Introduction to Astrophysics II, 29:120
Spring, 2009
Sixth Homework Set...March 6, 2009. Due March 12, 2009

(1) Cyclotron Radiation is emitted from the surface of a $1 M_{\odot}$ neutron star.
(a) At what frequency is it emitted in the frame of the neutron star surface?
(b) At what frequency is it observed by a very distant observer (like on Earth)? In what part of the spectrum does this occur?

(2) Calculate the intervals corresponding to the following pairs of events. One of the events is right now, where you are sitting, and the other event is
(a) The star Arcturus, 200 years ago.
(b) The galaxy M81, 5 million years ago.
(c) The galaxy M87, 30 million years in the future.

(3) Demonstrate by a specific example the validity of the statement in the book that the interval squared $(\Delta s)^2$ is a Lorentz invariant. To do this consider two events A and B in two frames. In the first frame, (the ``lab frame'') the two ``events'' are the motion of an object from $(x,y,z)=(0,0,0)$ at t=0 to the coordinates $(x,y,z)=(l,0,0)$ at a constant speed $V$ in the x direction. The second frame moves along the x axis at speed V.

(4) A radio telescope transmits a radio pulse to a spacecraft which is along the Sun-Earth line at a distance $R$ from the center of the Sun. The spacecraft immediately retransmits the signal back to Earth.
(a) Derive an expression for the round-trip propagation time, including the General Relativistic correction.
(b) Calculate the magnitude of the relativistic correction if $R = 4 R_{\odot}$.
Hint: The solution to this problem requires use of the Schwarzschild Metric.

(5) A binary star system consists of an invisible, presumably degenerate star, an accretion disk, and a K0III star. The period of the binary is 155.3 hours, and the radial velocity of the K0 giant varies between $\pm 208$ km/sec. What type of object is the unseen, degenerate object? Obviously you have to present arguments and calculations to support your answer.

(6) Problem # 17.18 from textbook

(7) Look at Table 1 in Remillard and McClintock, Annual Reviews of Astronomy and Astrophysics, vol 44, p49, 2006. Consider the case of 0422+32. The mass function is given as $1.19 M_{\odot}$, but the mass of the degenerate object is given as $3.7 - 5.0 M_{\odot}$. What factors must be accounted for to calculate the mass of the degenerate object?

(8) Problem # 17.24