Stellar Structure
Section 2: Dynamical Structure
Lecture 3 – Limit on gravitational energy
Limit on mean temperature inside stars
Contribution of radiation pressure
Virial theorem
Properties of polytropes
Limits on conditions inside stars:
gravitational potential energy
• The magnitude of the gravitational potential energy has a lower
limit given by Theorem III:
 
GM
2
s
2 Rs
• For the Sun this is ~1041 joules.
• (for proof, see blackboard, and Handout).
(2.14)
Limits on conditions inside stars:
temperature
• For an ideal gas, with constant μ, and neglecting radiation
pressure, the mean temperature satisfies Theorem IV:
T 
1  GM
6 
s
Rs
• (for proof, see blackboard, and Handout).
• For Sun, lower limit is ~4106 K.
(2.15)
Pressure in a star
T ~ 4  10 K,  ~ 10 kg m
6
3
3
imply (see blackboard) that
• the material behaves like an ideal gas
Neutral gas – particles overlap
Plasma – separation >> size
.
.
.
.
• radiation pressure is much less than gas pressure.
Limits on conditions inside stars:
radiation pressure
• Radiation pressure can be shown (for proof see reference in
Lecture Notes) to satisfy Theorem V:
• If the mean density at r does not increase outwards then, in a
wholly gaseous configuration, the central value of the ratio of
radiation pressure to total pressure satisfies
1 – βc ≤ 1 – β*
(2.16)
where β* satisfies the quartic equation (where μc is the mean
molecular weight at the centre of the star):
1
M
s
6
 
 
4
2
   3 1  * 1 



   a  4 G3

*
 c 
.
(2.17)
• Radiation pressure < 10% of total pressure for M < ~6 Msun (more
detailed numbers in Table in Lecture Notes).
Virial Theorem
• The internal energy of a non-relativistic ideal gas is ½kT per
degree of freedom per particle.
• Relating this to , the ratio of specific heats at constant pressure
and constant volume, we can find (see blackboard) an integral
expression for the total internal energy, U.
• Using Theorem II, if  is constant throughout the star, we can then
prove the Virial Theorem:
(2.20)
  3 (   1)U  0 .
• This can be used to show (see blackboard) that a self-gravitating
gas has a negative specific heat.
• For  = 5/3, half the energy released by a contracting star goes
into heating up the star; the other half is radiated away.
Polytropes – simple models for stars
For details see Lecture Notes
• The force balance and mass conservation equations were
derived long before anything was known about the nature of stellar
material, and are independent of that nature.
• To form a closed set of equations, early workers introduced
polytropes – models that had a power-law P(ρ) relation, with an
index n (see blackboard and Lecture Notes); n = 0 corresponds to
a liquid star, so these are mathematical generalisations of that.
• Combining the three equations gives a single second-order
differential equation, the Lane-Emden equation of index n.
• This produces model stars with a finite radius for 0 ≤ n < 5.
• For polytropes, explicit expressions can be obtained for the
potential energy and the mean temperature (see blackboard).