Physics 321
Astrophysics II:  Lecture #7
Prof. Dale E. Gary

Stellar Interiors -- I

Clusters and Main Sequence Fitting

Stellar Interiors Composition
Stars are made of relatively simple stuff.  By mass, our Sun is 73% hydrogen, 26% helium, and only 1% of higher Z (atomic number) atoms.  We write these quantities in terms of the mass fraction, as
Pressure is force/unit area, F / A.  The atmospheric pressure at sea level is about 14 lbs/square-inch, or 10 Pascal, or 1000 millibars (mm of mercury).  Pressure in the interior of a star arises from the weight of the atmosphere directly above the point--that is, from the force of gravity--balanced by the outward pressure due to energy release inside the star.  Nearly all stars are static, meaning that they are not expanding or contracting significantly.  Such a state is called hydrostatic equilibrium, and like all equilibrium situations, it is characterized by force balance.  Consider a column of gas of cross-sectional area A.  The forces on a section (parcel) of the gas is shown in the figure on the left:
Temperature and the Equation of State
The temperature, density, and pressure are not independent, but are joined by the equation of state, which describes the macroscopic manifestation of particle interactions.  One well-known example of an equation of state is the ideal gas law:
PV = NkT
where P = pressure, V = volume, N = number of particles, k = Boltzmann's constant, and T = temperature.  We can write this more simply in terms of the number density n = N / V as
P = nkT.
Above in equation (1), we saw that the number density is related to the mass density by
n = r/mmH
so the equation of state becomes
P = rkT / mmH .                       (3)
Inserting (3) into (2), and assuming for the moment that temperature T does not change with height (an isothermal atmosphere), we obtain an expression for the density:
dr/dr = - (mmHg / kT) r            (4a)
which is a first-order differential equation with the solution:
r = ro exp(-r / H)
where H = kT / mmH is called the scale height. and ro is called the base density since it is the value of the density at r = 0.  Note that if we had solved for r in equation (3) and substituted it for r on the right-hand side of equation (2), we would obtain a similar differential equation for the pressure:
dP/dr = - (mmHg / kT) P            (4b)
with the solution:
P = Po exp(-r / H).
Therefore, the pressure and density vary with the same scale height.  What is the physical interpretation of the scale height?  It means that the pressure and density drops with height by the same factor of 1/e = 0.368 for each increase in height of a distance H.  Note that a hot atmosphere (large T) has a larger scale height than a cool atmosphere.  Also, increasing gravity lowers the scale height.  It is of interest to calculate the scale height for some atmospheres.

Example 1: Scale height at Earth's surface.  H = kT / mmHg, where T ~ 300 K, g = 9.8 m/s2, but what is m?  The atmosphere is mostly (4/5) molecular nitrogen, N2, which has a mean molecular weight of 28.  Using these values, we obtain:

H = kT / mmHg = (1.38x10-23 J/K)(300 K) / [28(1.67x10-27 kg)(9.8 m/s2)]
    = 9.0 km
In fact, the Earth's atmosphere rapidly grows cooler with height, so the scale height is somewhat smaller.

Example 2: Scale height in the photosphere of the Sun.  Here, T = 5770 K, m ~ 0.6, but what is g?  Remember that mg is the force of gravity, which is also

F  = mg = GMm / R2
so g = GM / R2.  At the surface of the Sun, we would use M = 1.989x1030 kg, R = 6.96x108 m (from Appendix 7 in the text), so
g = (6.67x10-11 N m2 / kg2)(1.989x1030 kg)/(6.96x108 m)2 = 296 m/s2. ~ 30 g
H = kT / mmHg = (1.38x10-23 J/K)(5770 K) / [0.6(1.67x10-27 kg)(296 m/s2)]
    = 270 km
Other Equations of State
In the atmospheres of stars and planets, the ideal gas law is a pretty good approximation.  However, in the interiors of stars some new effects can come into play.  In any star, near the core, especially, the light itself can exert significant pressure.  A photon has momentum p = hn/ c, and when it is absorbed or reflected it exerts a force F = dp/dt.  Consequently, electromagnetic radiation has an associated pressure.  It may come as no surprise that the pressure is related to the total radiant energy flux, which we saw from the Stefan-Boltzmann law was F = sT4.  One can show that
Prad = 1/3 aT 4
where a = 4s/c = 7.56591 x 10-16 J m-3 K-4 is the radiation constant.  In this case, the total pressure, equation (3), is replaced by
P = rkT/ mmH + 1/3 aT 4                      (3')
Finally, when relativistic and quantum effects are included to derive the equation of state, a very different pressure equation is found that applies to extremely dense states of matter such as found in white dwarfs and neutron stars.  We will look into this later in the course.