• Planets and Worlds, Part 3: Moons
The Hill Sphere
The Hill Sphere radius is a relative value dependent on the mass of the body in question and the mass of the nearest more massive body with which it has a gravitational interaction. For example, the Hill Sphere of a moon orbiting a planet is limited by the mass of the planet it orbits; the Hill Sphere of the planet is limited by the mass of the star it orbits; and, finally, the Hill Sphere of the star is limited by the nearest more massive object to it (usually another star or other massive object in its neighborhood.) Beyond an object's Hill Sphere (or Hill Radius), its gravity is overwhelmed by that of the neighboring more massive object. So, another planet with the same mass as Earth but whose star is two times as massive as the Sun would have a different Hill Sphere radius than Earth’s.
The equation for calculating the Hill Sphere radius of a body is:
Earth's Hill Sphere
If we multiply this result by the number of meters in an Astronomical Unit (1.496E+11), we get Earth's Hill Sphere radius in meters:
The distance to the Moon is about 60.336 Earth radii; if we divide Earth's Hill Sphere radius by this value, we see that 234.853 / 60.336 = 3.892, so the Moon orbits at just over ¼ of the Earth’s Hill Sphere radius. Put another way, the farthest distance an object can orbit the Earth is almost four times farther than the distance to the Moon; so, the Moon is in no danger of being stolen away from Earth by the Sun. (In fact, in one sense, the Moon, itself, can be said to be directly orbiting the Sun, but more on that in another post).
Does the mass of the Moon alter the calculation of the Earth’s Hill Sphere?
In a word: no.
As we've defined, the calculation of a body’s Hill Sphere is a function of the mass of the body and the next nearest more massive body. Since the Moon is less massive than the Earth, its only relationship to Earth’s Hill Sphere is whether or not it orbits within Earth’s Hill Sphere, which we’ve shown above that it does certainly do.
It is theoretically possible for the Moon to have its own moon, so long as any such body orbited within the limits of the Moon's own Hill Sphere radius. We can determine what this critical distance is by using the same equation as above; however, if we define both masses in terms of the mass of the primary, then:
The Roche Limit
Conversely, the Roche Limit tells us the closest orbit any object can have to the body it orbits. However, four points should be noted:
- The Hill Sphere is calculated for a body by comparing its mass and that of its gravitational primary;
- The Hill Sphere limit is true for all bodies orbiting the object in question. That is to say, any satellite of the body which moves beyond its Hill Sphere radius will escape the its gravity
- The Roche Limit, however, is calculated for satellites, held together only by their own gravity, orbiting the body in question;
- The Roche Limit is also a relative value, but is entirely dependent on the densities—not the masses—of the bodies involved; the Roche Limit will be different for any two bodies of different densities.
So, while the Hill Sphere for a body is "constant" (in the sense of applying equally to all less massive bodies orbiting it), the Roche Limit is dependent on the density of the orbiting body, and is different for every orbiting body (unless they have exactly the same density, of course).
Generally speaking, if the two bodies are of similar densities, the Roche Limit will be about 2.44 times the radius of the larger body.
The fundamental equation is:
Note, also, that the densities may be expressed in any units desired, so long as both densities are in the same units.
When all the values are expressed in terms of the planet's mass and density, the equation simplifies to:
For simplicity, let's express the densities in terms of Earth masses and the radius in terms of Earth radii:
Thus, combining the Hill Sphere and the Roche Limit calculations, we can say that the "legal" limits of the range of orbits the Moon could inhabit runs from ~2.88 Earth radii out to ~235 Earth radii.
How close could it safely approach?
Keeping the values for Earth the same, our inputs are:
Let's reverse the scenario: What if the Moon were orbiting Mars; what would its closest possible orbit be, then?
We'll set the values for the primary in terms of Mars:
The Moon’s Roche Limit for Earth is 2.8834 Earth radii, and its Roche Limit for Mars is 2.5757 Mars radii; if we subtract 1.0 from each value, we get the number of planetary radii above the surface the Moon is orbiting. Then, we multiply by each planet’s radius in meters to get a result in meters above the surface (called the object's altitude).
For Earth and the Moon: (2.8834 - 1.0) × (6.371E+06) = 1.8834 × (6.371E+06) = 1.19991E+07 meters, or 11,999.14 kilometers
For Mars and the Moon: (2.5757 - 1.0) × (3.39E+06) = 1.5757 × (3.39E+06) = 5.341E+06 meters, or 5,340.84 kilometers
So, the Moon in its closest possible orbit would have to orbit 2.247 times higher in altitude above Earth than above Mars in order to remain intact.
The Earth As A Moon
Jupiter has a larger radius than Earth, but a lower density—1.33 g/cm³ compared to 5.51 g/cm³, so: