Astronomy & Celestial Mechanics
the sky, computed
Concept

Eccentricity — where it appears

The number that fixes the shape of a conic orbit, being zero for a circle, below one for an ellipse and above one for a hyperbola. It fixes the shape and says nothing about the size, and an orbit at 0.02 is visually indistinguishable from a circle while being physically quite different.

Named by 54 essays across 5 fields — each of them below, with the objects they name alongside it.

An orbit at eccentricity 0.6. An orbit of eccentricity 0.6. The primary sits at a focus, offset from the centre by 0.6 of the semi-major axis, and the closest and furthest points differ by a factor of 4.00.

The orbit is an ellipse, and the Sun is not in the middle of it

Kepler's first law is usually drawn wrong. The interesting content is not the ellipse — it is the focus, and the fact that one of the two is empty.

orbits · The ellipse Equal areas in equal times, at eccentricity 0.65. Positions of an orbiting body at equal intervals of time, obtained by solving Kepler's equation. The two shaded sectors span the same interval and enclose the same area — a long thin one at the far end, a short fat one at the close approach.

Equal areas in equal times, which is angular momentum in disguise

Kepler's second law is a statement about the area a radius line sweeps. It looks like an odd thing to have noticed, and it turns out to be a conservation law arriving eighty years early.

orbits · Angular momentum What an eccentricity does to the shape, and what it does to the offset. The fractional flattening 1 − b/a and the fractional focal offset c/a, against eccentricity. The offset is first order in e and the flattening is second, so at Earth's e = 0.0167 the outline is 0.014% from a circle while the Sun sits 1.67% of the semi-major axis off the centre — a factor of 120.

An orbit can look exactly like a circle and still not be one

Earth's orbit departs from a circle by fourteen parts in a hundred thousand. The Sun's offset from its centre is a hundred and twenty times larger, and everything interesting is in the offset.

orbits · The ellipse The three anomalies at E = 1.15 rad. The auxiliary circle construction. The eccentric anomaly E is measured at the centre, the true anomaly ν at the focus, and the mean anomaly M is time expressed as an angle. Here E = 1.150, ν = 1.827 and M = 0.602 radians, related by M = E − e sin E.

The position that has no formula, and is computed anyway

Kepler's equation relates where a body is to when it is there. It cannot be solved in elementary functions, Kepler said so, and nobody has managed it since — which has stopped nothing.

orbits · The ellipse Period against size for the planets, around the Sun. Orbital period against semi-major axis on logarithmic axes. The line has slope exactly three-halves — the harmonic law — and the measured bodies sit on it.

The law that links period to size, and weighs everything

Kepler found that the square of the period goes as the cube of the orbit. Newton found the constant of proportionality, and that constant is a mass — which is how every mass in astronomy has been obtained since.

orbits · Harmonic law Every orbit one force allows. Circle, ellipse, parabola and hyperbola, all sharing a focus and a closest approach. The eccentricity alone decides which one a body is on, and whether it returns.

Every orbit one force allows, and the number that picks between them

Circle, ellipse, parabola, hyperbola. A single inverse-square force permits exactly these four, and one number decides which — including whether the body ever comes back.

orbits · Conic sections An orbit at i = 42°, Ω = 35°, ω = 55°. The three orientation elements. The orbit is generated in its own plane and rotated by the standard sequence, so the inclination, the node line and the argument of periapsis are the angles that produced the drawn curve rather than labels applied to it.

Six numbers that fix an orbit for all time, and the sixth is the awkward one

Five of the orbital elements describe a curve that never changes. The sixth says where on it the body is, and it is the only one that has to keep being measured.

orbits · Orbital elements The energy budget of an orbit at e = 0.7. Kinetic, potential and total energy per unit mass against distance from the primary, in units where GM = 1. The total is a horizontal line — it depends only on the semi-major axis — and the vis-viva relation is that statement solved for the speed.

One equation for the speed anywhere, and the eccentricity is not in it

The vis-viva relation gives the speed at any point of any orbit from two numbers. What it leaves out is the surprise — the shape of the orbit does not appear at all.

orbits · Vis-viva Three rotations, applied in order. An orbit of eccentricity 0.45 carried into space by the three orientation angles, one panel per rotation: the argument of periapsis ω = 40°, then the inclination i = 42°, then the longitude of the ascending node Ω = 55°. The last panel applies the same three angles in the reverse order and arrives somewhere else, because rotations about different axes do not commute.

Three rotations that put an orbit in space, and they do not commute

An orbit's orientation takes three angles. Give the same three angles in a different order and the orbit ends up somewhere else — which is why the convention is part of the data.

orbits · Orbital elements 1I/ʻOumuamua: an orbit at e = 1.201, and the angle it turned through. The open branch of a conic at eccentricity 1.201 and periapsis 0.2559 AU, with the Sun at the occupied focus. Two numbers fix everything else: |a| = q/(e−1) = 1.273 AU, an impact parameter b = |a|√(e²−1) = 0.847 AU, an asymptote at ν∞ = arccos(−1/e) = 146.37° from periapsis, and a deflection of δ = 2ν∞ − 180° = 112.74° — which is the same statement as sin(δ/2) = 1/e. The asymptotes cross at |a|e = 1.529 AU from the Sun on the periapsis side, where a bound orbit's centre would be on the other. The speed left over at infinity is √(μ/|a|) = 26.40 km/s. Schematic in one respect: the drawing runs at about 119 px to the AU, so the Sun's own disc would be far smaller than its marker.

The orbit that has no period

Above an eccentricity of 1 the conic is open, the energy is positive and the semi-major axis is negative — and the vis-viva relation survives the sign change without a single alteration. What replaces the period is a speed, and that speed is what says where a visitor came from.

orbits · Hyperbolic orbits Ceres from five directions and no distance. Ceres seen five times over 41 days, from an Earth on a circular orbit, reduced in the plane. Each sighting gives a direction and no range, so the object is somewhere on its sight line; the five lines here span 1.37° of geocentric arc altogether, and Earth's own motion supplies the only baseline there is — 0.403 AU of its 0.691 AU of travel lies across the sight lines. A planar orbit is four numbers, so five angles over-determine it and one orbit comes out: a = 2.7658 AU, e = 0.0785. That is the answer and not an input — the sightings were generated at a = 2.7658 AU and e = 0.0785, and the solve, which sees only the directions and the dates, returns them to 3e-12. The two shaded sectors are what closes the determination: between the first and middle sightings the radius vector sweeps 0.2993 AU² in 21.0 days and between the middle and last 0.2853 AU² in 20.0 days, a ratio of 1.04918 against a time ratio of 1.04918. Slide all three crossings out along their sight lines together and that equality fails at once, so it fixes the distance by itself, with no propagation anywhere in the argument — and it gives a = 2.7658 AU over again. The two dashed curves are candidates that thread the same three sight lines at 85% and 108% of the recovered distance: a = 1.86 AU at e = 0.35, sweeping its areas in a ratio 4.5% wrong; and a = 5.61 AU at e = 0.49, sweeping its areas in a ratio 2.9% wrong. A few per cent in the distance is an orbit of another kind, which is the same fact the conditioning panel measures: one arcsecond of angle error moves a by 0.60% on this arc.

Five directions and no distance among them

An image of a moving point records an angle and throws the range away, so an orbit has to be assembled out of angles alone. How many angles are needed is not a detail of the method — it is the whole of what a determination is.

orbits · Orbit determination Four conics through one periapsis, drawn by one solve. Distance from the Sun against time for four orbits sharing a periapsis of 0.5 AU, at e = 0.6, e = 1, e = 1.0001, e = 1.4, over 900 days. Every point on every curve came from the same universal Kepler solve — no branch on the conic class anywhere in it — and each curve was then checked against the classical solution of its own kind at the midpoint: Kepler's equation at e = 0.6 agrees to machine precision; Barker's cubic at e = 1 agrees to machine precision; e sinh H − H at e = 1.0001 agrees to machine precision; e sinh H − H at e = 1.4 agrees to machine precision. The curve to read twice is e = 1.0001: over this arc it lies within 0.02% of the parabola and is indistinguishable from it, and it is the only one of the four whose fate the drawing cannot show. What the classical parameterisation costs there is arithmetic rather than impossibility: at the midpoint of this arc, e sinh H − H throws away 3.3 of its sixteen digits to cancellation, against 0.2 at e = 1.4 — enough to matter to an ephemeris and not enough to stop a plot.

The one solve that does not ask which conic it is

Kepler's equation is for ellipses, Barker's cubic for parabolas, and a hyperbolic sine for the rest — three parameterisations of one motion, each worst exactly where its neighbour takes over. The universal variable removes the question, and the removal is not a convenience.

orbits · Universal variables Four averages of one distance, and the two of them that are the semi-major axis. The average distance of a body from its primary, against eccentricity and in units of the semi-major axis, computed four ways: averaged over time, over true anomaly, over eccentric anomaly, and as the harmonic mean in time. Every curve is a quadrature over the orbit — 2,048 panels uniform in eccentric anomaly, with Kepler's equation supplying the time weight — and not a closed form. Two of the four are exactly a at every eccentricity, which is why they are drawn as one line: the eccentric-anomaly average, because the mean of cos E over a turn is zero, and the harmonic mean in time, because the time weight cancels 1/r at every node before the sum begins. The other two are not: the time average is a(1 + e²/2), which rises to 1.4050 a at e = 0.9, and the true-anomaly average is a√(1−e²) — the semi-minor axis — which falls to 0.4359 a there. So a is the average distance in two senses out of four, and the ordering b ≤ a ≤ ⟨r⟩ₜ holds at every eccentricity with equality only on the circle. At Earth's e = 0.0167 the four agree to 0.014%, and at Mercury's e = 0.2056 the spread is 2.14%. The distinction is invisible for the planets and unavoidable for a comet, and it is the reason a quoted "mean distance" has to say which mean.

The average depends on what is being averaged

Four ways of averaging one orbit's distance from its primary give four different numbers, and only two of them are the semi-major axis. Which two is not a matter of convention, and the same arithmetic decides how much sunlight a planet receives in a year.

orbits · Orbital averages Earth's eccentricity is a sum of 8 sinusoids. The eccentricity of Earth over 800 thousand years, from the Laplace–Lagrange solution for all eight planets — the secular matrix built from the JPL masses and semi-major axes, symmetrised, and diagonalised by Jacobi rotations. It runs between 0.0035 and 0.0436, and it has no period, because it is a sum of 8 incommensurable frequencies. The two largest contributions to this planet are the modes at 3.73 and 7.33 arcseconds per year, drawn as the flat lines: those are constants, and everything moving in the figure is their beat. The check is the quadratic form ½ΣΛe², which a symmetric secular matrix conserves exactly and which drifts by 6.7e-16 across the whole interval — computed from the same curves the figure draws and from nothing the eigenvalues were fitted to. The true angular momentum deficit, Σ Λ(1 − √(1−e²)), drifts by 7.1e-4, and that difference is not an error either: the two agree only to fourth order in e, and Mercury at 0.206 supplies almost all of the gap.

No planet has an eccentricity of its own

Strip the short-period terms out of the planetary equations and what is left is a linear system. Its eigenvectors are modes of the whole solar system, and the number a catalogue quotes for a planet's eccentricity turns out to be a reading of a clock rather than a property of the planet.

orbits · Secular theory Two bodies at a mass ratio of 3 to 1. Both bodies orbit their common centre of mass, on similar ellipses whose sizes are in inverse proportion to the masses — here 3 to 1, so the heavier body's path is 3 times smaller.

Neither body is still, and the wobble is how planets are found

A planet does not orbit its star. Both orbit a point between them, and the star's share of that motion is small, measurable, and the reason thousands of planets are known.

gravitation · The two-body problem 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 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 The two-body problem, and the one-body problem it is. Left: two bodies with mass ratio 0.4 on ellipses of eccentricity 0.5 about their common barycentre, the heavier one on the smaller orbit. Right: the same system as one body of the reduced mass on a single ellipse of the same eccentricity about a fixed centre, at the separation of the two. The right-hand curve is the point-by-point difference of the two left-hand curves, so the substitution is drawn, not merely stated.

Two bodies replaced by one that does not exist

The two-body problem is solved by turning it into a one-body problem about a fixed centre. The substitution is not an approximation — it is exact, and the body it invents has a mass no object in the system has.

gravitation · The two-body problem 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 The Kirkwood gaps. Asteroid numbers against semi-major axis, with the resonant radii marked. Each gap sits where the orbital period is a simple fraction of Jupiter's, and each of those radii is computed from the harmonic law rather than placed by eye.

Resonance clears a gap in one place and locks a moon in another

When two orbital periods are in a simple ratio, small tugs stop averaging away and start accumulating. Sometimes that empties a region entirely. Sometimes it holds three moons together for the age of the solar system.

gravitation · Resonance 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 PSR B1913+16: 45 years of periastron arriving 88 seconds early. The cumulative shift of periastron passage for PSR B1913+16, Δt = ½(Ṗ_b/P_b)T², over 45 years from its discovery in 1975. The orbital period is 0.322997449 days and is measured to be shortening at 2.423e-12 seconds per second — a change in the twelfth decimal place, which over a career accumulates to 87.5 seconds. That is the whole reason the effect is measurable. Ṗ_b is taken here as a measured quantity and no radiated power is derived from it; the parabola is the general-relativistic prediction as published, and the points are that prediction scaled by the published ratio of observed to predicted decay, 0.997 ± 0.002 (Hulse & Taylor 1975; Weisberg & Huckins 2016), which is how the agreement is quoted. The error bars are drawn: at the last point the bar is 1.25 px tall against a dot 9 pixels across, so they are invisible, and their invisibility is the result. The campaign has run 45 years.

An orbit measured to be shrinking

A 7.75-hour orbital period that shortens by 68 nanoseconds each turn is beyond any single measurement, and unmissable after fifty thousand of them, because the shift accumulates as the square of the elapsed time. Forty-five years of pulse arrival times have made it 87.5 seconds, which is why the rate is a measured quantity rather than an inferred one.

gravitation · Relativistic orbits The Sun's altitude through the day at latitude 52°. Solar altitude against the hour of the day, at one latitude, for the solstices and the equinox. Where a curve crosses zero is sunrise or sunset, and the width between the crossings is the length of the day.

The Sun's path, and the tilt that makes the seasons

Summer is not when the Earth is closest to the Sun — that happens in January. It is when the Sun climbs higher and stays up longer, and both come from a 23.4° tilt.

sky · Seasons Phases are a viewing angle, not a shadow. A satellite at eight points of its orbit. Exactly half of it is lit at every one of them; what changes is how much of the lit half faces the centre. Nothing is in shadow except at an eclipse.

Phases are not shadows, and eclipses are

Half the Moon is lit at every instant of every month. The phase is which part of the lit half faces the Earth — and confusing that with a shadow is the commonest error in astronomy.

sky · Phases and eclipses The equation of time, and its two causes. The difference between a sundial and a clock over the year, in minutes, computed from Kepler's equation and the tilt. The eccentricity term has one cycle a year and the obliquity term has two; their sum runs from −14.2 to 16.4 minutes.

The Sun is a bad clock, by up to sixteen minutes

Solar noon and twelve o'clock are not the same instant, and the discrepancy runs through a fixed annual cycle. It has two causes, one from the shape of the orbit and one from the tilt of the axis.

sky · Equation of time Mars through 8.3 months of sky. The geocentric ecliptic longitude and latitude of Mars over 252 days — 8.3 months — centred on opposition, computed as the direction of P − E with both orbits taken as circles: the Earth's of radius 1 AU, the planet's of 1.5237 AU inclined 1.850°. The motion reverses for 72.7 days and backs up 15.94° of longitude, and that interval is centred on opposition to within 0.0000° of longitude. The track closes on itself: over 149 days the planet visits the same point of the sky twice, and the loop it encloses is 15.9° long and 2.94° tall. Longitude and latitude are at different scales: a degree of latitude is drawn 4.6 times a degree of longitude, because the sweep of 51° in longitude and 4.5° in latitude will not share a scale on one page. Circular orbits mean one loop for every apparition, and that is the real cost of this figure: Mars is 0.525 AU away at the opposition drawn here, its true distance at opposition varies with its eccentricity, and the real loops differ in size from one apparition to the next because of it. The marks are 20 days apart, and they crowd where the motion stops.

The loop a planet does not make

Mars stops in the sky, backs up for ten weeks, and goes on. Neither orbit reverses anywhere — the loop belongs to the difference of two position vectors seen from one of them, and its width and duration are fixed by the ratio of the two radii and by nothing else.

sky · Apparent motion A gravity assist with a 70° turn. The velocity triangle of a flyby. In the planet's frame the spacecraft's speed is unchanged and only its direction turns; adding the planet's own velocity converts that turn into a gain in speed measured from the Sun.

Stealing speed from a planet, which does not notice

A flyby cannot change a spacecraft's speed relative to the planet. It changes its direction — and adding the planet's own motion back turns that into free velocity.

spaceflight · Gravity assist A transit of a planet 0.103 of its star's radius. The star's brightness through one transit, computed by integrating the limb-darkened stellar disc over the region the planet covers. The depth is 1.26%, deeper than (Rp/R⋆)² = 0.01055 because the planet crosses a limb-darkened disc whose centre is brighter than its average. The four contact points are where the two discs are externally and internally tangent, at separations 1 ± 0.103 stellar radii.

Four contact points, and what they fix

The depth of a transit gives a radius ratio. The shape gives the impact parameter, and then — through nothing but Kepler's third law — the mean density of the star being crossed.

exoplanets · Transits Everything close in is circular, and nothing else has to be. Orbital eccentricity against period for fifteen real planets, with the tidal circularisation boundary computed from τ_e = (2/21)(Q′/n)(M_p/M⋆)(a/R_p)⁵ for a Jupiter with Q′ = 1e+6 and an age of 3 billion years. It falls at 4.5 days, and the reason it is a wall rather than a slope is the fifth power: at half the period the timescale is 91 times shorter. Nothing inside it has a measurable eccentricity, and outside it eccentricities run to 0.95 — which is the number to hold on to, because a planet on a 0.93 orbit at 111 days comes within 0.030 AU of its star at periastron, closer than Mercury, and is being circularised as it is observed.

A planet where one cannot form

A Jupiter at four days orbits inside a region that was too hot to hold ice and too small to hold the material. It did not form there — and the distribution of eccentricities says which of two journeys brought it in.

exoplanets · Planet migration 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 The best-fitting eccentricity of a circular orbit. What a fitted eccentricity comes out at when the orbit's true eccentricity is 0 and each component of the eccentricity vector carries an error of 0.03. The distribution is not centred on the truth and cannot be: an eccentricity is the length of the vector (e cos ϖ, e sin ϖ), lengths are not negative, and a quantity bounded below by zero whose components scatter symmetrically has a distribution pushed away from the bound. For a circular orbit the most likely fitted value is exactly one error bar, 0.0300 here, and the mean is 0.0376 — 1.2533 error bars, which is √(π/2) and comes from geometry rather than from any property of the data. The practical consequence is a catalogue of small eccentricities that are all measurements of their own error bars, and the fix is not a better fit but a different question: an upper limit rather than a value.

An eccentricity that cannot be zero

Fit an orbit to noisy data and the eccentricity that comes back is never zero, not even when the orbit is a perfect circle. The reason has nothing to do with the data and everything to do with the fact that a length cannot be negative.

orbits · Orbit determination A transit that lasts 4.0 times longer at one end of the orbit than the other. The duration of a transit, relative to what a circular orbit of the same period around the same star would give, against the orientation of the orbit. A planet transiting near perihelion is moving fastest and its transit is shortest; one transiting near aphelion is slowest and its transit is longest. The two extremes are exact reciprocals — the circular duration is their geometric mean, whatever the eccentricity — and at e = 0.6 they differ by a factor of (1+e)/(1−e), which is 4.0. That is an enormous, easily measured effect, and it means a transit duration is not a stellar density unless the orbit is circular. Turned round, it is a measurement: given a stellar density from asteroseismology or from a parallax and a spectrum, the duration anomaly gives the eccentricity — from photometry alone, with no radial velocities at all.

A duration that measures an eccentricity

A transit's length is a measurement of how fast the planet was moving when it crossed, and that speed depends on where it was on its orbit. For a circular orbit the duration gives the star's density; for an eccentric one it gives the density times a factor of up to four — and if the density is known independently, the factor is the eccentricity.

exoplanets · Transits June sunlight at 65°N, against where perihelion sits. Daily-mean insolation at latitude 65 degrees north on the June solstice, at an obliquity of 23.44 degrees, against the longitude of perihelion measured from the March equinox — the angle that precesses right round in about twenty-one thousand years. Three eccentricities are drawn. The swing is ±10.0 per cent at e = 0.05 and ±1.0 per cent at e = 0.005, in proportion to the eccentricity, because the Sun–Earth distance on a fixed date carries e cos of the precession angle and that is first order. Over the same range of eccentricity the annual mean at this latitude moves by 0.12 per cent, because the annual mean carries 1/√(1−e²) and that is second order. The two together are the whole of the precession term in Milankovitch's theory: the eccentricity does almost nothing to how much sunlight the Earth receives and a great deal to when it arrives, and the ice sheets of the northern hemisphere respond to the summer they might melt in rather than to the year's total. It also explains why the precession signal disappears when the orbit is nearly circular: multiply a large angular swing by a vanishing eccentricity and there is nothing left, which is what the innermost curve here is.

An average that precession cannot move

Sunlight arrives as the inverse square of the distance and time passes as its square, so the two cancel exactly in a year's integral. The longitude of perihelion therefore changes the annual mean insolation at every latitude by precisely nothing — and changes June at 65°N by ten per cent.

orbits · Orbital averages The equation of time, and its two causes. The difference between a sundial and a clock over the year, in minutes, computed from Kepler's equation and the tilt. The eccentricity term has one cycle a year and the obliquity term has two; their sum runs from −14.2 to 16.4 minutes.

Two fictitious suns in sequence

The equation of time is usually described as one quantity with two causes. It is better read as two separate reductions, each with its own imaginary body — one that moves uniformly along the ecliptic and one that moves uniformly along the equator — and the second is not the first.

sky · Equation of time 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 Mars: a sundial 40 minutes ahead and 51 behind. The equation of time on Mars over one of its years, 668.6 sols long, against sols since the northern spring equinox, computed from Kepler's equation for an orbit of eccentricity 0.0934 and an axis tilted 25.19°, with perihelion at solar longitude 250.87°. The solid curve is the difference between true and mean solar time: it runs from −51.1 local minutes, 611 sols after the equinox, to +39.9, 385 sols after it — −52.5 to +41.0 in Earth minutes, since a local minute is a 1,440th of a sol. The dashed curves are its two parts. The eccentricity term, one cycle a year, swings by ±42.9 minutes; the obliquity term, two cycles, by ±11.4; the first is 3.74 times the second. Perihelion falls 485 sols after the equinox, marked, and the vertical lines are the equinoxes and solstices.

On Mars the orbit outweighs the tilt

The equation of time is the sum of two terms, one from the shape of the orbit and one from the tilt of the axis, and on the Earth they are nearly the same size. On Mars the orbit's term is almost four times the tilt's, a sundial runs from fifty-one minutes behind the clock to forty ahead, and the figure-of-eight the Sun traces in the Earth's sky becomes a teardrop. Nothing about the two terms is different; only their ratio is.

sky · Equation of time A Sun that runs backwards for 8.1 days. The rate at which the Sun moves across the sky of a planet with a 3:2 spin–orbit ratio and eccentricity 0.2056, in degrees of hour angle per Earth day, against days from perihelion, over one 87.97-day orbit. The rate is the spin rate minus the rate at which the Sun's direction turns because the planet moves along its orbit, and by Kepler's second law that orbital rate peaks at perihelion, at 1.551 times its mean. The spin is 1.5 times the mean orbital rate, so for 8.1 days around perihelion, from −4.0 to +4.0 days, the orbital rate wins, the rate is negative, and the Sun moves backwards across the sky by 1.11 degrees before resuming. At perihelion it is moving at 0.21 degrees a day in the wrong direction. Away from perihelion the Sun crosses the sky at up to 3.4 degrees a day, and a whole solar day, noon to noon, takes 175.9 days — two orbits.

A Sun that stops and runs backwards

On Mercury the equation of time is not a correction but a reversal. The planet turns three times for every two orbits, and near perihelion its orbital motion briefly outruns its spin, so the Sun halts, backs up by a degree over eight days and resumes. From one longitude that is three noons in a week; from another, a sunrise, a sunset and a second sunrise — and a slightly rounder orbit would have stopped it happening at all.

sky · Equation of time An eccentricity of 0.4 swings the surface by 126 K or by 0.1, depending on the length of the year. The peak-to-trough swing in surface temperature over one orbit, against orbital period, for a planet with eccentricity 0.4 receiving on average the flux the Earth does, for surface layers of 1, 10, 50 metres of water. The dashed line is the 126 K the surface would swing through with no heat capacity. Every curve rises from near zero at short periods, where the orbit is over before the layer can respond and the planet feels only the average flux, towards the full swing at long periods, where every part of the orbit lasts long enough to be felt in full. The crossover is where the orbital period is comparable with the layer's thermal time. The vertical marks are the orbital periods of the Earth-flux orbit round stars of 0.1 M☉ (7 days), 0.5 M☉ (0.2 years), 1 M☉ (0.9 years). At a fixed eccentricity a planet in the habitable zone of a small star is thermally averaging almost regardless of how much water it has, and one round a Sun-like star is not unless it has an ocean — the ordering is set by the star through the period, which is the one quantity the flux-averaged habitable zone discards.

A year too short to feel its own eccentricity

A planet on an eccentric orbit can have a comfortable average and murderous extremes, and the habitable zone is drawn from the average. Whether the surface lives on the average or on the extremes is not decided by the flux at all — it is the ratio of how long the surface takes to change temperature to how long the year lasts, and the star sets the year.

exoplanets · Habitable zone The relative orbit when the target's own is not circular. Free relative motion of a chaser about a target, over one revolution, for four eccentricities of the target's orbit — drawn by propagating two exact Keplerian orbits and differencing them in the target's own rotating frame rather than by any linearised solution. The chaser differs from the target by 0.0035 in eccentricity, so the two semi-major axes are identical. Along-track distance runs across the page. At e = 0 the path is the familiar closed ellipse, 2.00 times as long along the track as across it — recovered here from an exact propagation, which is the check that the calculation is right. Raise the target's eccentricity and the path stops being an ellipse and starts breathing: the separation varies by a factor of 2.00 over one revolution at e = 0 and 2.26 at e = 0.7, because the frame itself is now turning at a rate that changes through the orbit and the chaser is swept around it fastest at periapsis. Every one of them still closes exactly, and that is the one statement of the circular theory that survives — two orbits of equal semi-major axis have equal periods and cannot drift apart, whatever else differs between them.

The relative orbit that breathes

Every result in the close-approach frame — the two-to-one ellipse, the burn that moves the wrong way, the eight kilometres behind — assumes the target's own orbit is circular. Around an eccentric target none of the geometry survives. One statement does, and it is about energy rather than shape.

spaceflight · Rendezvous A ceiling the outer orbit imposes. The greatest eccentricity a cycle can drive an inner orbit to, against the mutual inclination it started at, for four ratios of the inner orbit's angular momentum to the outer one's. The bottom curve is the test-particle case and it is the familiar closed form, √(1 − (5/3)cos²i₀), reaching one at ninety degrees — recovered here by a solver that knows nothing about it, which is the check. Every other curve is lower. The reason is conservation: the cycle works by moving angular momentum out of the inner orbit, and it has nowhere to put it but the outer one. A test particle can give up all of its angular momentum because the outer orbit is infinite by comparison; an inner binary carrying 1 times as much as its companion cannot, because tilting the outer orbit by the required amount changes the mutual inclination and shuts the cycle down. At ninety degrees the ceiling falls from 1.000 to 0.866. The pericentre a mechanism can deliver is bounded by what the perturber can absorb, and the test-particle formula is an upper bound rather than a prediction.

A ceiling the outer orbit imposes

The closed form for how far a cycle can drive an eccentricity assumes the inner orbit is a test particle, so the angular momentum it gives up goes nowhere. A real inner binary of comparable mass has to be paid for: the outer orbit absorbs what the inner one loses, tilts, and changes the very inclination the cycle depends on.

orbits · Kozai–Lidov Circularised at twice the closest approach. Eccentricity against semi-major axis, measured in units of the planet's Roche distance 2.16 Rₚ(M⋆/Mₚ)^⅓, for planets delivered on orbits of eccentricity 0.97 with periastra of 1.2, 1.6, 2.2, 3 Roche distances. Tides raised on the planet dissipate orbital energy at periastron and exert almost no torque over the rest of the orbit, so the orbit's angular momentum is nearly conserved while its energy is removed: a(1 − e²) stays fixed and each track runs down and to the left along it. Each ends on a circle of radius q(1 + e₀), which for e₀ near one is twice the periastron — 1.2 → 2.36, 1.6 → 3.15, 2.2 → 4.33, 3 → 5.91. The shaded region above the curve is where the periastron is inside the Roche distance and the planet is stripped on its first close passage. So the channel cannot deliver a circular orbit inside twice the Roche distance: the inner edge of the hot Jupiters is a prediction, stated in units that depend on each planet's own radius and mass rather than on its period.

A planet circularised at twice the distance that would destroy it

A giant planet thrown onto an orbit of eccentricity 0.97 by a distant companion loses its orbital energy to tides at every close passage and keeps almost all of its angular momentum. The orbit therefore shrinks to a circle at exactly twice its closest approach — and since a closest approach inside the Roche distance destroys the planet, the hot Jupiters delivered this way should begin at twice the Roche distance, measured in units of each planet's own size.

exoplanets · Planet migration One eccentric planet, drawn as two circular ones. Top: the velocity of a star with one companion on an orbit of eccentricity 0.25 (argument of periastron 0.9 radians), in units of its semi-amplitude K, over two orbits — and on top of it the sum of two circular orbits, one at the same period and one at half of it, fitted by taking the first two terms of the exact curve's Fourier series. The second orbit's amplitude comes out at 0.246 of the first, which is the eccentricity to first order. Bottom: the difference between the two, on a scale 12 times magnified; its largest excursion is 0.0889 K and its rms 0.0473 K, second order in the eccentricity. For a planet with K = 30 m/s that is 1.42 m/s rms — below the noise of most instruments — so an eccentric single planet and a pair of planets in a 2:1 period ratio are the same measurement unless the data are good enough to see a term of order e².

One eccentric planet, or two circular ones

The velocity curve of a star with one planet on an eccentric orbit is, to first order in the eccentricity, exactly the curve of two planets on circular orbits with periods in the ratio two to one. The difference between the two readings is a term in the eccentricity squared, and for the modest eccentricities most planets have, it is smaller than the noise.

exoplanets · Reflex velocity A hotter summer that is also a shorter one. Daily-mean sunlight at 65°N through one year beginning at the March equinox, for an orbit of eccentricity 0.0167 and obliquity 23.44°, drawn twice: with perihelion at the June solstice, when the northern summer is closest to the Sun, and with perihelion at the December solstice, half a precession cycle later. The first summer peaks 6.9% higher, and it is also shorter, because the Earth moves fastest near perihelion: days above 275 W m⁻² number 139 against 144. The energy delivered on those days — the shaded area above the line — is 5.055 against 4.989 GJ m⁻², a difference of 1.3%. The second law makes the intensity and the duration trade against each other almost exactly, so a summer measured by how much energy it delivers barely feels precession at all.

A hotter summer that is also a shorter one

When the Earth passes closest to the Sun in northern summer, the northern summer is brighter — and, by Kepler's second law, shorter by exactly the right amount to deliver almost the same energy. Measured by the energy it delivers above a melting threshold rather than by its brightest day, a summer barely feels the precession cycle at all and feels the tilt of the axis several times more strongly.

orbits · Orbital averages 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 The same tilt, and ice ages of two different lengths. Top: the obliquity anomaly in degrees from the five largest terms of the standard series, over 2 million years of model time; it is dominated by the 41,000-year cycle and has no power at all near 100,000 years. Bottom: ice volume in a threshold model driven by it — the ice grows steadily, and it collapses whenever volume plus a multiple of the obliquity anomaly crosses a threshold, marked by dots. For the first 1 million years the threshold is low and the ice collapses at nearly every obliquity maximum, on average every 43 thousand years. At the vertical line the threshold is raised, and nothing else changes: the ice now survives one or two obliquity maxima and collapses at the second or third, every 108 thousand years on average. The forcing is identical on both sides; the ice sheet counts it differently.

An ice sheet that skips a beat of the tilt

The last eight hundred thousand years of ice ages come roughly every hundred thousand years, and the Earth's orbit has no forcing of any strength at that period. A model in which an ice sheet collapses only when it is both large and warmed by a high-obliquity summer reproduces the rhythm with nothing but the forty-one-thousand-year tilt — the ice sheet grows through one or two maxima and falls on the next — and a slow rise in how large the ice must grow turns the older forty-one-thousand-year world into the newer one without any change in the orbit.

orbits · Orbital averages Four transiting planets, one star, and a density that must agree. The stellar density implied by each of four transiting planets of one simulated star, each fitted as though its orbit were circular, relative to the star's true density, against orbital period, both logarithmic, with one-sigma errors of 8, 10, 14, 20 per cent from the transit fits. Three planets have eccentricities of a few hundredths and agree with one another and with the truth (dashed line: their weighted mean, 0.950). The planet at 15.9 days has e = 0.2 with its periastron 70° from the line of sight, which makes it cross the star faster than a circular orbit would, and its implied density is 2.10 times the star's — 5.2 standard deviations from the others. A single planet's density can be reconciled with any star by choosing its eccentricity; several planets must share one star, and the odd one out is caught.

Every planet of a star must agree on its density

A single transiting planet gives the density of its star only if its orbit is known to be circular, and an eccentric orbit can disguise almost any star as almost any other. A star with four transiting planets has four such measurements of one number. They must agree — and the way they disagree measures how eccentric the orbits are, catches the candidate that orbits some other star, and pins down the star better than its own spectrum does.

exoplanets · Transits A fall into the Sun from a standing start. The distance of a body released at rest 1 AU from the Sun, as a fraction of its starting distance, against time as a fraction of the fall's total duration, 64.6 days: the exact radial orbit (solid) and a fall at the constant acceleration felt at the start (dashed). The exact fall begins exactly like the constant one and finishes far faster, because the pull grows as the inverse square of the shrinking distance: it reaches the Sun at 79 per cent of the time the constant pull would take. The curve is a cycloid, and the total time is a fifth of the circular orbital period at the starting distance — exactly 1/(4√2) of it — because a straight fall is half of an ellipse whose semi-major axis is half the starting distance.

A fall is half an orbit with no width

Stop a planet dead and it falls into the Sun, in a time that is not a quarter of its year or a half but exactly one over four times the square root of two. The fall is an ellipse squeezed to a line, it spends four-fifths of its duration in the outer half of its path, and in a uniform cloud every shell of it arrives at the centre at the same moment — which is where the free-fall time of star formation comes from.

orbits · Conic sections Two ways out of a transfer orbit. The eccentricity against the semi-major axis of a spacecraft spiralling out from a geostationary transfer orbit — perigee 300 km up, apogee at geostationary height — under a continuous thrust of 0.2 millimetres a second squared, integrated through hundreds of revolutions, for two steering laws. Thrust always along the velocity (red) raises the orbit efficiently but leaves its shape almost unchanged, reaching the geostationary semi-major axis after 64 days still with an eccentricity of 0.59 — an orbit swinging between 17,000 and 67,000 km. A law that points the thrust to reduce a weighted distance from the target in both elements (blue) circularises as it raises, and reaches a circular geostationary orbit in 129 days.

The thrust that rounds an orbit turns once a revolution

An electric engine pushes gently for months, and the obvious way to point it — along the velocity — raises an orbit as fast as possible and leaves its shape almost untouched. Starting from an elongated transfer orbit, that is the wrong job. Rounding the orbit off needs the thrust forward at apogee, backward at perigee and sideways in between, a full turn of the thrust vector every revolution, and the law that does it can be written down from the equations for how each element changes.

spaceflight · Low-thrust transfer The kick speeds read back out of the periods: median 100 m/s against the 100 m/s put in. The distribution of kick speeds recovered from the fragments' periods alone, by p(v) = −2v g′(v) applied to the along-track kicks, for 300 and 3000 fragments (points), against the log-normal distribution actually used, median 100 m/s and width 0.3 dex (line), on a logarithmic speed axis with the density per logarithmic interval. With 3000 fragments the recovery follows the shape of the truth and puts the median at 100 m/s, though differentiating a measured distribution twice smooths its peak and roughens its tails; with 300 it is noisy but already in the right place. The inversion uses one assumption, that the kicks were isotropic, and one measurement per fragment, its period — the quantity tracking networks determine first and best.

A kick speed read off the slope of a histogram

Throw fragments at one speed in random directions and the component of each kick along the orbit is spread evenly between plus and minus that speed — Archimedes' theorem about the area of a sphere, applied to debris. Each fragment's period records that component, so the histogram of periods is a stack of flat bands, one per speed, and its slope gives the speeds back: a break-up weighed by its own violence from a catalogue of orbits, if the event was isotropic, and fragment by fragment if it was not.

orbits · Vis-viva Two eccentricity distributions for the planets that transit alone. Two hypotheses for the orbital eccentricities of planets that transit their stars alone, as probability densities on a logarithmic axis. Solid: a mixture of two populations, 70 per cent at σₑ = 0.04 and 30 per cent at σₑ = 0.25 — a cold majority like the planets in multi-planet systems and a hot minority. Dashed: a continuum, each planet's Rayleigh scale drawn evenly in its logarithm between 0.015 and 0.34, chosen as the continuum whose densities most resemble the mixture's. The two differ most between eccentricities of 0.1 and 0.15, where the mixture dips between its two populations and the continuum does not: 5.6 and 12.0 per cent of planets respectively. A trough there is what a separate hot population would leave, and it is the feature a measurement has to find.

A gap in eccentricity a density can only just see

Planets that transit alone have more eccentric orbits than planets with transiting siblings, and the excess can be fitted as a separate hot population or as a smooth spread with no gap. The two eccentricity distributions look plainly different. The stellar densities through which they are measured look almost the same, and how many planets it takes to tell them apart is set less by the star than by the transit.

exoplanets · Transits How much a dying star stirs its companion's orbit, against their separation. The rms change in orbital eccentricity of a 0.8 M☉ companion when a 2 M☉ star becomes a 0.6 M☉ white dwarf, against the binary's separation before the loss, both logarithmic. Blue: the mass loss alone — twelve thermal-pulse episodes 500 years wide and a superwind 10,000 years wide, each adding a kick of random sign whose size falls off exponentially once it lasts longer than about a sixth of an orbit. Below 100 au the orbit is slow to every episode and the eccentricity is untouched. The pulses switch on near 229 au, and by 3,000 au the change is 0.26 and the superwind too has become sudden. Dashed: the change a velocity kick to the white dwarf of 0.05 or 0.2 km/s would make on its own, which grows as the square root of the separation. A 0.05 km/s kick outweighs the pulses out to 251 au; a 0.2 km/s kick outweighs the pulses at every separation drawn.

A kick that writes over the record of the pulses

A star that becomes a white dwarf loses half its mass in pulses, and a companion far enough away feels each pulse as a sudden jolt to its orbit's shape. The wide binaries of white dwarfs should therefore carry the pulses' record in their eccentricities. They would, except that the white dwarf itself is born moving at a few tenths of a kilometre a second, and a kick that small writes more eccentricity than all the pulses together.

orbits · Effective potential The summer sunshine at Mars's north pole over the last million years. The sunshine reaching Mars's north pole at the summer solstice, relative to its average, over the last 1,000 thousand years (blue), from a forcing built from the main periods of Mars's recent orbit: the tilt varying by about 3.5 degrees in 120,000 years and 1 degree in 99,000, the eccentricity by 0.03 in 95,000 years and 0.02 in 2.4 million, and the perihelion going round the seasons once in 51,000 years. The part due to the tilt alone is dashed. The sunshine swings by a factor of 2.06 between its extremes: the tilt sets a slow swell with a period of 120,000 years, and the precession, whose strength the eccentricity sets, cuts a faster ripple of 51,000 years into it, strongest when the orbit is most eccentric. These two periods, and their ratio of 2.35, are what a record of the sunshine should carry.

Layers that keep the rhythm and lose the dates

Mars's polar caps are stacks of ice and dust laid down while the planet's tilt and orbit changed the summer sunshine, and the question is whether they can be read as a record of that sunshine. They can — for the rhythm. Even when the ice accumulates faster under weak sunshine than under strong, the cycles of the tilt and the precession survive in the depth of the stack in their correct proportion. What does not survive is the date of any one layer.

sky · Obliquity Which pairs of planets the angular momentum deficit can bring together. For each adjacent pair of planets, the system's angular momentum deficit divided by the critical deficit at which that pair's orbits could touch, on a logarithmic scale; below one the pair cannot collide under secular evolution, whatever happens. Blue: using only the deficit of the pair's own subsystem — the four terrestrial planets (7.8 × 10⁻⁹ in units of a solar mass times √AU) or the four giants (4.3 × 10⁻⁶). Orange: using the whole Solar System's deficit, which the giants hold almost entirely. On their own budget the inner pairs sit at 0.21, 0.12, 0.34 — safe. On the whole budget they sit at 112, 65, 187. The giants are safe either way, at 0.05, 0.18, 0.36. The terrestrial planets are stable because almost none of the giants' deficit reaches them, not because there is too little of it.

A deficit the giants hold and Mercury could borrow

The angular momentum the planets' orbits lack of circular motion is conserved by every secular interaction, to every order, so it is a budget no planet can overdraw. Compared with what it would take to bring a pair of orbits into contact, it proves without an integration that the terrestrial planets cannot collide among themselves — and shows that they are safe only because the giants, who hold five hundred times as much, lend them almost none. A loan of half a per cent of Jupiter's would be enough to unsettle Mercury.

orbits · Secular theory Where each test can name the host. The plane of a pair's separation and brightness difference, divided by which measurement can say which star a 1000-ppm transit crossed. Above Δm ≈ 4 a sub-Neptune on the companion is diluted below detection and there is nothing to assign. Imaging resolves the pair to the right of an illustrative contrast curve, Δm = 5 log(s/0.04″). The centroid's shift, read against the pair's separation and flux ratio, names the host to the right of s = σ(1 + f)/δ for a precision σ of 30 μas, about 0.05″ for near twins. The density from the transit's shape names it between Δm ≈ 1.1, below which the two stars are too alike, and the detection limit — at any separation, but only for strong transits on nearly circular orbits. The corner of close, nearly equal pairs, under about 0.04″ and 1.1 magnitudes, is reached by none.

A transit that says which star it crossed

When a planet transits one star of an unresolved pair, the catalogue usually assumes the brighter star. The light curve itself can say otherwise, in two ways: the transit's shape measures the density of the star crossed, and a red dwarf is three times denser than the Sun; and the centre of the pair's light moves during the transit, away from whichever star was dimmed. Each test works where the other fails, and between them they leave a corner of close, nearly equal pairs that no measurement of the transit reaches.

exoplanets · Detection bias

Named alongside it

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

Angular momentumObliquityPeriapsisSemi-major axisInsolationKepler's equationAnalemmaAnomalyEquation of timeKepler's second lawOrbital elementsMilankovitch cycles

All concepts