the structure and dynamics of the solar interior steve tobias (leeds) 5th potsdam thinkshop, 2007

The structure and dynamics of the Solar

Interior

Steve Tobias (Leeds)

5th Potsdam Thinkshop, 2007

Solar Observations: A brief history 1.

• 1223 BC: First Eclipse record. Clay tablet in Ugarit, Babylonia.

• 8th C BC. Babylonians systematic record of eclipses.

• ~800 BC. First sunspot observation– “A dou is seen in the Sun”,

Book of changes, China.

For more details on Solar history see: http://www.hao.ucar.edu/public/education

Solar Theory: A brief History 2• The Aristotelian View• Aristotle (384-322 BC).

Earth at centre of Universe

• Ptolemy (100-170 AD)• ~200 BC. First

calculation of distance to Sun (Aristarchos of Samos)– Got EM/ES = 19– True value EM/ES=397

Solar Theory: A brief History 3

• 968 AD – First mention of Corona (Diaconus)– “At the fourth hour of the day…darkness covered the Earth and all

the bright stars shone forth. And it was possible to see the disk of the Sun, dull and unlit, and a dim feeble glow like a narrow band shining in a circle around the edge of the disk”.

• 1128 AD – First Sunspot drawing (John of Worcester)– “…from morning to evening, appeared something like two black

circles within the Sun, the one in the upper part being bigger, the one in the lower part smaller”

Solar Theory: A brief History 4

• 1543 Copernicus moves the Sun to the centre, with all planets orbiting in circular orbits

• Kepler (1609) Sun at one focus of an ellipse.

• Galileo (1610) First telescopic observations of Sunspots

Solar Theory: A brief History 5

• Descartes (1644). Sun but one of many stars, each of which having formed at the centre of a primaeval vortex.

• 17th C. Sunspots vanish – Maunder Minimum (see lecture 2).• Origin of Sunspots: Herschel (1738-1822)• Sunspots openings in Sun’s luminous atmosphere, allowing a view of the

underlying cooler solar surface.• 1796 – Laplace. Nebular hypothesis. Sun and solar system formed from

gravitational collapse of slowly-rotating, diffuse cloud of gas.

Solar Theory: A brief History 6

• 1800 – Herschel discovers infrared radiation.

• 1817 – Fraunhofer – solar spectral lines

• 1907 – Hale – Zeeman splitting of spectral lines magnetic fields in sunspots.

The Sun as a star• Sun is a G2 Main-

sequence star.• Its activity and

structure can be related to that of many other stars “solar-type” stars.

• As it has spun-down owing to magnetic braking its magnetic properties have changed.

HR-diagram

Solar Structure

Solar Interior

1. Core2. Radiative Interior3. (Tachocline)4. Convection Zone5. Photosphere

Visible Sun

1. Photosphere2. Chromosphere3. Transition Region4. Corona5. (Solar Wind)How do we know?

A star is a self-gravitating mass of gas that radiates energy

Quick overview of the Sun’s properties

Mass pressure temperature heat luminosity

Sun – our closest starGlobal properties:

mass M 1.99 x 1030 kgradius R 6.96 x 108 mluminosity L 3.83 x 1026 W

Sun-Earth mean distance 1 Astronomical Unit (A.U.) = 1.50 x 1011 m

How are these quantities determined?

Distance: Kepler’s 3rd law (P2 / D3) relative scale of solarsystem but not absolute scale; then e.g. radar-ranging to VenusEarlier methods: transit observations; Greek astronomy

Radius: Angular size of Sun + distance

Mass: Orbital motions of planets + distance GM to high precision

θSun

Age of the Sun

Only known indirectly: radioactive dating of rocks;computed evolutionary models of the Sun. ~ 4.6 x 109 years

d

Luminosity: Measure flux (energy per unit time per unit area)at Sun-Earth distance. Useinverse-square law: f = L / (4d2 )( d = 1 A.U. )

solar “constant” ' 1368 W m-2

d1

d2

L

f

Chemical composition of the Sun

Similar to typical composition in the universe:

Hydrogen ~70% by mass X

Helium ~30% Y

Heavier elements ~1-2% Z O, C, N, Ne, Fe, … in order of abundance

Observational data: solar spectrum, meteorites

Assumptions

•Sun’s structure is spherically symmetric

Define radial coordinate r -- distance from centre

•Sun’s properties change so slowly that can neglect the rate of change

with time of these properties

•Start with equation of hydrostatic balance (which is a good

approximation)

Asphericity ~ 10-5

Hydrostatic equilibrium

gdr

dp pressure gravity

mass m(r)

Two differentialequations describing the structure of thesolar interior-- but 3 functionsm(r), p(r), ρ(r)

rONow...

So...

But by definition of m(r)

In order to make progress, we need to relate the pressure to the density (and temperature and the constution of the gas!)Hence we need to know something about energy...

Energy: How does the Sun shine?

Could Sun’s energy source be gravitational energy? -- No.

Total available gravitational energy = G M2 / R

So could sustain present luminosity for time (G M2 / R ) / L 107 yrs

By virial theorem, thermal time (if Sun were shining by cooling down)Is the same to within a factor 2.

Neither can explain how Sun has shone for > 109 yrs

Thermal timescale (Kelvin-Helmholtz timescale)

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 sun for

tnuc ~ 0.007 M c2 / L 1011 yr

Note tdyn << tK-H << tnuc

We’ll come back to this later

Some simple estimates

Energy transport

Opacity depends on density, temperature and chemical abundances(in solar interior arises mainly due to bound-free absorption)

Note: numerical value is not great, but functional dependence is qualitatively right!

Note2: Opacity is very sensitive to temperature

That’s it really...except• At some stage the

temperature gradients may become large enough that the energy can not be carried by radiation (and convection sets in)

• Energy production (fusion) can only take place if the temperature is high enough.

• Where these occurs depends on the mass, age (etc) of the star

Basic equations:

Composition characterized by abundances X, Y, Z of H, He and the rest

Plus models of convective processes, when temperature gradientsget large enough...

Sources included if templarge enough

Solar Core

Central 25% (175,000 km)Temperature at centre 1.5 x 107 K Temperature at edge 7 x 106 KDensity at centre 150 g cm-3 Density at edge 20 g cm-3

Temperature in core high enough for nuclear reactions. ENERGYp-p chain: 3 step process (above) leads to production of He4 andneutrinos ().Missing neutrinos (not as many detected as thought).Neutrino mass

The Radiative Zone

Extends from 25% to ~70% of the solar radius.Aptly-named: Energy produced in core carried by radiation photon radiationDensity drops: 20 g cm-3 to 0.2 g cm-3

Temperature drops: 7 x 106 K to 2 x 106 K.

The Convection Zone

Extends from: 70% of the solar radius to visible surface.Radiation less efficient as heavier ions not fully ionised(e.g. C, N, O, Ca, Fe).Fluid becomes unstable to convection (which adiabatically mixes the fluid). Highly turbulent. Motion on large range of scalesTemperature drops: from 2 x 106 K to 5,700 K.Density drops exponentially to 2 x 10-7 g cm-3

Convection visible at the surface (photosphere) as granules and supergranules (see later).

radiative

Temperature

Density

PressureT

(106

K)

15

0

r / R

0 0.5 1.0

convective

r / R0 0.5 1.0

r / R0 0.5 1.0

2

0

150

p

(1016

Nm

-2)

ρ (

103

kg m

-3)

Structure of Sun according to a Standard solar model

0

Hydrogen abundance

Luminosity

Energy generation rate

X

0.7

0.4

r / R

0 0.5 1.0

r / R0 0.5 1.0

r / R0 0.5 1.0

2

0

4

ε (

10-3 J

s-1kg

-1)

L (

1026

W )

The Photosphere

Visible surface of the Sun (100km)

Limb darkening

Photospheric features can be seenin white light. sunspots granules supergranules faculae

Sun rotates differentially at thesurface. (see Lecture 2)Equator ~ 24 daysPoles ~ 30 days.

The Photosphere: SunspotsDark spots on Sun (Galileo)cooler than surroundings ~3700K. Last for several days(large ones for weeks)

Sites of strong magnetic field(~3000G)

Dark central umbra (strong B)Filamentary penumbra.(inhibit convection)

Arise in pairs with oppositePolarity

Part of the solar cycle (Lecture 2)

The Photosphere: Granules

Convection at solar surface can be seen on many scales.

Smallest is granulation.

Granules ~ 1000 km across

Rising fluid in middleSinking fluid at edge (strong downwards plumes)

Lifetime 20 mins

Supersonic flows (~7 kms-1)

The Photosphere: SupergranulesCan also see larger structuresin convection patterns

(Mesogranules) and Supergranules

Seen in measurements of Dopplerfrequency.

Cover entire Sun

Lifetime: 1-2 days

Flow speeds: ~0.5kms-1

Magnetic flux swept to edgesChromospheric Network.

The Photosphere: FaculaeNot all magnetic fields appear dark at solar surface.

Small concentrations of strongmagnetic field seen at limbappear bright.

Actually win out over sunspotsOver the solar cycle

Sun appears brighter at solar maximum. Important for climate

Different on other stars.

So in summary...

• The solar interior conditions are determined largely theoretically.

• Can be checked to a certain extent using helioseismology.

• The solar interior determines all the dynamics of the Sun-Earth system, by providing all the energy.

• The activity of the Sun is all generated by the magnetic field which is generated by a hydromagnetic dynamo located in the solar interior.

• With thanks to HAO, JCD, MJT