Star Model - Isothermal Core

astrop

1. The problem statement, all variables and given/known data
Let’s make an idealized star model with two parts: an isothermal helium core, and pure
hydrogen layers outside the core. The core represents the part where hydrogen has already been burned. Mathematically this represents the simplest model that might resemble a partially-evolved star.

Neglect radiation pressure, and assume that the gas is fully ionized. The constant for the
ideal gas law is [tex]R_{0} [/tex] in the core (helium) and [tex]R_{1} [/tex] in the outer envelope (hydrogen). The
main parameter values at the center, [tex]r=0 [/tex], are [tex]T_{0} , \rho_{0} ,[/tex] and [tex] P_{0}[/tex].

(a) Evaluate the ratio [tex]R_{1}/R_{0} [/tex]

(b) Specify and explain the matching conditions at [tex] r_{1}[/tex], the boundary between core and envelope.

(c) If we adopt a particular pair of length and mass measurement units [tex]r_{x}[/tex] and [tex]m_{x}[/tex], then the hyrdostatic and mass equations in the isothermal core can be expressed simply as :

[tex]d\phi/dx = \mu/x^{2}[/tex]
[tex]d\mu/dx = x^{2}e^{-\phi}[/tex]

Briefly explain the meaning of the variables [tex]x[/tex], [tex]\mu(x)[/tex], and [tex]\phi(x)[/tex], then evaluate the reference constants [tex]r_{x}[/tex] and [tex]m_{x}[/tex] in terms of the model parameters [tex]T_{0}[/tex] and [tex]\rho_{0}[/tex].

(d) Specify the central conditions for [tex]\phi[/tex] and [tex]\mu[/tex] at x = 0. Work out the first two terms in a series solution for [tex]\phi(x)[/tex] near the center where x << 1. Do the same for [tex]\mu(x)[/tex] .


2. Relevant equations
[tex]\frac{dP}{dr} = -\rho\frac{G m}{r^{2}}[/tex]
[tex]\frac{dP}{dm} = -\frac{G m}{4 \pi r^{4}}[/tex]
[tex]\frac{dm}{dr} = 4\pi r^{2} \rho[/tex]
[tex]\frac{dr}{dm} = \frac{1}{4\pi r^{2} \rho}[/tex]
[tex]\frac{dT}{dr} = - \frac{3 \kappa \rho F}{4 a c T^{3} 4 \pi r^{2}}[/tex]
[tex]\frac{dT}{dm} = -\frac{3 \kappa F}{4 a c T^{3} (4 \pi r^{2})^{2}}[/tex]
[tex]\frac{dF}{dr} = 4 \pi r^{2} \rho q[/tex]
[tex]\frac{dF}{dm} = q[/tex]
[tex]P = \frac{R}{\mu_{I}}\rho T + P_{e} + \frac{1}{3}a T^{4}[/tex]
[tex]\kappa = \kappa_{0} \rho^{a}T^{b}[/tex]
[tex]q = q_{0} \rho^{m}T^{n}[/tex]


3. The attempt at a solution
(a) Here I assumed the answer was [tex]\approx[/tex] 1/4, since the ration of specific gas constants should reduce to the ratio of their molar masses.

(b) Temperature and Pressure should both be continuous between the core and the envelope, whereas the density may be discontinuous.

If there was a discontinuity in temperature, the core would either be absorbing or radiating heat and the isothermal assumption would not be valid. If the pressure were discontinuous, the star would expand or contract and we would not have hydrostatic equilibrium.

(c) I know that x is the radial variable in the model, ie [tex]r = x r_{x}[/tex]. I also know that [tex]\mu(x)[/tex] is the scaled mass function, ie [tex]m = x \mu(x)[/tex]. I am not sure of [tex]\phi(x)[/tex], but I think it is something like the scaled gravitational potential.

The part I am really having trouble with is evaluating the reference constants. I was thinking of taking the dimensionless differential equations given, rewriting them as regular, unit-having differential equations and then try to group the constants together and call that [tex]r_{x}[/tex] or [tex]m_{x}[/tex] (depending on the equation), but even if that works I not sure how to work in the temperature.

(d) If I was correct in part (c) then [tex]\phi(0) = 0[/tex] and [tex]\mu(0) = 0[/tex]. I'm still working on the series solutions.