Roche limit

 File:Roche limit (far away sphere).PNG Consider an orbiting mass of fluid held together by gravity, here viewed from above the orbital plane. Far from the Roche limit the mass is practically spherical. File:Roche limit (tidal sphere).PNG Closer to the Roche limit the body is deformed by tidal forces. File:Roche limit (ripped sphere).PNG Within the Roche limit the mass' own gravity can no longer withstand the tidal forces, and the body disintegrates. File:Roche limit (top view).PNG Particles closer to the primary orbit more quickly than particles farther away, as represented by the red arrows. File:Roche limit (ring).PNG The varying orbital speed of the material eventually causes it to form a ring.

The Roche limit 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.

The Roche limit should not be confused with the concept of the Roche lobe, which is 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.

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 natural satellites which are able to hold together despite being within their fluid Roche limits. They hold together partly because of their tensile strength, and partly because they are not actually fluid. In such cases, it is possible for an object resting on the surface of such a satellite 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. A weaker satellite, 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 satellite can coalesce out of smaller particles within that limit. Indeed, all known planetary rings are located within their Roche limit. 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.

Determining the Roche limit

The Roche limit depends on the rigidity of the satellite. At one extreme, a rigid satellite will maintain its shape until tidal forces break it apart. At the other extreme, a highly fluid satellite gradually deforms with increasing tidal forces until it breaks apart.

For a rigid spherical satellite, the cause of the rigidity is neglected, in that the material constituting the satellite is still treated as though held together only by its own self-gravity. Other effects are also neglected, such as tidal deformation of the primary, and rotation of the satellite. The Roche limit, $\displaystyle d$ , is then the following:

$\displaystyle d = R\left( 2\;\frac {\rho_M} {\rho_m} \right)^{\frac{1}{3}} \approx 1.260R\left( \frac {\rho_M} {\rho_m} \right)^{\frac{1}{3}}$

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.

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:

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

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

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

Rigid satellites

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

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

where $\displaystyle R$ is the radius of the primary, $\displaystyle \rho_M$ is the density of the primary, and $\displaystyle \rho_m$ is the density of the satellite. As described below, this rigid-body approximation does not take into account the deformation of the satellite's spherical shape due to tidal effects and is only an approximation of what a real satellite's Roche limit would be.

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 $\displaystyle u$ on the surface of the satellite closest to the primary. There are two forces on this mass $\displaystyle u$ : 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 $\displaystyle F_G$ on the mass $\displaystyle u$ towards the satellite 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 primary with radius $\displaystyle R$ and a distance $\displaystyle d$ between the center of the two bodies 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:

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

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 $\displaystyle M$ can be written as:

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

And likewise:

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

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

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

which can be simplified to the Roche limit:

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

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:

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

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

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

where $\displaystyle c/R$ is the oblateness of the primary.

Roche limits for selected examples

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

Sun 1,400 695,000,000
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 satellites can easily be calculated. The average density of comets is 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 flexibility, and will be somewhere between the rigid and fluid Roche Limits given below.

Body Satellite Roche limit (rigid) Roche limit (fluid)
Distance (m) R Distance (m) R
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

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, the Sun-Earth Roche Limit indicates that the Earth would collide with the Sun before disintegrating due to tidal forces.

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. (Note Naiad in particular, which may in fact be quite close to its actual Roche limit.)

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% 233%
Jupiter Metis 186% 93%
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%