Concept

Rotating frame — where it appears

A reference frame turning with two orbiting bodies, in which both sit still at the price of two fictitious forces. The two fictitious forces are the centrifugal one, which is a gradient and can be folded into a potential, and the Coriolis one, which cannot.

Named by 8 essays across 2 fields — each of them below, with the objects they name alongside it.

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

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 $Zero-velocity curves, mass fraction 0.15. Level sets of the Jacobi constant in the frame that rotates with the two bodies. A particle with a given value of C is confined to the side of its own curve where the kinetic energy would be positive; as C falls the curves open, first at the inner point between the bodies, then behind the smaller one, and finally around the whole system.$

The curve that says where a body cannot go

The restricted three-body problem has no solution and one conserved quantity. That quantity is enough to draw a boundary the body can never cross — without integrating anything, without knowing where it started, and for all time.

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

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

three revolutions, on a turning Earth. The ground track of a circular orbit at 420 km and 51.64° inclination, over 3 revolutions, on an equirectangular graticule. The latitude is a sine wave bounded by ±51.64° exactly — sin φ = sin i sin u, so the inclination is the highest latitude the orbit ever passes over, and it is reached twice per revolution. Each successive pass is displaced west by (ω⊕ − Ω̇) × 92.90 min = 23.61°, of which 0.32° is the orbital plane's own regression and the rest is the planet turning underneath: the vehicle comes back to nearly the same place in inertial space and the place has moved. The period used is the nodal one, 92.899 min against the Keplerian 92.970: J₂ makes the two differ by 4.31 s, which is 0.018° of walk per revolution and 102° in a year — the difference between a repeat track and a track that used to repeat. The map is equirectangular and therefore wrong about area everywhere; what it is right about is longitude difference, which is the whole of what this figure measures.

The line under a satellite

A ground track is an orbit seen from a frame that is turning, so every pass lands west of the last one. The track closes only when two periods are commensurable — which turns "look at the same place every day" into a condition on the altitude.

spaceflight · Ground tracks

How far across the ground a satellite can be seen from, at 0, 10, 30° masks. The footprint of a satellite — the largest angle at the Earth's centre between the point beneath it and a station that can still see it above an elevation mask — against altitude, on a logarithmic axis. λ = arccos(R cos ε / (R + h)) − ε. At low altitude it grows as the square root of the height, λ ≈ √(2h/R), drawn dashed for the horizon mask, so doubling a low orbit's altitude widens its footprint by only about forty per cent; far out it saturates, and no altitude sees past 90° − ε. At 420 km (a space station) the 0° footprint is 20.2°, a circle 2,254 km in radius holding 3.1 per cent of the Earth's surface. At 20,180 km (a navigation satellite) the 0° footprint is 76.1°, a circle 8,472 km in radius holding 38.0 per cent of the Earth's surface. At 35,786 km (geostationary) the 0° footprint is 81.3°, a circle 9,050 km in radius holding 42.4 per cent of the Earth's surface. Raising the mask costs most at low altitude: at 420 km a 10° mask takes 38 per cent off the footprint's radius and 62 per cent off its area, because most of what a low satellite could see is near the horizon, while at 35,786 km the same mask takes 20 per cent of the area.

The circle a station can see

A ground station can talk to a satellite only while the satellite is above its horizon, and the region it can do that from is a circle drawn round the point beneath the spacecraft. The circle grows as the square root of the altitude and then stops growing, the time spent inside it diverges at one altitude, and the stations that get the most contact are not where anyone would first put them.

spaceflight · Ground tracks $The quantity a capture has to change. The Jacobi constant along three trajectories through the same encounter with the secondary, integrated in the rotating frame at mass fraction 0.0009543. The undissipated one holds its value to 3e-11 over the whole passage — that is not an approximation but an identity of the equations of motion, and the residual is the integrator's. A body that enters through the neck at L₁ with the neck open leaves through it, because C has not changed and the neck is still open; a purely gravitational two-body encounter is time-reversible and cannot end in a bound orbit. The other two runs remove energy in the rotating frame. A drag proportional to velocity raises C continuously at a rate 2k v², and a single impulse against the motion raises it in a step. Both finish above C(L₁) = 3.039, which is the line at which the neck shuts and the body is trapped in the secondary's lobe. Which mechanism operated for a given satellite is not recoverable from its orbit today — but it is partly recoverable from the population, because the three leave different distributions behind.$

A moon that two bodies could not have caught

A body that falls into a planet's Hill sphere under gravity alone leaves it again, because the quantity the rotating frame conserves is the same on the way out as on the way in. Every captured moon is therefore a record of something that removed energy — and which of the three candidates did it is written in the inclinations rather than in any one orbit.

gravitation · Hill sphere

A loop round the Earth that the Earth does not make. The path over one year, in the frame that turns with the Earth (Earth at the centre, the Sun far to the left, the Earth's motion upward), of bodies on heliocentric orbits with the Earth's period and eccentricities of 0.05, 0.1, 0.15, started at the Earth's mean longitude. Each traces a retrograde loop round the Earth, about 2ae long in the direction of motion and ae wide towards the Sun: for e = 0.1, 0.395 by 0.199 AU. The small circle is the Earth's Hill sphere, 0.010 AU in radius, inside which the Earth's gravity dominates the Sun's tide. Every loop lies far outside it. The loop is a difference between two ellipses with one period, and would be traced the same way if the Earth had no mass at all.

A moon the Earth does not hold

Seen from the Earth, the small asteroid Kamoʻoalewa circles the planet once a year, backwards, between forty and a hundred times as far away as the Moon. It is not a moon. It orbits the Sun, with the Earth's period and a slightly different eccentricity, and the loop is the difference between two ellipses — one the Earth would trace even with no mass at all. What the Earth's gravity does is subtler: it keeps the loop from drifting away.

gravitation · Lagrange points

A way home that needs no engine

A spacecraft can leave the Earth so that the Moon's gravity turns it round and sends it home with no further burn. Drawn in the frame that turns with the Moon the path is a figure of eight. It was flown to the Moon by every early Apollo mission for one reason: if the engine fails, the crew comes back. That safety is exquisitely narrow. A tenth of a metre a second at departure moves the perilune by tens of kilometres and the return by thousands, and the perilune it allows is not one that most landing sites can use.

spaceflight · Patched conics

Named alongside it

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

Hill sphere Lagrange points Jacobi constant Restricted three-body problem Three-body problem Ground track Inclination Irregular satellites Mass ratio Orbital stability Sun-synchronous orbit Zero-velocity curve

All concepts