lecture 1 - the solar interior

January 19, 2006 Lecture 1 - The Solar Interior

Lecture 1 - The Solar InteriorLecture 1 - The Solar Interior

o Topics to be covered:

o Solar interioro Core

o Radiative zone

o Convection zone


The Solar Interior - “The Standard Model”The Solar Interior - “The Standard Model”

o Coreo Energy generated by

nuclear fusion (the proton-proton chain).

o Radiative Zoneo Energy transport by

radiation.

o Convective Zoneo Energy transport by

convection.


The Solar InteriorThe Solar Interior

o Christensen-Dalsgaard, J. et al., Science, 272, 1286 - 1292, (1996).


The Solar CoreThe Solar Core

o R: 0.0 - 0.25 Rsun

o T(r): 15 - 8 MK

(r): 150 - 10 g cm-3

o Temperatures and densities sufficiently high to drive hydrogen burning (H->He).

o Ultimate source of energy in the Sun and Sun-like stars.



o What is the temperature and pressure in the core?

o Assume hydrostatic equilibrium:

and mass conservation:

o Divide to cancel ’s =>

o Therefore, LHS =>

and RHS =>

€

dP

dr= −

GMρ

r2

€

dM

dr= −4πr2ρ

€

dP

dr/dM

dr=

dP

dM= −

GM

4πr4

€

−dP

dMdM = PC − PS0

M

∫

€

GM

4πr4dM

0

M

∫ =GM 2

8πr4

PC = pressure at core PS = pressure at surface

€

∴PC = PS +GM 2

8πr4



o Assuming PS << PC and setting r = R,

o Using the Ideal Gas Law

k = Boltzmann’s const

n = number density atoms/cm3

= density = M/4R3

o The core temperature is therefore

€

PC ~GM 2

8πR4

€

TC ~GMmH

kR

€

PC = nkT =ρkT

mH

o Which gives Tc ~ 2.7 x 107 K (actual value is ~1.5 x 107 K).



o Coulomb barrier between protons must be overcome for fusion to occur.

o To overcome Coulomb barrier, particles must have sufficient thermal kinetic energy to exceed Coulomb repulsion:

o Particles have Maxwell-Boltzmann distribution:

o There is a high-energy tail, but not sufficient … need quantum mechanics.

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€

=1010K!€

3

2kT >

e2

rnuc

€

=>T >2e2

3krnuc

€

P(E)dE ∝ Ee−

E

kT dE



o From Heisenberg Uncertainty Principle a proton of a given (insufficient) energy may be located within nucleus of neighbouring proton.

o Combined with high-energy M-B tail, we get the Gamow Peak.

o So protons in 3-10 keV energy

range can overcome the Coulomb

barrier (i.e., T>15MK).

o Fusion can therefore occur.

€

(ΔxΔp ≥ h /2)




Proton-proton cycleProton-proton cycle

o The p-p cycle occurs in three main steps.

Step 1: 1H + 1H 2H + e+ + (Q = 1.44 MeV)

o Might then expect a 2H + 2H reaction, but because of the large numbers of 1H, the following is more probable:

Step 2: 2H + 1H 3He + (Q = 5.49 MeV)

o 3He can then react with 1H, but the resultant 4Li is unstable (i.e. 3He + 1H 4Li 3He + 1H).

o The final step is then:

Step 3: 3He + 3He 4He + 21H + (Q = 12.86 MeV)

o The net result is: 4 1H 4He + 2e+ + 2 (Q = 26.7 MeV)


Proton-proton cycle (cont.)Proton-proton cycle (cont.)

o ~99% of the Sun’s energy is produced via the p-p cycle. o The remaining ~1% is produced by the Carbon-Nitrogen-Oxygen (CNO) cycle.o CNO cycle is more important in more massive stars.


Proton-proton vs. CNOProton-proton vs. CNO




The Radiative ZoneThe Radiative Zone

o R: 0.25 - 0.8 Rsun

o T(r): 8 - 0.5 MK

(r): 10 - 0.01 g cm-3

o Hydrogen burning cuts off abruptly at r ~ 0.25 Rsun

.

o Interior becomes optically thin or transparent as density decreases.

o Energy transported radiatively.

o Photons cannot be absorbed in the radiative zone as the temperature are too high to allow atoms to form. Therefore no mechanism for the absorption of photons.


The Radiative ZoneThe Radiative Zone

o For T = 15MK Wien’s displacement law implies max = 0.19 nm i.e., the center of the Sun is full of X-rays.

o Photons do 3D random walk out of Sun.

o Assume photon moves l between interactions (mean free path) and takes a total number of steps N.

o On average it will have moved a distance

o As tdifusion = N l / c and

=> tdiffusion >104 yrs!

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d = l N

€

R = l N => tdiffusion = R2 / lc


Solar InteriorSolar Interior

o Total radiative energy inside Sun is: J

where a = 4/c is the radiation constant.

o Can thus estimate solar luminosity from, W

o Which gives, L ~ 3 x 1026 W.

o Actual value is actually 4 x 1026 W.

€

E = aT 4 4

3πR3 ⎛

⎝ ⎜

⎞

⎠ ⎟

€

L =E

tdiffusion

=16π

3σT 4Rl


The Convective ZoneThe Convective Zone

o R: 0.8 - 1 Rsun

o T(r): 0.5 MK - 6000 K.

<0.01 g cm-3

o Photons now absorbed as temperature is sufficiently low to allow atoms to form. Gas is optically thick or opaque.

o Continuous absorption of photons by lower layers causes a temperature gradient to build up between the lower and upper layers.

o Plasma become convectively unstable, and large convective motions become the dominant transport mechanism.

TH > TC

TH

TC

r


The Convective ZoneThe Convective Zone



lecture 1 - the solar interior

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