Tidal forces and the Roche limit

Gravity is not uniform across an object

A point mass experiences one gravitational acceleration at its location. An extended body, such as a moon or planet, has different parts at different distances from the attracting object. The near side feels stronger gravity than the center, while the far side feels weaker gravity.

This difference in gravitational acceleration is a tidal force. Tidal forces stretch objects along the line connecting them and compress them in perpendicular directions.

Estimating tidal acceleration

For a body of radius $R$ at distance $r$ from a mass $M$, the gravitational acceleration is approximately

g(r)=rac{GM}{r^2}

The difference across the body is roughly the derivative times $R$:

ight|R$$ Since $$rac{dg}{dr}=-rac{2GM}{r^3}$$ we get $$Delta gsim rac{2GMR}{r^3}$$ Tidal effects grow rapidly as distance decreases because of the $1/r^3$ dependence. ## Ocean tides Earth's ocean tides are caused mainly by the Moon, with the Sun also contributing. The Moon's gravity pulls more strongly on the near side of Earth and less strongly on the far side. In the rotating Earth-Moon system, this produces two tidal bulges. The full ocean tide pattern is complicated by continents, ocean depth, coastline shape, Earth's rotation, and friction. But the basic cause is differential gravity. ## Tidal locking Tidal forces can create torques that change rotation. Over time, internal friction dissipates energy and can synchronize a body's rotation period with its orbital period. This is tidal locking. The Moon is tidally locked to Earth, so it keeps nearly the same face toward us. Many close-in exoplanets are expected to be tidally locked to their stars. ## Tidal heating If a moon's orbit is eccentric, tidal forces vary during the orbit. The body flexes repeatedly, and internal friction converts mechanical energy into heat. Jupiter's moon Io is strongly tidally heated, powering intense volcanism. Tidal heating can also maintain subsurface oceans in icy moons such as Europa and Enceladus. ## Roche limit If a satellite gets too close to a massive body, tidal forces can overcome the satellite's self-gravity and tear it apart. The critical distance is called the Roche limit. For a fluid satellite of density $ ho_s$ orbiting a planet of radius $R_p$ and density $ ho_p$, an approximate Roche limit is $$d_Rapprox2.44R_pleft(rac{ ho_p}{ ho_s} ight)^{1/3}$$ Rigid bodies can survive somewhat closer because material strength helps hold them together. ## Rings and disruption Planetary rings often lie within or near Roche limits, where material cannot easily accrete into a large moon. Tidal disruption can also occur when stars pass near massive black holes, producing tidal disruption events that release enormous energy. ## The big idea Tidal forces come from differences in gravity across extended bodies. They cause ocean tides, tidal locking, internal heating, orbital evolution, and disruption near massive bodies. The Roche limit estimates where tidal stress overwhelms self-gravity, shaping rings, moons, and extreme astrophysical events.