Based upon your knowledge of the ideal gas and radiation pressure equations of state (see the blackbody formula sheet handed out in class), estimate the ratio of the average gas pressure to the average radiation pressure, Pgas/Prad, in the Sun.

Pr

Pgas

=

1

3

aT4

R

m

rT

=

ma

3R

¥

T3

r

= 7 ¥10-4

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