Zachary was born and raised in Hilo, Hawaii. He graduated from Kamehameha Schools Hawaii and is a senior mathematics major at the University of Hawaii at Hilo. After graduating with his B.A., Zach plans to apply to graduate school and is interested in pursuing a Ph.D. in applied statistics.

**Home Island:**Â Hilo, Hawaii

**High School:** Kamehameha Schools Hawaii

**Institute when accepted**: University of Hawaii at Hilo

*Definite-Integral Root-Finding Algorithm to Solve Kepler’s Equation*

*ProjectÂ Site: Air Force Research Laboratory (AFRL)*

*Mentor: Paul W.Â Schumacher Jr., Ph.D.*

**Project Abstract:**

One mission of the United States Air Force is to conduct space surveillance of everything in orbit around the Earth. Predicting the position of a satellite on its orbit is essential to the operation of tracking every satellite. Keplerâ€™s equation is the key analytical relation for predicting the position of a satellite because it relates the angular position of a satellite traveling on its orbital path with time. Keplerâ€™s equation gives a great deal of analytical insight for the root-finding problem and serves as a foundation for many orbital mechanics problems. When the value of time is given, Keplerâ€™s equation must be solved for the value of the angle variable with a numerical root-finding algorithm because the equation is transcendental. The purpose of this project is to investigate a particular and rarely used root-finding method in orbital mechanics involving a definite integral to solve for the angular variable in Keplerâ€™s equation. The definite integral was selected because it is one of the few root-finding methods that can be parallelized. MATLAB was used to formulate the definite integral as a serial computational algorithm to verify the validity of the method. We compared our results with test cases constructed from known particular solutions of Keplerâ€™s equation. Results will be presented. In the future, the definite integral method should continue to be investigated to understand its performance in a parallel algorithm.