Astronomy & Celestial Mechanics
the sky, computed
Gravitation

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

A planet does not orbit its star. Both orbit a point between them, and the star's share of that motion is small, measurable, and the reason thousands of planets are known.
10 min read 6 figures Two bodies, and no moreMeasuring the unreachable

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.

Two bodies at a mass ratio of 3 to 1Both bodies orbit their common centre of mass, on similar ellipses whose sizes are in inverse proportion to the masses — here 3 to 1, so the heavier body's path is 3 times smaller.barycentrethe line joining them always passes through it
Fig. 1 Two bodies at a mass ratio of three to one. Both trace similar ellipses about their common centre of mass, and the heavier body’s orbit is smaller by exactly the mass ratio. The line joining them passes through the barycentre at every instant.

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:

a1a2=m2m1.\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.

Two bodies at a mass ratio of 1 to 1Both bodies orbit their common centre of mass, on similar ellipses whose sizes are in inverse proportion to the masses — here 1 to 1, so the heavier body's path is 1 times smaller.barycentrethe line joining them always passes through it
Fig. 2 Equal masses. Both orbits are the same size, the barycentre is exactly halfway between the two bodies, and there is no sense in which either one is the primary.
Two bodies at a mass ratio of 14 to 1Both bodies orbit their common centre of mass, on similar ellipses whose sizes are in inverse proportion to the masses — here 14 to 1, so the heavier body's path is 14 times smaller.barycentrethe line joining them always passes through it
Fig. 3 A ratio of fourteen to one. The heavy body’s orbit has shrunk to a small loop while the light one sweeps the same figure fourteen times larger — and this is still a far less extreme ratio than any star and planet.

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,

μr¨=−Gm1m2r2r^,μ=m1m2m1+m2,\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 μ\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, μ→m2\mu \to m_2, the lighter mass, and the heavy body might as well be nailed down. When the masses are equal, μ=m/2\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.

The wobble of a star with a circular companionThe star's velocity along the line of sight, over two orbits, computed from the companion's orbit. A circular orbit gives a sine wave; an eccentric one gives a skewed curve whose shape encodes the eccentricity.00.511.52-101orbitstoward the observeraway from the observercircular: a pure sine wave
Fig. 4 A star’s velocity along the line of sight when its companion is on a circular orbit. The curve is a pure sine wave; its period is the orbital period and its amplitude depends on the companion’s mass and the orbit’s size.

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 10810^8; 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 wobble of a star, companion at eccentricity 0.55The star's velocity along the line of sight, over two orbits, computed from the companion's orbit. A circular orbit gives a sine wave; an eccentric one gives a skewed curve whose shape encodes the eccentricity.00.511.52-101orbitstoward the observeraway from the observere = 0.55: skewed, and the skew is the measurement
Fig. 5 The same measurement when the companion’s orbit is eccentric. The curve is skewed rather than sinusoidal, and fitting the skew recovers both the eccentricity and the orientation of an orbit nobody can see.

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 msin⁡im\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≈90°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.

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.

An orbit at eccentricity 0.5An orbit of eccentricity 0.5. The primary sits at a focus, offset from the centre by 0.5 of the semi-major axis, and the closest and furthest points differ by a factor of 3.00.empty focusrperiapsisapoapsis
Fig. 6 The relative orbit — the path of one body as seen from the other. This is the ellipse Kepler’s laws describe, and it is not the path of either body through space.

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 ladder from here

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.

Where this goes next

0.321.03.210320.100.321.03.21032100316semi-major axis (AU)MercuryVenusEarthMarsJupiterSaturnUranusNeptuneslope 3/2 — P² ∝ a³period (years) The law that links period to size, and weighs everything Kepler found that the square of the period goes as the cube of the orbit. Newton found the constant of proportionality, and that constant is a mass — which is how every mass in astronomy has been obtained since. fast hereslow here16 equal intervals of time, one full orbit Equal areas in equal times, which is angular momentum in disguise Kepler's second law is a statement about the area a radius line sweeps. It looks like an odd thing to have noticed, and it turns out to be a conservation law arriving eighty years early. toward the sourcethe near side is pulled harder than the centrethe far side is pulled less — and so falls behindnot to scale: the source is far outside this frame The tide is a difference, which is why there are two of them The Moon pulls the ocean toward it. That explains one bulge. The second one, on the far side, is the whole of the physics — and it comes from subtracting.