Neither body is still, and the wobble is how planets are found

The phrase “the Earth orbits the Sun” contains a small and consequential error. The Earth orbits a point 450 kilometres from the Sun’s centre, and the Sun orbits the same point. Both of them move; neither is still.

For the Earth and the Sun the correction is negligible — 450 km is a thousandth of the Sun’s own radius, so the barycentre is buried deep inside it. For Jupiter and the Sun it is not: their common centre sits just outside the solar surface, and the Sun genuinely circles a point in empty space once every twelve years. That motion is small, it is periodic, and detecting the equivalent motion in other stars is how the first extrasolar planets were found.

The point that does not move #

Newton’s third law forces the arrangement. Whatever pull the star exerts on the planet, the planet exerts an equal and opposite pull on the star. Equal forces on unequal masses give unequal accelerations, in inverse proportion to the masses, and always in opposite directions.

The consequence is that one particular point — the centre of mass, or barycentre — has zero net acceleration and either sits still or drifts in a straight line forever. Everything else in the system moves relative to it.

Both bodies orbit that point on similar ellipses, meaning ellipses of identical eccentricity and orientation but different size. The sizes are in inverse proportion to the masses:

$$\frac{a_1}{a_2} = \frac{m_2}{m_1}.$$

The two bodies stay diametrically opposite one another across the barycentre at every instant, which is what the drawn line in the figure records. They reach periapsis together, apoapsis together, and share one period.

The three figures are the same computation at three ratios, and the sequence is the whole idea. At equal masses there is no primary. At fourteen to one there is nearly one. At a thousand to one — the Sun and Jupiter — the heavy body’s motion is a wobble, and at 333,000 to one, the Sun and the Earth, it is buried inside the star.

The trick that makes it solvable #

The two-body problem has six coordinates and looks like it should be twice as hard as the one-body problem. It is not; it is exactly as hard, because of a change of variables that decouples it completely.

Instead of tracking both positions, track the barycentre and the separation vector between the bodies. The barycentre moves in a straight line and can be transformed away entirely by choosing to sit on it. What remains is a single equation for the separation,

$$\mu \ddot{\mathbf{r}} = -\frac{Gm_1m_2}{r^2}\hat{\mathbf{r}}, \qquad \mu = \frac{m_1m_2}{m_1+m_2},$$

which is the equation for one body of mass $\mu$ — the reduced mass — orbiting a fixed centre. That is the problem Kepler’s laws already solve.

So the two-body problem is the one-body problem wearing a disguise, and the ellipse everyone draws is the relative orbit: the path of one body as seen by an observer riding on the other. Both individual orbits are that same ellipse, scaled down by the mass fractions and reflected through the barycentre.

The reduced mass has a useful limiting behaviour. When one body dominates, $\mu \to m_2$, the lighter mass, and the heavy body might as well be nailed down. When the masses are equal, $\mu = m/2$, and neither is. That single expression covers a planet round a star and a pair of equal stars without changing form, which is why the two-body problem is regarded as solved and the three-body problem is not.

Reading it in the light #

The star’s share of the motion is the observable, and there is no need to see the companion at all — the whole measurement is a shift in wavelength.

A star moving toward the observer has its spectral lines shifted to shorter wavelengths, and away, to longer. Measuring that shift over months traces the line-of-sight component of the star’s own small orbit. The period comes straight off the curve; the amplitude gives a combination of the companion’s mass and the orbit size, which Kepler’s third law then separates.

The amplitudes involved are absurdly small. Jupiter makes the Sun move at 12.5 metres per second — a brisk walk — over twelve years. The Earth makes it move at 9 centimetres per second. Detecting the first requires measuring a wavelength shift of four parts in $10^8$; the second is forty times harder and remains out of reach.

That measurement was achieved in 1995, and the first planet found was nothing like what anybody expected: a Jupiter-mass body orbiting 51 Pegasi every 4.2 days, closer to its star than Mercury is to the Sun. It should not have been there. The discovery was believed because the wobble was unmistakable and periodic, and because a second team confirmed it within a week.

The shape of that curve carries more than the period. Its asymmetry encodes the eccentricity — the second law in a measurement, since the star swings quickly through the close approach and lingers at the far end — and the phase at which the peak occurs gives the orientation of the orbit in its own plane. Two more numbers, from the shape of a wiggle.

What the method cannot give #

The radial-velocity technique has a limitation built into its geometry, and it is worth stating precisely because it shaped two decades of exoplanet statistics.

Only the line-of-sight component is measured. An orbit seen face-on produces no radial velocity at all, however massive the companion; an orbit seen edge-on produces the full amplitude. Every intermediate inclination produces something in between, and the inclination is unknown.

So the method returns not the companion’s mass but $m\sin i$ — a lower bound, and one that the third law cannot repair, because the law needs the same inclination the measurement is missing. A reported “half a Jupiter mass” could be half a Jupiter mass seen edge-on, or a brown dwarf seen nearly face-on. Only when a transit is also seen — which fixes $i \approx 90°$ — does the true mass follow, and the combination of the two methods is why transiting planets are the ones with densities attached to them.

There is a second and subtler bias. The amplitude grows with companion mass and shrinks with orbital distance, so the method finds massive planets on short orbits far more easily than anything else. The early catalogue was full of hot Jupiters, and it took years of careful work to establish that this said more about the instrument than about the galaxy.

What was actually measured #

Nine centimetres per second is a strange number to build an instrument around, and the way it is reached is worth setting out, because the limit is no longer the instrument.

A spectrograph does not measure a velocity. It measures the position of a set of absorption lines on a detector, and converts a displacement in pixels into a Doppler shift. One metre per second corresponds to a shift of about $3\times10^{-9}$ of the wavelength — for a spectrograph with 4,000 pixels across the visible range, roughly a thousandth of a pixel. Every effect capable of moving a spectrum by a thousandth of a pixel is therefore a source of systematic error: a change in the instrument’s temperature, in the air pressure inside it, in how the starlight illuminates the entrance slit, or in which part of the fibre the seeing happened to fill.

The solutions were successive. Early work passed the starlight through a cell of iodine vapour, imprinting a dense forest of reference lines on the same detector, in the same optical path, at the same instant — so any shift of the instrument moved star and reference together. Later spectrographs were placed in vacuum tanks held to a thousandth of a degree and fed by scrambled fibres, and calibrated against thorium–argon emission lamps. The current generation uses a laser frequency comb: a spectrum of evenly spaced lines whose frequencies are locked to an atomic clock, which supplies a calibration ruler with no drift of its own at all. ESPRESSO on the Very Large Telescope reaches about 25 centimetres per second on the instrument side.

And the Earth is still not detectable, because the star is in the way. A stellar surface is a boiling granulation pattern in which rising cells are hot, bright and blueshifted while sinking lanes are cool, dark and redshifted. The two do not cancel exactly, and the residual moves at the level of about a metre per second on timescales of minutes to days. Starspots rotating across the disc add a periodic signal at the rotation period, which has repeatedly been mistaken for a planet. Magnetic cycles add a decades-long drift.

That floor is called stellar jitter, and it is a property of the star rather than of the observation, so it does not average down in any straightforward way. The 9 cm/s signal of an Earth around a Sun sits an order of magnitude below it. Every current effort to reach that signal is an effort to model the star’s surface well enough to subtract it, which is a very different problem from building a better spectrograph.

The same factor, in an atom #

The reduced mass looks like a bookkeeping convenience for orbits. It is more general than that, and the place it shows up next is one nobody would have predicted from celestial mechanics.

The hydrogen atom is a two-body problem. An electron and a proton attract each other by an inverse-square force, both move about their common centre of mass, and the energy levels depend on the reduced mass exactly as an orbit’s period does. The proton is 1,836 times the electron’s mass, so the correction is about one part in 1,836 — small, and entirely measurable.

Now replace the proton with a deuteron, which is twice as heavy. The reduced mass changes by about one part in 3,700, and every spectral line of deuterium sits at a slightly different wavelength from the corresponding line of ordinary hydrogen. Harold Urey found those displaced lines in 1931 in a sample of hydrogen that had been evaporated down to concentrate whatever heavy component it might contain, and identified deuterium from a wavelength shift of about a tenth of a nanometre. He had discovered an isotope by measuring a two-body correction.

The connection runs onward into astronomy, which is the part that closes the circle. The relative strengths of those two sets of lines measure the deuterium abundance in interstellar gas, and the deuterium abundance is set by nuclear reactions in the first few minutes after the Big Bang — it is one of the tightest constraints on the density of ordinary matter in the universe. A correction derived to make binary stars tractable is the reason that number is known.

Where else the wobble shows #

Astrometry — measuring the star’s position on the sky rather than its speed — traces the same orbit from a different angle, and it has the opposite bias: it favours distant companions, whose orbits are larger and whose angular wobble is bigger. The two methods are complementary, and Gaia’s astrometric catalogue is now producing planets that radial velocity could never have found.

Astrometry and radial velocity between them also supply the masses that stellar astronomy is calibrated on: a few hundred well-measured binaries, and every other stellar mass in the literature is an inference from them.

Pulsar timing is the extreme case. A millisecond pulsar is a clock accurate to nanoseconds, and a companion’s pull moves the pulsar toward and away from the Earth, changing the arrival time of the pulses. The first confirmed extrasolar planets, in 1992, were found this way — around a pulsar, two years before the first around an ordinary star, and detectable at masses far below anything the Doppler method could reach.

The wobble seen rather than inferred #

Everything above measures the star’s motion along the line of sight. The same orbit can be measured across it, and the method has spent a century being almost good enough.

Astrometric detection means watching the star’s position on the sky trace its small ellipse. The angular size of that ellipse is the orbit’s physical size divided by the distance, so unlike the Doppler method it favours wide orbits and nearby stars. Jupiter moves the Sun by about 0.0005 arcseconds as seen from 10 parsecs — beyond Hipparcos, comfortably within Gaia.

The method’s advantage is that it measures the orbit in two dimensions, so the inclination falls out and the mass is a true mass rather than $M\sin i$. Its history is unfortunate: several claimed astrometric detections in the twentieth century, most famously around Barnard’s Star in the 1960s, turned out to be instrumental artefacts from telescope maintenance.

Gaia’s astrometric catalogue has now produced its first confirmed planets, in the regime radial velocity finds hardest — massive companions on orbits of years.

Where the model stops #

Two bodies. Everything here assumes exactly two. A third changes the picture qualitatively, and multi-planet systems produce radial-velocity curves that are sums of terms and are unpicked with difficulty.

Point masses. Close binaries raise tides on each other, transfer mass, and stop obeying any of this.

Constant masses. A star losing mass in a wind, or a binary transferring it, has an orbit that evolves.

No radiation of energy. Two neutron stars in a tight orbit radiate gravitational waves and spiral together. The Hulse–Taylor binary’s period is shortening by 76 microseconds a year, matching general relativity to better than a part in a thousand — the first evidence gravitational waves existed at all.

The figures have their own limit, and it is the usual one. All three barycentre diagrams are face-on and to scale in shape but not in size: at the real ratio for a star and a planet, the heavy body’s orbit would be a dot. Drawing a mass ratio of a thousand to one honestly means drawing one of the two orbits at a size the page cannot resolve, which is exactly why the wobble was not detected until instruments could measure metres per second.

The barycentre as a working quantity #

The last observation is not merely a curiosity, because the solar system’s barycentre is a place that has to be located precisely for one of the measurements described above to work at all.

Pulsar timing compares the arrival times of pulses against a uniform time scale, and an observatory on the Earth is not a uniform platform: it is moving at thirty kilometres a second around the Sun, which changes the light travel time from a pulsar by up to five hundred seconds over a year. Removing that requires referring every arrival time to the solar system’s barycentre.

Which requires knowing where the barycentre is, relative to the Earth, at every instant — to within metres, because a metre of error is three nanoseconds of timing and the timing precision is tens of nanoseconds. That means knowing the positions and masses of all the planets, since each contributes to where the centre of mass sits, and it means knowing them well enough that the residual error is below the measurement.

The consequence is a genuine coupling between two subjects that have no other contact. An error in the mass of a giant planet displaces the computed barycentre, which puts a periodic signal at that planet’s orbital period into every pulsar’s timing residuals. Comparing residuals across many pulsars separates that common signal from each pulsar’s own behaviour — and the planetary masses have been improved this way, from timing rather than from tracking.

A point in empty space that nothing occupies is a quantity with an error bar, and the error bar is small enough to matter.

The construction is the same at every mass ratio and every eccentricity, and it is worth drawing at two combinations the essay has not used.

The ladder from here #

The extreme case of the same construction is a star wobbling under a planet a thousandth its mass, where the barycentre lies inside the star itself and the reflex motion is metres per second.

Later rungs: the reduced-mass derivation in full. Visual binaries, and how the mass ratio is read off the two apparent orbits. Spectroscopic binaries, single- and double-lined. The mass function, and what a single-lined system can and cannot yield. Eclipsing binaries, the one configuration that gives radii as well as masses. The inclination problem and how transits solve it. Astrometric detection with Gaia. Pulsar timing. And the relativistic two-body problem, where the orbit decays and the decay was measured before the waves were.

The barycentre of the solar system spends most of its time outside the Sun, dragged about by Jupiter and Saturn. The Sun has been orbiting a point in empty space for its entire existence, and nobody noticed until there was a reason to look for the same thing elsewhere.