Stellar Interior

Solar Facts

• Radius:

– R = 7 105 km = 109 RE

• Mass :

– M = 2 1030 kg

– M = 333,000 ME

• Density:

– = 1.4 g/cm3

– (water is 1.0 g/cm3, Earth is 5.6 g/cm3)

• Composition:

– Mostly H and He

• Temperature:

– Surface is 5,770 K

– Core is 15,600,000 K

• Power:

– 4 1026 W

Solar Layers

• Core

– 0 to 0.25 R

– Nuclear fusion region

• Radiative Zone

– 0.25 to 0.70 R

– Photon transport region

• Convective Zone

– 0.70 to 1 R

– Fluid flow region

Equilibrium

• A static model of a star can be made by balancing gravity against pressure.

– Need mass density and pressure

FgFb

Ft

ghPP

ghAAPPA

PAF

APF

ghAmgF

b

t

g

0

0

0

0)(

)(

Particles and States

• The particles in a star form a nearly ideal fluid.

– Classical ideal gas

– Quantum fluid

• The particles quantum states can be found by considering the particle in a box.

– Dimension L

– Wave vector (kx, ky, kz)

),,(),,( zyxzyx nnnL

kkkk

zyx dkdkdkL

kdkg3

3)(

dpph

Vdppg 2

3

4)(

dkkL

dkkg 23

8

4)(

Vp

13 note:

Internal Energy

• The internal energy depends on the quantum states.

– Density of states g(p)dp

– Energy of each state p

– Number in each state f(p)

• The distribution depends on the type of particle

– Fermion or boson

– Reduces to Maxwell

42222 cmcpp

0

)()( dppgfE pp

0

)()( dppgfN p

11)(

kTpFD

pef 1

1)( kT

pBEpef

kTp

pef )(

Pressure

• The energy is related to the thermodynamic properties.

– Temperature T

– Pressure P

– Chemical potential

• The pressure comes from the energy.

– Related to kinetic energy density

dNPdVTdSdE

0

)()( dppgfdV

d

V

EP p

p

V

ppc

dV

dp

dp

d

dV

d

p

pp

3

2

V

p

dV

d pp

3

ppvV

NP

3

Relativity Effects

• The calculation for the ideal gas applied to both non-relativistic and relativistic particles.

• For non-relativistic particles • For ultra-relativistic particles

m

p

V

NP

23

2 2

m

p

V

NP

23

2

42222 cmcpp 222 cpp

mpvp cvp

Ideal Gas

• A classical gas assumes that the average occupation of any quantum state is small.

– States are g(p)dp

– State occupancy gs

– Maxwellian f(p)

• The number N can be similarly integrated.

– Compare to pressure

– Equation of state

– True for relativistic, also

0

23

4dpp

h

VgepvP s

kTp

p

nkTkTV

NP

0

)()(3

1

3dppgfpv

Vpv

V

NP ppp

0

23

4dpp

h

Vgee

V

kTP s

kTkT p

0

23

4dpp

h

VgeeN s

kTkT p

Particle Density

• The equation of state is the same for both non-relativistic and relativistic particles.

– Derived quantities differ

• For non-relativistic particles • For ultra-relativistic particles

23

2

2

h

mkTnQ

n

ngkTmc Qsln2

23

32

2

mkTh

VgeN s

kTmc 3

3

8

c

kT

h

VgeN s

kT

3

8

hc

kTnQ

n

ngkT Qsln

Electron Gas

• Electrons are fermions.

– Non-relativistic

– Fill lowest energy states

• The Fermi momentum is used for the highest filled state.

• This leads to an equation of state.m

np

m

pnP F

1023

2 22

Fp

s dpph

VgN

0

23

4

313

8

3

nh

pF3

33

8Fp

h

VN

Fp

sp dpph

VgE

0

23

4

m

pmcNE F

10

3 22

35322

8

3

5n

m

hP

Relativistic Electron Gas

• Relativistic electrons are also fermions.

– Fill lowest energy states

– Neglect rest mass

• The equation of state is not the same as for non-relativistic electrons.

423

2 cnp

m

pnP F

Fp

s dpph

VgN

0

23

4

313

8

3

nh

pF3

33

8Fp

h

VN

Fp

s dpph

VpcgE

0

23

4

4

3 cpNE F

3431

8

3

4n

hcP

Electron Regimes

• Region A: classical, non-relativistic

– Ideal gases, P = nkT

• Region B: classical, ultra-relativistic

– P = nkT

• Region C: degenerate, non-relativistic

– Metals, P = KNRn5/3

• Region D: degenerate, ultra-relativistic

– P = KURn4/3

1015

105

1010

1025 1030 1035 1040 1045

T(K)

n(m3)

A

B

C D

Hydrogen Ionization

• Particle equilibrium is dominated by ionized hydrogen.

• Equilibrium is a balance of chemical potentials.

n = 1

n = 2

n = 3

peH n

p = p2/2m

n

n

n

H

QpHHn n

ngkTcmH ln2

p

Qppp n

ngkTcmp ln2

e

Qeee n

ngkTcme ln2

Saha Equation

• The masses in H are related.

– Small amount n for degeneracy

• Protons and electrons each have half spin, gs = 2.

– H has multiple states.

• The concntration relation is the Saha equation.

kT

Qe

n

pe

n nen

g

nn

Hn )(

nepH cmcmcmn

222

24)( ngggHg penn