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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.
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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)
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Given Values:
The Roche Limit = 2.423
Earth's radius = 1
Earth's density = 5.5
The moon's density = 3.34
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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
made of water ice)
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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
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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)
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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
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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!)
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Given Values:
The Roche Limit = 2.423
Earth's radius = 1
Earth's density = 5.5
Density of imaginary moon made of iron = 8
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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.)
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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
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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.
Return to Calculating Orbits for Your
Moons
Picture of Saturn is from a
NASA photograph
©
1998, 2003. Elizabeth Anne Viau. All rights reserved. This material
may be used by individuals for instructional purposes but not
sold. Please inform the author if you use it at eviau@earthlink.net.
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