Astronomy & Celestial Mechanics
the sky, computed
Concept

Apsidal precession — where it appears

The slow turning of an orbit's long axis within its own plane, produced by any departure from a pure inverse-square attraction. Mercury's is 574 arcseconds a century, of which 43 remain after every Newtonian contribution has been removed.

Named by 14 essays across 5 fields — each of them below, with the objects they name alongside it.

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

The third thing that is conserved

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

orbits · Conic sections An eccentricity and an inclination trading, at 65° of mutual tilt. The secular equations integrated from a nearly circular orbit (e = 0.02) inclined at 65° to a distant perturber's plane, over three oscillations. Above, the eccentricity; below, the inclination, with the constant √(1−e²)cos i drawn as the flat line it is. The eccentricity climbs to 0.8380 and the inclination falls to 39.25° at the same instant, and neither is a coincidence: the product is fixed, so one can only rise as the other falls. That floor is the same for every starting tilt — at maximum eccentricity j = √(5/3)Θ, so cos i = √(3/5) and the inclination arrives at 39.23° whether the orbit began at 50° or at 89°. The closed form for a circular start is e_max = √(1 − (5/3)cos²i₀) = 0.8380, which contains nothing about the perturber — not its mass, not its distance. Those set the clock and not the amplitude, and the period here is 4.83 Kozai times. What the figure cannot show is what happens at the top of the cycle in a real system: at e = 0.838 the pericentre is 0.1620 of the semi-major axis, where tides, general relativity or a stellar surface all intervene, and the quadrupole picture ends.

An inclination that turns into an eccentricity

A distant companion cannot change an orbit's size or its energy. It can take a circular orbit tilted past 39.23 degrees and drive it to an eccentricity near one, and back, over and over — and the companion's mass and distance set only the clock.

orbits · Kozai–Lidov 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 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 Love number against central condensation: 3/2 for a uniform body, 2.4e-3 at n = 4. How willingly a body deforms, drawn against how concentrated it is. The horizontal axis is the polytropic index, which is a proxy for the run of density inside — n = 0 is uniform, n = 1.5 is a non-relativistic degenerate gas, n = 3 is a radiative star like the Sun — and the vertical axis is the fluid Love number k₂ on a logarithmic scale. The curve is the Radau equation integrated over each polytrope's own density profile, and its two ends are exact rather than fitted: a uniform incompressible body has k₂ = 3/2 exactly, and a body with all its mass at the centre has k₂ = 0, because a point mass has no quadrupole to offer. Everything real lies between. The fall is steep — four orders of magnitude across the family — which is what makes the number diagnostic: k₂ is not a mild function of structure, it is a sensitive one, and measuring it to ten per cent constrains the interior far better than measuring a mean density to the same precision. The Sun, at n ≈ 3, sits near 2.9e-2. Two conventions collide here and the figure uses one of them: the planetary literature's k₂, for which a uniform body gives 3/2. The stellar literature's apsidal-motion constant is half of this at every point, so a uniform body gives 0.75, and the two are the same quantity. What the picture cannot show is rigidity — every body on it is a fluid, and for anything smaller than a planet that assumption fails badly.

How much a world gives

A body pulled on from one side deforms, and how much it deforms is a single dimensionless number. That number is three halves for a uniform fluid, three hundredths for the Sun, and two thousandths for a moon made of ice — so measuring it is a measurement of what is inside.

gravitation · Love numbers How far the returning debris swings round, against the hole's mass. The relativistic advance of pericentre, per orbit, of the most bound debris from a disrupted star of 1 solar radius and 1 solar mass, passing at the tidal radius, against the hole's mass. Δω = 6π GM / c² a(1 − e²), and for these nearly parabolic orbits a(1 − e²) is about twice the pericentre, so the angle is set by how many gravitational radii the pericentre is — and that falls as the mass to the minus two thirds, because the tidal radius grows as the cube root of the mass while the gravitational radius grows as the mass. At 10⁵ solar masses the pericentre is 219 gravitational radii and the stream swings round by 2.5°; at 10⁶ solar masses the pericentre is 47 gravitational radii and the stream swings round by 11.6°; at 10⁷ solar masses the pericentre is 10 gravitational radii and the stream swings round by 53.4°. A light hole barely bends its debris's orbit, and a heavy one bends it by tens of degrees.

The debris that returns fastest lights up last

The debris of a star torn apart by a light black hole comes back within two weeks, in greater excess of what the hole can swallow than for any heavier hole. It should make the promptest flare, and it may make the slowest — because to shine, the returning stream has to crash into itself, and where it does so is decided by how far relativity swings its orbit round. Round a light hole the swing is a few degrees, and the streams meet only near the far end of their orbit, moving slowly.

galaxies · Tidal disruption A limit with a notch cut in the middle of it. The stability limit for a satellite, in units of its planet's Hill radius, against the inclination of its orbit to the planet's. The two flat stretches are the coplanar answers — 0.5 prograde and 0.7 retrograde, the difference between them being the Coriolis asymmetry that a rotating frame imposes and no potential can hold. Between them the limit is not an interpolation. Inside the shaded band, from 39.23° to 140.77°, the Sun's averaged perturbation exchanges the orbit's inclination for its eccentricity, and a satellite whose semi-major axis is comfortably inside the lobe has an apocentre that is not — so the boundary is set by a(1 + eₘₐₓ) rather than by a, and it falls to exactly half the coplanar value at ninety degrees. The band edges are cos²i = 3/5 exactly, with no fitted quantity anywhere in them, and the notch is the reason a quoted stability limit in Hill radii is a statement about a coplanar orbit and not about a satellite.

A stability limit with a notch cut in it

The two numbers usually quoted for how far out a moon may orbit are coplanar numbers. A tilted orbit is not somewhere between them: inside a band of inclination fifty-six degrees wide the Sun converts the tilt into an eccentricity, the apocentre rather than the semi-major axis has to stay inside the lobe, and the limit falls to half. The band is empty in the sky.

gravitation · Hill sphere A cycle and a radiation, racing for the same pericentre. Two integrations of the same compact triple, in units of the cycle's own timescale at the starting separation. The dashed curves are the semi-major axis and the solid ones the eccentricity. Radiation alone shrinks the orbit slowly and smoothly, reaching a tenth of its starting size after 1136 cycle times — which the closed form for a circular inspiral predicts to within a few per cent, and which is the check that the integrator is right. With the cycle running, the distant companion drives the eccentricity to 0.9104, the pericentre falls by two orders of magnitude, and the radiation — whose rate goes as the inverse seventh power of one minus the square of the eccentricity — empties the orbit in 18. That is a factor of 62, obtained without changing a single parameter of either mechanism. The two do not merely add: the cycle manufactures, at its own peak, exactly the configuration in which the radiation is most efficient, and the radiation then shrinks the orbit until the relativistic precession switches the cycle off.

A cycle that builds what the wave needs

Gravitational radiation empties an orbit at a rate that goes as the inverse seventh power of one minus the square of the eccentricity. A distant companion drives exactly that eccentricity, over and over. The two are not independent mechanisms that happen to act on the same binary — the first manufactures, at its own peak, the configuration the second is most efficient in.

orbits · Kozai–Lidov The eccentricity the planets force on an asteroid, and where it diverges. The forced eccentricity of a body in the inner Solar System against its semi-major axis, from the linear secular theory with all eight planets — its two giant-planet frequencies set to the measured 4.257 and 28.245 arcseconds a year — on a logarithmic axis: the largest value it reaches over the planetary cycles (solid) and its present value (dashed). Across most of the asteroid belt it is a few hundredths — 0.048 at 2.8 AU — the part of every asteroid's eccentricity that belongs to Jupiter and Saturn rather than to the asteroid. Near 2.04 AU it diverges: there the asteroid's own apsidal precession rate equals the frequency of the planetary mode dominated by Saturn, the denominator A − g₆ vanishes, and the theory's forced eccentricity goes to infinity. That is the ν₆ secular resonance. Real asteroids near it do not reach infinite eccentricity; they reach eccentricities large enough to cross the orbits of Mars and the Earth, and are removed. The location is for a body of low eccentricity and inclination; the full theory curves the resonance to larger distances at higher inclination, which is why the belt's inner edge is not a single number.

A denominator that emptied the inner belt

The eccentricity Jupiter and Saturn force on an asteroid is a fraction, and its denominator is the difference between the asteroid's own precession rate and one of the planets'. At two astronomical units that difference is zero. Where the zero sits depends on where Saturn is — and the belt's quiet orbits say Saturn did not take long to get there.

orbits · Secular theory Five clocks around the black hole at the centre of the Galaxy. Timescales against distance from a black hole of 4.3·10⁶ solar masses at the centre of a cluster whose stellar mass inside r is 10⁶ (r/1 pc)^1.5 solar masses, made of stars of 1 solar mass. Ordinary two-body relaxation (orange) is slow everywhere, 8.4·10¹⁰ years at 0.01 pc. Orbits that close are nearly Keplerian and keep their orientation until they precess — through the stars' own mass farther out, through relativity close in — and over that coherence time the torques between them add rather than random-walk. That makes scalar resonant relaxation (blue) faster than two-body relaxation inside about 0.57 pc, and vector resonant relaxation (green), which turns orbital planes and is not stopped by precession within a plane, faster still: 3.5·10⁷ years at 0.1 pc, against the 6·10⁶-year age of the young stars there. Coefficients of order one are set to one; the slopes are the content.

Torques that add because the orbits hold still

Near a massive black hole every star's orbit is almost a fixed ellipse, repeated thousands of times before it precesses. The torques the other stars exert on it therefore point the same way orbit after orbit, and they add in proportion to time instead of cancelling in a random walk — so angular momentum near the hole relaxes hundreds of times faster than ordinary encounters allow, and orbital planes faster still.

gravitation · Two-body relaxation One parabola, two causes. Transit times of a planet on a 1.0914-day orbit over 12 years, minus the best straight-line ephemeris, with timing errors of 30 seconds — the shape a period shrinking by 29 milliseconds a year produces, a parabola 116 seconds deep. The curve is drawn once because it is the same curve for both explanations: an orbit decaying at that rate, or a star accelerating towards the observer at 2.9e-6 m/s², which shortens the light-travel time to each successive transit by the same steadily growing amount. Nothing in the transit times can tell them apart.

A clock that slows, or a star that falls

Transit times that curve away from a straight line can mean a planet's orbit is shrinking — the tides draining it into its star — or that the star itself is accelerating along the line of sight under a distant companion's pull. Both draw the same parabola, and the transit times cannot choose. The star's velocities can, but only for the shortest orbits: the acceleration that would fake a given decay grows as one over the period, and beyond a day or two an unseen Saturn can mimic more decay than the tides produce.

exoplanets · Transit timing Every orbit turns by about the same angle, whatever its size. How far the Earth's equatorial bulge turns an ellipse with a 300-km perigee in one revolution, in degrees, against its semi-major axis, logarithmic: the line of apsides (solid) and the node (dashed), for inclinations of 28.5, 51.6, 98°. The turn per orbit depends on the semi-latus rectum, which for an elliptical orbit lies between one and two perigee radii, so it changes by less than a factor of four from a near-circular orbit to one reaching the Moon's distance: at 28.5°, 0.56° of apsides per orbit at 8,000 km and 0.20° at 300,000 km. Per day the large orbit turns far less, because it completes so few revolutions; per revolution, almost the same.

An orbit lost to a burn that did not fire

A weak stage leaving the Earth splits its departure into several burns at perigee, coasting round a staircase of ever larger ellipses between them. If one burn does not fire, the stage goes round once more. That costs a revolution of time — an hour and a half early in the staircase, more than a day at the top — and a revolution of turning by the Earth's bulge, which comes out at nearly the same angle for every step. The angle is cheap to undo at the slow apogee of a large orbit. The time is not, when the target is the Moon.

spaceflight · Orbital transfer The first-order perigee rate is right for a slower Sun, and half wrong for ours. The rate at which an orbit's perigee turns, in turns per orbit, against the ratio m of the Sun's motion to the orbit's, from separate integrations of a satellite with the Moon's eccentricity under a solar tide on a circular path. The first-order rate (3/4)m² gives 0.0042 at the Moon's m of 0.0748, a perigee period of 18 years. The integrations give 0.0085, 8.9 years; the Moon's perigee takes 8.85. At m = 0.01 the integration gives 8.3e-5 against a first-order 7.5e-5. So the disagreement is not a matter of neglected forces: it depends on how large m is.

A perigee that moved twice as fast as Newton

The Sun's tide on the Moon should turn the line of its perigee once in eighteen years. It takes less than nine. For sixty years that was the outstanding failure of the law of gravitation, and in 1747 Clairaut told the Paris Academy that the inverse-square law would have to be amended. Two years later he withdrew the remark. The law was right and the calculation was unfinished: the series for the perigee has a small parameter that is not small, and its second term is seventy per cent of its first.

orbits · Perturbations

Named alongside it

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

General relativityKozai lidov mechanismSecular perturbationEccentricityHierarchical tripleHigh-eccentricity migrationInverse square lawBinary pulsarCritical inclinationEffective potentialHot jupiterIrregular satellites

All concepts