What if Earth had two moons?
The Earth-Moon system is a very stable system. After all, it is theorized that our natural satellite was formed when a Mars-sized body collided with the forming Earth about 4.5 billion years ago and spewed a lunar amount of molten rock into orbit [1]. Without the Moon, Earth may not have the axial tilt or rotation that it does today, which both act to regulate what we consider the climate [2]. Another relatively stable planetary-lunar system is Mars and its two moons, Phobos and Deimos. Phobos has a mass about 7 times that of Deimos, and orbits Mars at a height nearly equivalent to Mars's diameter (lowest lunar orbit in the solar system, see figure below), while Deimos orbits about 3 times farther [3]. Mars's lunar system is a fascinating display of the orbital mechanics between planets and large satellites, but what are other forms of a stable dual-lunar system? More specifically, how stable would the Earth-Moon system behave if the Moon suddenly captured one of its own? and what kinds of orbit patterns might be most stable?
Relevant physics
The relevant physics of this system includes all of the interactions between the Earth, Moon, and Metamoon. In this system, the Moon and Metamoon move with respect to Earth. The moons will have positions, and as time evolves, they will change positions and velocities. The dominant force driving the orbital mechanics and change in velocities here is Newton's law of universal gravitation.
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I have chosen a system where the Earth is stationary with respect to the Moon and Metamoon. This implies that the Earth will not have any forces acting on it, and there will only be forces that it acts on the moons, and the forces that the moons act on each other. This means there are 4 gravitational interactions contained in gravitational forces.
Computationally modeling the orbits
The method of computation here is to create a function that takes a state and time range, and returns the change in state with respect to time, then apply a solver that determines how the state unfolds. The state at a given time consists of the positions of each moon, their velocities, and their accelerations (note that Newton's second law of motion can be simplified to define a force on a mass by its mass times acceleration, F = ma & a = F/m). Specifically, this function encapsulates 4 first-order ordinary differential equations for each moon; the change in both x and y positions over time, and the change in both x and y velocities over time. It's important to note that the values assigned to the gravitational constant G, Earth mass M, and both Moon masses are significantly altered so that the computations are easier to manage by my computer. While not cohesive with the fictitious gravitational constant, the mass values are set with proper ratios in mind. The mass of the Earth is set at a value of 8, which following the Earth-to-Moon mass ratio, the Moon is 1/6 the Earth's mass, and the Metamoon mass is 1/6 that of the Moon's.
My function's components
Within the function, which takes state and time arguments, I've created columns with the x and y positions of each moon and the x and y velocities of each moon that will be filled the system evolves. Once the position and velocity columns are defined, the force equations from above need to be defined. These force equations have a dependence on direction and magnitude of distance between the relevant masses. The function returns the change in state for a change in time. In this system's case, the position of each mass is changing with respect to time (velocity), and the velocity is changing with respect to time (acceleration). The function must then return the velocities at each time, and the acceleration at each time. The velocities are captured by the columns defined as vs, and as mentioned, the acceleration is defined by Newton's second law of motion. The moons' accelerations at each time will be the sum of forces on each moon divided by the respective moon's mass. The Pandas concatenate function is utilized so that the returned array is 1 dimensional and compatible with the solver.
Results and summary
After manipulating the initial positions and velocities of each moon, I've found 4 different types of orbits that are stable for a relatively large time scale. The two simplest stable systems exist when one of the moons orbit close and the other orbits far similar to Mars with the close orbit of Phobos and more distant orbit of Deimos. The second type of stable system, which I call a "coupled" system, is one where the Metamoon orbits the Moon as the Moon follows a similar path to its present-day orbit. The system I've labeled "drop off & pick up" has the behavior of the coupled system but the Meta moon drops to a lower orbit while the Moon makes a larger orbit till it makes a revolution and picks up the Metamoon again. When the moons start on opposite sides of the Earth, I've discovered two types of orbits; one where the Metamoon looks as though it is chasing the moon but comes short of getting grabbed, and one where each moon orbits the Earth as if the other moon wasn't in the system at a position directly across the Earth from each other in syzygy, orbiting with a phase-lag of pi.
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Reflection
This is the first computational modeling project that solves a system of ordinary differential equations. I was able to gain insight into a thought provoking world of possibilities. My solution process here can be applied to many other systems whose behavior is described by ordinary differential equations.
References
[1] solarsystem.nasa.gov/moons/earths-moon/in-depth/
[2] solarsystem.nasa.gov/moons/earths-moon/overview/
[3] solarsystem.nasa.gov/moons/mars-moons/in-depth/
[4] www.jpl.nasa.gov/news/traffic-around-mars-gets-busy
[5] Carlson, Brant. “ODE solution examples” Introduction to Computational Physics, 2022, Carthage College, Kenosha, WI.
[2] solarsystem.nasa.gov/moons/earths-moon/overview/
[3] solarsystem.nasa.gov/moons/mars-moons/in-depth/
[4] www.jpl.nasa.gov/news/traffic-around-mars-gets-busy
[5] Carlson, Brant. “ODE solution examples” Introduction to Computational Physics, 2022, Carthage College, Kenosha, WI.
My key takeaways from this project
- This computational physics project belongs in my portfolio because it displays my ability to break down a system into its fundamental parts, encode the physical interactions into a function, apply a solver to simulate the system, and make visualizations to answer a question.