Determining Orbital Radius and Capture
Date: Fall 2012
What determines a moon's distance from a planet? I understand that this would vary depending on how the moon got there in the first place, but the most commonly hypothesized ways (like a chunk blown off of the planet or an asteroid caught in the planet's gravitational pull) still require the moon to have a certain trajectory when it first becomes trapped in the planet's gravity well. So if this is the case, what stops the moon from eventually crashing into the planet or eventually flying off into space?
Q: What determines the satellite’s distance from the planet?
A: The altitude (distance) of the satellite from the planet is determined by its linear speed, the gravitational force (pull) of the planet, and the radius (or diameter) of the planet
The linear speed of the satellite that allows the satellite to keep “falling” (being pulled down by gravity) over the side of the planet will be the satellite’s orbital altitude above the planet…
The linear speed (in contrast to its circular speed) of the satellite, the gravitational force (pull) of the planet, and the radius (or diameter) of the planet determines if it will fall into the planet, orbit the planet, or fly off into space.
The satellite is constantly falling towards the planet in a (nearly) constant gravitational field.
If the satellite is not going fast enough (linear speed) to fall past the corner of the planet, it will fall into the planet.
If the satellite is at a linear speed that is equal to its fall towards its planet, it will orbit the planet.
If the satellite’s linear speed is going faster than its fall towards the planet, it will fly into an ever higher orbit maybe even beyond the gravitational field of the planet and off into space.
The determination of that linear speed parameter depends on the altitude of the satellite above the planet, the radius of the planet, and the gravitational force (pull) on the satellite.
It is quite simple geometry.
If we want to increase the altitude of a space shuttle’s orbit, we simply increase its linear speed.
If we want to decrease the altitude of a space shuttle’s orbit, we simply decrease its linear speed.
Unfortunately, this is a question with a very complex answer. Data from the lunar laser ranging experiment (the time of flight of laser light onto reflectors placed on the Moon during the Apollo 11 mission is used to calculate the distance of the Moon to the Earth) sets the Moon as receding from the Earth at about 3.8 centimeters per year. Thus any theory on the Earth-Moon history has to account for this (and other) data. However, using these data to extrapolate what the Earth-Moon system was like or what it will be has to take into account other factors that can perturb or change the movement of the two bodies. For example, the Moon has the effect of causing tides on Earth, but those same tides also affect the Moon (this is why the Moon has come to continually face the Earth), the continued gravitational pull will certainly slow down the recession of the Moon, the transfer of momentum between the two bodies could stabilize the two so that they eventually settle into stable orbits that have both the Earth and Moon always presenting the same face to each other, etc. The bottom line is that we know that there are factors that can cause a moon to eventually crash into its planet, but we also know of factors that can cause a moon to eventually separate from its planet - it all depends on the balance of these factors.
Greg (Roberto Gregorius)
The masses, radius and angular velocity largely determine that. It is the same as for a planet. Bear in mind that the math may be a bit more rigorous for elliptical orbits.
Let us use our Moon, assumed as a perfect circular orbit as an example. To make it clear: the Moon IS constantly falling into the Earth, but the trajectory is such that it never reaches Earth. That is because of the angular momentum(L) and the radius(r) from Earth.
The angular momentum does not change because there is no torque applied to the Moon. The mass(m), tangential velocity(v) are also constant, so the radius does not change. r = L/mv
Now, things could get real interesting if the Moon collides with something so that a force is applied to or a mass shift occurs to the Moon.
Happy orbiting! Peter E. Hughes, Ph.D, Milford, NH
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Update: December 2011