Roche limit or radius of Roche

Where is the Roche limit?

All agglomerates of a celestial body are held together by their own gravity. The Roche limit and radius of Roche is the distance at which a small celestial body will disintegrate due to tidal forces of another celestial body whose gravitational pull than the self-attraction of the small body. In other words, the Roche Limit is the minimum distance from the center of the planet, which allows the material to come together to form, for example, a large enough moon.
Indeed tidal forces prohibit the formation of a massive planet near the body. A certain distance is necessary for dust and small debris « stick together » and form a very massive object. This distance is called the Roche Limit of the French mathematician and astronomer Édouard Albert Roche (1820-1883) who calculated the theoretical limit in 1848.
Below this limit, an object starts to break because the action of tidal forces takes over the forces of cohesion of the materials constituting the object.
Beyond this limit, tidal forces produce only friction between the materials of the satellite and the planet. That product, generally a bead on the surface of objects.
The Roche limit for rigid body is known, it is located for two bodies of the same density, ≈ 2.42 times the radius of the planet. For body fluids it is located for two bodies of the same density, ≈ 1.26 times the radius of the planet.

Table: distance of the rings of the solar system relative to the Roche limit. How do you read this table?
A ring of Saturn is 1.44 times the Roche limit for rigid bodies and 0.75 times, i.e. on the inside of the Roche limit for fluid bodies. The ring by definition are not robust and are considered as fluid bodies. Agglomerates of dust and small stones can not come together because the tidal force of Saturn will prevent, any « bonding ».
Calculations done by  astronoo.com

Image: Saturn rings (A, B, C, D, E, F, G) are located inside or near the Roche limit for fluid bodies. Prometheus and Pandora are within the Roche limit for fluid bodies, but beyond the Roche limit for rigid bodies. In this picture we see Prometheus create strange currents in the F ring of Saturn. This small moon revolves around Saturn inside the thin F ring and near the inner edge of the ring every 15 hours. The low gravity field of Prometheus attracts fine particles of ice surrounding dust, causing the dark waves, lack of material. Prometheus creates a new stream for each passage, so that sometimes several of them are visible simultaneously.
The gravity field of the tiny moon creates dark waves visible in this image of the ring F.
Credit image: Cassini Imaging Team, ISS, JPL, ESA, NASA

How to calculate the Roche limit?

How to calculate the Roche limit for rigid bodies?
Roche limit for rigid bodies:
d = 2.422849865 x R x ₃√ρM/ρm
d = Roche limit
R = radius of the planet
ρM = density or volumetric mass density of the planet
ρm = density or volumetric mass density of the moon
Excel formula to calculate:
d=(2.422849865*R)*(ρM/ρm)^(1/3)

How to calculate the Roche limit for fluid bodies?
Roche limit for fluid bodies:
d= 1.26 x R x ₃√ρM/ρm
Excel formula to calculate:
d=(1.26*R)*(ρM/ρm)^(1/3)

Tidal forces exerted by the planet slowly slow the satellite when it is inside the Roche limit. The moon is gradually losing altitude and can dislocate reaching the Roche limit and thus form a new planetary ring. Conversely, beyond the Roche limit, tidal forces accelerate very slowly the satellite and move away, is the case of the Moon moves away from the Earth of 3.78 cm per year.
But several moons in the solar system are perilously close to the Roche limit their planet, their end of life is programmed. If they do not dislocate themselves, to approach the Roche limit, they will ignite in the atmosphere of their planet. This is particularly the case of Phobos (moon of Mars), Amalthea (moon of Jupiter), Prometheus and Pandora (moon of Saturn), Cordelia and Ophelia (moons of Uranus) and Galatea, Thalassa, Despina or Naiad (moons of Neptune).  

NB: The density, or more precisely, the volumetric mass density of a substance is the mass per unit volume. The symbol most often used for density is ρ (the Greek letter rho). Mathematically, the density is defined as the weight divided by the volume.

Table: distances (semi-major axis) of the closest moons of the Roche limit of their planet.
How do you read this table?
Phobos, moon of Mars is 1.72 times the Roche limit for rigid bodies and 0.89 times, i.e. inside the Roche limit for fluid bodies. The moons are rigid bodies, they were able to assemble because the tidal force at this point is lower than their own gravity.

Image: When the moon is near the Roche limit, tidal forces exerted by the planet, slow slowly. The moon then gradually loses altitude and dislocated through the Roche limit, it is one case of possible scenarios for the formation of planetary rings. Although the origin of the planetary ring is not known with certainty, there are three scenarios.
1) planetary rings formed from the outset, from of the protoplanetary disk material. The material is in the Roche limit of the planet, it can not come together to form moons.
2) The planetary rings formed from the debris of a moon that was hit by another object.
3) planetary rings formed from the debris of a moon that was shattered by tidal forces inside the Roche limit of the planet.
Credit image: astronoo.com