Astronomy & Celestial Mechanics
the sky, computed
Concept

Titan — where it appears

Saturn's largest moon, the only moon with a thick atmosphere, and a body with a subsurface ocean. Its outward migration of about eleven centimetres a year is far faster than a constant tidal quality factor allows, pointing to resonant dissipation in Saturn.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

How far the Sun has to sink before the sky darkens, on four worlds. The dimming of the zenith sky relative to sunset, in magnitudes, against the Sun's depression, for single scattering in an isothermal atmosphere with the stated scale height and vertical optical depth, scattering isotropically: the Earth (R 6371 km, H 8.5 km, τ 0.097); Mars (dust) (R 3389.5 km, H 11.1 km, τ 0.5); Titan (haze) (R 2574.7 km, H 50 km, τ 4); Pluto (haze) (R 1188 km, H 50 km, τ 0.02). These are representative values for the layer that does the scattering — air on the Earth, dust on Mars, haze on Titan and Pluto — not full models of those atmospheres. The angle over which a sky darkens is set by the height of the scatterers against the size of the planet, √(2H/R), because that is the depression at which the shadow over the zenith has risen one scale height. The Earth dims by ten magnitudes at 8.6° of depression, which at the equator of a world whose solar day is 24 hours takes the Sun 35 minutes. Mars (dust) dims by ten magnitudes at 13.4° of depression, which at the equator of a world whose solar day is 24.66 hours takes the Sun 55 minutes. Titan (haze) dims by ten magnitudes at 29.8° of depression, which at the equator of a world whose solar day is 382.7 hours takes the Sun 31.7 hours. Pluto (haze) dims by ten magnitudes at 43.9° of depression, which at the equator of a world whose solar day is 153.3 hours takes the Sun 18.7 hours.

The air's height against the planet sets the twilight

On the Earth the sky overhead fades over about nine degrees of the Sun's descent. That angle is not a property of air or of sunlight. It is the square root of twice the height of the scattering layer divided by the radius of the planet, and on a small world with a tall haze it grows to tens of degrees — so that twilight on Titan lasts more than a day.

sky · Twilight Six moons, and how long each takes to move. The time each of Saturn's six largest moons would take to move outward by its own orbital radius, a divided by its rate of change, on a logarithmic scale. Bars: a constant tidal quality factor tuned to the inner moons' measured motion, under which the time grows steeply with distance and falls with mass — 6 billion years for Enceladus, 210 billion for Titan. Dashed: Titan's measured migration, 11 centimetres a year, which is 11 billion years — 19 times faster than the constant-Q prediction and within a factor of two of the inner moons, where the constant-Q picture puts them a factor of 33 apart. Resonance locking predicts exactly that: one time scale for every locked moon, set by the planet rather than by the moon.

A migration timed by the planet and not the moon

Titan is Saturn's most distant large moon, and by the standard account of tides it should be the slowest to move — a two-hundred-billion-year journey across its own orbit. It is moving eleven centimetres a year, twenty times faster, on the same time scale as the small moons close to the planet. A quality factor cannot do that. An oscillation inside Saturn, tuned to each moon in turn and dragged along by the planet's own slow evolution, can.

gravitation · Tidal heating

Named alongside it

The objects these essays reach for when they reach for this one.

Dimensional analysisDynamical tideEarth shadowEnceladusLove numbersMarsOptical depthOrbital resonancePlanetary oscillationsPlutoResonance lockingScale height

All concepts