The star moves, and the mass is a lower bound

The Sun does not sit still while Jupiter goes round it. Both bodies orbit their common centre of mass, and since the Sun is a thousand times the heavier, its own orbit is a thousand times the smaller — a circle of about 0.005 AU, roughly the Sun’s own radius, traversed once every twelve years at 12.5 metres per second.

That last number is the entire method. Twelve and a half metres per second is a fast bicycle. It is also, measured in the wavelength of a spectral line, four parts in $10^8$ — and it is measurable.

What the amplitude contains #

Neither body is still: they orbit a barycentre that divides the line between them in the inverse ratio of their masses. The star’s orbital speed is therefore the planet’s, scaled down by the mass ratio.

Only the component along the line of sight is measurable, because that is what a Doppler shift records. So the observed semi-amplitude is

$$K = \left(\frac{2\pi G}{P}\right)^{1/3}\frac{m_p \sin i}{(M_\star + m_p)^{2/3}}\frac{1}{\sqrt{1-e^2}},$$

which for a planet much lighter than its star is more usefully written

$$K \approx 28.4\ \text{m/s}\ \left(\frac{m_p \sin i}{M_J}\right)\left(\frac{M_\star}{M_\odot}\right)^{-2/3}\left(\frac{P}{1\ \text{yr}}\right)^{-1/3}.$$

Jupiter around the Sun gives 12.5 m/s. A Jupiter at 4 days gives 140 m/s. The Earth around the Sun gives 9 cm/s. The period comes out of the same fit, and with the period comes the orbital distance through the harmonic law. So a radial-velocity orbit delivers $P$, $e$, the argument of periapsis, and the product $m_p\sin i$ — everything except the one factor that would turn the last of those into a mass.

The shift itself #

The velocity is read from the positions of absorption lines in the star’s spectrum, displaced by $\Delta\lambda/\lambda = v/c$. At 12.5 m/s that is a shift of $4\times10^{-8}$, which for a line at 500 nm is 0.00002 nm — about a thousandth of the width of the line itself, and around a ten-thousandth of a pixel on the detector. Nothing about that is achievable one line at a time, and the method does not try. A cross-correlation of the whole spectrum against a template built from thousands of lines beats the single-line precision down by roughly the square root of the number of lines — and then the residual problem is not photon noise but the instrument, which must have its wavelength scale stable to the same four parts in $10^8$ over years.

The lines themselves are the ones a stellar spectrum shows in absorption, and the method is indifferent to which elements produce them — what it needs is many narrow features whose rest wavelengths are known, which is why cool stars with forests of metal lines are far better radial-velocity targets than hot stars with a handful of broad hydrogen lines. A star’s suitability for this method is decided by its spectral type before any planet is involved.

Two solutions were found, and both are in use. The first passes the starlight through a cell of iodine vapour, imprinting a dense forest of reference lines on the same detector pixels at the same moment, so that any drift affects star and reference identically. The second stabilises the spectrograph itself — in vacuum, in a temperature-controlled room, fed by an optical fibre that scrambles the illumination — and calibrates it against a thorium–argon lamp or, now, a laser frequency comb. The second is what HARPS did to reach 1 m/s and ESPRESSO to reach a few tens of centimetres per second.

Six numbers, of which this method supplies five #

It is worth setting out exactly which of the elements that fix an orbit a velocity curve delivers, because the accounting is unusually clean.

The period and the epoch of periapsis come from the timing. The eccentricity and the argument of periapsis come from the shape of the curve. The semi-major axis follows from the period and the stellar mass. That is five, and the sixth — the inclination — is absent, along with the longitude of the ascending node, which no line-of-sight measurement can ever see because rotating the whole orbit about the line of sight changes nothing about it.

So the method is not vague about the orbit. It is precise about everything except the orientation of the plane, and that single gap is what the rest of this essay is about.

The eccentricity is in the shape #

A circular orbit gives a sine wave. An eccentric one does not: the star moves fastest near periapsis and slowest near apoapsis, so the curve becomes skewed, and both the eccentricity and the orientation of the orbit in its own plane are readable from the skew.

The speed variation itself is Kepler’s second law seen edge-on: the radius vector sweeps equal areas in equal times, so the star races through periapsis and dawdles through apoapsis, and the projection of that onto the line of sight is the skew in the curve. This is genuine and it matters, because eccentricity is a fossil of a system’s history — the record of whether a hot Jupiter arrived gently or violently is written in the eccentricity distribution and nowhere else.

It is also the method’s most notorious trap. An eccentric single-planet signal and a pair of planets on circular orbits near a 2:1 period ratio produce very similar velocity curves, and at least a dozen published eccentric planets have later been reinterpreted as two circular ones. Where the two planets are genuinely present and genuinely near a commensurability, they also perturb each other, and what “unsolvable” means for three bodies starts to matter: the orbits are no longer fixed ellipses, and a model of two independent Keplerian signals is fitting a system that does not have them. The ambiguity is not a failure of the instrument; two different systems really do produce nearly the same one-dimensional signal, and only more data or a different method separates them.

The angle that is never measured #

Here is the flaw, and it is structural rather than technical: the amplitude carries $m_p\sin i$, and $i$ is the angle between the orbital plane and the sky. Nothing in a radial-velocity measurement constrains it. An orbit seen face-on has $\sin i = 0$ and produces no signal at all, however massive the planet.

So every mass from this method is a minimum mass. The true mass is $m_p\sin i$ divided by $\sin i$, and the divisor is unknown and at most 1.

The saving grace is statistical and it is stronger than it first looks. Randomly oriented orbits are uniform in $\cos i$, which means edge-on orientations are far more likely than face-on ones — a face-on orbit occupies a vanishing solid angle. The median inclination is 60°, where the correction is 15.5 per cent. Only 13 per cent of systems need a correction of two or more, and only 1.5 per cent need a factor of five.

For a population, then, $m\sin i$ is nearly as good as a mass. For a single object it is not, and the distinction matters most where it matters most: an object with $m\sin i$ of 12 Jupiter masses is a planet if the orbit is anywhere near edge-on and a brown dwarf if it happens to be seen at 20°, and the difference is a difference of kind — deuterium burns above about 13 Jupiter masses, so the boundary being straddled is a threshold in nuclear physics rather than a matter of taste.

The escape is to measure $i$ another way. A transit does it directly — a transiting planet has $i$ within a few degrees of 90°, so $\sin i$ is 1 to within a fraction of a per cent, and the two measurements together give a true mass and therefore a density. Astrometry does it by mapping the star’s reflex motion on the sky in two dimensions rather than one. Both are the same move: add a second projection of the same motion.

The shape of the curve is the eccentricity and the orientation, and it is worth seeing both extremes drawn at the same amplitude.

What each survey can see #

The scaling $K \propto m_p P^{-1/3}$ is a boundary in the plane of planet mass against orbital distance, and it is a rising one: the further out a planet is, the more massive it must be to produce the same wobble.

This has a consequence that is worth stating plainly, because it was misread for a while. The first planets found were massive and close in, and that is what a 10 m/s survey watching for a few years is able to find. It says nothing about how common such planets are until the sensitivity boundary is divided out, and doing that division carefully is a whole subject of its own.

What was actually measured #

51 Pegasi b, 1995. Michel Mayor and Didier Queloz, using the ELODIE spectrograph at Haute-Provence, found a 4.23-day sinusoid of semi-amplitude 59 m/s in a Sun-like star. That gives $m\sin i = 0.47$ Jupiter masses at 0.05 AU — a giant planet eight times closer to its star than Mercury is to the Sun, which no theory of planet formation had predicted and several had excluded.

The reception is part of the measurement’s history. A period of 4.23 days was so far from expectation that the alternative explanations — non-radial stellar pulsation, in particular — were taken seriously for two years, and were only closed off when the line shapes were shown not to vary in the way pulsation demands. The distinction between a star that is moving and a star whose surface is changing shape is the permanent occupational hazard of this method.

The precision ladder. ELODIE reached about 13 m/s in 1995; HARPS reached 1 m/s in 2003; ESPRESSO reaches about 25 cm/s today. The Earth’s signal on the Sun is 9 cm/s, and the gap between 25 and 9 is now dominated not by instruments but by the star.

Proxima Centauri b, 2016. An 11.2-day signal of amplitude 1.38 m/s in the nearest star, giving $m\sin i = 1.27$ Earth masses in the habitable zone of an M dwarf 4.24 light years away. The signal is smaller than the star’s own activity-induced velocity variations, and the detection rests on separating the two by their different behaviour with wavelength and with the star’s 83-day rotation period.

And the one that was not there. Peter van de Kamp reported planets around Barnard’s Star from astrometry in 1963, refined over two decades. The wobble was real in the data and was an artefact of the telescope: it changed when the plate holder was modified in 1949 and again in 1957. The lesson the field took from it is the one that shapes every claim above — a periodic signal is a signal about the apparatus and the star and the planet, and the planet is the last of the three to be established.

When there is more than one #

A single planet produces one periodic curve. Two produce the sum of two, and the sum is where a distinctive class of error lives.

Fitting a multi-planet system means finding several periods, amplitudes, eccentricities and orientations at once, from a time series sampled whenever the telescope was available and the weather held. The parameters are not independent, and one degeneracy in particular has produced published planets that later evaporated.

An eccentric orbit at period $P$ looks very much like two circular orbits at $P$ and $P/2$. That is not a coincidence: expanding an eccentric Keplerian curve as a Fourier series gives a fundamental at the orbital period and a harmonic at half of it, with the harmonic’s amplitude growing with the eccentricity. So a single planet on a moderately eccentric orbit and a pair of circular planets in a 2:1 period ratio produce nearly the same velocities, and separating them requires either very high signal-to-noise or a longer baseline than the data have.

The two interpretations are not equally likely, and the argument between them is usually settled outside the data. A 2:1 pair is dynamically special — it is a mean-motion resonance, which is a configuration that has to be built by migration rather than arrived at by chance — so a fit preferring that solution is claiming something about the system’s history as well as its architecture. Several announced resonant pairs have been reinterpreted as single eccentric planets, and at least one single eccentric planet has gone the other way.

The general shape of the problem is that a periodic model with enough parameters can absorb almost anything, including the star’s own rotation and the sampling of the observations. That is why a modern analysis compares models rather than fitting one, and why the number of planets in a system is itself a fitted quantity with an uncertainty attached.

The remedy that settles such cases is almost always a different observable rather than more of the same one: a transit, an astrometric detection, or a direct image, each of which constrains the geometry in a way the velocities cannot.

Where the picture stops #

The star is not a rigid body with a velocity. Its surface is a boiling convective layer covered in magnetic spots, and both move spectral lines. Granulation shifts line centroids by a few m/s; a spot rotating across the disc removes light from an approaching or receding limb and mimics a velocity of tens of m/s on the rotation period; the magnetic cycle changes the average convective blueshift over years. This “jitter” is the floor for every current instrument, and separating it from a planet relies on the fact that activity signals vary with wavelength and change line shapes, while a real Doppler shift does neither.

Only the radial component exists. A face-on system is undetectable. Systems are not found; systems with a suitable geometry are found.

Long periods need long records. A signal must be observed through at least one full period, and preferably several, before it is a period rather than a trend. The Jupiter analogues in the current census exist because a few programmes have run continuously for twenty-five years.

The generalisation #

The technique is older than exoplanets by a century, and the arithmetic is unchanged.

A single-lined spectroscopic binary gives exactly this: a period, an eccentricity, and a mass function $f(m) = (m_2\sin i)^3/(M_1+m_2)^2$, which is a lower limit on the companion’s mass and became the standard tool of stellar astronomy. A double-lined binary — where both spectra are visible — gives the ratio of the masses directly, because both amplitudes are measured, and the inclination remains the missing factor until an eclipse supplies it. Reduced mass is the formal statement of why the two-body problem collapses to one.

The same measurement at a larger scale is how the mass of everything else is known. The orbital velocities of stars around the centre of the Galaxy give the enclosed mass; the velocity dispersion of a galaxy cluster gives its mass; the reflex motion of a star at the Galactic centre around an invisible companion gives four million solar masses in a region smaller than the solar system. In every case the observable is a Doppler shift and the inference is Kepler’s, and in every case a projection factor has to be argued about.

Two more settings separate the two things the curve’s amplitude and shape are doing.

Where this goes next #

The obvious next question is what happens when both a wobble and a transit are available for the same object — because the first gives a mass with an unknown factor and the second removes the factor, and the pair gives something neither can: a density, and with it a statement about what the planet is made of.

Later rungs on this anchor: the mass function and what a single-lined orbit cannot say. Stellar activity, and how a spot imitates a planet. Line-shape diagnostics and the bisector span. The iodine cell against the stabilised spectrograph. Laser frequency combs. Aliasing, sampling, and planets that were periods of the Earth’s year. The eccentricity–multiplicity degeneracy. Radial velocities of the Sun as a star. And the 9 cm/s problem, which is where the method’s future is decided.