Unit 6 - Some extensions

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As you are well aware we have had to deal with a VERY simplified version of the original problem. So far in our study we have not taken account of some of the following:

• The inclination of orbits.
• The whole issue of time, that is at what time can we expect the vehicle to be at certain points in an orbit, and given a certain time where is the vehicle?
• The fact that the orbits do not have to lie with their major axis along the horizontal axis.
• The whole idea that in the solar system we are really dealing with orbits within orbits, that is consider the moon - it is in an orbit around the Earth, but the Earth in turn is in an orbit around the sun, so what does the path of the moon look like relative to the Sun?
• Finally we have of course not taken any account at all of the various forces in the solar system that make the system less than the ideal one we have been working with.

Extensions

1. Orbits that do not have their major axes along the same horizontal (but still in the same plane)

This problem is resolved using an important principle in mathematics - the shifting of axes.

• First determine the general equation for ellipses whose focus is still at the origin, but whose major axis no longer lies along the horizontal axis (as in our previous discussions). Since we have been dealing with only Earth or Sun centered orbits so far this new equation would be sufficient.
• If we wanted to draw the orbit of a satellite on an Earth centered orbit but with respect to the Sun for various positions of the Earth we would then need to modify our equation developed above to cope with a focal point not at the origin. Determine the equation needed.

2. Ground tracks

If you have ever seen a picture of Mission Control (Houston) for the shuttles you will have noticed that on one of the walls there is a map of the world with the path of the shuttle (the ground track) superimposed on it. Furthermore you will have noticed that the shape of this path is "sinusoidal" - and yet in from our discussions you will be aware that the shuttle travels in a plane. This strange observation is caused by among other factors; the inclination of the orbit and the inclination of the Earth.

• Develop a detailed explanation for this observation. You may even wish to build a model to help with the explanation. A good place to start.
• You explanation should deal with the following:
  • the phenomenon of precession,
  • the fact that the amplitude of the the "sinusoidal" curves varies as a function of how the shuttle is launched,
  • you should discuss whether or not the curve is in fact sinusoidal - consider the Molniya orbits used by many Russian communications satellites.

3. The issue of time - this is a critical aspect not yet dealt with

This is a fascinating topic. There are really two aspects to this problem: 1) given a certain time can we predict where the object will be in its orbit, and 2) can we predict at what time objects will reach particular points in their orbits?

The solutions to these two questions rely on Kepler's second law and a clever trick that involves circles, what is particularly interesting is that the process is really one of approximation - certainly some very good approximations, but nevertheless approximations.

• Determine the basis on which the approximations are made,
• Write a computer program that will take the parameters of the orbit, and the time or place in which you are interested and then determine the best approximation. (If you can get hold of the book: Space Mathematics - A Resource for Secondary School Teachers, NASA, 1985 you will find some useful discussion on pages 151 - 153)

4. Rendezvous between objects

We began this whole lesson series with the setting of getting the Mars Pathfinder from Earth to Mars. If you have dealt with the previous problem you are now in a position to finally deal with the original problem.

• Determine the various orbital elements for Earth and Mars including their times of perihelion passage and some actual times at which they are at their respective perihelions.
• Design an actual Hohmann transfer to take an object from Earth to Mars, giving the date of launch, date of landing and the various parameters.
• The problem you are dealing with is still a simplification - that is you will not be taking into account the orbit shape determined by the parameters of the vehicle etc. It is enough, to begin with, that you simply state if on day x we launch the vehicle into orbit y then it will rendezvous with Mars on day z (provide sufficient evidence that Mars will be in the right place on day z)!

5. Orbits within orbits

This challenge is suited to both a geometrical investigation using GSP or by means of algebraic investigation using polar coordinates (in the latter case it would be a good idea to first do extension activity 1). The principle is simply this - as the Earth orbits the Sun, so the Earth's moon orbits the Earth - what path does the Earth's moon trace with respect to the Sun.

• Develop either a GSP sketch which animates the movement described above or determine the polar equations of the movement.
• You will need to get certain data on the bodies you choose: in addition to their orbital elements, you will need to know such data as their periods of revolution.
• In the event that you choose to use GSP you will probably need to treat the orbits as being circular.

References

Edwards, C. H. and Penney, D. E. , (1985). Calculus and Analytic Geometry. New Jersey, Prentice Hall.

Jet Propulsion Laboratories, (1995). Basics of Space Flight Learners' Workbook. California, JPL.

National Aeronautics and Space Administration, (1985). Space Mathematics - A Resource for Secondary School Teachers. Washington, NASA.

Resources

Internet links:

• Liftoff Academy
• Live from Mars
• NASA Core Instructional Materials
• NASA Spacelink Search
• The Space Educators' Handbook