PARAMETRIC REPRESENTATION OF KEPLERIAN ORBITS

Peter M. Lewis

govo@azstarnet.com

Proposal for summer research funding

Summer 2000

In cooperation with Professor Edward Alexander.

The principle goal of my project is to develop a parametric method of representation of Keplerian orbits principally for use in computer display applications. This representation will include effects of large bodies on an orbiting body (with possible extensions with multiple star systems).

Before now I have proven to my satisfaction that the basic bare-bones equations are possible to determine. Edward Alexander has discussed this with me, he knows of an existing solution, but my goal is to make the equations in such a fashion that they are easily integrated onto a computer plotting program, so that basic orbits are very easily manipulated. So far I have been able to create models that describe the forces acting on a body as a function of position, and will get that down to describing position. After I can prove that the basic engine works, it can be used for various applications in modeling multiple body problems. (Three-body systems, such as 2 large planets and one star, cannot be treated, but restricted three-body systems such as a double star with an orbiting planet will be relatively easy to model. The only 3-system situations are star-planet-moon systems where the effect of one of the three bodies on another of the three is negligible). The major incentive for modeling in this fashion is that most of the complications in traditional modeling arises from the machinery used to deal with more complicated situations.

My basic procedure will be as follows:

1. Start with a very simple set of equations describing motion of a single planet around a single star. This model will ignore gravity and will only deal with a 2-dimensional system (in the plane of the orbit).

2. I will then add to this model the effects of gravity onto the original equation, allowing user-entered variables to be entered for the masses of the objects (so the model can be adapted for all similar situations).

3. * I will integrate a third dimension into the equations to allow for a third body to be out of the plane of the orbit.

4. * I will then take the basic engine and in effect tack it onto the existing equation. The first deals with one body and the second deals with the second body (add extra stars simply onto the end of existing equations).

5. * I will develop another set of equations to deal with multiple stars interactions with each other in terms of position only and substitute each part of the equations (one part for each star) into the corresponding position variable of the equations for the orbit of a single planet.

6. I will enter the equations into a parametric equation plotting software that will allow click-of-the-button changes in all aspects of the orbit (i.e. number of stars*, mean radius of orbit, eccentricity, starting position, starting velocity, mass of star, mass of orbiting object).

I intend to spend many hours a day on the project while I am in town (I will be gone for 3-4 weeks during the summer in New York, Colorado, and Maryland). This will be my main focus and I will only divide my time between this project and the Student Satellite Project. I am very excited about this project, and I already have someone in the Planetary Sciences department interested in the possibility of finding interesting anomalies in plotting multiple star systems.

I do not know the depth in which I am expected to write, but I assume you will require a little personal information about me:

I am 17, and a sophomore double majoring in Physics and Math at the UofA, I enjoy the beauty of math greatly. During the Fall '00 term I will be taking MATH 254H and 323 along with PHYS 142H and 242H.