1. Stellar structure and equilibrium
In Assignment #1, we studied a power law spherical density distribution. Here, we will write
it in slightly different form to make the calculations easier:
$\rho = Br^{-\alpha}$
where B is a positive constant and $\alpha < 3$ (recall the constraints determined from
Assignment 1). The gas obeys a polytropic equation of state:
$P = A\rho^\gamma$
where gamma is a positive known constant - typical values are 1/2, 4/3, or 5/3, but leave it
as y for generality.
a. Use the equations of stellar structure to determine the value of $\alpha$ required for
equilibrium. Your expression for $\alpha$ will depend on $\gamma$, but nothing else. [Hint: match
power laws: if $Cr^\beta = Dr^\delta$, then $\beta = \delta$.]
b. Show that your value of $\alpha$ is consistent with the Singular Isothermal Sphere from
Assignment 1 in the special case that the equation of state is isothermal.
c. Determine an expression for B in terms of A, $\gamma$, and constants.
