Tidal Force Calculator (Differential Gravity)

How it works

Δa (stretch) ≈ 2 G M r ÷ d³ · Δa (squeeze) ≈ G M r ÷ d³ · g = G M ÷ d² · Δa ÷ g = 2 r ÷ d

Gravity weakens with the square of distance, so the near side of a body is pulled slightly harder than its centre and the far side slightly less; in the body's own frame that shows up as a stretch along the line to the attractor and a squeeze at right angles to it. Differentiating the inverse-square law gives the inverse-cube dependence, which explains the everyday puzzle of why the small Moon raises bigger ocean tides than the Sun: the Sun pulls Earth 180 times harder, but its tidal effect is only 46% of the Moon's because it is 390 times farther away. The same physics heats Io, circularises orbits, locks rotations and, near a stellar-mass black hole, would stretch an infalling astronaut long before the horizon.

Worked example

The Moon (7.34 × 10²² kg) at 384,400 km exerts a tidal acceleration of 2 × 6.674 × 10⁻¹¹ × 7.34 × 10²² × 6.371 × 10⁶ ÷ (3.844 × 10⁸)³ ≈ 1.10 × 10⁻⁶ m/s² across Earth — about 0.11 millionths of g, roughly a 0.5 m equilibrium bulge. The Sun's figure is 5.05 × 10⁻⁷ m/s², 46% of the Moon's.

Frequently asked questions