Purpose and Goals

One purpose of this course is to explore the use of computational methods to find solutions to otherwise difficult analytical problems. The 3-body system is a prominent example. It is relatively straightforward to solve the equations of motion of a 2-body problem analytically, but far more difficult to do so for 3 bodies. The introduction of computational methods have allowed for fairly straightforward ways of analyzing 3-body motion.

A prominent example is that of the Sun, the Earth, and Jupiter, bound by the force of gravity. Since Jupiter is about 0.1% of the solar mass, it is the next biggest influence on Earth’s orbit in the Solar system. Using computational methods to analyze Jupiter’s effect would be far more efficient. We seek to create a model of the orbits of the Earth and Jupiter bound to the sun–and interacting with each other–via the inverse square law of gravity.

For this project we plan to make use of sections 4.1 and 4.4 in Giordano’s book. 4.1 outlines a method for calculating the motion of a planet orbiting the sun, whereas 4.4 opens the model to include Jupiter. The program will generate position data for each planet given their acceleration due to gravity, using the Euler-Cromer Method to approximate each new position based on this acceleration. We will start with a simplified model, with the Sun fixed to the origin, then graduate to a true 3-body model, where the origin would be fixed at the system’s center of mass.

Physics

We start with Newton’s Law of Gravitation, where Ms is the solar mass, Mp is the planet Mass, Fp is the gravitational force experience by the planet due to the Sun, and G is the gravitational constant:

Fp=G*Ms*Mp/r^2

Solving for acceleration (given Newton’s 2nd Law: Fp=Mp*a), or d^2x/dt^2:

a=G*Ms/r^2

This solution for acceleration is a second order differential equation, appropriate for use with the Euler Cromer Method.

Foundational Code (Routines provided in Giordano)

Planets: A routine to model the orbit of a planet around the sun as described by the inverse square law of gravitational force, with the sun at the origin. The Euler-Cromer method is used to approximate position. The Euler Method by itself would not be sufficient, as with all oscillatory motion it would introduce energy into the system and the planet would move away from the Sun. The Euler-Cromer Method guarantees a conservation of energy over one orbit.

Jupiter-Earth: Expands the above system to model Jupiter and Earth orbiting the sun. This method will also use the Euler-Cromer method for the same reasons. The suns mass will affect the orbits of the other two planets but the sun itself will sit at the origin of the system and not move.

True 3-body: This model is similar to the Jupiter-Earth model but instead uses the center of mass of the system as the origin. This model will be used when we are altering the mass of jupiter to be close to that of the sun, therefore causing changing orbits for each planet in the system. In this case, we want the sun to be able to move to better describe the full motion.

Initial Resources

http://astro.geo.tu-dresden.de/~klioner/celmech.pdf

http://arxiv.org/pdf/astro-ph/0411043v1.pdf

http://www.iumj.indiana.edu/docs/27030/27030.asp

Computational Physics, Giordano and Nakanishi. Sections 4.1 and 4.4.

Timeline

Week 1 (4/6-4/12)

During the first week we hope to work on organizing sources and solidifying understanding of background materials and physics. We will also use this week to read the Giordano text to begin understanding the setup of the MATLAB routines. We will begin writing the code for the Planets routine. This routine does not directly inform the 3-body problem but is the foundational code for the project. For the entire first week we will split the workload together since this work constitutes the base understanding of our project.

Week 2 (4/13-4/19)

Since the Planets routine should be well defined and operational by this week, we will use this time to finish off the other routines. This should be manageable since both the Jupiter-Earth and True 3-body routines are derivations of the Planets routine. Nadav will work on the Jupiter-Earth model while Matteo tackles the True 3-Body routine. This week will mostly be coding based, as we hope have all models complete by this time.

Week 3 (4/20-4/26)

By now, all routines should be complete. We will use this week to explore the implications of our models. This exploration will consist of multiple demonstrations of the effects of the changed mass of Jupiter on the system and will also include qualitative conclusions. We will also begin to work on writing up our results, with each of us taking charge of specific simulations.

Week 4 (4/27-5/3)

This final week will consist of finishing up our written results and finalizing the report. We will also leave space here for potential further exploration of our system, given appropriate time. Possible extensions include adding in another body to the system or analyzing in more detail the stability of orbits (Section 4.2, Giordano).