State” theory is now popular again (for the

very early universe). . .

Introduction P 10

DETERMINATION OF THE DISTANCE SCALE

Traditional “step by step” approach: the distance ladder

• Solar system: planetary radar, tracking of spacecraft and pulsar timing

gives value of AU to a few metres.

• Nearby stars: use parallax. Hipparchos satellite measured parallaxes

to 0.001 arcsec (expect a further factor of 100 improvement soon).

Thus establish luminosity as a function of spectral type for main

sequence stars.

• Apply to more distant stars, especially clusters. Find luminosity of

bright “standard candles” (in particular Cepheid variables, for which the

absolute luminosity is well correlated with the period of oscillation).

Introduction P 11

THE DISTANCE SCALE II

• Observe Cepheids in nearby galaxies to establish distances to them

and thus obtain the absolute luminosity of still brighter objects: globular

clusters, H+ regions and whole galaxies.

• Extend these to find distances to galaxies that are sufficiently far away

that the overall expansion of the universe dominates over random

motions. This should provide the value of the Hubble constant H0.

• Beyond that the recession velocity is used as the indicator of distance,

though the true form of the distance–redshift relation remains to be

determined.

• The brightest standard candles available are Type Ia supernovae. They

arise from the ignition of a white dwarf star following accretion of matter

from its normal companion in a binary system. Ignition occurs at a

definite mass 1.4 M�, and there is usually not much absorbing material

in the way, since the lifetime of these stellar systems is very long and

they are likely to have moved away from the dense regions in which

they were formed. These standard candles have extended the distance

scale to high redshift, but calibration will continue to require refinement.

Introduction P 12

HUBBLE DIAGRAM FROM TYPE IA SUPERNOVAE

• Hubble diagram from Perlmutter & Schmidt (2003).

• A distance modulus of 40 corresponds to a distance of 1 Gpc, and

increase of 5 in the distance modulus is equivalent to an factor of 10

increase in distance. (There are lots of technical issues here.)

Introduction P 13

COSMIC ABUNDANCES OF THE ELEMENTS

• Only hydrogen and helium were formed in the Big Bang.

• Binding energy graph shows that Li, Be, B will be rare.

• C, O, Ne are formed in normal stars, post-Fe arise from supernovae.

Introduction P 14

THE SOLAR SYSTEM

• The Sun

- Contains 99.9% of the mass of the solar system

- Rotation period: 25 days at equator, 30 days at poles.

- The angular momentum of the rotation is only 2% of that of the

solar system (most is in the orbital motion of Jupiter).

- The differential rotation creates magnetic field, typically 10−4 T, but

0.3 T in sunspots.

- Optical photosphere has Teff = 6000 K (emits continuum).

- Chromosphere has T = 4500 K (absorption lines).

- Whiplash effect of convection cells heats the low density corona to

T = 106 K.

- Generates solar wind 400–700 km s−1, with spiral sector structure.

- Heat is generated by conversion of hydrogen to helium in the

centre, where the temperature is 1.5 × 107 K and the density is

1.6 × 105 kg m−3.

- Heat is transported outwards by radiative diffusion, except for the

outer 1%, where the decreasing density makes the fluid develop

convection cells.

- Clues to interior: solar oscillations; neutrinos. Has enabled us to

make very detailed measurements of the solar structure and

elemental composition.

Introduction P 15

THE SOLAR SYSTEM II

• Planetary system

- Orbits are approximately circular and lie in a plane — must have

been formed from a gaseous disc.

- Composition shows temperature gradient: inner planets have

ρ ≈ 5 × 103 kg m−3; outer gas giants have ρ ≈ 103 kg m−3

• Earth

- Differentiated: Iron/Nickel core; silicate mantle and crust.

- Heated by radioactive decay, causing geological activity.

- Age of oldest surface rock is about 3.8 Gyr.

• Mars and Venus

- Similar histories to Earth.

- Differences in atmosphere can be understood from temperature and

gravity.

• Moon and Mercury

- Geological activity ceased before the end of asteroid bombardment.

- Age of oldest Moon rock is 4.5 Gyr.

Introduction P 16

THE SOLAR SYSTEM III

• Asteroids

- Smaller bodies, mostly lying between Mars and Jupiter, sizes from

< 1 km to a few hundred km..

- Total mass < 10−3 that of the Earth.

- Interactions and collisions generate meteors which we can study

directly. Differentiated: stony; stony/iron; irons. Some stony

meteors contain “chondrites” — appear to be pre-solar material

(contain SiC which cannot form in the presence of O).

• Outer planets

- Formed from “ices” — H2O, NH3, CH4.

- Massive enough to keep most of the H and He.

- Systems of moons — formation similar to planetary system as a

whole?.

- Ring systems: dust to bolder-sized particles inside the Roche limit.

• Comets

- Made from“Dirty ices”. There are probably about 1010 comets.

- Highly eccentric orbits — regular visitors must have had interaction

with planets that circularised their orbits.

- Probably originate in the Oort cloud at 105 AU

Introduction P 17

STARS AND THEIR CONSEQUENCES

• There are large clouds of molecular hydrogen orbiting in the plane of

the Galaxy. As compression takes place in the shocks of the spiral

density waves, star formation can occur.

• Stars are born when a hydrogen gas cloud collapses under gravity.

Newly formed stars (T Tauri) have jets and other outflows.

• A collapsed gas cloud becomes a star when the temperature in its core

becomes high enough to ignite nuclear burning (107 K).

• The mass range for stars is 0.075 M� to 100 M� and their initial

luminosity is approximately L ∝ M 3.

• Main-sequence stars burn hydrogen in their cores and remain stable

until the fuel is exhausted. This takes 10 Gyr for a solar-mass star.

• Red giants, planetary nebulae, white dwarves, Type II supernovae and

neutron stars are later stages in the evolution of stars.

• Many star systems are binaries and there can be very interesting

consequences as a result of mass transfer. These include novae and

other variable stars, Type IA supernovae, X-ray binaries and other

exotics.

• Supernovae and other mass loss from stellar systems recycle material

into the gas of the Galactic disc and enrich the interstellar medium with

heavy elements.

Introduction P 18

STAR CLUSTERS, GALAXIES AND CLUSTERS

• Star clusters

- Globular clusters: massive (106 M�); old; dynamically relaxed.

- Open clusters: smaller; younger; dynamically evolving and

evaporating.

- Particularly important because all stars are at about the same

distance and presumably of about the same age.

- Massive stars form in clusters.

• Galaxies Limited range of types:

- Ellipticals: similar to globular clusters; masses range from 108 M�

to 1014 M� (found in clusters – arise through cannibalism). Some

galaxies have active nuclei, arising from accretion onto massive

black holes.

- Spirals: pattern is a density wave in the interstellar gas — moves

with respect to the stars; gas and dust are concentrated in the spiral

arms; find that young stars emerge from the arms.

- Irregulars: disruption by interactions, mergers; huge burst of star

formation.

• Clusters of galaxies

- Large aggregations (thousands) of galaxies, often with a very large

elliptical in the centre. Contain hydrostatic X-ray emitting hot gas

(108 K), which strips out the gas from the individual galaxies as

they travel through it.

- There are also larger superclusters and immense voids in the

expanding universe.

Introduction P 19

ASTROPHYSICAL OBJECTS — MASS VERSUS RADIUS PLOT

• Over a very wide range of masses (factor of 1054) objects have a

density of order 103 kg m−3.

• Narrow range of masses (1027–1031 kg with wide range of densities.

• On larger scales see dynamical groups of stars.

Introduction P 20

THE FATE OF COLD MATTER IN THE UNIVERSE

• Gravity tries to concentrate matter.

• Other forces resist:

1) Coulomb force can do the job for M < 2 × 1027 kg

2) Degeneracy pressure i.e. Pauli principle for identical fermions

a) Electrons in white dwarf stars

are sufficient for 2 × 1027 kg < M < 2 × 1030 kg

b) Neutrons in neutron stars (pulsars)

manage for 2 × 1030 kg < M < 1031 kg

• Above this limit gravity must win in the end.

• Collapse is postponed by:

3) Entropy

a) Compression + opacity → heat → pressure.

b) Compression + turbulence → heat → pressure.

c) If temperature T > 107K → nuclear fusion.

Introduction P 21

HOW COMPRESSIBLE ARE ATOMS?

• Recall Bohr model for hydrogen atom:

Forces:e2

4πε0r2= meω

2r

⇒ ω2 =e2

4πε0mer3

Quantum: h̄ = meωr2⇒ ω =

h̄

mer2

• Hence radius of orbit a0 is given by a0 =4πε0h̄

2

e2me

= 5 × 10−11 m.

• Energy is (using Virial theorem):

E = −12

e2

4πε0a0

= −e4me

32π2ε20h̄2

= −13.6 eV

.

• Atoms contain a large amount of

energy and resist compression

very strongly.

• We can estimate the pressure that

atoms can resist as approximately

equal to the energy density ∼ E/(2a0)3∼ 2 × 1012 Pa.

• This estimate is an upper limit: if exceeded the electrons will certainly

not be bound to the protons.

Introduction P 22

HOW COMPRESSIBLE ARE ATOMS? II

• We can make a simpler model of an atom: ignore the proton and

suppose the electron is in a box of side πa, so that its momentum is

p =h̄

aand the kinetic energy is

p2

2me

=h̄2

2mea2.

• Now remember the proton and find the total energy

E(a) =h̄2

2mea2−

e2

4πε0a

E(a)

a

• Treat a as a variable parameter

and locate the minimum energy

as a is varied:

amin =4πε0h̄

2

e2me

= a0.

• The minimum energy itself is Emin = −e4me

32π2ε20h̄2

as before.

• I admit I fiddled that — you can also invoke the uncertainty principle if

you like. . .

• The general point is that, as you try to compress an atom, the kinetic

energy will increase and eventually the total energy will be positive. At

this point the electrons are not bound to individual protons and the

hydrogen becomes metallic (predicted in 1935).

• According to our simple model this occurs when a = 12a0; i.e. when

compressed by a factor of 8.

Introduction P 23

GRAVITY: THE ATOM-CRUSHER

• The gravitational energy of neighbouring atoms in hydrogen is about

Gm2p

2a0

∼ 10−35eV — a factor of 1036 times weaker than the

electrostatic energy.

• But gravity is always attractive and, as you add more atoms,

gravitational effects become more important. . .

• Suppose we have a body containing N atoms.

• The electrostatic forces on an atom are no

stronger than before — the electric field is

shielded on scales larger than a0.

• An atom will feel the gravitational effect of all the other atoms in the

body, but we have to allow for the increased separation.

• For an incompressible body the average separation is ∼ a0N1/3.

• The gravitational energy of the atom is nowNGm2

p

2a0N1/3; i.e. ∝ N2/3

• Although gravitation is 1036 times weaker than electromagnetism at

the atomic level, it will dominate when the number of atoms exceeds

(1036)3/2 = 1054.

• This is a mass of1054mp ≈ 1027 kg — about the size of Jupiter.

Introduction P 24

PROPERTIES OF SOLID HYDROGEN

• Density of solid hydrogen is low: the molecular volume is 22.7 cm3.

• As solids go it is rather weak – it compresses by a factor of two in

volume under a pressure of “only” 2 × 104 atmospheres (3 GPa).

• Resistivity and molecular volume have now been measured up to

pressures of 300 GPa, which is about the central pressure of Jupiter.

• The resistivity drops dramatically and hydrogen becomes “metallic”

when compressed by a factor of 9.