**Undergraduate #348**

**Discipline:**Physics

**Subcategory:**Astronomy and Astrophysics

**Session:**4

**Room:**Exhibit Hall A

**Alden Andrei Fernandez**

**- Kapiolani Community College**

**Co-Author(s):**Kiyomi Sanders, Kapiolani Community College, Honolulu, Hawaii.

Kepler’s first law of planetary motion states that orbiting masses travel in an elliptical path, with the object being orbited positioned at one of the ellipse’s foci. Seeing how Kepler derived this law by using measurements of Mars’ positions, this knowledge can also be used to predict the location of unexplored orbiting systems which would benefit space exploration to other planets and celestial bodies. The window to travel from different planets or bodies vary in time and frequency, thus it is integral to have precise knowledge on the value of the eccentricity of the elliptical path of a body that one plans to travel to. The purpose of this research is to obtain the eccentricity value of the Moon’s orbit and in turn verify its elliptical property by using measurements gathered by the Apache Point Observatory Lunar Laser Operation (APOLLO) regarding the Moon’s position relative to Earth. First, the differential equation containing the dynamical properties of the Moon orbiting Earth was derived using Newton’s second law. Then, using vector analysis, the conservation of the angular momentum and the total energy of the orbiting Moon was verified. By further using vector analysis techniques, the functional form of an ellipse was obtained as the solution of the differential equation originally derived. Measured in irregular intervals of time, the distances between the surface of the Earth and the Moon were obtained from the APOLLO experiment over the course of a year. The theoretical equation of the ellipse previously derived involves distances as a function of angular displacement. Hence, the angular displacements corresponding to each time of measured distances had to be obtained in order to fit the ellipse equation. To achieve this, the time elapsed since the first collected data point needed to be calculated in minutes. The period of the lunar cycle was then converted into minutes, then divided by 360° to obtain a unit conversion between angular displacements and times. The derived theoretical equation of the ellipse was then used to perform a non-linear regression in Cartesian coordinates on the reformatted experimental data from APOLLO. Gnuplot was utilized to perform the fit on a periodic function, and the eccentricity was obtained in this manner. The eccentricity value calculated using APOLLO’s data was 0.0545572 +/- 0.00014, aligning closely with the theoretical eccentricity value of the Moon’s orbit, thus verifying Kepler’s law. Future research looks to apply the same test for other orbiting bodies, such as Phobos, one of Mars’ moons. As human missions to Mars are being proposed, studying the position of Phobos may facilitate space exploration to the Red Planet. The Faulkes Telescope Project is being considered as a means of obtaining measurements of Mars and Phobos. Additionally, continuing research on the topic may lead to investigating the source of errors in calculating the eccentricity of a celestial body’s orbit.

**Funder Acknowledgement(s):** This research was supported and funded by the B2B grant, as part of LSAMP.

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

**Role:** I was tasked with deriving the functional form of an ellipse through the use of vector analysis. I then had to obtain the data from APOLLO and interpret it to provide angular displacement as to align with the previously-derived function of an ellipse. Finally, I had to take the data we collected and plot it using gnuplot to obtain the eccentricity value by performing a non-linear regression on the resulting function.