PHYS-1500 PHYSICAL MODELING
Class 16: Transfer Orbit
FALL 2006
NAME _________________________________
A space shuttle is to place a communication satellite in a geosynchronous circular orbit. The
shuttle is in a circular orbit of radius RA, relatively close to Earth, and the geosynchronous orbit
has a much larger radius RB. The shuttle crew places the satellite in an elliptical "transfer" orbit,
which is RA from Earth's center at perigee (closest point), and RB from Earth's center at apogee
(farthest point). When the satellite arrives at apogee, its speed is increased to inject it into the
geosynchronous orbit. The diagram shows the shuttle's orbit, the geosynchronous orbit, and the
transfer orbit. Earth is the shaded circle.
The satellite is launched from the shuttle at point P. If nothing else is done, the satellite will
continue on the elliptical transfer orbit, and come back to point P. However, at point Q, its speed
is increased, and its goes into the geosynchronous orbit.
1. Find an expression for the speed that the satellite must have at point P if it is to go into an
elliptical orbit with a perigee distance of RA and an apogee distance of RB. Your answer
should be in terms of RA, RB, G, and M, where G is the gravitational constant and M is the
mass of Earth.
Hint: This can be solved using conservation of mechanical energy and conservation of angular
momentum. The mechanical energy (kinetic plus potential) of the satellite at points P and Q
must be the same. Also, the angular momentum of the satellite about Earth's center must be
equal at points P and Q. At points P and Q, the satellite's velocity is perpendicular to the line
connecting it to Earth's center. Therefore, at point P, its angular momentum can be written as
LP = mvP RA, where vP is its speed at point P. Similarly, at point Q, LQ = mvQ RB.
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2
mv P2 
GMm 1
GMm
 2 mv Q2 
RA
RB
and
mv P R A  mv Q R B
2. If RA = 7.00 10 m and RB = 4.22 ×10 m, find vP.
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vP = __________________ _____
units
3. What is the difference between vP, the satellite's speed in the transfer orbit at point P, and the
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shuttle's speed in its circular orbit of radius RA = 7.00 10 m?


 vP = __________________ _____
units
4. If RA = 7.00 106 m and RB = 4.22 ×107 m, find vQ.
vQ = __________________ _____
units
5. What is the difference between vQ, the satellite's speed in the transfer orbit at point Q, and the
speed the satellite will have in the circular orbit of radius RB = 4.22 ×107 orbit?


vQ = __________________ _____
units
Now, let us complicate things a bit more, and try a problem that involves time. The attached
modification to the orbit program may be of help in this problem. It allows you to specify a
“transfer obit” by choosing the starting distance from Earth, R0, and the final distance, Rf. The
program then calculates the speed needed at R0 to get into that orbit, vt, and the period of the
transfer orbit, T,
Suppose the space shuttle is in an orbit of radius RA, and wishes to rendezvous with a
communication satellite in an orbit of radius RB. Suppose the shuttle fires its engines when at
point P, and goes into the transfer orbit. The shuttle is to arrive at point Q at the same time as the
communication satellite. Since the satellite is in a higher orbit, its speed is lower, and the shuttle
must allow it a “head start”. The picture illustrates the situation.
6. The satellite is at point S when the shuttle fires its engines at point P. It is ahead of the shuttle
by an angle . Find the angle , which will result in the satellite and the shuttle arriving at Q at
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the same time. As before, RA = 7.00 10 m and RB = 4.22 ×10 m.
 = _____________°