Chapter 2 Stellar Structure Equations 1 Mass conservation equation

• The nuclear burning timescale: This is the timescale for burning
all the nuclear fuel of a star at constant luminosity and is therefore
equal to τ nucl ≈ Mc 2 ∆/L, where ∆ ≈ 10 −3 is the typical binding
energy of a nucleon divided by the rest energy of the nucleon. For our
Sun, τ nucl ≈ 10 11 yr. The nuclear burning timescale is a rough estimate
of the lifespan of a star. (You may notice that the nuclear burning
timescale overestimates the lifespan of a star. Do you know why?)
Since τ ff ≪ τ KH ≪ τ nucl , we know once again that our assumption
of LTE is a good approximation to study the interior structure and
evolution of a star.
7 Equation of state
So far, we have used the conservation of mass, energy and momentum
to construct three equations for stellar structure. Yet, we have P , T , m, ρ,
L, ϵ, as well as the chemical composition as a function of r for variables.
To further our analysis, we have to look for equations that tell us specific
properties of the material in a star. One such equation is the equation of
state (EOS), namely, an equation expressing the pressure P as a function of
T , ρ and chemical compositions.
To model a star, astrophysicists may use some extremely accurate and
hence complicated EOSs. I shall not do so in this introductory course for the
following reason. Although those EOSs more accurately model the material
behavior, their predictions on the interior structure of a star are in most cases
not significantly different from the simple EOS we will use in this course. The
simplest EOS, which at the same time turns out to be the one we will use in
this course, is the ideal gas law that you have learned in high school. That
is, P = nkT where n is the number density of gas particle.
Ideal gas law is a good approximation to describe the properties of matter
in a star. To see why, I first have to introduce the Saha equation discovered
by M. Saha in 1920.
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