The Roche Limit
A satellite cannot approach its primary too closely (the Roche limit) or stray too far (the
instability limit) without breakup or loss to the system respectively. The Roche Limit is the
point within which a satellite would be torn apart by tidal forces if the only force hloding it
together is its selfgravity. Taking a primary mass M, radius R, a distance d from its satellite
mass m radius r (where M>>m) we have:
If the satellite is large enough (r greater than about 500km) its selfgravity dominates other
cohesive forces. It will be torn apart if it approaches the primary closer than:
which is known as the Roche Limit, after the mathematician who first derived it. For our Moon d
= 2.44(5.5/5.3)
1/3
or about 2.9R
E
 around 18,500 km.
Roche's calculation was complicated by the fact he took into account the tidal distortions in the
satellite just before breakup. We can get a similar (though not the "full") term if we just consider
a
rigid
spherical satellite. Then the centripetal acceleration of the orbit is w
2
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 Fall '10
 EmilyHoward
 Astronomy, Neptune, Natural satellite

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