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Incelestial mechanics, theRoche limit, also calledRoche radius, is the distance from a celestial body within which a second celestial body, held together only by its own force ofgravity, will disintegrate because the first body's tidal forces exceed the second body'sself-gravitation.[1] Inside the Roche limit,orbiting material disperses and formsrings, whereas outside the limit, material tends tocoalesce. The Roche radius depends on the radius of the second body and on the ratio of the bodies' densities.
The term is named afterÉdouard Roche (French:[ʁɔʃ],English:/rɒʃ/ROSH), theFrenchastronomer who first calculated this theoretical limit in 1848.[2]
The Roche limit typically applies to asatellite's disintegration due totidal forces induced by itsprimary, the body around which itorbits. Parts of the satellite that are closer to the primary are attracted more strongly by gravity from the primary than parts that are farther away; this disparity effectively pulls the near and far parts of the satellite apart from each other, and if the disparity (combined with any centrifugal effects due to the object's spin) is larger than the force of gravity holding the satellite together, it can pull the satellite apart. Some real satellites, bothnatural andartificial, can orbit within their Roche limits because they are held together by forces other than gravitation. Objects resting on the surface of such a satellite would be lifted away by tidal forces. A weaker satellite, such as acomet, could be broken up when it passes within its Roche limit.
Since, within the Roche limit, tidal forces overwhelm the gravitational forces that might otherwise hold the satellite together, no satellite can gravitationally coalesce out of smaller particles within that limit. Indeed, almost all knownplanetary rings are located within their Roche limit. (Notable exceptions are Saturn'sE-Ring andPhoebe ring. These two rings are formed from particles released from the moonsEnceladus andPhoebe due tocryovolcanic plumes and meteoroid impacts, respectively.)
The gravitational effect occurring below the Roche limit is not the only factor that causes comets to break apart. Splitting bythermal stress, internalgas pressure, and rotational splitting are other ways for a comet to split under stress.
The limiting distance to which asatellite can approach without breaking up depends on the rigidity of the satellite. At one extreme, a completely rigid satellite will maintain its shape until tidal forces break it apart. At the other extreme, a highly fluid satellite gradually deforms leading to increased tidal forces, causing the satellite to elongate, further compounding the tidal forces and causing it to break apart more readily.
Most real satellites would lie somewhere between these two extremes, with tensile strength rendering the satellite neither perfectly rigid nor perfectly fluid. For example, arubble-pile asteroid will behave more like a fluid than a solid rocky one; an icy body will behave quite rigidly at first but become more fluid as tidal heating accumulates and its ices begin to melt.
But note that, as defined above, the Roche limit refers to a body held together solely by the gravitational forces which cause otherwise unconnected particles to coalesce, thus forming the body in question. The Roche limit is also usually calculated for the case of a circular orbit, although it is straightforward to modify the calculation to apply to the case (for example) of a body passing the primary on a parabolic or hyperbolic trajectory.
Therigid-body Roche limit is a simplified calculation for aspherical satellite. Irregular shapes such as those of tidal deformation on the body or the primary it orbits are neglected. It is assumed to be inhydrostatic equilibrium. These assumptions, although unrealistic, greatly simplify calculations.
The Roche limit for a rigid spherical satellite is the distance,, from the primary at which the gravitational force on a test mass at the surface of the object is exactly equal to the tidal force pulling the mass away from the object:[3][4]
where is theradius of the primary, is thedensity of the primary, and is the density of the satellite.
This represents the orbital distance inside of which loose material (e.g.regolith) on the surface of the satellite closest to the primary would be pulled away, and likewise material on the side opposite the primary will also go away from, rather than toward, the satellite.
Since the limit depends only on the density, this implies, that the satellite will be torn entirely, if it consists of loose dust or separate rocks bound only by gravity.
A more accurate approach for calculating the Roche limit takes the deformation of the satellite into account. An extreme example would be atidally locked liquid satellite orbiting a planet, where any force acting upon the satellite would deform it into a prolatespheroid.
The calculation is complex and its result cannot be represented in an exact algebraic formula. Roche himself derived the following approximate solution for the Roche limit:
However, a better approximation that takes into account the primary's oblateness and the satellite's mass is:
where is theoblateness of the primary.
The fluid solution is appropriate for bodies that are only loosely held together, such as a comet. For instance,comet Shoemaker–Levy 9's decaying orbit around Jupiter passed within its Roche limit in July 1992, causing it to fragment into a number of smaller pieces. On its next approach in 1994, the fragments crashed into the planet. Shoemaker–Levy 9 was first observed in 1993, but its orbit indicated that it had been captured by Jupiter a few decades prior.[5]
