PHYS-1500 PHYSICAL MODELING
FALL 2006
10/27/2006, Class 17: Earth Orbits and Solar Orbits
NAME _________KEY___________________
Now that you have a working spreadsheet model of the orbits, here are a few more problems.
1. Suppose that two spacecraft are in the same orbit, of radius R = 8.37 10 m, (2000 km above
the surface of Earth), but one leads the other by 45°. The trailing spacecraft wishes to
rendezvous with the spacecraft up ahead. The trailing spacecraft must reduce its speed in
order to catch up.
a) To what value must the spacecraft’s speed be reduced to produce a rendezvous?
6570 m/s
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b) What will be the relative speed of the two spacecraft when they meet?
333 m/s
c) How long will it take to rendezvous? That is, how much time will pass from the moment
the trailing spacecraft achieves the speed calculated in (a) until the two meet?
6666 s = 111 min = 1.85 h
2. The “transfer orbit” model that you used to get from a low orbit to a high orbit can be used to
plan interplanetary trips, approximately. For example, from Earth to Jupiter, or Venus. For
these trips, you would need to change a few numbers in your spreadsheet. Everything orbits
the sun, so replace the mass of Earth with the mass of the Sun. Also the radii of the orbits
about the sun will be the radii of the planetary orbits. The numbers of interest are:
MS = 1.99 ×1030 kg
RE = 1.50 ×1011 m
RJ = 7.78 ×1011 m
RV = 1.08 ×1011 m
a) What speed would a spacecraft, in Earth’s orbit need to get to Jupiter? How long would it
take?
3.85 ×104 m/s
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8.62 ×10 s = 998 days = 2.73 years
b) What speed would a spacecraft, in Earth’s orbit need to get to Venus? How long would it
take?
2.72 ×104 m/s
1.26 ×107 s = 146 days
c) Why are these calculations not really accurate?
The gravity of the planets is ignored.
3. The orbit of Earth about the Sun is not a perfect circle. It is an ellipse with an eccentricity of
r r
0.0167. Eccentricity is defined as e  max min . In addition, we know that the period of
rmax  rmin
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Earth about the Sun is T = 3.16 ×10 s. Use this information, and your model, to calculate
Earth’s closest approach to the Sun, and its farthest distance.
Earth’s average distance from the Sun is rave = 1.50 ×1011 m, where rave is defined as
r r
rave  max min
2
If the equations for e and rave are solved for rmax and rmin we get,
rmax = rave (1 + e), and rmin = rave (1 – e).
rmax = (1.50 ×1011 m)(1 + 0.0167) = 1.525 ×1011 m
rmin = (1.50 ×1011 m)(1 – 0.0167) = 1.475 ×1011 m
4. Pluto is was once described as the farthest planet from the sun in our solar system. Now, it is
just another dwarf planet. However, when most of you were born, Pluto was closer to the sun
than Neptune. That is because Pluto has a very eccentric orbit, and Neptune has one of the
most circular (only Venus has a less eccentric orbit). Pluto’s average distance from the sun is
greater than Neptune’s, but it spends some time inside Neptune’s orbit. Use your computer
model, and the following data to calculate the requested properties of Pluto’s orbit. For
simplicity, consider Neptune’s orbit to be a perfect circle (its eccentricity is only 0.0086).
Period of Pluto’s orbit:
Pluto’s average distance from the sun:
248 years
5.9 ×1012 m
(1 year = 3.16 ×107 s)
Pluto’s eccentricity:
0.250
eccentricity, e 
Radius of Neptune’s orbit:
4.5 ×1012 m
rmax  rmin
rmax  rmin
A. Find rmax, the farthest Pluto gets from the sun.
rmax = (5.9 ×1012 m)(1 + 0.250)
rmax = __7.375 ×1012 m __
units
B. Find rmin, the closest Pluto gets to the sun.
rmin = (5.9 ×1012 m)(1 – 0.250)
rmin = __4.425 ×1012 m __
units
C. Find how much time Pluto spends inside Neptune’s orbit each period.
From the program, t = 6.18 ×108 s = 19.6 y
t = ____19.6 y _________
units
From U. Arizona web site: “Pluto's orbit is highly eccentric. At times it is closer to the Sun than
Neptune (as it was from January 1979 thru February 11 1999).”