Astr 345 Assignment 6
Due before midnight Fri, Oct 10, 2008 (full credit), before midnight Fri Oct 17, 2008 (half credit)
1. Hartmann, chapter 3, problems 6, 7, 8, 10, 11, 16, 17, 18, 19
2. Supplemental problem: The goal of this problem is to identify Kirkwood gaps in the main belt asteroids.
Download the Statistical Asteroid Model (SAM-I) from NASA’s Planetary Data System internet site:
http://www.psi.edu/pds/SAM-I/. You will want the Known Asteroid Model (file name
SAM-CompDB.txt). Column 2 of this text file contains the semi-major axis lengths of 8,538 main belt
asteroids. Extract this data, and form a histogram of the number of asteroids versus semi-major axis
length. You will want pretty fine divisions in semi-major axis length, no greater than 0.1 AU.
Once the histogram is plotted, identify on it the locations of 4 or more obvious gaps in the distribution.
Then use Kepler’s 3rd law to determine the orbital periods at these locations. Finally, express these orbital
periods as simple fractions of Jupiter’s orbital period.
3. Supplemental problem (Carroll & Ostlie P2.3)
(a) Starting with Kepler’s 2nd law, derive the following expression for the angular velocity of an object in
an elliptical orbit in terms of orbital parameters:
vθ=2πa
P
1 + ecos θ
√1−e2
(b) Starting with the polar equation for the ellipse, show that the radial velocity of an object in an
elliptical orbit is:
vr=2πa
P
esin θ
√1−e2
(c) Using the above two expressions, verify the vis-visa equation from v2=v2
r+v2
θ.
4. Supplemental problem (Carroll & Ostlie P2.9): In general, an integral average of some continuous
function f(t)over an interval τis given by
< f(t)>=1
τZτ
0
f(t)dt.
Beginning with the expression for the integral average, prove that
< U >=−GMµ
a.
You may use the following integral:
Z2π
0
dθ
1 + ecos θ=2π
√1−e2.
HINT: you will have to transform the integral from one of dt to one of dθ using Kepler’s 1st and second
laws.
Grading:
•You can share concepts, but all work must be completely original
•Write neatly and legibly
•Line up equal signs in a straight vertical column, and never have more than one equal sign on a line
•Define all non-standard variables
•Do not skip essential lines of algebra
•Develop ideas logically from start to finish
•Include a statement at the end of each problem interpreting the result
•Label your diagrams; all plots must be computer plots
•Take pride in your work
•All assignments are out of 30 points
