Introduction to Astrophysics II, 29:120
Winter, 2006-2007
Second Homework Set...January 26, 2007. Due February 1, 2007

Show calculations and give reasons for your answers. Don't go around confused and despondent; if you do not know how to get started, ask me for help. The purpose of problem sets is to promote thinking and lead to understanding, not produce a confiteor of revealed truth.

(1) Problem 10.16 from textbook

(2) Assume that the pressure everywhere in a star is given by

$$p=K\rho^{\gamma}$$ (1)

where $\rho$ is the density. Show that the equations of stellar structure yield a non-coupled differential equation for $\rho(r)$, where $r$ is the radial coordinate. By ``non-coupled'', I mean an equation for $\rho$ alone as a function of $r$. Derive that equation. You are not required to solve it. Hint: In this case, you don't need all of the equations of stellar structure to answer this problem.

For the following problems, consult Appendix I, p A-54, which gives a stellar model for a $1 M_{\odot}$ star. Use finite differences to approximate the derivatives which appear in the equations of stellar structure.

(3) Use the luminosity equation (from the equations of stellar structure), plus information in the table, to obtain the energy generation rate $\epsilon$ at a radial distance $r=4.90 \times 10^9$ cm.

(4) Using equations in the textbook (and discussed in lecture), calculate $\epsilon_{pp}$ (the energy generation rate due to the proton-proton cycle) at $r=4.90 \times 10^9$. Compare it with your answer from # 3.

(5) Show that hydrostatic equilibrium is satisfied by the model at $r=9.17 \times 10^9$ cm, by calculating both sides of the equation of stellar structure that describes hydrostatic equilibrium, and showing that they are equal.

(6) At $r=9.17 \times 10^9$ cm, is the star radiative or convective? Show your work.

(7) Calculate $\frac{d \rho}{dr}$ at $r=1.06 \times 10^{10}$ cm. Use this number to calculate the radius of the star, if the density continued to decrease linearly with $r$. Compare this number with the true radius of a $1 M_{\odot}$ star.

(8) Use the data from the table to calculate $\frac{d^2 \rho}{dr^2}$ at $r=1.06 \times 10^{10}$ cm. Use this number, plus $\frac{d \rho}{dr}$ from problem #7, to get a 2nd estimate for the radius of the star. Discuss the degree of improvement (or detriment) that occurs.