Introduction to Astrophysics I, 29:119
Fall, 2008
Seventh Homework Set...October 24, 2008. Due October 30, 2008

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. All problem numbers refer to the first edition of the book.

(1) You are measuring radiation coming in from the sky, which has the following characteristics. It has a blackbody spectrum, and is centered on $\theta = 0$ ($\phi$ is the complementary azimuthal angle):

1. For $\theta < 10^{\circ}$, T=10,000K
2. For $10^{\circ} < \theta < 30^{\circ}$, T=3000K
3. For $70^{\circ} < \theta < 110^{\circ}$, T=5000K

Pretty weird, eh? Make a plot showing the intensity of radiation at $\lambda=500$ nm as a function of $\theta$. Both axes on the plot must be properly labeled and have the right numbers.

(2) For the radiation field in problem # 1, calculate the mean intensity and the flux.

(3) The local interstellar medium has a density of about 0.2 atoms/cm$^3$ ($2 \times 10^5$ atoms/m$^3$). You may consider the gas to be entirely hydrogen. Calculate the collisional mean free path and collision time.

(4) The electron density in the solar corona as a function of heliocentric distance $r$ is given by

$$n_e(r) = \left[ 365 \left(\frac{r}{R_{\odot}} \right)^{-4.3}... ...r}{R_{\odot}} \right)^{-2} \right] \times 10^5 \mbox{ cm}^{-3}$$ (1)

You will have to convert the density to SI units or do the calculation in cgs. It will be good for you.
(a) What is the optical depth of the corona due to Thomson scattering (or electron scattering)? Formally, this is a scattering optical depth rather than an absorption optical depth, but calculate it anyway.
(b) What does your number tell you about the amount of sunlight that is scattered by the corona as opposed to passing through unscattered?
Hint: If the approach to this problem doesn't immediately come to you, check the textbook from such additional information.

(5) Problem 9.5

(6) Problem 9.7

(7) Problem 9.8

(8) Problem 9.12