The large rockets and extensive technology associated with putting a satellite into orbit introduces a brute- force element, leaving the impression that satellites and their movements can be arranged entirely to suit our convenience. This is not so. After the rocket power has been shut off, the motion of a satellite is determined by the effects of the earth's gravitational field, and this places severe restrictions on what is possible. The outcome is exactly the same as the effect of the sun's gravitational field on the orbits of the planets, and in this way the large number of satellites now orbiting the earth is quite similar to a small solar system. This assignment deals with the similarity between the two systems, and the inherent natural laws which they must obey.
Johannes Kepler derived, early in the 17th century, the geometry of the planetary orbits. Of particular interest here is the Harmonic Law, which may be written in algebraic form as :
P
2 k a
3 where P is the period of revolution of the planet, a is the semi--major axis of the planet's orbit about the sun, and k is a constant for all the planets set by the sun's gravity. In the case of the solar system, k

1 when the period of the sun orbiting satellite is stated in year s and the semi-major axis is measured in astronomical units . The same algebraic form applies to the orbits of satellites around the earth, but in that case the constant k will have a different value because the source of gravity is the earth rather than the sun. Simply substitute ‘satellite' for ‘planet' and ‘centre of the earth' for ‘sun'. (The actual value of the constant k is irrelevant for this lab exercise but for those interested, it can be determined simply by calculating the slope of the line in Graph 1.)
The relation between satellite period and distance from the centre of the earth is shown graphically in
Graph 1. Recall that the average radius of the earth is 6371km.
The line on Graph 1 illustrates the behavior of the system of satellites around the earth. Each point on that line represents a possible satellite either natural (like the moon) or artificial (like a space shuttle). Use the graph to answer the following questions. When explaining any of your answers, be short and concise. Three or four sentences should suffice for an answer.
In the case of an object orbiting one that is much larger, we can also use
3
2
To give us the mass (M) of the larger object.

1000
100
10
1
1 10 100 1000
Distance from the Earth's Centre (x1000km)
Questions
1.
A satellite was placed in an orbit somewhat above the earth's atmosphere, at a distance of 15,000 km from the centre of the earth. Determine the orbital period (in hours) of the satellite.
2.
If the shuttle orbits about 300 km from the surface of the Earth and Earth’s radius is 6500 km, how long would it take for the shuttle to go around Earth? (remember the orbital distance is measured from the centre of Earth)
3.
The moon orbits the earth every 27.32 days (sidereal period). Based upon this information and using Graph 1, what is the distance to the moon? If the distance to the moon was only half of the distance you have just determined, what would be the corresponding period? (Obviously it is NOT a linear relation between distance and period - not surprising given Kepler's third law.)
4.
The ideal setup for a communications satellite is to have it in a geo-stationary orbit, that is, to have its orbital motion synchronized with the rotation of the Earth, so that it remains above the same location on the Earth continuously. What would be the orbital period of such a satellite?
What would be its orbital radius (semi-major axis)?.
5.
Using Kepler's third law, determine the distance from the sun of an object that has a sidereal period of 12 years.
6.
If the distance of this object from the sun had been 19 AU, what is it’s period?

7.
For the moons of Jupiter, we can use their orbital periods and distances from Jupiter to calculate the mass of Jupiter in terms of Earth’s mass. The periods of the moons must be in years and the distance to Jupiter in A.U. (1 A.U. = 150,000,000 km) Fill in the numbers you convert on the table on the answer page. Do the calculation for each moon, but you only have to show one example of what you did. Is the mass from one of the moons not the same as the others?

Name: Student Number: .
Answer Sheet - Assignment # 2
5. Distance of object?
6. Period of object ?

Name: Student Number: .
Answer Sheet - Assignment # 2
7.
Moon
Io
Europa
Period
(in days)
1.8
3.6
Ganymede 7.3
Callisto 14.5
Distance
(in km)
421,600
670,900
Period
(in Years)
1,070,000
1,883,000
Distance
(in A.U. )
Jupiter mass (in Earth masses)