EQUATION OF STATE FOR STARS

NOTES CH 3

LOCAL THERMODYNAMICAL EQUILIBRIUM=LTE

1.The interior of a star is a very optically thick medium:

where is the mean free path of photons between scatterings and/or absorptionlph

⌧ ' R

lph� 1 e.g. ⌧Sun ' RSun

< lph >⇡ 1011

In a region of several mean free path but << R matter and radiation are in thermal equilibrium and can be described by the same local temperature

~1 cm

Note: thermal equilibrium LTE6=

we can calculate microscopic thermodynamical properties as a function of the local temperature, density and composition

mixing locally is very effective: well defined local composition

In each position within the star, radiation has a black body spectrum with that local temperature

EOSTHE EQUATION OF STATE

• Def: A relation between the pressure exerted by a system of particles of known composition, temperature and density:

P = P (T, ⇢, X)

EOSIDEAL OR PERFECT GAS

• Def: an ensemble of free, non-interacting particle so that the internal energy is given by the sum of the particles’ kinetic energy

• Good description for ionised gas in a star with T~106 K. One can show that the ratio

of Coulomb interaction energy per particle and mean kinetic energy per particle is

• Note: Cristallization can happen in cooling white dwarfs (3.6.1)

✏Ckb < T >

⇡ 1% see 3.1 in Prialnik’s book and section 3.6.1 notes

• Note: gas is completely ionised in the hot stellar interiors but in cooler (<10 6 K) atmospheric layers there maybe partial ionization (3.5)

THERMODYNAMICAL QUANTITIES

definition of perfect gas

definition of n(p) =momenta distribution

P= flux of momentum

We need the particle energy ✏p = ✏�mc2 and velocity vp =pc2

✏

Relation between U & P

NR limit: p ⌧ mc

< pvp >=

⌧p2

m

�= 2 < ✏p > P =

2

3n < ✏p >=

2

3U

vp = c

ER limit p � mc

}< pvp >=< pc >=< ✏p >✏p = pc P =

1

3U

”

“n(p)dp

LET’S REFRESH OUR KNOWLEDGE OF STATISTICAL MECHANICS

TO COMPUTE THE EOS WE NEED

CLASSICAL IDEAL GASConsider:

•Gas of identical, non-interacting particles (perfect gas), with mass “m”, non relativistic (p << mc)•At LTE, at temperature T

the momentum distribution is a Maxwell-Boltzmann distribution:

vp =p

mwith

P = nkbT

NOTE: true also for a relativistic Maxwellian distribution

MIXTURE OF IDEAL NON DEGENERATE GASES

ne = ⌃iZini = ⌃iZiXi

Ai

⇢

mu⌘ 1

µe

⇢

mu

mean molecular weight per free electron

nion

= ⌃ini = ⌃iXi

Ai

⇢

mu

⌘ 1

µion

⇢

mu

mean atomic mass per ion

mu =1

12m12C

Pgas

= Pion

+ Pe

=

✓1

µion

+1

µe

◆R⇢T

R = kb/mu = the universal gas constant

=R

µ⇢T

MEAN MOLECULAR WEIGHT

Fully ionised hydrogen Z=A =1 gas while fully ionised He and metals Z/A ~1/2

1

µ=

1

µion

+1

µe= ⌃i

(Zi + 1)Xi

Ai

1

µ⇡ 1

2X + 32Y + 1

2Z⇡ 0.6

it is the inverse sum of the mean atomic mass per ion and the mean molecular weight per electron

X=0.7, Y=0.28, Z=0.02

QUANTUM MECHANICAL STATISTICAL DESCRIPTION OF GAS

•Perfect gas, with mass “m” and spin “s”•At, at LTE temperature T•Fermions

More generally, then in the previous case, we state that the distribution of its numerical density as a function of its linear

momentum is

total energy

μ chemical potential = is the work necessary to change the particle number by dN: dW/dN

Fermi-Dirac distribution

g = 2s+1 energy level degeneracy

dn(p)

dp=

4⇡g

h3

p2

e✏�µkbT � 1

✏ = (pc)2 +m2c4

DONEC QUIS NUNC

==> classical Maxwell-Boltzmann distribution

WDS ARE MADE OF “COLD” GAS

Degenerate gas of fermions at T—>0

Here the chemical potential is called Fermi energy

The corresponding Fermi momentum

Fermi temperature

“zero” temperature distribution can be assumed if a gas has T << TF (i.e. KT << μ)

dn

dp=

4⇡g

h3p2⇥{ 0, ✏ > µ

1, ✏ µ

pF = mc

s✓EF

mc2

◆2

� 1

kbT = EF �mc2

particle density determines Fermi Energy

simply counting particles in a sphere with

radius pF

THERMODYNAMICAL PROPERTIES: NUMBER DENSITY

n =

Zn(p)dp

THERMODYNAMICAL PROPERTIES: PRESSURE

let’s now specialise to WD typical conditions...

DEGENERATE ELECTRON PRESSURE IN WDS&

for typical density and number of free electron per nucleon (Ye), electron are mildly relativistic XF ~1.

DEGENERACY TEMPERATURE FOR ELECTRONS

Degeneracy if T is << TF

Typical central temperature ~107 K

electrons are fully degenerate in a WD

NUCLEONS ARE NON DEGENERATE IN WDS

TF / n2/3/m

TF,e ⇠ 2000⇥ TF,p/n

inversely proportional to mass

In a WD the condition Te << TF,e is attained well before Tn/p << TF,e/p .

In fact, in WD only electrons are degenerate

TF,p/n = 3⇥ 106K < TWD ⌧ TF,e ⇡ 6⇥ 109K

DEGENERATE ELECTRON PRESSURE IN WDtwo extreme regimes:

xF � 1

xF ⌧ 1

⇢ � 106g cm�3

⇢ ⌧ 106g cm�3

P ⇡ 3⇥ 1020 Ba (Ye/0.5)5/3

✓⇢

106g/cm3

◆5/3

P ⇡ 5⇥ 1020 Ba (Ye/0.5)4/3

✓⇢

106g/cm3

◆4/3

WD’s pressure does not depend on temperature => does have to be hot to be in hydrostatic equilibrium

P / ⇢�

RADIATION

photon are bosons with gs = 2 and µ = 0

✏p = pc = h⌫

and completely relativistic

Planck function

Bose-Einstein distribution

BLACK BODY RADIATION

A MIXTURE OF GAS AND RADIATION

P = Prad

+ Pion

+ Pe�

Pgas

= Pion

+ Pe� = �P

Prad = (1� �)P

RADIATION PRESSURE

Prad =1

3U =

1

3aT 4

U = aT 4 energy density

T/⇢1/3 ⇡ 107µ�1/3Prad = Pgas,classical

SUMMARY

from O.R. Pols

P rad=P ga

s

Pe,d,N

R=

Pe,d,U

R

P gas=P e

,d,N

R