Astronomy & Celestial Mechanics
the sky, computed
Concept

Inclination — where it appears

The tilt of an orbit's plane against a chosen reference plane, and the quantity a spectroscopic orbit alone can never supply. It is what a transit or an astrometric orbit supplies and a radial velocity does not, which is why a velocity measurement gives a minimum mass rather than a mass.

Named by 15 essays across 6 fields — each of them below, with the objects they name alongside it.

The mass correction against the mass. Each planet's departure from the massless harmonic law, against its own mass in solar units, on logarithmic axes. The exact law puts every point on the diagonal. Jupiter and Saturn are the only planets whose mass correction is larger than the perturbations from everything else, and Saturn's measured departure has the opposite sign.

The third law is wrong by the mass of the planet

Kepler's harmonic law says the square of the period goes as the cube of the size. Newton's version has one more term in it, and the term is the orbiting body's own mass — negligible for a planet, decisive for a binary star, and the reason a period can be converted into a mass at all.

orbits · Harmonic law Two radial-velocity curves, and one mass ratio. The line-of-sight velocity of each star through one orbit of AI Phoenicis. Both curves are computed from the two masses and the period; what a spectrograph delivers is the reverse. The ratio of the amplitudes is the inverse ratio of the masses — 48.2 to 50.3 kilometres a second, so the heavier star moves more slowly — and the sum of the amplitudes with the period gives the mass sum, 2.437 solar masses, once the inclination is known from the eclipses.

The only stars whose masses are known

A star's mass cannot be measured by looking at it. It can be measured by watching two stars pull on each other, and if the pair also eclipses, the same observations give both radii as well — with no stellar model anywhere in the chain. A few hundred such systems calibrate everything else.

stars · Binary stars The light curve of AI Phoenicis, computed from its elements. Total light against orbital phase, computed by overlapping two discs of radius 1.805 and 2.9303 solar radii at an inclination of 88.5°, each weighted by its own surface brightness. The two eclipses hide the same area of sky and have different depths — 48.0 and 19.1 per cent — because what is lost is the light of whichever star is behind, and the ratio of the depths is therefore the ratio of the two surface brightnesses. Two things are left out and both matter to a real solution: this is the bolometric light rather than the light in a filter, and the discs are uniform, where a real one is limb-darkened and so has a deeper, rounder eclipse than the flat-bottomed one drawn here.

Two radii, from a light curve alone

The radius of a star is not measured. It is inferred, from a temperature and a luminosity, through a model. There is one exception — a pair of stars that eclipse each other, whose light curve and velocity curves between them give both radii, both masses and the ratio of temperatures with no model of a stellar interior anywhere in the chain.

stars · Binary stars 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 The wobble of a star with a circular companion. The star's velocity along the line of sight, over two orbits, computed from the companion's orbit. A circular orbit gives a sine wave; an eccentric one gives a skewed curve whose shape encodes the eccentricity.

The star moves, and the mass is a lower bound

A planet is found by watching its star fall towards it. The wobble gives a mass multiplied by the sine of an angle nobody has measured — and the shortfall is not a rounding error.

exoplanets · Reflex velocity Relativity switches the cycle off, halving its reach at a ratio of 0.80. The greatest eccentricity a Kozai–Lidov cycle reaches, against the strength of the orbit's own relativistic pericentre precession, measured in units of the cycle's own precession rate at zero eccentricity. The horizontal axis is logarithmic and spans three decades. At the left the relativistic term is negligible and the cycle reaches 0.838, which is the closed-form value for a start at 65 degrees and is what the integration is checked against. At the right it is gone. The mechanism depends on the pericentre staying put while the outer body pulls on the same side of the orbit for a whole cycle, and the relativistic precession is a competing rotation of that same pericentre; when it is faster, the pull averages away. The threshold sits near one by construction and the transition is sharp rather than gradual, with the reach halved at 0.80. What makes it matter is where the relativistic term is largest: it grows as the pericentre falls, so it strengthens exactly as the cycle drives the orbit inward, and it therefore sets a floor on the pericentre distance that this mechanism can deliver a body to.

The precession that switches the cycle off

A distant companion can trade an orbit's inclination for its eccentricity, over and over, and drive a pericentre almost onto the central body. General relativity's own precession competes with the mechanism for the same pericentre, and when it wins the cycle stops — sharply, at a ratio of one, which puts a floor on how close anything can be delivered.

orbits · Kozai–Lidov A star 1.24 times wider than it is tall, and 19 per cent brighter pole-on. Left, the meridional section of a star rotating at ω = 0.9257 of its critical angular velocity, computed from the Roche potential rather than sketched: the equator sits at 1.245 polar radii, and at the critical rate that ratio is exactly 1.5 whatever the star is made of. The same rotation expressed as a fraction of the critical equatorial speed is 0.768, and the two conventions differ by the distortion itself — a figure that prints one under the other's name is wrong by an amount that looks like rounding. Effective gravity at the equator is 0.329 of its polar value, so von Zeipel's flux law makes the pole hotter than the equator by a factor 1.320 at the theoretical exponent 0.25 and 1.232 at the 0.188 that interferometric imaging actually fits. Right, the apparent bolometric brightness against viewing inclination, integrated over the visible gravity-darkened surface: pole-on the star is 1.19 times brighter than edge-on, and the apparent temperature falls with it. The consequence is that a rapid rotator's place on the Hertzsprung–Russell diagram is partly a statement about the observer's position, which no spectrum taken alone can undo.

A temperature that depends on where the observer stands

A star turning near its break-up rate is half again as wide as it is tall, and its equator is thousands of degrees cooler than its poles. Neither of those is a small correction to a spectrum — the effective temperature and the luminosity such a star appears to have are partly statements about which way its axis happens to point.

starlight · Gravity darkening One amplitude, and every point on this curve fits it. The set of distances and inclinations that produce the same measured amplitude in one detector. A binary seen face-on radiates most strongly along its spin axis, so it can be twice as far away as an edge-on binary and still arrive with the same strain — the curve is (1 + cos²ι)/2 and the factor between its ends is exactly two. Nothing in a single detector's data distinguishes the two ends. What makes this worse rather than merely awkward is that the prior pulls the other way: an isotropically oriented population has half its members beyond 60 degrees, so most binaries really are closer to edge-on, and a posterior that combines a flat likelihood along this curve with that prior returns a distance biased low with an error bar that understates the range. The whole business of standard-siren cosmology is the business of cutting across this curve.

A distance tangled with an angle

A gravitational wave carries its own distance, with no ladder underneath it and nothing to calibrate. What it also carries, inseparably, is the orientation of the orbit that made it — and one detector cannot tell a nearby binary seen edge-on from one twice as far away seen face-on.

gravitation · Gravitational waves 15 revolutions of a 185 km swath. The band an instrument 185 km wide sweeps over 15 revolutions of a 233/16 repeat orbit at 98.2° and 699.6 km, drawn on an equirectangular map. Each band's edges are placed 92.5 km either side of the track on the sphere, so the bands really are the same width everywhere and only look wider towards the poles because this projection stretches longitude by 1/cos φ. Successive passes are 24.72° of longitude apart, and across the track at the equator that is 2691 km — so neighbouring passes leave strips about 2,506 km wide unimaged between them at the equator, while near ±82° the same bands lie on top of one another many times over. Filling the tropical strips is what the rest of the 16-day cycle is for.

A swath is sized at the equator

An imaging satellite's tracks crowd together towards the poles and spread apart towards the equator, so whether a swath leaves gaps is settled at the equator and nowhere else. The order in which a repeat cycle then closes those gaps is not orbital mechanics at all. It is a theorem about points on a circle.

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 figure of eight a geosynchronous orbit draws at 5°, 15°, 30° of inclination. The ground track over one sidereal day of a circular orbit whose period is exactly a sidereal day, at inclinations of 5, 15, 30°, centred on its own mean longitude. It is not a point. The latitude swings to ±i and back twice a day, and the longitude falls behind and then runs ahead of the Earth's rotation, because the rate at which an inclined orbit gains longitude is u̇ cos i / cos²φ: slowest at the nodes, where part of the motion is north–south, and fastest at the extremes of latitude, where all of it is eastward and a degree of longitude is shorter — so the track closes as a figure of eight. Its half-width in longitude is ±0.109° at 5°, ±0.993° at 15°, ±4.117° at 30° — exactly arcsin(tan²(i/2)) — against the small-inclination form i²/4 = ±0.109°, ±0.982°, ±3.927°: quadratic in the inclination, so the eight is tall and very thin. The longitude axis is stretched 8 times relative to the latitude axis, and without that stretch every one of these curves would be drawn as a vertical line. This is why a few degrees of inclination, which a geostationary operator spends most of its propellant preventing, moves the satellite a long way north and south and almost not at all east and west.

A stationary satellite that draws a figure of eight

A satellite with a period of exactly one sidereal day returns over the same ground every day, but only an orbit in the equator with no eccentricity returns over a single point. A few degrees of tilt draw a figure of eight, a little eccentricity a swing in longitude, and the two together draw the figure the Sun draws in the sky over a year.

spaceflight · Ground tracks A 5/5/1 constellation at one instant: where no satellite is above 0°. The points beneath the 5 satellites of a 5/5/1 Walker delta pattern at 43.5°, at 11,605 km, at one instant, with each satellite's 0° footprint — a circle 69.2° in radius at the Earth's centre — drawn round it, on an equirectangular map that swells the circles towards the poles. The shaded cells are ground with no satellite above the mask: none at this instant. Everywhere else is seen by between 1 and 3 at once. The largest empty circle at this instant is 67.6° in radius, computed exactly from the satellites' directions, inside the footprint, which is why there are none.

Five satellites and not four

No single orbit keeps a satellite above every point on the Earth. How many are needed is a question about the largest empty circle among their directions at the worst instant, and it has sharp answers. Four can never do it, five can from 11,605 kilometres up, and past that the arrangement of the orbits matters as much as their number.

spaceflight · Ground tracks A tumble removed with a coil and a compass, at 500 km. The rotation rate of a small spacecraft — principal moments 0.0067, 0.041, 0.043 kg m², the proportions of a three-unit cubesat — tumbling at 8.8° a second after release, against orbits at 500 km, with nothing to control it but magnetic coils driven by the B-dot law: a dipole opposite to the rate of change of the field measured aboard, capped at 0.2 A m². The field is a dipole tilted 9.2° from the Earth's axis and turning with the Earth. In a polar, 97.4°, orbit the rate settles at 0.12° a second over the last orbit drawn, passing 1° a second after 0.58 orbits; in a 51.6° orbit the rate settles at 0.12° a second over the last orbit drawn, passing 1° a second after 0.32 orbits; in an equatorial orbit the rate settles at 2.44° a second over the last orbit drawn. The law needs no knowledge of the spacecraft's attitude: a tumbling body sees the Earth's field swing round in its own frame, and a dipole opposing that swing produces a torque that removes the part of the spin perpendicular to the field. The polar orbit does not reach zero. It settles at 0.94 of twice the orbital rate, 0.13° a second, and twice the orbital rate is how fast the field direction itself turns round a polar orbit: a body turning with the field sees little change to oppose. An equatorial orbit keeps the field pointing nearly the same way all the way round, so the spin about it is reached only through the dipole's tilt and the Earth's turning, and 28 per cent of the starting rate is still there at the end.

A tumble stopped by the field it tumbles through

A small satellite leaves its deployer tumbling, and the first thing most of them do is stop, using nothing but a magnetometer and three coils. The law they run needs no idea where the satellite is pointing. What it cannot do, at any instant, is touch the spin about the local field line — so how much tumble survives is decided by how much the field's direction changes along the orbit, and the stillness it reaches is defined by the field rather than by the stars.

spaceflight · Attitude control Where the axes of rapid rotators point, in a sample chosen by brightness, at a darkening exponent of 0.19. The distribution of rotation-axis inclinations — 0° pole-on, 90° equator-on — for gravity-darkened stars whose axes point at random in space, surveyed down to a limit in apparent brightness, with the surface temperature following the local gravity to the power 0.19. The dashed curve is random orientation, which puts 13.4 per cent of stars within 30° of pole-on. At 0.9 of the critical rotation rate a star looks 1.093 times as luminous pole-on as equator-on, the survey reaches correspondingly further for the pole-on ones, and 14.4 per cent of the sample lies within 30° of pole-on; the nearly pole-on stars are 1.09 times as common as random orientation would make them. At 0.98 of the critical rotation rate a star looks 1.188 times as luminous pole-on as equator-on, the survey reaches correspondingly further for the pole-on ones, and 15.3 per cent of the sample lies within 30° of pole-on; the nearly pole-on stars are 1.19 times as common as random orientation would make them. Averaged over random orientations the apparent luminosity of each star equals its true luminosity to better than half a per cent, as it must; the tilt towards pole-on comes entirely from choosing stars by how bright they look.

The pole-on stars a brightness limit prefers

A rapidly rotating star looks brighter and hotter from its pole than from its equator, so a survey that picks stars by how bright they look should pick more of them pole-on. It does — and the surprise is how little. Averaged over random orientations, a gravity-darkened star's apparent luminosity is exactly its true one, because every photon goes somewhere; a brightness limit restores only a fraction of a per cent of bias, while the error in any single star is ten times larger and of either sign.

starlight · Gravity darkening One square root that raises the orbit and turns it. The velocity budget from a 300 km circular orbit to geostationary, against the plane change carried out along the way. Edelbaum's closed form — √(v₁² + v₂² − 2v₁v₂cos(½πΔi)) with Δi in radians — puts the whole continuous manoeuvre in one square root, and at Δi = 0 it collapses to |v₁ − v₂| = 4.651 km/s, which is the spiral's cost with no plane change in it. The two-impulse curve puts its rotation into the circularisation burn at apoapsis, where the vehicle is moving at 1.608 km/s and a rotation is cheap. At 28.5° the continuous transfer costs 5.951 km/s against the impulsive 4.256 — the plane change adds 1.300 to one and 0.363 to the other. That is the opposite of the usual claim that low thrust turns for free. It turns continuously, which is not the same thing: the gain is that the propellant is not the budget, and the Δv is worse here as it is everywhere else.

One square root that raises the orbit and turns it

Edelbaum put the plane change inside the same radical as the raise, and the half-pi in its cosine is the whole result — a continuous turn costs π/2 times an impulsive one below 140° and less above it.

spaceflight · Low-thrust transfer

Named alongside it

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

Ground trackSun-synchronous orbitLimb darkeningBarycentreCommensurabilityCritical rotationEccentricityEclipsing binaryEffective temperatureElevation maskFootprintGeostationary orbit

All concepts