🌊 Ocean Tides

Tides arise from the difference in gravitational pull between the near and far sides of Earth, not from gravity itself. Differentiating Newton's 1/r² force law with respect to r introduces an extra factor of 1/r, giving the 1/r³ dependence used in this simulation's a_tide = 2GMR/r³ formula. That steep fall-off is why the much closer Moon dominates over the far more massive Sun.

The near side of Earth is pulled toward the Moon more strongly than Earth's center is, while the center is pulled more strongly than the far side. Relative to Earth's center this produces two outward-stretching accelerations — one toward the Moon and one directly away from it — creating bulges on opposite sides at the same time, which is why the simulated tide gauge sees roughly two highs and two lows during every Earth rotation.

The simulation combines the Moon's and Sun's tidal vectors using H_amp = a_moon + a_sun·(3cos²θ−1)/2, where θ is the angle between the Sun and Moon as seen from Earth. When θ≈0° or 180° (new or full Moon) the two vectors reinforce, producing spring tides; near θ≈90° (first/last quarter) they partly cancel, giving smaller neap tides — exactly what the Regime readout and Moon-phase indicator track as the Moon orbits.

Tidal strength depends on M/r³, not on mass alone. The Sun's mass (1.989×10³⁰ kg) is about 27 million times the Moon's (7.342×10²² kg), but the Sun's distance (1.496×10¹¹ m) is roughly 390 times the Moon's (3.844×10⁸ m). Cubing that distance ratio overwhelms the mass ratio, leaving solar tides at about 46% of lunar strength, matching the simulation's Solar tidal a readout.

It plots gaugeHeight = H_amp·(cos²Δ − 0.5), where Δ is the angle between the observer's longitude and the current bulge axis, converted to metres and recorded for the last 30 simulated days. As Earth rotates and the Moon orbits, this produces the characteristic twice-daily tidal rhythm modulated by the slower spring–neap cycle.