Astronomy & Celestial Mechanics
the sky, computed
Concept

Inverse square law — where it appears

The falloff of a field or a flux as the reciprocal of the squared distance, which follows from a conserved quantity spread over a sphere. It is the reason a spherical shell attracts an outside body as a point and an inside body not at all, which is a coincidence of the exponent rather than a general fact.

Named by 9 essays across 4 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 Why a shell pulls a point inside it not at all. A double cone from a point inside a uniform shell. In the narrow-cone limit the far patch is 2.80 times further away and 2.80 times wider — the same number, checked to a part in a thousand before this figure is drawn. Its mass is greater by the square of that ratio and its pull weaker by the same square, so the two cancel in every direction.

A sphere pulls exactly like a point, and the proof is a pair of cones

Every orbit ever computed treats the Sun as a dot. That is not an approximation — for a spherical body it is exact, and the reason is a cancellation between two patches of a shell.

gravitation · Shell theorem 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 The magnitude scale, plotted. The logarithm of the received light against magnitude from -27 to 32, with the vertical scale left unlabelled because only its slope matters. The relation is a straight line of slope −0.4, which is what makes five magnitudes exactly a hundredfold — and the numbers run backwards, so brighter is smaller. 9 landmarks are marked, from the Sun to the deepest exposures, spanning a factor of 1.2·10²³ in received light.

A scale that runs backwards, multiplies, and works

Brighter stars have smaller magnitudes, and five steps is a factor of a hundred. A scale invented by eye in the second century BC turned out to be logarithmic, because eyes are.

starlight · Magnitudes The distance modulus. The difference between apparent and absolute magnitude against distance, on a logarithmic distance axis. It is a straight line of slope five per decade, passing through zero at ten parsecs — the definition of the absolute magnitude. Reading a distance off it requires the absolute magnitude, which is never measured and always inferred.

A brightness is a distance only if something is known

The inverse square law turns a brightness into a distance in one line. The line contains a quantity that has never been measured for any object outside the solar system.

starlight · Magnitudes What distance does, and does not, do to a galaxy. Three quantities against distance, each relative to its value at 5 Mpc, on logarithmic axes so that a power law is a straight line and its exponent is the slope. Flux falls with slope −2 and angular size with slope −1, both of which are ordinary. Their ratio has slope zero: a galaxy of surface brightness 23.5 magnitudes per square arcsecond has that surface brightness at every distance, and a sky of 22 is brighter than it at every distance too. The contrast against the sky — the quantity that decides whether the thing is detectable at all — is -1.5 magnitudes wherever it is put.

The brightness distance cannot touch

Flux falls as the inverse square of distance and so does solid angle, so their ratio does not fall at all. A galaxy's surface brightness is the same number wherever it is put, which means whole populations can be undetectable at any distance whatever.

galaxies · Surface brightness The pull of a ring, a shell and a point, in the plane. Radial force in the plane of a thin ring of radius 1, against distance from its centre, in units where the same mass at a point would pull with one over the distance squared; positive is outward. Inside a spherical shell of the same radius and mass the force is exactly zero. Inside the ring it is not: it points outward, toward the nearer part of the ring, at 0.34 at half the radius and 2.75 at nine tenths, growing without limit toward the rim. Outside, the ring pulls inward harder than a point would, by 25 per cent at twice its radius and 9 per cent at three times, because the near side of the ring is closer than the centre. The shell is indistinguishable from the point everywhere outside it.

Inside a ring the pull points outward

A hollow sphere exerts no force on anything inside it, and the proof is a pair of cones whose areas grow exactly as fast as gravity weakens. Cut the sphere down to a ring and the arcs grow only as the distance, the nearer side wins, and a planet set in the middle of a rigid ring falls into it within a handful of orbits — which is why Saturn's rings cannot be solid, and why a flat galaxy spins faster than its own enclosed mass allows.

gravitation · Shell theorem A valley of limits, and how much of it each experiment holds. The strength α a Yukawa term could have, against its range, with both axes logarithmic and everything above the curve excluded at about 95% confidence. The curve is a rounded envelope of published limits: laboratory torsion balances below a metre, tower and mine gravimetry to a few kilometres, the orbits of geodetic satellites near 10⁷ metres, lunar laser ranging near 4×10⁸ metres at a few parts in 10¹¹, and planetary ranging beyond. It does not rule out a term at ranges under about ten micrometres or above about 10¹³ metres; the limits are weakest where no pair of masses of the right size can be arranged.

A force that can hide at any one distance

The inverse-square law is tested by comparing the strength of gravity at two distances, and every instrument that does so is sensitive over about one decade of range. A Yukawa force with a range outside that decade is invisible to it: much shorter and it never reaches, much longer and it only rescales the constant. So testing the law is not one experiment made more precise but a chain of unrelated ones, from torsion pendulums to the orbits of planets, and the chain has weak links exactly where no pair of masses of the right size can be arranged.

gravitation · Gravitational constant 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.

Apsidal precessionAbsolute magnitudeApparent magnitudeDistance modulusEffective potentialGauss's lawParsecPhotometryShell theoremAstronomical unit (AU)Binary pulsarCentral force

All concepts