**Undergraduate #352**

**Discipline:**Physics

**Subcategory:**Astronomy and Astrophysics

**Session:**4

**Room:**Exhibit Hall A

**Kiyomi Sanders**

**- University of Hawaii Manoa**

Kepler’s first law states that systems subjected to a central force orbit in an elliptical motion around one of the two foci. A fundamental quantity characterizing the uniqueness of the type of motion of orbiting satellites, planets, stars or galaxies is eccentricity. Being able to predict its value is essential in orbital dynamics as it dictates whether a system follows a circular, elliptical, parabolic or hyperbolic path.The purpose of this research is to verify the eccentricity of the Moon’s orbit around the earth using NASA?s APOLLO experiment. The Lagrangian method was first used to verify the law of conservation of angular momentum and then to derive the differential equation that describes the dynamical properties of the moon. To solve the differential equation, a clever change of variable was employed, which led the original equation to be rewritten as a well-known differential equation with constant coefficients describing a simple harmonic oscillator (SHO). Changing the variable back again from the solution of the SHO led to the equation of an ellipse. NASA?s APOLLO experiment provided measurements at irregular time intervals of the moon’s distance from earth. In order to use the equation of the ellipse and fit it with the experimental distances, leading in turn to the value of the eccentricity, the data needed to be reformatted to angle vs. distance by identifying the corresponding angle for the time of each data point. Additionally, the experiment did not specify when the vertex of the ellipse occurred in the data, so an additional alignment was required. Using distances measured in the APOLLO experiment, gnuplot graphing utility was used to plot the data with polar coordinates and apply a nonlinear regression to the data using the derived equation of the ellipse to obtain the eccentricity of the moon around the earth. This value was found to be 0.049 +/- 0.021. Being able to obtain the eccentricity of rotating objects is fundamental to astrophysics modeling since along with measurement of periods and semi-major axes, it allows precise estimation of the mass of observed objects in the universe. Now that the kinematics of celestial objects has been quantified and verified, the next step in the research is to obtain the masses of observed celestial objects using Kepler’s Third Law, the orbital period, and the observed semi-minor axes.

**Funder Acknowledgement(s):** I would like to thank the Tribal Colleges and Universities Program (TCUP) and the Hawai'i Pre-Engineering Education Collaborative Phase II (PEEC II).

**Faculty Advisor: **Herve Collin,
herve@hawaii.edu

**Role:** I performed the calculations that allowed us to verify the law of conservation of angular momentum and the differential equation, and solve this equation for the equation of an ellipse. I reformatted the experimental data on Microsoft Excel by converting the time to the corresponding angle measurement for the first thirty data points. I learned how to use the gnuplot graphing utility to plot both the experimental and theoretical data in polar.