Chapter 2 Stellar Structure Equations 1 Mass conservation equation

Coulomb energy per particle is about ¯Z 2 e 2 /d where ¯Z is the average atomic
number of particles. In contrast, virial theorem tells us that the gravitational
potential energy and internal energy of a star are of the same order. Consequently,
the ratio of Coulombic energy ( M ) to internal energy of a star
( GM 2
R ) is about ¯Z 2 e 2
¯Z 2 e 2
Ām H d
. (73)
G(Ām H ) 4/3 M
2/3
Note that ¯Z and Ā are of order of 1. Therefore, the above ratio is about 10 −2 ,
which is much less than 1. Thus, electrostatic potential energy contribute
only a small fraction of the internal energy of a star; and most of the internal
energy are in the form of heat. More specifically, the thermal K.E. of a
typical particle inside a star is much greater than the electrostatic potential
energy it experiences.
Well, you may also ask if quantum effects may alter the ideal gas EOS
in a star. We may estimate the uncertainty in position ∆x to be of order of
d. Similarly, the uncertainty in momentum ∆p is of the order of √ kT Ām H .
Thus, for all living stars, ∆x ∆p is much larger than h. Hence, quantum
effects play little role in affecting the ideal gas EOS of a star. (Note: the
situation is completely different for dead stars such as white dwarfs and
neutron stars. Because the density of dead stars is high, ∆x ∆p for some
particle species inside these stars are of order of h and hence quantum effect
drastically changes their EOS.)
To summarize, ideal gas EOS is a good approximation in the study of
the interior of most living stars. (However, quantum mechanical effect is
important in some other “stars” such as brown dwarfs and supernovae. Radiation
pressure also plays an important role in the structure and stability
of a super-massive star. We shall come across them later on in this course.)
From now on, we model the EOS of a star by ideal gas law. More precisely,
we assume that the gas pressure of a star is given by P gas = nkT . Since the
temperature and luminosity of a star are high, we conclude that electrons,
ions, and photons are the three most important constituents of a star. The
electron pressure is given by
where n e is the electron number density.
P e = n e kT (74)
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