World Builders™ World Builders™ Session One   --  Astronomy Session One   --  Astronomy The Roche Limit           The Roche Limit

If your moons are too close to your planet, they will disintegrate!  Don't let this happen!

How close to your planet should you put your moons? If you put them too close to your planet, the gravity of your planet will break them up. Then the moon or moons will become a cluster of individual stones of various sizes. Over time the stones may spread out to become a ring around the planet.

The Roche Limit gives us a formula that tells us how close the satellite will be when gravitational forces start to pull it apart. You can have rings inside this limit, but not moons.

***  Be sure to locate your moons outside the Roche Limit. ***

If the moon and the planet have the same densities, the limit is at 2.423 of the planet's radius, measuring from the center of the planet to the center of the moon. ( if you don't want to do the math, put your moon out at least two and a half times the planet's radius from center to center of the planet and moon.)

If the densities are not the same, this number changes. A moon that is denser than the planet can be closer to the planet: a moon that is less dense than the planet needs to be farther away.

Here is a formula for calculating the limit when the planet and the moon have different densities:

Here are some examples of how to use this formula:

Example One:   (This uses the real numbers for our own earth and moon)

 Given Values: The Roche Limit = 2.423 Earth's radius = 1 Earth's density = 5.5 The moon's density = 3.34

Simplify this result to 2.9 earth radii.

Notice that our moon, which is less dense than the earth, needs to orbit at least 2.9 earth radii (center to center) from the earth. Actually, the moon is about 60.27 radii from the earth right now. At one time it was much closer to the earth. and one day it will be even further away.

Example Two:   (Using our earth and an imaginary (and very unlikely) moon

 Given Values: The Roche Limit = 2.423 Earth's radius = 1 Earth's density = 5.5 Density of imaginary moon made of water = 1.0

Simplify this result to 4.4 earth radii.

A low density moon needs to be further away from the planet. Compare this with Example 1.

Example Three:   (Imagine that the earth and the moon have the same
density)

 Given Values: The Roche Limit = 2.423 Earth's radius = 1 Earth's density = 5.5 Density of imaginary moon the same density as the earth = 5.5

Note that this comes out to the constant 2.423 because the density of the planet and the moon are the same.

Example Four:   (The earth has a moon made of iron (Not possible,

but let's try it!)

 Given Values: The Roche Limit = 2.423 Earth's radius = 1 Earth's density = 5.5 Density of imaginary moon made of iron = 8

This dense, heavy moon can be closer to the planet without breaking up.

Example Five:   (The earth is made entirely of iron (unlikely!). The moon is
earth's normal moon.)

 Given Values: The Roche Limit = 2.423 Earth's radius = 1 Earth's density if earth is made of iron = 8 Density of our normal moon = 3.34

A denser planet has a stronger gravitational field, so the moon must be further out to keep from being broken up.

Compare this to Example One.

REMEMBER!!!!! Put your moons outside the Roche Limit!

Moons can be further away, but not closer than the Roche Limit.