Tidal Force Calculator

Tides on Earth are not caused by the Moon pulling up on the ocean. They are caused by the difference in the Moon’s gravitational pull between the near side and the far side of Earth. This difference is the tidal force.

The formula

For a body of half-size R at distance d from a mass M:

a_tidal = 2GMR / d^3

This is the differential gravitational acceleration (in m/s^2) across the extended body. It stretches the body toward and away from the source mass, and squeezes it sideways.

For the actual tidal force on a test mass m:

F_tidal = m x 2GMR / d^3

Moon-Earth example

Moon mass M = 7.342 x 10^22 kg. Distance d = 384,400 km. Earth radius R = 6,371 km.

a_tidal = 2 x 6.674e-11 x 7.342e22 x 6.371e6 / (3.844e8)^3 ≈ 1.11 x 10^-6 m/s^2

This tiny acceleration — one millionth of surface gravity — is enough to raise oceans by a meter because tides act over the entire planet simultaneously.

The Roche limit

The Roche limit is the distance at which tidal forces overcome a satellite’s self-gravity, tearing it apart. For a fluid body:

d_Roche ≈ 2.44 x R_primary x (rho_primary / rho_satellite)^(1/3)

Saturn’s rings sit inside the Roche limit of Saturn — material there cannot coalesce into a moon. The rings are the remnants of moons or comets that strayed too close.

Spaghettification

Near a stellar-mass black hole, tidal forces become extreme. An astronaut falling feet-first toward a stellar black hole (a few solar masses) would be stretched to death before crossing the event horizon. Near a supermassive black hole like M87*, tidal forces at the horizon are mild enough to cross without spaghettification.