What If the Moon Vanished? 2D Tide Diagram, Tilt and Night Sky

Setting up the lab bench...

Presets

Live readouts

Graphs

Sea level at the gauge: this scenario, with the Moon (dashed) and Sun only

Obliquity (axial tilt) over 10 million years with and without the Moon

Day length against geological time

Night brightness against Moon phase, compared with starlight

What is happening now

Starting.

Numerical checks

running...

How to use

The diagram looks down on the Earth's north pole. The blue ring is the equilibrium ocean: two bulges toward and away from the Moon plus a smaller pair toward the Sun. The yellow dot is a tide gauge that rotates with the Earth; its sea level is the left column and graph 1. Try: (1) Today and watch the bulges align (spring tide) at new and full Moon and cross at the quarters (neap); (2) Moon vanishes at day 7: the lunar bulge fades, leaving only the solar tide and no spring-neap cycle; (3) drag the Moon distance: the tide force falls as 1/d^3; (4) pick a coast to see how much of the range is lost there; (5) slide the obliquity run time to compare the tilt with and without the Moon, and the geological time for day length. The lower strip shows how dark the night gets.

The science

Tides. The tide-generating force from a body of mass M at distance d scales as M / d³ (the difference of gravity across Earth, not gravity itself). Moon: 7.342×10²² kg at 384,400 km; Sun: 1.989×10³° kg at 149.6 million km. Ratio Moon : Sun = (Mm/dm³)/(Ms/ds³) ≈ 2.18. Without the Moon only the solar tide remains: about 31% of today's spring range and 46% of the average range, and the 14.8-day spring/neap beat (two close frequencies, M2 12.42 h and S2 12.00 h, drifting in and out of phase) disappears.

The water level is a sum of tidal constituents h(t) = G · Σ A_i cos(ω_i t + φ_i) with equilibrium amplitudes M2 0.244 m, S2 0.114, N2 0.047, K1 0.142, O1 0.101, P1 0.047 m (Cartwright-Doodson type values). The site gain G (resonance of the local basin) is calibrated so today's spring range matches the coast you choose. Lunar terms scale as (d0/d)³; the Moon's frequencies follow Kepler's third law (period ∝ d^3/2). The 3D bulge is the equilibrium tide, exaggerated roughly a million times.

Axial tilt. The spin axis s obeys ds/dt = α (s·n)(s×n), where n is the orbit normal and α the precession constant: about 50.3 arcsec/yr today, of which the Sun gives ~15.9 and the Moon ~34.4 (so α falls to ~16 without the Moon). The orbit normal wobbles because of the planets, here as six sinusoids with the real nodal frequencies (-5.6, -7.1, -17.7, -18.9, -26.4, -3.0 arcsec/yr) and rough amplitudes. With the Moon the model reproduces the 41,000-year cycle and a ~2 degree range; without it the precession rate falls into the band of planetary frequencies and resonances stir the tilt. Published results: Laskar, Joutel and Robutel (Nature 1993) found a moonless obliquity wandering chaotically between about 0 and 85 degrees over millions of years, versus 22.1-24.5 degrees with the Moon; Lissauer, Barnes and Chambers (Icarus 2012) found much milder variations (a few tens of degrees at most) for a moonless Earth with fast rotation. This simulation is a simplified illustration, not a full N-body integration - at forcing x1 it lands nearer the milder results.

Day length. Tidal bulges are dragged ahead of the Moon, giving a torque that slows Earth's spin and pushes the Moon outwards (angular momentum is conserved): the Moon recedes 3.8 cm/yr and the day lengthens ~2.3 ms per century from tides (the net observed change is ~1.7 ms/century because post-glacial rebound partly offsets it). Solar-tide torque scales as (1/2.18)² of the lunar one, so the remainder would be small. Past day lengths come from fossil coral and bivalve growth bands (~400 days per year about 400 Myr ago).

Night sky. Moon illuminance follows E = 10^{-0.4(m + 14.18)} lux with m = -12.74 + 0.026|α| + 4×10^-9α⁴ (α = phase angle in degrees; Krisciunas and Schaefer 1991 type formula), reduced by atmospheric extinction (airmass). Full Moon at the zenith ≈ 0.25 lux; a moonless clear night (starlight plus airglow) ≈ 0.002 lux, about 100 times darker. The Moon's angular diameter (0.52 degrees mean) is almost equal to the Sun's (0.53), which is why total eclipses exist; without the Moon there would be no lunar and no total solar eclipses.

What is simplified or dramatised. Sudden disappearance is a thought experiment. Screen brightness at night is boosted, like dark-adapted eyes; the lux numbers are the real values. Space view distances are compressed unless "True scale" is on; the Sun is drawn not to scale. Continents are procedurally generated (not real geography). The "close Moon" preset (tide force 100x, ~83,000 km) is indicative only; the equilibrium tide model is not valid at that strength. The ocean response after the Moon vanishes is modelled as a simple exponential decay. "Moon twice as far" is hypothetical: at 3.8 cm/yr recession this would take far longer than the Sun's lifetime.

Sources: Laskar et al. 1993, Nature 361, 615 and 361, 608; Lissauer et al. 2012, Icarus 217, 77; Stephenson, Morrison and Hohenkerk 2016 (length of day); Williams 2000 (palaeorotation); Doodson/Cartwright tidal potential amplitudes; Krisciunas and Schaefer 1991 (lunar brightness).