# 7.12 The Roche Limit

Tidal forces are very strong when a body is close to a planet. For example, the Moon’s tidal force on the Earth causes our ocean tides. We also know that tidal forces distort a satellite close to a planet more than one farther away. If a satellite comes within a critical distance from a planet, tidal forces can actually tear it apart. That critical distance, at which a satellite will be destroyed, is called the Roche limit.

Description of the creation of ring systems from a celestial body entering the Roche limit of a planet. Click here for original source URL

The exact location of the Roche limit depends on the density, strength and shape of the bodies, but generally it’s about two to three planetary radii from the center of a planet. Inside this limit, it’s difficult for a large satellite to exist. Spaceships and small satellites less than a kilometer or so in size are more likely to survive inside the Roche limit. If they consist of metal, unfactured rock, or ice, they will have enough strength to resist being pulled apart. Larger bodies may stretch plastically, until the stresses are so great that they break apart.

If a massive swarm of particles were in orbit around a planet outside the Roche limit, they might aggregate into a moon. Inside the Roche limit, however, they could not accrete, but would remain as a dispersed ring. This lets us understand why rings persist once particles are established close in to a planet: the tidal forces are too strong to allow the particles to combine into a solid body. Well-understood forces then keep such particles orbiting in a thin disk around the equator of the planet.

These seemingly abstract principles are supported by direct observation. All ring systems are inside the Roche limit for their planets. Some ring systems, such as Saturn's, have outermost edges very near the Roche limit, at about 1.8 to 2.5 times the planetary radius from the center of the planet. Why aren't the edges of some ring systems at larger or smaller multiples of the planetary radius?

Illustration for the derivation of the Roche limit. Click here for original source URL.

Edouard Roche. Click here for original source URL.

You may be amazed to find you already know enough science to predict the position of ring systems' outer edges! It’s actually fairly easy to predict this basic fact of nature: it all goes back to Newton's universal law of gravitation. Edouard Roche, a French mathematician, first worked it out in 1850. Imagine that there are two tiny ring particles at a distance X from a planet. Each particle has mass m and radius r (the two particles have the same mass and size). The planet has mass M and radius R. Suppose the two particles are touching. Then the gravity force pulling them together would be (from Newton's law of gravity):

F = G m^{2} / (2r)^{2}

Now think about the gravitational force pulling either particle toward the planet. It would be:

F = G Mm / X^{2}

According to Newton’s law of gravity, the planet will exert more gravitational force on the nearer particle than the farther particle. Differential calculus allows us to find the difference in the forces at the locations of the two particles. We can write the expression for this differential force (if you haven’t studied calculus yet, you will have to take this statement on faith!):

Δ F = (-2GMm / X^{3} ) Δ r

The Greek capital letter Δ (delta) is often used in mathematics and physics for a small increment or difference. So Δ F is a small change in force, and Δ r is a small change in distance. The minus sign means that the differential force gets bigger as the distance from the planet gets smaller. Since the difference in distance is just 2 particle radii, we have:

Δ F = -4GMmr / X^{3}

The differential force Δ F is the force pulling the two small particles apart, and the gravity between them, F, is the force pulling them together. From our study of rings, we can predict that the outer edge of the ring will be where these two forces are equal. Outside that distance, gravity will keep the particles together, and they will clump into a single satellite. Inside that distance, tidal forces will pull the particles apart, so they will orbit separately as a ring system. So let's set the two forces equal and solve for the distance X. We have:

Gm^{2} / 4r^{2} = - 4GMmr / X^{3}

Or,

X^{3} = -16Mr^{3} / m

The minus sign is no longer important because we are only interested in the size of the distance. The mass of the planet is M = volume × density = (4/3 πR^{3}) × the planet’s density. The mass of a particle is m = volume × density = (4/3 πr^{3}) × the particle’s density. If we substitute these two expressions for M and m, you will notice that the factors of 4/3 and π cancel out. If the density of the particle is equal to the mean density of the planet, the densities cancel out, too! The expression then becomes much simpler:

X^{3} = 16 R^{3}

Or,

X ≈ 2.5 R

This last simple relation says that when the two forces are equal, the particles are located at about 2.5 times the planet radius. Closer to the planet than this distance, the gravitational force between the particles will not be strong enough to hold them together. In other words, a ring can exist only inside 2.5 planet radii. This is essentially what we observe!

As Roche himself concluded in 1850, rings are defined by tidal forces, which act to keep the ring particles apart. If the particle densities are not the same as the planet’s density, the outer edge of the ring occurs at different distances - which is also observed. The theory can be made more sophisticated by considering variations in particle density, strength, and shape. The astonishing fact is that we can look through a telescope, observe the dimensions of Saturn's rings, or other planets’ rings, and then sit down with a pencil, paper, and Newton's law of gravity (dating from 1687), and explain why the rings look the way they do!