The law that links period to size, and weighs everything

Kepler published the third law in 1619, in a book largely about musical harmony, and treated it as one result among many. It is the one that turned astronomy into a science of measurement, and the reason is not the law itself but a constant that Kepler could not have known was in it.

The law says that for anything orbiting the Sun, the square of the period is proportional to the cube of the semi-major axis. Newton showed sixty-eight years later that the constant of proportionality is $4\pi^2/GM$ — that it contains the mass of the thing being orbited. Which means: watch something go round, time it, measure the size of its orbit, and the mass of whatever it is going round falls out.

Every mass in astronomy has been obtained that way. The Sun, the planets, the stars in binaries, the black hole at the centre of the galaxy, the dark matter in a galaxy’s halo. There is no scale in space, and this is the substitute.

Why three-halves #

The exponent comes out of two lines of algebra for a circular orbit, and the circular case is enough, because the full derivation replaces the radius with the semi-major axis and changes nothing else.

A body in a circular orbit is accelerating toward the centre at $v^2/r$, and gravity supplies that:

$$\frac{GM}{r^2} = \frac{v^2}{r}.$$

The speed is the circumference over the period, $v = 2\pi r/P$. Substituting and rearranging,

$$P^2 = \frac{4\pi^2}{GM}\,r^3.$$

The three-halves power is therefore a consequence of the inverse square and of nothing else. A different force law gives a different exponent — an inverse cube would give $P \propto r^2$ — so a measurement of the slope is a measurement of the exponent in the force law, made across the whole solar system at once, without ever measuring a force.

Plotting logarithms is what makes that legible. Taking logs of $P^2 \propto a^3$ gives

$$\log P = \tfrac{3}{2}\log a + \text{constant},$$

a straight line whose slope is the exponent and whose intercept is the mass. Both quantities of interest are read off a line, which is why nearly every relation in astronomy ends up on log axes.

The pair of figures is the argument. Two systems with nothing in common but a central body, both on lines of slope 3/2, with intercepts differing by the ratio of the masses. Galileo saw those moons in 1610 and used their motions as a clock; that they obey the same law as the planets, with a different constant, is the observation that says gravity is universal and that the constant means something.

The law as a balance #

Read Newton’s version backwards and it becomes an instrument:

$$M = \frac{4\pi^2 a^3}{G P^2}.$$

Time an orbit, measure its size, and the mass of the central body follows. There is no other practical way to weigh anything in space, and there never has been.

The Sun’s mass comes from the Earth’s orbit: one year, one astronomical unit. Jupiter’s mass comes from Io’s orbit: 1.77 days, 422,000 km. The Earth’s mass comes from the Moon, and more precisely from any artificial satellite. The mass of the black hole at the centre of the Milky Way — four million solar masses — comes from twenty years of tracking a single star called S2 through a 16-year orbit, and the measurement is the same measurement Kepler made of Mars, with better instruments and a considerably stranger primary.

Two cautions come with it. The mass obtained is the total mass of the system, $M + m$, so the method is only a measurement of the primary when the orbiting body is much lighter — fine for a planet round a star, useless for a pair of comparable stars, where it gives the sum and something else must supply the ratio. And the result is a dynamical mass: it says how much gravity there is, not how much light. When the dynamical mass of a galaxy came out several times its luminous mass, the discrepancy did not go away, and it is still there.

What had to be measured first #

The law relates a period to a size, and the two are not equally easy to obtain.

Periods are the easiest quantities in astronomy. They are timings of a repeating event, they improve with the length of the record rather than with the quality of the instrument, and they were known to several decimal places long before anything else was. Kepler’s periods were essentially modern.

Distances are the hardest. Kepler’s third law gave ratios of distances beautifully — everything in units of the Earth–Sun distance — and the absolute scale not at all. The solar system was a complete and accurate map with no scale bar for two centuries.

Fixing the scale required measuring one distance directly, and the effort that went into it is a fair measure of how badly it was wanted. The transits of Venus in 1761 and 1769 sent expeditions to Siberia, Tahiti and Hudson Bay to time a black dot crossing the Sun from widely separated places; the parallax of the transit gives the distance. Cook’s first voyage was one of them. The result, about 153 million kilometres, was within 3% of the modern value, and it converted every ratio in the solar system into kilometres at a stroke.

The modern value comes from radar: bounce a signal off Venus, time the echo, and the astronomical unit follows from the speed of light. That is the only rung of the distance ladder that involves no assumptions at all, and everything above it is calibrated on it.

That the law uses $a$ rather than the actual distance is the subtle part of Kepler’s third law and the part most often mis-stated. A comet with $a$ equal to Neptune’s has Neptune’s period, however wildly its distance swings during the orbit. Period depends on the orbit’s size, not its shape — the same independence from eccentricity that makes the vis-viva relation so clean.

What was actually measured, at the hardest place to measure it #

The claim that the same law weighs a black hole is easy to make and worth checking against what was actually done, because it is the most demanding orbit determination ever carried out.

The star is called S2. It orbits the compact object at the centre of the Milky Way with a period of 16.05 years, a semi-major axis of about 970 astronomical units, and an eccentricity of 0.885 — so it swings from 120 AU at closest approach out to 1,800. At periapsis it is moving at about 7,650 kilometres per second, or 2.5% of the speed of light.

None of that is inferred. The orbit was traced by imaging: two independent groups followed S2 from the early 1990s, first with speckle imaging and then with adaptive optics on 8-metre telescopes, recording its position among a crowded field of stars twice a year. The angular size of the whole orbit is about 0.2 arcseconds, so the measurement is a stellar position to a few hundred microarcseconds, repeated for a quarter of a century until the star had gone round once and the ellipse closed. Later, the GRAVITY interferometer combined the four VLT telescopes and reached about 50 microarcseconds, which resolved the periapsis passage of 2018 in detail.

Kepler’s third law then supplies the mass: $4.3 \times 10^6$ solar masses, inside a region smaller than the orbit of Uranus. Nothing made of stars can be that dense and remain stable, which is the argument that the object is a black hole.

There is a systematic worth being explicit about. The semi-major axis is measured as an angle, so converting it to a length requires the distance to the galactic centre, and the mass depends on the cube of that. An error of 1% in the distance is an error of 3% in the mass. For most of the measurement’s history the distance was the dominant uncertainty, which is why the same maser and stellar-orbit work that pinned $R_0$ to 8.28 kiloparsecs mattered so much: the mass of the black hole is known as precisely as the distance to it is, and not one digit better.

The orbit has since done something the law cannot describe. In 2020 the periapsis was shown to advance by about 12 arcminutes per orbit, in the direction and by the amount general relativity predicts — the same Schwarzschild precession that shows up as Mercury’s 43 arcseconds per century, scaled up by a factor of a few thousand.

The slope as an experiment #

The line in the first figure is drawn at slope exactly 3/2 rather than fitted, which makes the figure a test rather than a summary. It is worth asking how good a test.

The exponent in the slope is $(n+1)/2$ for a force law going as $r^{-n}$, so measuring the slope measures $n$. Doing it with the planets alone is already a strong constraint: the periods are known to many significant figures, the semi-major axes to a part in $10^{8}$ or better from radar and spacecraft tracking, and the range spans a factor of 78 in distance. Any departure from $n = 2$ large enough to see would have shown up centuries ago.

The sharper tests come from measuring a departure directly rather than a slope. Lunar laser ranging bounces pulses off retroreflectors left on the Moon and measures the Earth–Moon distance to about a millimetre, over a distance of 384,000 kilometres; fitting the resulting orbit constrains any additional force with a range near that scale to below a part in $10^{11}$ of gravity. Planetary ephemerides, fitted to decades of spacecraft ranging, place comparable bounds at solar-system scales. A hypothetical extra interaction — a fifth force, a modification of gravity, an extra dimension revealing itself at some length — is excluded across an enormous range of parameters purely by orbits continuing to obey the harmonic law.

The interesting consequence is where the tests run out. All of them probe accelerations far larger than the ones that govern the outskirts of galaxies, where the discrepancy discussed below actually appears. The solar system says nothing about that regime, and it is a mistake to treat its precision as if it did.

Used forwards #

The law is a design tool as much as a measuring one. Getting from one orbit to another means moving onto an ellipse that touches both, and the journey time is half that ellipse’s period. Since its semi-major axis is the average of the two radii, the third law returns the flight time in one line. Eight and a half months to Mars, six years to Saturn, and those numbers were known before rockets existed.

The same equation fixes the geostationary orbit. Set the period to one sidereal day and solve for $a$: 42,164 km from the Earth’s centre, and every communications satellite in that belt is there because the third law only permits one radius.

And it defines the resonances. Two bodies whose periods are in a simple ratio meet in the same relative position over and over, so their small mutual tugs accumulate instead of averaging away. The Kirkwood gaps in the asteroid belt sit exactly at the radii where the third law makes the period a simple fraction of Jupiter’s, and the gaps are empty because the accumulation eventually threw everything out. A law about periods becomes a statement about which distances can hold anything at all.

Period, speed, and the same law twice #

The harmonic law is a statement about periods, and a period is a distance divided by a speed. Written the other way it becomes a speed law.

Combining $P^2 \propto a^3$ with $v = 2\pi a/P$ gives $v \propto a^{-1/2}$. Mercury moves at 47 km/s and Neptune at 5.4, and the ratio is the square root of the ratio of their distances. That is why the inner solar system is a fast place and the outer one is not, and why a transfer outward is mostly a matter of waiting. The same square-root falloff appears in a galaxy’s rotation, and there it fails. Stars far from the centre of a spiral galaxy should orbit more slowly than those nearer in, by exactly this law. They do not — the rotation curve stays flat — and the discrepancy is the dynamical-mass problem restated as a picture. Kepler’s third law is how dark matter was noticed.

Two more systems have periods and distances good enough to draw, and both were measured by spacecraft rather than by telescopes.

The exponent that is not three-halves #

The slope is three-halves because the orbiting body is outside all the mass. Where it is not, the exponent changes, and the way it changes is a diagnostic of how the mass is arranged.

Take a body orbiting inside a uniform sphere. Only the mass interior to its own radius pulls on it, and that mass grows as the cube of the radius — so the attraction grows linearly with distance rather than falling as its square. A linear restoring force is a harmonic oscillator, and a harmonic oscillator’s period does not depend on its amplitude at all.

So inside a uniform sphere every orbit has the same period, whatever its size, and the exponent is zero. A tunnel bored through such a body would return an object to its starting point in that same period regardless of where it was dropped.

Now take a body orbiting in a galaxy whose rotation curve is flat — where the orbital speed is the same at every radius. The period is the circumference over the speed, so it is proportional to the radius, and the exponent is one rather than three-halves.

Those three cases bracket the possibilities, and the exponent is a direct readout of how the enclosed mass grows with radius. Constant enclosed mass gives three-halves; enclosed mass growing as the cube gives zero; enclosed mass growing linearly gives one.

That is the sharper way of stating the discrepancy the essay’s rotation-curve section ends on. A galaxy’s outer stars do not merely orbit faster than expected; they obey a period–radius relation with the wrong exponent, and the exponent says the enclosed mass is still growing where the light has stopped.

Kepler’s law is a special case selected by geometry, and measuring which case applies is how the mass distribution of anything larger than a solar system is determined.

Where the model stops #

Two bodies. The law in Kepler’s form assumes the orbiting body is massless. Newton’s correction replaces $M$ with $M+m$, which matters for Jupiter at the part-in-a-thousand level and matters enormously for binary stars.

Point masses. A satellite low enough to feel the Earth’s equatorial bulge does not obey a pure third law; its period differs from the spherical prediction by an amount that has to be modelled explicitly.

No other perturbers. In a system of several bodies the semi-major axes themselves change slowly, so the law holds instant by instant rather than forever.

Newtonian gravity. For a star skimming a black hole the relativistic corrections are large, and general relativity supplies a modified law that S2’s orbit is now precise enough to test — the same regime where escape speed stops being a speed.

The figures have a limit too, and it is one worth naming because logarithmic plots invite it. On log axes, a straight line looks like a good fit almost regardless of the data, and four decades of range compress a factor-of-two error into a barely visible offset. The planets sit on that line to far better than the line’s own width — Uranus’s period is known to a fraction of a day out of eighty-four years — and none of that precision is visible in the picture. A log plot is the right way to show a power law and a poor way to show how well it holds.

And two readings of the correction the law does not contain.

The ladder from here #

Later rungs: Newton’s derivation from the inverse square. The two-body correction and what it does to binaries. The astronomical unit’s history, from Aristarchus through the transits to radar. Weighing the Galilean moons, and Roemer’s discovery of the speed of light in the timing of their eclipses. Resonances and the Kirkwood gaps. The Titius–Bode pattern, and why an apparently similar numerical coincidence turned out to be nothing. Dynamical masses of galaxies, and the discrepancy that will not close. And the relativistic third law, now measurable at the galactic centre.

Kepler found the law on 15 May 1618, wrote that he had been carried away by “sacred frenzy”, and put it in a book whose main argument was that the planets sing six-part harmony. The frenzy was justified; the harmony was not.