Astronomy 110 PRINT Name   __________________________
Fall 2005   Section 006

Homework 4 : Orbits

(Due Thursday, Oct 6, 2005)

This homework asks you to compute some of the orbital properties of artificial satellites of the Earth by comparing with the Earth's natural satellite --- the Moon --- using Kepler's third law. Although Kepler formulated this law for the planets orbiting the Sun, it is also valid for any orbit situation where gravity is the force keeping the satellites in orbit. Comparing an unknown situation with a known one (in this case, the orbit of the Moon) is a powerful technique that makes the algebra simpler. However, you will have to be careful about units, e.g. all orbital radii in kilometers, say, and all orbital periods in days. It doesn't matter which units you choose, as long as they are consistent among the different orbits.

Start by writing down the parameters of the Moon's orbit --- its average size and its period. These are your known quantities. You can find these in the textbook (try the appendices). The quantities you want are the distance of the Moon from the center of its orbit (the center of the Earth) and the period of its orbit.

Write these in here :   (Remember the UNITS!)

Size of orbit  (dmoon)   :   ___________________     Period   (Pmoon)   :   ___________________

Is this the sidereal period or the synodic period? _______________


Low Earth Orbit (Space Shuttle)     The first thing you'll calculate is the period of a satellite, such as the Space Shuttle, that (usually for reasons of economy) is in an orbit quite close to the surface of the Earth. Assume the average height of the satellite is 300 km above the surface of the Earth. Remember that it's distance from the center of the Earth that's important, however, so you'll need to add the 300 km to the radius of the Earth to get the size of the satellite orbit. This is also one of the known (or assumed) quantities.

Write this in here :   (Remember to use the SAME UNITS as for the Moon!)

Size   (dshuttle) = 300 km + radius of Earth   :   ___________________     Period   (Pshuttle)   :   unknown

To solve for Pshuttle,the method is to write down Kepler's third law, as an equality, twice, once for the Moon and once for the shuttle :

P2moon = K x d3moon        and          P2shuttle = K x d3shuttle


The trick is to divide one equation by the other so that you end up with one equation for what you want to know (the period of the space shuttle's orbit) in terms of the other three known quantities. Do that here:

Notice that the constant K goes away so you don't need to worry about its value.

Use this equation to solve for the value of Pshuttle . You will need a calulator that calculates square roots. Remember the UNITS.

What is the answer in minutes? ________________________________

Notice that this is less than a day. A satellite in an orbit with a period of one day is called a geostationary satellite. Such a satellite stays over the same spot on Earth all the time.

Do you expect that a geostationary satellite will orbit closer to the Earth than the low-Earth satellite you just calculated, or further away? _______________


Will it be closer than the orbit of the Moon? _______________


Consider a satellite in a very low orbit, one just grazing the surface of the Earth (you'd have to clear away mountains and tall buildings). Explain why such a satellite doesn't orbit once in 24 hours, even though the ground just underneath it rotates once round in 24 hours.


Can you use your equation to calculate the properties of the orbit of, say, Mars? that is, can you directly compare in this way the orbit of the Moon and the orbit of Mars?