Tidal Forces (2D)

Differential gravity: why a tide is a difference, not a pull

The Moon pulls every part of the Earth, but not equally. The near side is closer, so it is pulled harder than the centre, and the centre harder than the far side. What deforms the Earth is the difference from the pull at the centre. Expanding the Newtonian pull to first order in R/d gives the tide-raising acceleration at a point with unit vector n on a body of radius R, with u the direction to the attracting mass M at distance d:

It stretches along the line to M (twice as strongly, 2GMR/d³) and squeezes across it (GMR/d³), so there are two bulges, one on each side, and two high tides a day. It falls off as 1/d³, which is why the nearby Moon (0.0123 Earth masses at 60 R) beats the Sun (333,000 Earth masses at 23,500 R) by a factor of about 2.2.

Equilibrium ocean tide

If the ocean could respond instantly the surface would follow the tidal potential, a second-degree Legendre shape:

where ψ is the angle between the point and the body. For the Moon ζ₀ = 0.357 m and for the Sun 0.164 m, giving a high-to-low range of 1.5 ζ₀ = 0.53 m and 0.25 m. When the Moon and Sun line up (new and full Moon) the ranges add, 0.78 m: a spring tide. At the quarters they are at right angles and subtract, 0.29 m: a neap tide. At the equator the combined range over a month is 1.5 √(ζ_M² + ζ_S² + 2 ζ_M ζ_S cos 2Δ), where Δ is the angle between Moon and Sun. Because the Moon’s orbit is an ellipse (356,500 to 406,700 km) the lunar tide changes by about 40% between perigee and apogee. The lunar day is 24 h 50 min, so tides come about 50 minutes later each day.

Roche limit

A moon holds itself together by its own gravity G m / r_m² at the surface. The planet’s tidal stretch on the same surface is 2 G M r_m / d³. Setting them equal for a rigid sphere gives the rigid Roche limit; allowing the body to flow and elongate into a prolate shape gives the larger fluid Roche limit:

Only the density ratio matters. Inside the limit, material at the sub-planet point is pulled off faster than the body can hold it, and the body spreads into a stream and then a ring: this is the usual explanation of planetary rings (and of the fate of Comet Shoemaker-Levy 9 near Jupiter in 1992). A real rubble pile, which has no tensile strength but does have contact friction, breaks somewhere between the two limits.

Tidal locking

The tide raised on a body by its partner is dragged by the body’s spin, and the resulting torque slows the spin until it matches the orbit. The standard estimate for the time to lock a body of mass m, radius R, moment of inertia 0.4 m R², rigidity expressed through k₂ and Q, spinning at ω at distance a from a mass M is

Locking time grows as a⁶: halving the distance makes it 64 times faster, which is why the Moon always shows us one face while the planets farther from the Sun are not locked.

What the page computes

• Ocean tides: the sum of the Moon’s and Sun’s second-degree tides, with the Moon on a real 27.32-day sidereal orbit, the Sun in the equatorial plane, new Moon at day 0 and a rotating Earth (sidereal day 23 h 56 min). The 3D bulge is exaggerated by a factor shown on the page (the real one is about 0.36 m on a 6,371 km radius).
• Falling body and Roche rubble pile: up to 260 equal-mass particles with softened mutual gravity and a damped contact spring, moving in the point-mass field of a uniform-density planet (R = 1, G = 1, planet density 1) and integrated with a fixed-step leapfrog. The body starts as a relaxed, synchronously rotating ball on a circular (or slower, plunging) orbit. Stretch ratio is the root-mean-square extent along the line to the planet divided by that perpendicular to the orbital plane, measured on the densest clump.
• Physical units: one time unit is 1/√(G · 4π/3 · ρ_p), about 13 minutes for an Earth-density planet and 27 minutes for a Jupiter-density one.

Honest simplifications

• Equilibrium tide: the real ocean cannot respond instantly. Coasts, basin shapes and resonances change the height by an order of magnitude (the Bay of Fundy tide is about 15 m) and shift the timing; the page shows the idealised global tide, not a tide table.
• The Moon’s orbit is circular in the page apart from the distance slider, the Sun has no declination, and the solid-Earth tide (about 0.2 m) is ignored.
• The rubble pile uses equal spheres with no friction or cohesion beyond the contact spring, a planet that does not move, a circular start and no Sun. The breaking point of real moons depends on their strength, shape and spin, which is why the page reports a range between the rigid and fluid limits.
• Particles that fall into the planet are removed instead of colliding with it.
• The 3D and 2D pages use the same physics module, so the numbers agree. Drawn sizes (bulge, Moon distance, particle radii) are not to scale; the readouts are.

• Start with Spring tide: new Moon, then press Neap tide: first quarter. Watch the range in the readouts drop from about 0.78 m to 0.29 m and the point on graph 2 move down.
• Set the Sun’s strength to 0: the ocean makes exactly two bulges each lunar day. Set it to 2 and the spring/neap contrast grows.
• Tilt the Moon’s orbit by 28.5 degrees and move the tide gauge to 40 degrees latitude: the two daily highs become unequal.
• Choose Falling body: no self-gravity. Each particle follows its own orbit, so the cloud is stretched along the line to the planet and squeezed across it.
• Choose Rubble Moon outside the limit and then Rubble Moon just inside the limit. Drag the orbit radius between 2.0 and 3.6 planets radii and find where the pile breaks; compare with the rigid and fluid lines.
• Raise the density ratio (a fluffier moon) and the Roche limit moves outwards; lower it (a dense moon) and the moon survives closer in.