Astronomy & Celestial Mechanics
the sky, computed
Concept

Semi-major axis — where it appears

Half the longest diameter of an elliptical orbit, which fixes both its energy and its period. For an unbound orbit it is negative, which is a bookkeeping convention that keeps the vis-viva relation and the energy expression unchanged across every conic.

Named by 12 essays across 2 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 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 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 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 Flight time against semi-major axis, for a fixed 135° sweep. Lambert's theorem drawn: the time to fly between two points 1 and 1.524 AU out and 135° apart, against the semi-major axis of the orbit that does it. Nothing else about the orbit enters — not its eccentricity, not where its periapsis is, not how it is oriented — which is the content of the theorem and the reason a two-point transfer is a one-dimensional search rather than a six-dimensional one. Two branches: the lower one is the ellipse whose arc stays short of apoapsis, falling towards the parabolic floor at 103.2 days as a grows without limit; the upper one is the ellipse of the same size whose arc runs through apoapsis, rising without limit. They meet at a = s/2 = 1.2161 AU, 244.2 days, which is the minimum-energy transfer and the slowest ellipse available — every faster one is bigger. Each branch is monotone, checked point by point across the drawn range, so a horizontal line cuts each at most once: for a given pair of points and a given time there is exactly one ellipse, and at 260 days it is a = 1.2189 AU on the upper branch. The freedom a mission designer has is not in this picture: it is the choice of the two points, which is what a porkchop plot sweeps.

Two places and a clock decide the path

The time to fly between two points depends on the semi-major axis, the chord between them, and the sum of their distances — and on nothing else about the orbit. Not the eccentricity, not where periapsis is, not the orientation. Lambert's theorem is why an interplanetary launch date is the root of one equation.

orbits · Lambert's problem 5 transfers through the same two points in the same 1400 days. Time of flight against semi-major axis for every transfer through two points 135° apart at 1 and 1.524 AU, with the revolution count running from 0 to 2. Each count contributes two branches, and for N ≥ 1 the pair folds: the time has a minimum at a = 1.2426 AU for one revolution — only 2.2 per cent above the minimum-energy value of 1.2161, which is why the horizontal axis is the excess over that value and logarithmic — so a flight time above it is met twice and below it not at all. Reading the crossings of the 1400-day line off the drawn curves gives 5 of them — 0 revs high, 1 rev low, 1 rev high, 2 revs low, 2 revs high — which is 2N + 1 with N = 2, and the count is a property of the time rather than of the geometry. That is the practical content: a root-finder started from a single guess returns one of these 5 and gives no sign that the other 4 exist, and the cheapest of them is often not the one nearest the guess.

One time of flight and five ways round

Lambert's theorem says two positions and an interval fix the transfer. Allow the transfer to complete whole revolutions and that stops being true: the flight time folds, one number admits five arcs, and the cheapest of them is usually not the one a solver started nearest to.

orbits · Lambert's problem 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 doubling rule, and the planet that broke it. Each planet's real semi-major axis against the value the Titius–Bode rule 0.4 + 0.3 × 2ⁿ gives it, both in astronomical units on logarithmic axes; points on the diagonal are fitted exactly. The rule gives the Earth 1.0 by construction and fits Venus, Mars, Jupiter and Saturn within a few per cent. It needed the value n = 3 left empty until Ceres was found in its slot in 1801, and it predicted Uranus, discovered in 1781, within 2%. Neptune, discovered in 1846 by a calculation that assumed the rule, sits 22% inside its predicted place, and Pluto occupies the place Neptune should have had. Mercury is given n = −∞, which is a free choice made to fit it. The labels give each body's departure from the rule.

A spacing law that random orbits obey

The planets' distances roughly double from one to the next, and a rule written down in 1766 predicted the asteroid belt and Uranus before either was found. It failed on Neptune, and the reason it worked at all turns out to be that almost any set of orbits forbidden to crowd each other looks geometric on a logarithmic scale — as a test against thousands of random systems shows.

orbits · Harmonic law A cloud of fragments that pinches where it was born. 240 fragments thrown from a satellite on a circular orbit 800 kilometres up, with kicks of about 100 metres a second in random directions in the orbital plane, each then following its own Kepler ellipse, drawn at 0.5, 1, 4.5, 20 orbital periods of the parent, with the Earth's centre marked and the break-up point at the top. Radial offsets from the parent's circle are exaggerated 2.5 times. Half a period after the break-up the fragments are spread 28 degrees along the orbit and 1374 kilometres in altitude, the kicks having carried some to higher apogees and others to lower perigees. After one period the along-track spread has grown to 53 degrees but the whole cloud is only 112 kilometres thick, and the 83 fragments within eight degrees of the break-up point span 35: every fragment's orbit passes through the break-up point, and the cloud pinches there once a period. By 4.5 periods it is 231 degrees long and 863 kilometres thick overall; by 20 it is 352 degrees long, still pinched to 11 kilometres where it began.

Every fragment comes back through the same point

When a satellite breaks up, its pieces fly off in every direction and every one of them returns, an orbit later, to the place where it broke. The cloud pinches there once a period, stretches along the orbit at a rate set by one component of each kick, and becomes a ring in hours.

orbits · Vis-viva 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

Named alongside it

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

EccentricityKepler's equationConic sectionsPeriapsisAngular momentumCharacteristic energyFocusHyperbolic orbitInsolationLambert's problemMinimum-energy transferObliquity

All concepts