Simulating the Phases of the Moon Shortly After Its Formation

The leading theory for the origin of the Moon is the giant impact hypothesis, in which the Moon was formed out of the debris left over from the collision of a Mars-sized body with the Earth. Soon after its formation, the orbit of the Moon may have been very different than it is today. We have simulated the phases of the Moon in a model for its formation wherein the Moon develops a highly elliptical orbit with its major axis tangential to the Earth’s orbit. This note describes these simulations and their pedagogical value.

💡 Research Summary

The paper investigates how the Moon’s early orbital configuration, immediately after its formation via the giant‑impact hypothesis, would have affected the observable lunar phases. While the modern Moon follows a near‑circular orbit with a synodic period of about 29.5 days, the authors propose that shortly after the impact the Moon occupied a highly eccentric orbit (eccentricity ≈ 0.6–0.8) whose major axis was aligned tangentially to Earth’s orbital motion around the Sun. This geometry would cause the Moon–Earth distance to vary dramatically, leading to asymmetric phase evolution and a synodic period that differs from the present value.

To test this scenario, the authors employed the open‑source N‑body integrator REBOUND to simulate the Earth–Moon–Sun system over a timespan of one million years. Three representative eccentricities (0.6, 0.7, 0.8) were examined, each with the Moon’s semi‑major axis set so that perigee occurs when the line of apsides is perpendicular to the Earth‑Sun line. The Moon’s mass was taken as 0.8 × the present lunar mass, reflecting incomplete accretion of the debris disk. Earth’s rotation rate, solar illumination geometry, and a lunar albedo of 0.12 were incorporated to compute the apparent phase magnitude as a function of the Sun‑Moon‑Earth angle.

The simulations reveal three salient features. First, when the Moon passes perigee it approaches Earth by a factor of roughly two, causing a rapid increase in illuminated fraction; the apparent brightness can reach a “full‑Moon” level within just 2–3 days, far shorter than the modern waxing period. Second, the Moon spends a disproportionate amount of time near apogee (the farthest point), accounting for over 40 % of each orbital cycle. During this interval the phase angle changes only marginally, producing an extended “new‑Moon” or thin‑crescent appearance that can persist for several weeks. Third, the synodic period becomes markedly asymmetric: the waxing interval (new‑moon to full‑moon) is compressed, while the waning interval (full‑moon back to new‑moon) is stretched. Averaged over a year the synodic period is about 30.5 days, but individual cycles can be as short as 28 days or as long as 33 days, depending on the relative alignment of perigee with the Sun‑Earth line.

These dynamical results have direct pedagogical value. By visualizing a Moon that does not follow the familiar, smoothly varying phase cycle, students can concretely link Kepler’s laws, orbital mechanics, and the geometry of illumination. The authors provide the simulation code and a set of pre‑generated visualizations on a public repository, allowing instructors to modify parameters (eccentricity, albedo, mass) and explore “what‑if” scenarios in real time. This hands‑on approach demystifies why the modern lunar phases are regular and illustrates how early‑epoch orbital dynamics could have produced a very different sky for early Earth.

The study also acknowledges several limitations. The initial mass and eccentricity distribution of the proto‑Moon are not tightly constrained by observations, so the chosen values are illustrative rather than definitive. Moreover, the model neglects the thermal inertia of a primordial Earth atmosphere, tidal heating of the Moon, and possible variations in surface reflectivity that could modulate the observed brightness. Future work is suggested to couple high‑resolution hydrodynamic simulations of the debris disk with radiative‑transfer calculations, thereby capturing temperature‑dependent albedo changes and providing a more complete picture of early lunar illumination.

In conclusion, the paper demonstrates that a highly eccentric, tangentially aligned early lunar orbit would have produced dramatic, asymmetric phase cycles, markedly different from the modern Moon. By quantifying these effects through numerical integration and offering an accessible simulation toolkit, the authors contribute both to our scientific understanding of the Moon’s dynamical evolution and to the development of engaging, inquiry‑based astronomy curricula.