**Team:** 89

**School:** SANTA FE HIGH

**Area of Science:** Astrophysics

**Interim:** Team Number: 89

School Name: Santa Fe High

Area of Science: Astrophysics

Project Title: Asteroid Dynamics

Team Members: Luwen Huang

Project Mentor: Charles McClenahan, Ph.D.

Sponsoring Teacher: Anita Gerlach

Problem Definition:

The n-Body problem has, for decades, evoked many inherent difficulties when physicists tried to solve it for a system. With the development of fast computers however, the exact, conservative, solutions of an n-Body problem can now be computed given a set of conditions.

The n-Body problem will be applied to a group of asteroids, whose orbits will be affected by each other, Jupiter, Mars, Earth, and the sun. Occasionally, an asteroid will deviate from its normal orbit. What will be its trajectory, and most importantly, will it collide with the Earth?

Solution Plan:

The plan is to start with a 2-D, 2 body system, that of the sun and the asteroid, to safeguard against possible errors. From there, more bodies will continually be added along with a 3rd dimension, if time permits, as the program is checked for any possible errors with each addition.

We will employ a conservative integration algorithm. The system will first be described using equations in Newtonian mechanics set in polar coordinates. The Hamiltonian will then be derived for the system, and then it will be transformed into a new space (T(x)), integrated in that new space, and transformed back. This process will be iterated over time steps. After each iteration, the position and momenta of any single asteroid at any single time step can be derived.

We will program this in C++ and its libraries. A single class will be created to include the properties and actions of the asteroids (the objects).

Progress to-Date:

We have done extensive research in n-Body mathematics. Though many algorithms exist for the n-Body problem, the conservative integration scheme is the only one to exactly conserve energy. Even symplectic integration schemes lead to large errors due to small but accumulated losses with each iteration. Therefore, we have decided to use the conservative algorithm, combined with our own additions, as the plan for our code.

Expected Results:

The complete, finished program will predict any perturbed asteroid's trajectory, as well as the properties of any asteroid in 3-D space. It will run through each iteration relatively quickly without errors. The momenta of the asteroids will be exactly conserved to machine precision. It will be possible to display a graphics simulation of the system at each time step, and a continuous, dynamic graphics simulation.

References:

1. Kotovych, Oksana and John C. Bowman. "An Exactly Conservative Integrator for

the n-Body Problem." J.Phys. A:Math. Gen. (2001)

2. "Lagrangian." Wikipedia, The Free Encyclopedia. 14 Dec 2005, 15:23 UTC. 15 Dec 2005, 21:08 http://en.wikipedia.org/w/index.php?title=Lagrangian&oldid=3133303

3. "Hamiltonian mechanics." Wikipedia, The Free Encyclopedia. 5 Dec 2005, 02:49 UTC. 15 Dec 2005, 21:12 http://en.wikipedia.org/w/index.php?title=Hamiltonian_mechanics&old d=30180420

4. Graps, Amara. N-Body/ Particle Simulation Methods?? 20 Mar. 2000, 15 Dec. 2005 http://www.amara.com/papers/nbody.html

5. "N-body problem." Wikipedia, The Free Encyclopedia. 30 Nov 2005, 06:38 UTC. 15 Dec 2005, 21:21 http://en.wikipedia.org/w/index.php?title=N body_problem&oldid=29670843

**Team Members:**

Luwen Huang

**Sponsoring Teacher:** Anita Gerlach