# Roche limit

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The Roche limit, sometimes referred to as the Roche radius, is the distance within which a celestial body held together only by its own gravity will disintegrate due to a second celestial body's tidal forces exceeding the first body's gravitational self-attraction. Inside the Roche limit, orbiting material will tend to disperse and form rings, while outside the limit, material will tend to coalesce. The term is named after Édouard Roche, the French astronomer who first calculated this theoretical limit in 1848.

Typically, the Roche limit applies to a satellite disintegrating due to tidal forces induced by its primary, the body about which it orbits. Some real satellites, both natural and artificial, can orbit within their Roche limits because they are held together by forces other than gravitation. Jupiter's moon Metis and Saturn's moon Pan are examples of such satellites, which hold together because of their tensile strength. In extreme cases, objects resting on the surface of such a satellite could actually be lifted away by tidal forces. A weaker satellite, such as a comet, could be broken up when it passes within its Roche limit.

Since tidal forces overwhelm gravity within the Roche limit, no large satellite can coalesce out of smaller particles within that limit. Indeed, almost all known planetary rings are located within their Roche limit (Saturn's E-Ring being a notable exception). They could 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 should not be confused with the concept of the Roche lobe or Roche sphere, which are also named after Édouard Roche. The Roche lobe describes the limits at which an object which is in orbit around two other objects will be captured by one or the other. The Roche sphere approximates the gravitational sphere of influence of one astronomical body in the face of perturbations from another heavier body around which it orbits.)

## Determining the Roche limit

The Roche limit depends on the rigidity and density of the satellite. A rigid satellite is one extreme, and will maintain its shape until tidal forces break it apart. The Roche limit, $d$, is then the following:

$d = R\left( 2\;\frac {\rho_M} {\rho_m} \right)^{\frac{1}{3}} \approx 1.25992R\left( \frac {\rho_M} {\rho_m} \right)^{\frac{1}{3}}$

where $R$ is the primary's radius, $\rho_M$ is the primary's density and $\rho_m$ is the satellite's density.

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. The calculation is complex and its result cannot be expressed as an algebraic formula, but a close approximation is the following:

$d \approx 2.423R\left( \frac {\rho_M} {\rho_m} \right)^{\frac{1}{3}}$

which indicates that a fluid satellite will disintegrate at almost twice the distance from the primary as a rigid sphere of similar density.

Most real satellites are somewhere between these two extremes, with internal friction, viscosity, and tensile strength rendering the satellite neither perfectly rigid nor perfectly fluid.

### Rigid satellites

As stated above, the formula for calculating the Roche limit, $d$, for a rigid spherical satellite orbiting a spherical primary is:

$d = R\left( 2\;\frac {\rho_M} {\rho_m} \right)^{\frac{1}{3}}$

where $R$ is the radius of the primary, $\rho_M$ is the density of the primary, and $\rho_m$ is the density of the satellite.

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.

To calculate the "rigid body" Roche limit for a spherical satellite, the cause of the rigidity is neglected but it is assumed that the body does not deform from its spherical shape while being held together only by its own self-gravity. Other effects are also neglected, such as tidal deformation of the primary, rotation of the satellite, and its irregular shape.

#### Derivation of the formula

In order to determine the Roche limit, we consider a small mass $u$ on the surface of the satellite closest to the primary. There are two forces on this mass $u$: 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 $F_G$ on the mass $u$ towards the satellite with mass $m$ and radius $r$ can be expressed according to Newton's law of gravitation.

$F_G = \frac{Gmu}{r^2}$

The tidal force $F_T$ on the mass $u$ towards the primary with radius $R$ and a distance $d$ between the centers of the two bodies can be expressed as:

$F_T = \frac{2GMur}{d^3}$

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

$F_G = F_T \;$

or

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

Which quickly gives the Roche limit, $d$, as:

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

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

For a sphere the mass $M$ can be written as:

$M = \frac{4\pi\rho_M R^3}{3}$ where $R$ is the radius of the primary.

And likewise:

$m = \frac{4\pi\rho_m r^3}{3}$ where $r$ is the radius of the satellite.

Substituting for the masses in the equation for the Roche limit, and cancelling out $4\pi/3$ gives:

$d = r \left( \frac{ 2 \rho_M R^3 }{ \rho_m r^3 } \right)^{1/3}$

which can be simplified to the Roche limit:

$d = R\left( 2\;\frac {\rho_M} {\rho_m} \right)^{\frac{1}{3}}$

### Fluid satellites

A more accurate 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 it (into a prolate spheroid).

The calculation is complex and its result cannot be represented as an algebraic formula. Historically, Roche himself derived the following numerical solution for the Roche Limit:

$d \approx 2.44R\left( \frac {\rho_M} {\rho_m} \right)^{1/3}$

However, a better approximation that takes into account the primary's oblateness and the satellite's mass is:

$d \approx 2.423 R\left( \frac {\rho_M} {\rho_m} \right)^{1/3} \left( \frac{(1+\frac{m}{3M})+\frac{c}{3R}(1+\frac{m}{M})}{1-c/R} \right)^{1/3}$

where $c/R$ is the oblateness of the primary. The numerical factor is calculated with the aid of a computer.

The fluid solution is appropriate for cases in which the body is 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. [1]

## Roche limits for selected examples

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

Sun 1,408 696,000,000
Jupiter 1,326 71,492,000
Earth 5,513 6,378,137
Moon 3,346 1,738,100
Saturn 0,6873 60,268,000
Uranus 1,318 25,559,000
Neptune 1,638 24,764,000

Using these data, the Roche Limits for rigid and fluid bodies can easily be calculated. The average density of comets is taken to be around 500 kg/m3.

The table below gives the Roche limits expressed in metres and in primary radii. The true Roche Limit for a satellite depends on its density and rigidity.

Body Satellite Roche limit (rigid) Roche limit (fluid)
Distance (km) R Distance (km) R
Earth Moon 9,496 1.49 18,261 2.86
Earth average Comet 17,880 2.80 34,390 5.39
Sun Earth 554,400 0.80 1,066,300 1.53
Sun Jupiter 890,700 1.28 1,713,000 2.46
Sun Moon 655,300 0.94 1,260,300 1.81
Sun average Comet 1,234,000 1.78 2,374,000 3.42

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.

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. Both rigid and fluid body calculations are given. Note Naiad in particular, which may in fact be quite close to its actual break-up point.

In practice, the densities of most of the inner satellites of giant planets are not known. In these cases (shown in italics), likely values have been assumed, but their actual Roche limit can vary from the value shown.

Primary Satellite Orbital Radius vs. Roche limit
(rigid) (fluid)
Sun Mercury 104:1 54:1
Earth Moon 41:1 21:1
Mars Phobos 172% 89%
Deimos 451% 234%
Jupiter Metis ~186% ~94%
Amalthea 175% 88%
Thebe 254% 128%
Saturn Pan 168% 84%
Atlas 176% 88%
Prometheus 178% 89%
Pandora 178% 89%
Epimetheus 192% 95%
Janus 196% 98%
Uranus Cordelia ~154% ~79%
Ophelia ~166% ~86%
Bianca ~183% ~94%
Cressida ~191% ~98%
Desdemona ~194% ~100%
Juliet ~199% ~102%