Astronomy & Celestial Mechanics
the sky, computed
Concept

Tidal force — where it appears

The difference between the gravitational attraction at one part of a body and at its centre, which stretches it along the line to the attractor. It falls as the cube of the distance rather than the square, which is why the Moon raises a larger tide than the Sun despite the Sun's far greater pull.

Named by 9 essays across 4 fields — each of them below, with the objects they name alongside it.

The tidal field is a difference. The pull of a distant body at each point of a sphere, minus its pull at the sphere's centre. What remains stretches along the line to the source and squeezes across it — two bulges, not one.

The tide is a difference, which is why there are two of them

The Moon pulls the ocean toward it. That explains one bulge. The second one, on the far side, is the whole of the physics — and it comes from subtracting.

gravitation · Tides The tide across a moon, against the moon's own gravity. The tidal acceleration across a satellite and the satellite's own surface gravity, both in units of that surface gravity, against distance from the primary in planet radii. The tide falls as the inverse cube and the self-gravity does not fall at all, so they cross once — at 2.23 radii for the density ratio drawn. Inside the crossing the tide wins and a body held together only by its own weight comes apart.

The distance at which a moon stops holding together

The tide across a body falls as the inverse cube; the body's own gravity does not fall at all. There is therefore exactly one crossing, and Saturn's rings end within a few per cent of it.

gravitation · Tides Integrating the measured recession back: the Moon reaches the Earth 1.54 Gyr ago. The Earth–Moon separation and the length of the Earth's day, integrated backwards from the measured present recession rate of 3.83 cm per year. Constant-Q tidal friction makes a^(13/2) linear in time, so the history is a single line in a variable nobody plots, and it is calibrated to the laser-ranging measurement rather than to a modelled k₂/Q — the k₂/Q it implies is 0.0257, or Q = 11.6 for the Earth's k₂ of 0.299, which is a startlingly dissipative Earth. Run back at that rate the separation reaches zero 1.54 Gyr ago and crosses the Roche limit at 2.88 Earth radii only 4 years before it, so the drawing is cut off there rather than extrapolated. The Moon is 4.5 Gyr old, so this is a refutation and not a date: the present rate cannot have been the rate, and a mean Q of 34 — drawn dashed, reaching 4.51 Gyr — is the sort of value the age requires. Tidal rhythmites at 620 Myr put the day at 21.9 h and the Moon at 96.5 per cent of its present distance, and this history reads 20.1 h and 92.4 per cent — too fast and too close, which is the same failure the zero crossing is. Day length follows from total angular momentum, 23.93 h today, 9.84 h at half the present lunar distance and 4.97 h at the Roche limit, and depends on the separation alone: it is the same curve whatever Q is. The rate of lengthening the recession requires is 2.10 ms per century, against a tidal total of about 2.3 including the Sun's tide, which slows the Earth without moving the Moon, and an observed 1.75 from ancient eclipses and occultations — the shortfall being the Earth's moment of inertia falling as the mantle rebounds from the last glaciation.

A day five hours long

The tidal bulge leads, so the Earth's spin is being paid into the Moon's orbit. Run the measured payment backwards and two curves come out of one integration — a timeline that is refuted by the Moon's own age, and a day length that is refuted by nothing.

gravitation · Tides The Hill radius against the mass ratio, and where four planets keep their outermost moon. The Hill radius R_H = a(μ/3)^⅓ and the exact L₁ distance from the same quintic, both in units of the planet's own semi-major axis, against the mass ratio μ = m/(M+m) from 10⁻⁸ to 0.12. The two are the same line to the width of the stroke across the whole planetary range — the cube-root formula runs 0.33% short of the exact L₁ distance at the Earth's mass ratio and 10.0% short at the right-hand edge — so the departure between them is drawn on its own percentage scale at the right, which is the only place it can be seen. Below the curves are the numerically determined stability limits, 0.5 R_H for a prograde satellite and 0.7 R_H for a retrograde one, and the outermost known satellite of each of four planets, each at a Hill fraction computed from that planet's own mass and orbit: Moon 0.257, Sinope 0.451, Phoebe 0.198, Neso 0.424. Of the four, three are retrograde — Sinope, Phoebe, Neso — and they reach 0.451 of their planet's Hill radius against 0.257 for the one prograde satellite: 1.8 times as far out, in a diagram whose two stability limits stand in the ratio 0.7 to 0.5. No equilibrium argument predicts that, and it is what the drawing makes plain.

The region a planet may keep a moon in

A satellite is not held by orbiting but by orbiting inside the radius at which the Sun's tidal field would take it away. That radius is the same distance the innermost Lagrange point sits at, asked a different question — and the boundary the sky actually respects is smaller, by a factor that depends on which way the moon is going round.

gravitation · Hill sphere Two responses to the same transfer, turning round at q = 0.79 and q = 1. What conservative mass transfer does to the orbit and to the lobe, plotted against the mass ratio of donor to accretor on a logarithmic axis. Both curves are logarithmic derivatives with respect to the donor's mass, so a positive value means the quantity shrinks as the donor loses mass and a negative one means it grows. The orbit's response is exactly twice the mass ratio less one, which follows from holding the total mass and the total angular momentum fixed and nothing else, and it crosses zero at equal masses: transfer from the heavier star draws the orbit in, transfer from the lighter one pushes it out. The lobe's response adds to that the change in the lobe's shape, and it crosses zero earlier, at a mass ratio of 0.788. Between those two crossings the orbit is still widening while the lobe is already closing. To the right of both, a donor that loses mass finds its lobe shrinking around it, which is the runaway the essay is about: the transfer narrows the valve it is flowing through.

The flow that narrows its own channel

Two stars close enough share a surface, and the point where that surface pinches is a valve. What comes through it changes the orbit, and the orbit changes the valve — with a sign that reverses at equal masses, which is why some binaries transfer quietly for a hundred million years and others tear themselves apart in a thousand.

stars · Mass transfer Two lengths that cross at 1.1·10⁸ solar masses, above which nothing is seen. The radius at which a star of 1 solar radius and 1 solar mass is pulled apart by a black hole, and the hole's own horizon, both against the hole's mass and both on logarithmic axes. The tidal radius is the star's own radius times the cube root of the mass ratio, so it climbs with a slope of one third; the horizon is proportional to the mass, so it climbs with a slope of one. Two lines of different slope cross once, and this pair crosses at 1.14·10⁸ solar masses. Below that the star is torn apart outside the horizon, half of it is thrown out and half falls back, and the fallback is visible for months. Above it the star crosses the horizon while it is still a star, is swallowed whole, and produces no flare at all. The consequence is the reason these events are worth watching: a flare that is seen is an upper limit on the mass of the hole that made it, obtained without resolving anything, and it is the only such limit available for a hole that is not currently accreting.

A flare that puts a ceiling on a mass

A star torn apart by a black hole lights up for a year. The tidal radius grows as the cube root of the hole's mass and the horizon grows as the mass itself, so above about a hundred million suns the star is swallowed whole and nothing is seen — which makes the existence of a flare a measurement.

galaxies · Tidal disruption The Love number against central condensation: 3/2 for a uniform body, 2.4e-3 at n = 4. How willingly a body deforms, drawn against how concentrated it is. The horizontal axis is the polytropic index, which is a proxy for the run of density inside — n = 0 is uniform, n = 1.5 is a non-relativistic degenerate gas, n = 3 is a radiative star like the Sun — and the vertical axis is the fluid Love number k₂ on a logarithmic scale. The curve is the Radau equation integrated over each polytrope's own density profile, and its two ends are exact rather than fitted: a uniform incompressible body has k₂ = 3/2 exactly, and a body with all its mass at the centre has k₂ = 0, because a point mass has no quadrupole to offer. Everything real lies between. The fall is steep — four orders of magnitude across the family — which is what makes the number diagnostic: k₂ is not a mild function of structure, it is a sensitive one, and measuring it to ten per cent constrains the interior far better than measuring a mean density to the same precision. The Sun, at n ≈ 3, sits near 2.9e-2. Two conventions collide here and the figure uses one of them: the planetary literature's k₂, for which a uniform body gives 3/2. The stellar literature's apsidal-motion constant is half of this at every point, so a uniform body gives 0.75, and the two are the same quantity. What the picture cannot show is rigidity — every body on it is a fluid, and for anything smaller than a planet that assumption fails badly.

How much a world gives

A body pulled on from one side deforms, and how much it deforms is a single dimensionless number. That number is three halves for a uniform fluid, three hundredths for the Sun, and two thousandths for a moon made of ice — so measuring it is a measurement of what is inside.

gravitation · Love numbers The kick falls as b⁻² and the heating as b⁻⁴. Two quantities delivered by the same distant encounter, against impact parameter in units of the target's half-mass radius, both logarithmic and both scaled to cross near one. The upper line is the impulse itself, 2Gm/vb, which every star in a bound system receives almost equally — so it moves the system and changes nothing inside it. The lower line is what is left after the common part is subtracted: the difference of the impulse across the system, which is its gradient multiplied by the system's own size, one power of b smaller and squared in the energy. The slopes are −2 and −4 exactly, and they are measured off the drawn curves rather than quoted. What follows is the point. Encounters at impact parameter b arrive at a rate proportional to b db, so summing the impulse over all of them gives ∫b⁻¹ db, which diverges logarithmically and is the origin of the Coulomb logarithm that appears in every treatment of relaxation. Summing the heating gives ∫b⁻³ db, which converges: extending the population from ten half-mass radii out to 300 multiplies the summed impulse by 2.48 and the summed heating by 1.010. Distant encounters diffuse velocities and heat nothing, and a tidal-heating calculation therefore needs no cutoff at large impact parameter, where a relaxation calculation cannot proceed without one.

The part of a kick that heats nothing

Most of the impulse a passing mass delivers to a star cluster is delivered equally to every star in it, so the cluster moves and nothing inside it changes. What heats it is the difference across it — one power of the impact parameter smaller, squared in the energy — and that one distinction decides which encounters matter and which cannot.

gravitation · Impulse approximation The expansion rate and the acceleration, and which of them reaches inside an orbit. Two quantities per unit distance through cosmic time, both in units of today's H₀². The square of the expansion rate, H², falls steeply from the big bang and is 1.00 today by definition. The acceleration of the expansion, ä/a, is the sum of a matter-and-radiation part, −Ωₘ/(2a³) − Ωᵣ/a⁴, drawn dashed, and the cosmological constant's part, ΩΛ = 0.685, which is the same at every time. The sum was negative — the expansion decelerating — until the universe was 7.7 Gyr old, at a = 0.614 or redshift 0.63, and is 0.527 today. The equation of motion of anything orbiting inside a bound system carries ä/a and never H: the rate at which distant galaxies recede does not appear in it at all. Of ä/a, the matter part is the mean density of the universe, which inside a galaxy or a planetary system is already counted in the mass that is doing the holding, many million times over. What is left is the constant: a fixed outward acceleration per unit distance, ΩΛ H₀², that does not grow with time and does not care how fast the universe is expanding.

An orbit feels the acceleration and never the rate

If space expands, it is natural to ask why the Earth's orbit does not. The answer is not that gravity resists the stretching. It is that the expansion rate never appears in the equation of motion of a bound orbit at all — only the acceleration does, and of that only the cosmological constant's part survives, as a fixed outward push that moves the Earth's orbit once, by twelve picometres, and never again.

cosmology · Expansion

Named alongside it

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

Roche limitAngular momentumTidal lockingAccretionAccretion discBlack holeEquipotentialGravity gradientLagrange pointsMass ratioOrbital stabilityTidal dissipation

All concepts