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Consider a mass of orbiting fluid held together by gravity. Far from the Roche limit, the mass is practically spherical.

Closer to the Roche limit, the body deformed by tidal forces.

Within the Roche limit, the mass' own gravity can no longer withstand the tidal forces, and the body disintegrates.

The Roche limit is the distance within which a satellite—orbiting a celestial body (typically a moon, planet or star) and held together only by its own gravity—will start to disintegrate due to tidal forces exceeding the satellite's gravitational self-attraction. The term is named after Édouard Roche, the French astronomer who first discovered this theoretical limit in 1848.

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). Jupiter's moons Adrastea and Metis are examples of natural bodies which are able to hold together despite being within their (non-rigid) Roche limits. However, an object resting on the surface of such a body can be pulled away by tidal forces, depending on where it is: tidal forces are repulsive along the axis to the central mass and attractive along the "equator" perpendicular to this: see tidal force. A weaker body, such as a comet, could be broken up when it passes within its Roche limit. Comet Shoemaker-Levy 9 passed within its Roche limit of Jupiter in July 1992 and fragmented into a number of smaller pieces before it finally crashed into the planet in 1994 after one last go around the Sun.

Since tidal forces overwhelm gravity within the Roche limit, no large body can coalesce out of smaller particles within that limit. And, 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.

Determining the Roche Limit

The roche limit depends on the rigidity of the satellite orbiting the planet. On one extreme, a force acting upon a rigid spherical body does not deform the body at all. In this case, the Roche Limit is

${\displaystyle d=R\left(2\;{\frac {\rho _{M}}{\rho _{m}}}\right)^{\frac {1}{3}}\approx 1.260R\left({\frac {\rho _{M}}{\rho _{m}}}\right)^{\frac {1}{3}}}$

Where ${\displaystyle R}$ is the primary's radius, ${\displaystyle \rho _{M}}$ is the primary's density and ${\displaystyle \rho _{m}}$ is the satellite's density. The other extreme is a satellite that deforms with very small or no resistance, like a liquid. The exact calculation is complex and cannot be solved exactly. A close approximation would be the following formula:

${\displaystyle d\approx 2.423R\left({\frac {\rho _{M}}{\rho _{m}}}\right)^{\frac {1}{3}}}$

Most real world satellites are somewhere between these two extremes, with internal friction reducing deformation and gravity increasing it.

Rigid Bodies

The formula for calculating the Roche limit, ${\displaystyle d}$, for a rigid spherical satellite orbiting a spherical planet is:

${\displaystyle d=R\left(2\;{\frac {\rho _{M}}{\rho _{m}}}\right)^{\frac {1}{3}}}$

Where ${\displaystyle R}$ is the radius of the planet, ${\displaystyle \rho _{M}}$ is the density of the planet, and ${\displaystyle \rho _{m}}$ 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 is only an approximation of what a real moon's Roche limit would be.

Notice that if the moon is more than twice as dense as the planet (as can easily be the case for a rocky moon orbiting a gas giant) then the Roche limit will be inside the planet and hence not relevant.

In order to determine the Roche limit, we consider a small mass ${\displaystyle u}$ on the surface of the smaller planet closest to the larger planet. There are two forces on this mass ${\displaystyle u}$: the gravitational pull towards the smaller planet and the gravitational pull towards the larger planet. Since the smaller planet is already in orbital free fall around the larger planet, the tidal force is the only relevant term of the gravitational attraction of the larger planet.

The gravitational pull ${\displaystyle F_{G}}$ on the mass ${\displaystyle u}$ towards the smaller planet with mass ${\displaystyle m}$ and radius ${\displaystyle r}$ can be expressed according to Newton's law of gravitation.

${\displaystyle F_{G}={\frac {Gmu}{r^{2}}}}$

The tidal force ${\displaystyle F_{T}}$ on the mass ${\displaystyle u}$ towards the larger planet with radius ${\displaystyle R}$ and a distance ${\displaystyle d}$ between the center of the two planets can be expressed as:

${\displaystyle F_{T}={\frac {2GMur}{d^{3}}}}$

The Roche limit is reached when the gravitational pull and the tidal force cancel each other out.

${\displaystyle F_{G}=F_{T}}$

or

${\displaystyle {\frac {Gmu}{r^{2}}}={\frac {2GMur}{d^{3}}}}$

Which quickly gives the Roche limit, d, as:

${\displaystyle d=r\left(2M/m\right)^{\frac {1}{3}}}$

However, we don't really want the radius of the smaller body to appear in the expression for the limit, so we re-write this in terms of densities.

For a sphere the mass ${\displaystyle M}$ can be written as:

${\displaystyle M={\frac {4\pi \rho _{M}R^{3}}{3}}}$ where ${\displaystyle R}$ is the radius of the larger planet.

And likewise:

${\displaystyle m={\frac {4\pi \rho _{m}r^{3}}{3}}}$ where ${\displaystyle r}$ is the radius of the smaller planet.

Substiting for the masses in the equation for the Roche limit, and cancelling out ${\displaystyle 4\pi /3}$ gives:

${\displaystyle d=r\left(2\rho _{M}R^{3}/\rho _{m}r^{3}\right)^{\frac {1}{3}}}$

which can be simplified to the Roche limit:

${\displaystyle d=R\left(2\;{\frac {\rho _{M}}{\rho _{m}}}\right)^{\frac {1}{3}}}$

Non-rigid Bodies

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 exact calculation is complex and cannot be solved exactly. Historically, Roche derived the following numerical solution for the Roche Limit:

${\displaystyle d\approx 2.44R\left({\frac {\rho _{M}}{\rho _{m}}}\right)^{\frac {1}{3}}}$

However, with the aid of a computer a better numerical solution is:

${\displaystyle d\approx 2.423R\left({\frac {\rho _{M}}{\rho _{m}}}\right)^{\frac {1}{3}}\left({\frac {(1+{\frac {m}{3M}})+{\frac {c}{3R}}(1+{\frac {m}{M}})}{1-{\frac {c}{R}}}}\right)^{\frac {1}{3}}}$

Roche Limits for selected examples

The table below shows the mean density and the equatorial radius for selected objects in our solar system.

Using these data, the Roche Limits for rigid and non-rigid bodies can easily be calculated. The average density of comets is around 500 kg/m. The true Roche Limit depends on the flexibility of the satellite and will be somewhere between the rigid and non-rigid Roche Limit. If the larger body is less than half as dense as the smaller one, the rigid-body Roche Limit is less than the larger body's radius, and the two bodies may collide before the Roche limit is reached (for example, the Sun-Earth Roche Limit). The table below gives the Roche limits expressed in metres and in primary radii.

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 non-rigid cases.

See also

• Hill sphere

References

• Édouard Roche: La figure d'une masse fluide soumise à l'attraction d'un point éloigné, Acad. des sciences de Montpellier, Vol. 1 (1847-50) p. 243

External link

• Detailed derivation of formulae for calculating the Roche limit