Introduction to Astrophysics I, 29:119
Fall, 2006
Second Homework Set...September 14, 2006.
Due September 21, 2006

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) 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 plot the rotation curve.

(2) Problem 2.9 from textbook.

(3) Problem 2.10

(4) Problem 2.11

(5) Problem 2.13 (I recommend you use MathCad in solution of this problem.)

(6) Problem 2.14 (I recommend you use MathCad in solution of this problem.)

(7) 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.

(8) Here is one to let you appreciate how special the inverse square law of gravity is for our existing here in the universe. An object is subject to a central force, and moves on an orbit given by

$$r = k \theta$$ (4)

What is the force law which would produce such an orbit? Hint: You need to generalize the arguments which led to the equation for $u(\theta)$ for the case of the inverse square law.