Star Model - Isothermal Core

[astrop]

Homework Statement

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 $$R_{0}$$ in the core (helium) and $$R_{1}$$ in the outer envelope (hydrogen). The
main parameter values at the center, $$r=0$$, are $$T_{0} , \rho_{0} ,$$ and $$P_{0}$$.

(a) Evaluate the ratio $$R_{1}/R_{0}$$

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

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

$$d\phi/dx = \mu/x^{2}$$
$$d\mu/dx = x^{2}e^{-\phi}$$

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

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

Homework Equations

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

The Attempt at a Solution

(a) Here I assumed the answer was $$\approx$$ 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 $$r = x r_{x}$$. I also know that $$\mu(x)$$ is the scaled mass function, ie $$m = x \mu(x)$$. I am not sure of $$\phi(x)$$, 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 $$r_{x}$$ or $$m_{x}$$ (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 $$\phi(0) = 0$$ and $$\mu(0) = 0$$. I'm still working on the series solutions.

 

[member38644]

Your response is off to a great start! Here are some additional thoughts on each part:

(a) Your reasoning for the ratio of specific gas constants is correct. However, it might be helpful to actually calculate the ratio using the values for R_0 and R_1 given in the problem. This will give you a more precise answer.

(b) Your explanation of the matching conditions is correct. It might be helpful to also mention that the temperature gradient should be continuous at the boundary as well.

(c) Your understanding of the variables x, \mu(x), and \phi(x) is correct. To evaluate the reference constants, you can use the equations given in the problem and substitute in the values for T_0 and \rho_0. For example, using the hydrostatic equation, you can solve for r_x in terms of T_0 and \rho_0. Similarly, using the mass equation, you can solve for m_x in terms of T_0 and \rho_0.

(d) Your central conditions are correct. To find the series solutions, you can use the Taylor series expansion for e^{-\phi} and solve for the first few terms. Similarly, you can use the Taylor series expansion for \mu and solve for the first few terms.