Unit 6 - Some extensions
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
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