Answer:
Pr
Pgas
=
1
3
aT4
R
m
rT
=
ma
3R
¥
T3
r
= 7 ¥10-4
Step-by-step explanation:
Equation of state in stars
Interior of a star contains a mixture of ions, electrons, and
radiation (photons). For most stars (exception very low mass
stars and stellar remnants) the ions and electrons can be
treated as an ideal gas and quantum effects can be
neglected.
Total pressure:
†
P = PI + Pe + Pr
= Pgas + Pr
• PI is the pressure of the ions
• Pe is the electron pressure
• Pr
is the radiation pressure
Gas pressure
The equation of state for an ideal gas is:
†
Pgas
= nkT
n is the number of particles per unit volume.
n = nI
+ ne, where nI and ne are the number
densities of ions and electrons respectively
In terms of the mass density r:
†
Pgas
=
r
mmH
¥ kT
…where mH is the mass of hydrogen and m is the average
mass of particles in units of mH. Define the ideal gas
constant:
†
R ≡
k
mH
†
Pgas
=
R
m
rT
Determining m
m will depend upon the composition of the gas and the state
of ionization. For example:
• Neutral hydrogen: m = 1
• Fully ionized hydrogen: m = 0.5
In the central regions of stars, OK to assume that all the
elements are fully ionized. Bookeeping task to determine
what m is.
Denote abundances of different elements per unit mass by:
• X hydrogen - mass mH, one electron
• Y helium - mass 4mH, two electrons
• Z the rest, `metals’, average mass AmH, approximately
(A / 2) electrons per nucleus
If the density of the plasma is r, then add up number densities
of hydrogen, helium, and metal nuclei, plus electrons from
each species:
Number density
of nuclei
Number density
of electrons
H He metals
†
Xr
mH
†
Xr
mH
†
Yr
4mH
†
2Yr
4mH
†
Zr
AmH
†
ª
A
2
¥
Zr
AmH
†
n =
r
mH
2X +
3
4
Y +
1
2
Z È
Î Í
˘
˚ ˙ =
r
mmH
…assuming that
A >> 1
†
m-1
= 2X +
3
4
Y + 2Z
Radiation pressure
Expression for the radiation pressure of blackbody radiation
is derived in Section 3.4 of the textbook. Result:
†
Pr
=
1
3
aT4
…where a is the radiation constant:
†
a =
8p5
k 4
15c 3
h3
=
4s
c
= 7.565 ¥10-15 erg cm-3 K-4
= 7.565 ¥10-16 J m-3 K-4
Conditions in the Solar core
A detailed model of the Sun gives core conditions of:
• T = 1.6 x 107 K
• r = 150 g cm-3
• X = 0.34, Y = 0.64, Z = 0.02 (note: hydrogen is almost
half gone compared to initial or surface composition!)
†
m-1
= 2X +
3
4
Y + 2Z
†
m = 0.83
Ideal gas constant is R = 8.3 x 107 erg g-1 K-1
Ratio of radiation pressure to gas pressure is therefore:
†
Pr
Pgas
=
1
3
aT4
R
m
rT
=
ma
3R
¥
T3
r
= 7 ¥10-4 Radiation pressure is not at
all important in the center of
the Sun under these conditions
In which stars are gas and radiation pressure important?
†
Pgas
=
R
m
rT
Pr
=
1
3
aT4
equal when:
†
T3
=
3R
am
r
log r
log T
slope 1 / 3
gas pressure
dominated
Using the virial theorem, we deduced that the characteristic
temperature in a star scales with the mass and radius as:
†
T µ
M
R
(see also textbook 2.4)
The average density scales as:
†
r µ
M
R3
The ratio of radiation pressure to gas pressure in a star is:
†
Pr
Pgas
=
1
3
aT4
R
m
rT
=
ma
3R
¥
T3
r
µ
M3 R3
M R3
µ M2
Gas pressure is most important in low mass stars
Radiation pressure is most important in high mass stars