Introduction to Astrophysics II, 29:120
Spring, 2007
Ninth Homework Set...April 13, 2007. Due April 19, 2007

(1) The famous (and bright) pulsar 0329+54 has a period and period derivative of $P=0.7145$ seconds and $\dot{P}=2.050 \times 10^{-15}$. What is the luminosity, in some form of energy, of this pulsar? Note of Interest: The period and period derivative of this pulsar are known to many more significant figures than given above. Pulsar timing measurements are some of the most precision measurements made in all of physical science.

(2) In class I derived a differential equation for $\phi$, the angular deflection due to the gravitational deflection of starlight. Use this equation to obtain a complete expression for the total deflection of a ray which approaches an object with an impact parameter $b$. In other words, starting with $x=- \infty$, solve for $\phi(x=+\infty)$. The mass of the object is $M$. Hint: For this problem, you may make the very good approximation that $\phi \ll 1$.

(3) Use the expression from (2) to obtain a numerical value (in arcseconds) for the deflection of a ray which passes by the Sun with an impact parameter $b=2 R_{\odot}$.

(4) Calculate the curvature of spacetime (here taken to be the radius of curvature of a deflected light beam) at the surface of a
(a) White Dwarf
(b) Neutron Star

(5) Type DA white dwarfs have hydrogen lines in their spectra. Take $H \alpha$ with a wavelength of 6563 Å. At what wavelength will we see this line when we observe a white dwarf from Earth? First identify the physical mechanism that causes the observed wavelength to be different.

(6) Show that equation 16.6 of the textbook, as derived, is consistent with 16.7.

(7) Assume that you are observing planets orbiting a neutron star. Describe how Gravitational time dilation will affect Kepler's 3rd Law. In other words, what will you observe instead of Kepler's 3rd Law (i.e. a different equation)?

(8) 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.

Which of these lies inside the forward or reverse light cones?