Homework 1
Mechanics 701, Fall 2011
Due: Friday, September 9
1
Problem 316
Prove that in a Kepler elliptic orbit with small eccentricity
e
the angular motion of a particle as
viewed from the
empty
focus of the ellipse is uniform to first order in
e
. The empty focus is the
focus that is
not
the center of attraction.
It is this theorem that enables the Ptolemaic picture of planetary motion to be a reasonably
accurate approximation. On this picture the Sun is assumed to move uniformly on a circle whose
center is shifted from Earth by a distance called the
equant
. If the equant is taken as the distance
between the two foci of the correct elliptical orbit, then the angular motion is thus described by
the Ptolemaic picture accurately to first order in
e
.
2
Problem 317
One classic theme in science fiction is a twin planet (“Planet X”) to Earth that is identical in
mass, energy, and momentum but is located on the orbit 180
◦
out of phase with Earth so that it
is hidden behind the Sun. However, because of the elliptical nature of the orbit, it is not always
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 Fall '11
 Bazaliy
 mechanics, Work

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