where d is the orbital radius, R is the radius of the planet, ρ_{M} is the density of the planet, and ρ_{m} is the density of the moon. This formula does not take into account the deformation of the moon's spherical shape due to tidal effects, and so is only an approximation of what a real moon's Roche limit would be.
Real satellites, both natural and artificial, can usually orbit within their Roche limits because they are held together by forces other than gravitation (primarily the tensile strengths of their materials). Jupiter's moons Adrastea and Metis are examples of natural bodies which are able to hold together despite being within their Roche limits. However, an object resting on the surface of such a body held in place only by that body's gravity could find itself pulled away into orbit by tidal forces.
The Roche limit is sometimes described in terms of "Roche lobes", a threedimensional mathematical surface which forms a twolobed figureeight shape with each of the two masses at the center of its respective lobe. This surface represents a set of points with equal gravitational potential. If you were to hypothetically fill these lobes completely with water a ship could sail from one mass to the other, passing through a point between the two where both planetwide oceans touch the tips of their sharp peaks together. When an object "exceeds its Roche limit", its surface extends out beyond its Roche lobe and the material which lies beyond "falls off" into the other object's Roche lobe. This can lead to the total disintegration of the object, since reducing its mass causes its Roche lobe to shrink.
Recurring novas are a white dwarf/red giant binary star where they are close enough together that the red giant overflows its Roche lobe and dribbles outer material down onto the white dwarf.
Planetary rings are always located within their Roche limit, which prevents their component particles from coalescing into one or more larger bodies. It is theorized that they form when a moon passes within its Roche limit and breaks apart.
The Roche limit is named after Edouard Roche, the astronomer who defined the term.
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