Introduction to Astrophysics II, 29:120
Spring, 2009
Eleventh Homework Set...April 26, 2009. Due April 30, 2009

(1) Consider a spiral galaxy, and a rotating, noninertial reference frame, in which a star has a closed orbit of the $(n,m) = (1,4)$ ``resonance'' form. Assume the galaxy is like the Milky Way (same rotation curve) and the star is at the galactocentric distance of the Sun.
(a) What is the pattern speed $\Omega_{lp}$ of this reference frame?
(b) Calculate out and plot the orbit of the star in this reference frame (Hint: Mathematica is useful for things like this).

For the questions below involving synchrotron radiation, assume that the pitch angle $\theta=\pi/2$

(2) A typical magnetic field in a cosmic source is $10^{-9}$ T. What is the energy of an electron that would have the peak of its synchrotron radiation occur at a wavelength of 2 Å?

(3) The synchrotron loss time (let's note it by $\tau_{1/2}$) can be defined as the time it takes an electron to radiate away half of its energy to synchrotron radiation. Show that $\tau_{1/2}$ decreases with increasing energy. Do this by obtaining the correct functional dependence of $\tau_{1/2}$ on $E$.

(4) Assuming a magnetic field of $10^{-9}$ T, what is the synchrotron loss time for an electron with a spectral peak at 2 Å? What is $\tau_{1/2}$ for an electron with a spectral peak at 5000 MHz?

(5) A typical radio galaxy has a redshift of $z=0.25$. What is its distance? Just use a simple estimate to answer this question; I do not want the cosmologically-correct, but more complicated expression which we (may) encounter later in the course.

(6) One of my favorite radio sources is 3C79. Some data on 3C79 are as follows (true stuff!). The flux densities at 1.446 and 4.885 GHz are 4.30 and 1.31 Janskys, respectively. The redshift of 3C79 is 0.256.
(a) What is the spectral luminosity (units of Watts/m$^2$/Hz) of 3C79 at 1.446 GHz?
(b) What is the spectral index $\alpha$?
(c) Write down an expression for the total luminosity, i.e. the luminosity integrated over all frequencies. What kind of difficulty arises in answering this question? What does this tell you about the overall shape of the radio spectrum of 3C79 (and every other similar galaxy)?

(7) Problem 27.13 from the book

(8) Problem 27.14 from the book