The third law is wrong by the mass of the planet

Kepler published the harmonic law in 1619, in the Harmonices Mundi, as a relation between two numbers that could both be read out of a table: the period of a planet and the size of its orbit. Newton derived it sixty-eight years later, and in the derivation it acquired a term Kepler had no way of noticing.

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

The sum in the denominator is the whole of the difference. Kepler’s law is the case $m = 0$, and every planet in the solar system is close enough to that for the discrepancy to be undetectable by seventeenth-century astronomy. But the term is not a correction to be applied and forgotten. It is the reason the law can be used as a balance.

Where the term comes from #

The two-body problem has one body at rest at a focus only in the limit where one mass is negligible. In general neither body is still: both orbit the barycentre, and the quantity that obeys a simple inverse-square equation is the separation vector between them.

Making that substitution is the reduced-mass trick, and its consequence for the harmonic law is direct. The separation traces an ellipse of semi-major axis $a$ about a fixed point, under an attraction whose strength is set by $G(M+m)$ rather than by $GM$. Everything else in Kepler’s derivation goes through unaltered, and the total mass appears where the primary’s mass used to be. For the Earth the term is $3.0 \times 10^{-6}$ of the Sun’s mass: a period longer by 1.5 parts in a million, or 47 seconds a year. For Jupiter it is $9.5 \times 10^{-4}$, worth about four and a half hours in a period of nearly twelve years. Neither is small by the standards of modern timing. Both are far below what Kepler’s data could reach — Tycho’s positions were good to a minute or two of arc, which corresponds to a period uncertainty of order a part in a thousand.

Why the planets cannot show it #

The hero figure looks like a failure of the law and is a failure of something else, and disentangling the two is the point of the essay.

A planet’s departure from the massless law would be its own mass fraction if the only thing acting on it were the Sun. It is not. Every other planet perturbs it, and those perturbations shift the mean period by amounts that have nothing to do with the planet’s own mass. For Saturn, the dominant term is the great inequality — the near 5 : 2 commensurability with Jupiter — whose contribution to the mean motion is larger than the mass term and, at the epoch these elements are quoted for, of the opposite sign.

So the picture shows two effects of comparable size and cannot separate them, and the honest conclusion is the one the figure is drawn to force: the mass term in the third law is not how any planetary mass has ever been measured. The perturbations that swamp it are, in fact, how the masses of Uranus and Neptune were first estimated, but that is a different calculation entirely.

For the inner four the situation is worse in a way that has nothing to do with physics. Mercury’s mass fraction is $1.7\times10^{-7}$, and the published mean elements are not quoted to enough digits to resolve a departure of that size. A measurement below the precision of its own input is not a measurement.

Where it is the entire measurement #

Change the mass ratio and the term stops being a correction. The sensitivity runs the other way from intuition. The correction to the period is only $\sqrt{1+m/M}$, so a companion a tenth of the primary’s mass changes the period by 4.9 per cent — which sounds small until it is read as a mass. Inverting the law, a 4.9 per cent error in a period becomes a 10 per cent error in the mass sum, and the error in an individual mass is larger again once the ratio is folded in. A quantity that is a correction in one direction is the leading term in the other, and which of the two a reader is looking at depends entirely on whether the period or the mass is the unknown.

A binary’s period and separation give the sum of the two masses and nothing else. To split the sum, a second measurement is needed, and it is the ratio of the two bodies’ distances from the barycentre — or equivalently the ratio of their speeds, which a spectrograph delivers directly. Two equations, two masses. That chain, and only that chain, is what makes a double-lined eclipsing binary the one place a stellar mass is measured without a stellar model.

What was actually measured — and it was Jupiter’s moons #

The way a planetary mass is actually obtained from the harmonic law is to apply the law to something orbiting it.

Io’s orbit around Jupiter has a semi-major axis of 421,800 kilometres and a period of 1.769 days. Substituting straight into the law, with Io’s own mass entirely negligible against Jupiter’s, gives

$$GM_{\rm J} = \frac{4\pi^2 a^3}{P^2} = 1.268\times10^{17}\ \mathrm{m^3\,s^{-2}},$$

against the currently adopted $1.26687\times10^{17}$. The agreement is to four significant figures, from two numbers a seventeenth-century observer could have measured — and did. The four Galilean moons were the first system after the solar system itself in which the law was checked, and it holds with a constant a thousandth of the Sun’s.

Two things about that measurement are worth saying plainly, because they set up the next essay but one.

The quantity that comes out is $GM$, not $M$. The two are not interchangeable: $GM_{\rm J}$ is known to about nine significant figures from spacecraft tracking, and $G$ to five, so Jupiter’s mass in kilograms is four digits worse than Jupiter’s mass in the only units the measurement actually delivers.

And the mass obtained is the sum, always. For Io and Jupiter the sum is Jupiter’s mass to within a part in twenty thousand and the distinction does not arise. For the Pluto–Charon system, where the ratio is about eight to one, it very much does: what the harmonic law gave when Charon was discovered in 1978 was the mass of the pair, which was five times smaller than the value that had been assumed for Pluto alone.

Newton did this in the Principia, and got it nearly right #

The reason to insist that the moons are the measurement rather than the planets is that Newton says so himself, in Book III of the Principia, and the passage is easy to miss because it reads as a corollary rather than as a result.

Having established the law with the mass term in it, he applies it twice — once to a planet round the Sun and once to a moon round its planet — and divides. The two central masses appear in the ratio

$$\frac{M_\odot}{M_{\rm J}} = \frac{a_\oplus^3 / P_\oplus^2}{a_{\rm Io}^3 / P_{\rm Io}^2},$$

in which $G$ has cancelled and every quantity on the right is an angle or a time. Newton’s value for the Sun-to-Jupiter ratio was 1067; the modern one is 1047. For Saturn he had 3021 against a modern 3498, the difference being that the Saturnian satellite distances were much worse. These are the first masses of astronomical bodies ever computed, and they were computed as ratios, by a method that requires no knowledge of $G$, no knowledge of the astronomical unit, and no absolute distance of any kind.

That is why the solar system’s mass scale was known to a few parts in a thousand two centuries before anybody knew how heavy the Sun was in pounds. The ratios are geometry. The absolute scale needs a laboratory.

The Moon’s mass, from a wobble in the Sun’s position #

The Earth–Moon system is the case where the mass term is neither negligible nor dominant, and the way its ratio was pinned down before spacecraft is one of the most economical measurements in the subject.

The Earth does not travel round the Sun; the Earth–Moon barycentre does, and the Earth swings about that barycentre once a month. The barycentre sits about 4,670 kilometres from the Earth’s centre — inside the Earth, but not by much — so the Earth’s own position oscillates by that amount every month, and the direction of the Sun as seen from the Earth oscillates with it.

The amplitude is 4,670 km seen from one astronomical unit, which is 6.4 arcseconds. That periodic term in the Sun’s apparent longitude has been called the lunar inequality since Tycho, and measuring it gives the ratio of the Earth’s distance from the barycentre to the Moon’s — which is the inverse ratio of their masses. Nineteenth-century determinations by this route gave the Moon at about 1/81 of the Earth, which is the value still quoted: 1/81.30.

So the Moon was weighed by watching the Sun move, and the observable was six arcseconds — a hundredth of the width of a lunar crater seen with the naked eye, extracted from a periodic signal in a quantity nobody was measuring for that purpose. It is the same inversion Le Verrier used on Uranus, done a century earlier on a smaller residual.

The mass function, and what it refuses to say #

There is a version of this arithmetic in which the answer is deliberately incomplete, and it is the version most often used.

If only one star’s spectrum is visible — the other is too faint — its velocity amplitude $K_1$ is measurable and the companion’s is not. Combining what is left gives the mass function:

$$f(m) = \frac{P K_1^3}{2\pi G} = \frac{(m\sin i)^3}{(M+m)^2}.$$

Everything on the left is observed. On the right are three unknowns entangled with each other and with the inclination, which a spectroscopic orbit cannot supply. The mass function is therefore not a mass. It is a lower bound on the companion’s mass, and it becomes a mass only when something else — an eclipse, an astrometric wobble, a statistical prior — fixes $i$. The mass function is also where the first stellar-mass black holes were identified. Cygnus X-1’s optical star shows a 5.6-day spectroscopic orbit with a mass function of about 0.25 solar masses, which is a hard floor on the companion whatever the inclination and whatever the primary’s mass — and combining it with the visible star’s own mass puts the companion above every limit a neutron star can respect. The argument is Kepler’s law with the mass term retained, an inequality rather than an equality, and it is strong enough to establish an object nobody can see.

The law can be drawn for five systems now and the point of drawing a fourth and a fifth is the intercept rather than the slope.

Whether the constant is a constant #

The law with its correction reads $P^2 = 4\pi^2 a^3 / G(M+m)$, and it contains one quantity that is asserted rather than measured in any orbit: $G$ is assumed to be the same number here as at Jupiter, and the same today as a billion years ago.

That is a testable assumption, and the solar system is where it is tested most sharply.

Suppose $G$ were slowly changing. Then the product $GM_\odot$ would change with it, and every orbit in the solar system would respond: a decreasing $G$ means a weakening grip, so orbits would expand and periods would lengthen, by a fractional amount equal to the fractional change in $G$ over the same interval.

Nothing about that would be visible in a single orbit, because $G$ and the mass appear only as a product and nobody weighs the Sun in kilograms independently. What is visible is the drift: an orbit expanding steadily over decades, common to every body and proportional to its distance in a specific way.

The measurements are among the most precise in the subject. Lunar laser ranging tracks the Earth–Moon distance to a few millimetres over fifty years, and planetary ranging tracks Mercury and Mars to metres over the same span. Neither shows a drift, and the resulting bounds on a fractional change in $G$ are of order a part in $10^{13}$ per year — which over the age of the solar system permits a change of less than a tenth of a per cent.

The difficulty is separating that signal from everything else that makes an orbit expand. The Moon’s distance is increasing by 3.8 centimetres a year from tides, which is enormous compared with any plausible gravitational drift; the Sun loses mass through radiation and its wind, which expands every planetary orbit by about a part in $10^{13}$ per year on its own. Each of those has to be modelled to better than the bound being quoted, which is why the limit is a fitted parameter of the ephemeris rather than a separate measurement.

That is the same structure the whole essay has: the term of interest is small, it enters through a product with something else, and isolating it means finding a system or a timescale where the competing effects can be pinned down independently.

It is also the reason the bound is quoted per year rather than as a total. What the data constrain is a rate over the interval they span, and extrapolating that to the age of the solar system assumes the rate has been constant — an assumption no measurement of the last fifty years can support and that some theories predicting a varying constant explicitly deny.

Which leaves the law in an unusual position for something so old: its form is exact, its correction term is measurable in the systems that matter, and the constant in front of it is the least well determined quantity in the whole expression.

What the picture cannot show #

A period is a sum, and no picture of one orbit can split it. Every figure in this essay reports $M+m$. The splitting always comes from somewhere else — a second spectrum, a resolved pair of images, an astrometric wobble against background stars — and where that second measurement is unavailable, so is the individual mass. This is not a limitation of technique but of information: the two-body problem with one observable orbit has one fewer equation than unknowns.

The corrected law is still a two-body law. It fails wherever three bodies matter, which is exactly where the hero figure fails. There is no version of $P^2 \propto a^3$ that absorbs a resonance.

A drawn orbit is a projection. Every semi-major axis in this essay is a true one, and what a telescope measures for a visual binary is the apparent ellipse on the sky, which is the true ellipse projected through an unknown inclination and an unknown orientation. The projection of an ellipse is an ellipse, so nothing looks wrong; what gives the deception away is that the primary is no longer at a focus of the projected curve, and the departure is exactly enough to solve for the projection. That is a piece of geometry the pictures here quietly assume has already been done.

And it is Newtonian. For a binary pulsar the post-Newtonian corrections to the relation between period and separation are measurable, which turns the same equation into a test of gravitation rather than a measurement of a mass — and the same is true, at a much smaller amplitude, of the forty-three arcseconds a century that Mercury’s perihelion moves and Newton cannot account for.

Two more readings show the law at the smallest central mass anybody has measured it around, and the correction term drawn without the case that dominates it.

Where the ladder goes next #

Two rungs open from here. One goes downward in mass ratio, to the case where the companion is so much lighter than the primary that the sum is useless and the wobble is the measurement — the reflex-velocity method that finds planets. The other goes upward in precision, to the systems where the two-body law is known well enough that its residual is the signal: binary pulsars, whose orbital periods are measured to a dozen significant figures and whose decay is the closest thing there is to a direct detection of a body radiating away its own orbital energy.