Astronomy & Celestial Mechanics
the sky, computed
Concept

Effective potential — where it appears

The radial potential obtained by folding a body's conserved angular momentum into the central potential, so that the motion becomes one-dimensional. The centrifugal barrier it produces is why a body with any angular momentum at all cannot reach the centre, however much energy it has.

Named by 11 essays across 4 fields — each of them below, with the objects they name alongside it.

A vector that does not change, drawn five times. The Laplace–Runge–Lenz vector A = v × L − GM r̂, constructed at five points of one orbit at e = 0.45 under a force ∝ r^−2, each drawn from the body rather than from the focus so that its length and direction can be compared point by point. Under the inverse square every one of them is the same vector: identical to machine precision in length and in direction. Its length is 0.4500, which is the eccentricity, and it points at pericentre — so the orbit's orientation is a conserved quantity and not a constant of integration, which is why the ellipse does not turn.

The third thing that is conserved

Energy fixes an orbit's size and angular momentum fixes its shape. Neither fixes which way it points — so a curve that closes needs a third conserved quantity, and only two force laws in the universe supply one.

orbits · Conic sections The effective potential, for three angular momenta. The radial motion of an orbiting body is one-dimensional motion in an effective potential: the attraction −GM/r plus the centrifugal term L²/2r² that the angular momentum contributes. The barrier at small radius is what stops a body with any angular momentum at all from reaching the centre, and the bottom of each well is the circular orbit.

The wall that angular momentum builds

A body falling toward a star almost never arrives. Sideways motion, which looks like a detail of the initial conditions, turns the attraction into a well with a wall around the middle of it — and the wall is why it costs more fuel to hit the Sun than to leave the solar system.

orbits · Effective potential The five Lagrange points at mass fraction 0.12. The five points at which a small body can keep station with two larger ones. The three on the line of centres are roots of a quintic and are unstable; the two forming equilateral triangles are stable for a sufficiently lopsided mass ratio.

Five places that keep station, in a problem with no solution

Three bodies under gravity cannot be solved. Restrict the problem slightly and five exact answers fall out anyway — three of them roots of a quintic, two of them perfect equilateral triangles.

gravitation · Lagrange points Three bodies, integrated. Three equal masses integrated forward under mutual gravity: the figure-eight choreography — all three bodies on one closed curve. Every point is a step of the equations of motion, and the total energy is conserved to 2.5e-11 across the run.

Three bodies, and what "no solution" actually means

The three-body problem is routinely called unsolvable. Trajectories are computed for it every day, exact periodic solutions are known, and both statements are true — the word is doing more work than it looks.

gravitation · The three-body problem Mercury's perihelion, term by term. The observed advance of Mercury's perihelion is 5600 arcseconds per century against the equinox. Almost all of it is the equinox: the coordinate frame itself turns, and removing it leaves 574. Subtracting the perturbations of the other planets — computed by Le Verrier in 1859 and refined many times since — leaves 42.98 arcseconds a century that nothing in Newtonian gravitation accounts for. The bars are logarithmic in nothing; they are the real proportions, which is why the residual is barely visible beside the frame term.

Forty-three arcseconds, after everything else

Mercury's perihelion moves through 5,600 arcseconds a century. Nearly all of that is the coordinate system turning, and almost all of the rest is the other planets. What was left over was 43 — under one per cent of the raw number, and the most consequential residual in the history of the subject.

gravitation · Relativistic orbits The horizon and the orbit that cannot come back, against the hole's spin. Two radii round a rotating black hole, in gravitational radii GM/c², against its spin a from −1 (an orbit against the rotation) to +1 (with it), for orbits in the equatorial plane. The lower curve is the horizon, 1 + √(1 − a²), which is the same whichever way the orbit goes. The upper one is the marginally bound orbit, 2 − a + 2√(1 − a): the closest a body falling in from far away can pass and still escape back out. For a hole with no spin the horizon is at 2 and the marginally bound orbit at 4 — twice as far out. With maximal spin the marginally bound orbit comes in to 1.09 for a prograde orbit (a = 0.998) and moves out to 5.83 for a retrograde one. A star whose tidal radius lies inside this curve is swallowed whole, so the curve, not the horizon, is the line a disruption flare is measured against — and it moves by a factor of 5.3 with the spin.

The line a star is swallowed at is not the horizon

A star torn apart by a black hole makes a flare, and a star swallowed whole makes nothing, so the heaviest hole that can produce a flare is a measurement. That ceiling is set not at the horizon but at the closest orbit from which infalling matter can still come back out — twice as far out for a hole that does not spin, and moved by a factor of five by the hole's rotation.

galaxies · Tidal disruption The two force laws whose orbits close. The apsidal angle — the angle swept from periapsis to the next apoapsis — against the exponent of the force law, for F ∝ r^p. The dashed curve is the near-circular limit π/√(3+p), which has a closed form; the solid curve is the same angle for an orbit of eccentricity 0.4, computed by quadrature of ∫(L/r²)dr/√(2(E−U)) between its two turning points, with U the effective potential. An orbit closes when the apsidal angle is a rational multiple of π, and an orbit closes at every eccentricity only where the two curves meet: p = −2 at exactly 180° and p = +1 at exactly 90°, which is Bertrand's theorem. The quadrature returns 180.0000° and 90.0000° at those two exponents and departs from the near-circular curve by 1.4° at p = 0. The angle diverges as p approaches −3, where the circular orbit stops being stable and there is no well left to oscillate in.

Only two force laws let an orbit come back

That a planet returns to the same point of its own path after one lap is not a fact about orbits. It is a fact about the exponent in the force, and out of the whole continuum of attractions only two — the inverse square, and a spring — bring every bound orbit back to where it started.

orbits · Effective potential The same well after the primary has lost 45 per cent of its mass. Two effective potentials for one body: the solid curve before the primary loses mass and the faint one after, both at the same angular momentum, because a central force of any strength exerts no torque. The well shallows and its floor moves out from r = 1.00 to 1.82. The body's own level moves with it — from E = -0.420 to -0.127 — and the two horizontal lines are drawn where the radial action is conserved, which puts the turning points at 0.71–1.67 before and 1.30–3.03 after. The ratio between them is 2.3333 in both, so the orbit is the same shape at a larger size: everything about the body's path has scaled and its eccentricity of 0.4 has not moved. That is what a slow change leaves behind, and it is not what a sudden one leaves.

The well moves, and the body keeps its share of it

When the Sun becomes a white dwarf it will throw away half its mass, and every planet's orbit will swell by the same factor. Their eccentricities will not change at all — provided the loss is slow, and the only meaning "slow" has here is slow compared with one orbital period.

orbits · Effective potential One history of mass loss, three orbits, three outcomes. The eccentricity of three planets' orbits against the fraction of the star's mass lost so far, as the star sheds 45 per cent of its mass in 6 equal pulses at irregular intervals. Each planet starts with e = 0.05. The pulses are the same events in the star's own time, and the planets differ only in their periods, so each pulse lasts 3, 0.3, 0.03 orbital periods respectively. For the innermost planet, to which every pulse is slow, the eccentricity ends at 0.050: the orbit swells and keeps its shape. For the outermost, to which every pulse is sudden, each pulse kicks the eccentricity by an amount set by where the planet happened to be, and it ends at 0.298. The middle planet ends at 0.090. The same total loss leaves three different shapes, and only one of them is predicted by either limit alone.

The size forgets how the mass was lost, and the shape does not

A dying star loses half its mass, and a planet round it keeps its orbit's shape if the loss is slow and has it scrambled if the loss is sudden. Real stars lose mass in pulses — each one sudden or slow depending on the planet — and what survives a train of pulses is neither answer: the orbit's size settles to the slow value quickly, while its eccentricity random-walks and keeps a record of the history.

orbits · Effective potential Where a spinning sphere of rubble is steepest. The slope of the surface of a spinning sphere of loose material against latitude — the angle between the local vertical and the combined pull of gravity and rotation — for spins of 30%, 60%, 80%, 90%, 97% of the critical rate, measured in its square, at which the equator would lift off: for a density of 1.19 g cm⁻³ the critical period is 3.03 hours and these are periods of 5.5, 3.9, 3.4, 3.2, 3.1 hours. The slope is zero at the pole and at the equator, where the pull and the vertical are aligned by symmetry, and steepest in between: at 30% it peaks at 10° at latitude 40°, at 97% at 70° at 10°. The band marks the angle of repose of loose rock, 30° to 40°. From 80% of the critical rate the steepest band is past it: the surface begins to slide well before the equator has lost its weight.

A surface that slides before the body sheds

A rubble pile spun towards its limit does not wait for its equator to lose its weight. Long before that, its mid-latitudes grow steeper than loose rock can stand, and the surface flows — downhill, which on a spinning body means towards the equator. The ridge that so many small asteroids wear is built in that interval, at spins a sixth slower than the one that would fling material off, and whether it is built at all depends on a cohesion of less than a pascal.

orbits · Rubble piles How much thicker the cable must be at the top. The ratio of an Earth elevator cable's cross-section at synchronous orbit to its cross-section at the ground, as a power of ten, against the specific strength of its material, for safety factors of 1 and 2. The taper is exp(ΔΦ/(σ/ρ)) with ΔΦ = 48.4 MJ/kg, the same exponential as the rocket equation. With a safety factor of 2: high-strength steel (0.25 MJ/kg) beyond 10³⁰; Kevlar (2.5 MJ/kg) about 6.7×10¹⁶; carbon fibre (3.6 MJ/kg) about 4.8×10¹¹; Zylon (3.8 MJ/kg) about 1.2×10¹¹; nanotube yarn, today (4 MJ/kg) about 3.3×10¹⁰; ideal nanotube (50 MJ/kg) about 7. Only a material with tens of megajoules per kilogram — near the ideal strength of a carbon–carbon bond, ten times anything yet made in bulk — gives a cable that is not absurd.

A cable with a rocket equation of its own

A cable hanging from synchronous orbit to the equator would let a climber reach orbit on electricity, with no rocket at all. But the cable has to hold its own weight, and a cable that holds its own weight at constant stress must thicken exponentially with the potential it spans. The exponent is the same 48 megajoules per kilogram a rocket must supply, divided by the material's strength per unit mass. The rocket equation has not been escaped; it has moved from the propellant to the rope.

spaceflight · Rocket equation

Named alongside it

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

Angular momentumEccentricityAdiabatic invariantApsidal precessionCentrifugal barrierImpulsive approximationInverse square lawLagrange pointsMass lossMass ratioOrbital stabilitySpecific orbital energy

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