Roche limit Guide, Meaning , Facts, Information and Description
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The Roche limit should not be confused with the concept of the Roche lobe which is also named after Édouard Roche and which describes the limits at which an object which is in orbit around two other objects will be captured by one or the other.
Some real satellites, both natural and artificial, can orbit within their Roche limits because they are held together by forces other than gravitation (primarily the tensile strengths of their materials). In such cases, it is possible for an object resting on the surface of such a body to be pulled away by tidal forces, depending on where it is: tidal forces are most repulsive along the line of centers between the satellite and primary. Jupiter's moon Metis and Saturn's moon Pan are examples of natural bodies which are able to hold together despite being within their fluid Roche limitss. (They hold together partly because of their tenslie strength, and partly because they are not actually fluid.) A weaker body, such as a comet, could be broken up when it passes within its Roche limit. 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.
Since tidal forces overwhelm gravity within the Roche limit, no large body can coalesce out of smaller particles within that limit. Indeed, all known planetary rings are located within their Roche limit. They could therefore either be remnants from the planet's proto-planetary accretion disc that failed to coalesce into moonlets, or conversely have formed when a moon passed within its Roche limit and broke apart.
Note that the Roche limit is defined only in terms of gravitational forces - tidal forces and self-gravity. In practice, the question of the stability of a moon could also depend on the centrifugal forces arising from the moon's spin.
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For a rigid spherical satellite, the cause of the rigidity is neglected, in that the material constituting the body is still treated as though held together only by its own self-gravity. The Roche limit is the following:
Determining the Roche limit
The Roche limit depends on the rigidity of the satellite. On one extreme, a rigid satellite will maintain its shape up until tidal forces break it apart. On the other extreme, a highly fluid satellite gradually deforms with increasing tidal force until it breaks apart.
where is the primary's radius, is the primary's density and is the satellite's density.
For a fluid satellite, tidal forces cause the satellite to elongate, further compounding the tidal forces and causing it to break apart more readily. The calculation is complex and cannot be solved exactly, but a close approximation is the following:
Most real satellites are somewhere between these two extremes, with internal friction, viscosity, and chemical bonds rendering the body neither perfectly rigid nor perfectly fluid.
Rigid satellites
As stated above, the formula for calculating the Roche limit, , for a rigid spherical satellite orbiting a spherical primary is:
Notice that if the satellite is more than twice as dense as the primary (as can easily be the case for a rocky moon orbiting a gas giant) then the Roche limit will be inside the primary and hence not relevant.
Derivation of the formula
In order to determine the Roche limit, we consider a small mass on the surface of the satellite closest to the primary. There are two forces on this mass : the gravitational pull towards the satellite and the gravitational pull towards the primary. Since the satellite is already in orbital free fall around the primary, the tidal force is the only relevant term of the gravitational attraction of the primary.
The gravitational pull on the mass towards the satellite with mass and radius can be expressed according to Newton's law of gravitation.
For a sphere the mass can be written as:
- where is the radius of the primary.
- where is the radius of the satellite.
Fluid satellites
A more correct approach for calculating the Roche Limit takes the deformation of the satellite into account. An extreme example would be a tidally locked liquid satellite orbiting a planet, where any force acting upon the satellite would deform the satellite. In this case, the satellite is deformed into a prolate spheroid.The calculation is complex and cannot be solved exactly. Historically, Roche himself derived the following numerical solution for the Roche Limit:
The table below shows the mean density and the equatorial radius for selected objects in our solar system.
Roche limits for selected examples
| Body | Density (kg/m3) | Radius (m) |
|---|---|---|
| Sun | 1,400 | 695,000,000 |
| Jupiter | 1,330 | 71,500,000 |
| Earth | 5,515 | 6,376,500 |
| Moon | 3,340 | 1,737,400 |
Using these data, the Roche Limits for rigid and fluid bodies can easily be calculated. The average density of comets is around 500 kg/m3. The true Roche Limit for a body depends on its flexibility, and will be somewhere between the rigid and fluid Roche Limits. If the primary is less than half as dense as the satellite, the rigid-body Roche Limit is less than the primary's radius, and the two bodies may collide before the Roche limit is reached. (For example, Sun-Earth Roche Limit indicates that the Earth would collide with the Sun before disintegrating due to tidal forces.)
The table below gives the Roche limits expressed in metres and in primary radii.
| Body | Satellite | Roche limit (rigid) | Roche limit (fluid) | ||
|---|---|---|---|---|---|
| Distance (m) | Radii | Distance (m) | Radii | ||
| Earth | Moon | 9,495,665 | 1.49 | 18,261,459 | 2.86 |
| Earth | Comet | 17,883,432 | 2.80 | 34,392,279 | 5.39 |
| Sun | Earth | 554,441,389 | 0.80 | 1,066,266,402 | 1.53 |
| Sun | Jupiter | 890,745,427 | 1.28 | 1,713,024,931 | 2.46 |
| Sun | Moon | 655,322,872 | 0.94 | 1,260,275,253 | 1.81 |
| Sun | Comet | 1,234,186,562 | 1.78 | 2,373,509,071 | 3.42 |
How close are the solar system's moons to their Roche limits? The table below gives each inner satellite's orbital radius divided by its own Roche radius, for both the rigid and fluid cases.
| Primary | Satellite | Orbital Radius : Roche limit | |
|---|---|---|---|
| (rigid) | (fluid) | ||
| Sun | Mercury | 104:1 | 54:1 |
| Earth | Moon | 41:1 | 21:1 |
| Mars | Phobos | 172% | 89% |
| Deimos | 451% | 233% | |
| Jupiter | Metis | 186% | 93% |
| Adrastea | 220% | 110% | |
| Amalthea | 228% | 114% | |
| Thebe | 260% | 129% | |
| Saturn | Pan | 174% | 85% |
| Atlas | 182% | 89% | |
| Prometheus | 185% | 90% | |
| Pandora | 185% | 90% | |
| Epimetheus | 198% | 97% | |
| Uranus | Cordelia | 155% | 79% |
| Ophelia | 167% | 86% | |
| Bianca | 184% | 94% | |
| Cressida | 192% | 99% | |
| Neptune | Naiad | 140% | 72% |
| Thalassa | 149% | 77% | |
| Despina | 153% | 78% | |
| Galatea | 184% | 95% | |
| Larissa | 220% | 113% | |
| Pluto | Charon | 14:1 | 7.2:1 |
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