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{\Large ASTR688S Homework \# 1}

1. Consider an alternate universe in which the strength of gravity is
proportional to r (i.e. ${\bf F} = -k {\bf r}$, with k a positive
constant).  This is Hooke's law, so we are in effect postulating that
gravity behaves as if planets and stars were connected to one another
by springs. \\
a) Derive new versions of Kepler's laws for this force. \\
b) Discuss some reasons why this would be a very strange universe!


2. Ellipse Geometry.\\
a) show that the projection of a circular orbit on the sky is an
ellipse.\\
b) Show that the projection of an elliptical orbit on the sky is an
ellipse. What are the limits on the projected quantities a'', b'' and
e'' in terms of the actual quantities? Where can the central body lie
in the projected ellipse? 


3. Consider the two methods of detecting extrasolar planets by
observing the effects of the planet's gravity: doppler measurements
and measurements of stellar position. Assume that the center of mass of
the star-planet system is at rest with respect to the Earth, that the
planet (of mass m) moves about the star (of mass M) on a circular
orbit, and that the system is inclined at angle I where I=0 denotes
the face-on geometry.\\
a) Doppler Measurements. Show, for circular orbits, that although m
and I cannot be determined uniquely, the {\it mass function} f can be.
How do elliptical orbits change this picture?

\be
  f_{doppler} = {(m \sin I)^3 \over (M+m)^2}
\ee

b) Stellar Position. Derive the equivalent {\it mass function} for the
case of measurements of stellar position. What happens if the star's
orbit is elliptical?
  
4. a) Derive the polar equation of an ellipse $r= a(1-e^2)/(1+e\cos
f)$ from the Cartesian equation $x^2/a^2 + y^2/b^2 = 1$. \\ 
b) Show that if $e^2$ terms are neglected, an elliptical orbit is a circle. \\
c) Derive an expression for $\psi$, the angle between
the radius and velocity vectors. 

5. Comet Hale Bopp is a recently-discovered comet with the following
orbital elements: [$q=a(1-e)=0.913993837, e=0.995048001$].\\
a) What is the orbital period of Hale Bopp?\\
b) How long will Hale Bopp spend inside 1AU? Inside 5AU? \\
c) Write a computer program to translate between the true anomaly $f$
mean anomaly $M$, and eccentric anomaly $E$. Routines like these are
at the heart of all ephemeris programs. Check you program by
translating $f \rightarrow M \rightarrow f$. Where will Hale Bopp be 2
years after its pericenter passage of April 1, 1997?

6. Show that, if $e^2$ terms 
are neglected, the Moon always keeps one
face toward the {\it empty} focus of its elliptical orbit.

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