The goal of this project is to investigate two problems in celestial mechanics for which a computational solution is quite valuable: the precession of the perihelion of Mercury and Kirkwood gaps, both found in Chapter 4 of Computational Physics by Giordano and Nakanishi. In the case of Mercury, the precession can be calculated analytically thanks to general relativity, but a computational solution is easier to obtain. Giordano and Nakanishi describe the process of obtaining the rate of precession, keeping in mind that this variable must be adjusted for practical computation times (that is, not 230,000 years), using the force law predicted by general relativity:
F_G ≈ (G M_S M_M)/r^2 (1+a/r^2 ),
Where M_M≡mass of Mercury, M_S≡mass of Sun ,a ≈1.1 ×〖10〗^(-8) 〖AU〗^2.
Part of the project will be to compare different computational methods for analyzing this procession, particularly the Euler-Cromer method and Runge-Kutta (which uses the Euler-Cromer method to update its values). Using a two-body system simulation, the value for rate of procession can be found by varying until it matches expected, observed rates.

The second computational problem covered is one for which an analytical solution is difficult, as the system is considered chaotic. In our solar system there are certain “gaps” in orbital radii from the Sun at which there are far fewer asteroids. Numerically it can be found that these gaps occur at certain resonances with Jupiter’s orbit, such as 2/1, 5/2, 7/3, etc. Running a simulation to find Kirkwood gaps and investigate asteroids’ paths near these gaps first involves construction of a multi-body model, at least a two-body system for the Jupiter and Sun, and for more accuracy (if time permits) a three-body system along with the Earth. Luckily the magnitude comparison between Jupiter and an asteroid makes their mass insignificant, meaning their gravitational force can be excluded from the simulation without much effect on accuracy of the program. An analysis of the asteroid’s orbits will also naturally include a discussion of chaos, described in Chapter 3 of Giordano and Nakanishi.

Timeline:
Week 1: Continue background research on physics and computational methods of analysis, write code for precession of perihelion of Mercury, begin (and hopefully finish) writing code for 2- or 3-body system in Kirkwood problem
Week 2: Adjust solar system model to include simulations of asteroids, their paths/orbits, finish all code, begin gathering data.
Week 3: Finish gathering data, finish all research, comparison to actual observed/computer values from literature sources.
Week 4: Finish data analysis, write-up for project, prepare presentation.

Sources:
Computational Physics, Giordano and Nakanishi, Chapters 3 & 4.
Mercury’s Perihelion from Le Verrier to Einstein, N.T. Roseveare.
Chaotic Dynamics in the Solar System, J. Wisdom.