Introduction to Astrophysics I, 29:119
Fall, 2008
Second Homework Set...September 04, 2008.
Due September 11, 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.

(1) Problem 2.3 from textbook.

(2) Problem 2.7

(3) Problem 2.9

(4) Problem 2.14

(5) A rotation curve of a galaxy is a plot of orbital speed versus galactocentric distance. Let's consider an elliptical galaxy, that can be roughly considered spherically symmetric. Assume the density as a function of galactocentric distance $R$ can be described as follows.

$$\rho(R) = \rho_0; \mbox{ if } R < R_c$$ (1)

$$\rho(R) = \rho_0 (\frac{R}{R_c})^{-2}; \mbox{ if } R \geq R_c$$ (2)

Write an expression for the orbital speed as a function of $r$, and graph (roughly, by hand) the rotation curve. You may assume that the orbit is circular.

(6) In lecture, I showed how elliptical orbits could be derived from Newton's Laws. In the course of that derivation, a vector $\vec{D}$ emerged, defined by the equation (check class notes)

$$\vec{v} \times \vec{L} = G M \mu \hat{r} + \vec{D}$$ (3)

In lecture I said that $\vec{D}$ pointed towards perihelion. Show that this is true (i.e. give a good mathematical argument). Suggestion: Draw a fairly eccentric orbit and think about what $\vec{D}$ would be at some special points.

(7) A comet orbits in the solar system (and therefore obeys Kepler's Laws). Its orbit has a semimajor axis of 10 astronomical units, and an eccentricity of 0.95. On January 1, 2009, it is at perihelion ( polar angular coordinate $\theta=0^{\circ}$). What will its coordinates be 10 years later? Strong Suggestion: This is a good one to solve with Mathematica or MathCad. You can do it with pencil and paper, but your life will be horrible.