One eccentric planet, or two circular ones

A star with a planet moves in a small orbit of its own, and its velocity towards and away from the observer rises and falls once per orbit. For a circular orbit the curve is a sine wave, and its amplitude and period give the planet’s minimum mass and its distance. For an eccentric orbit the curve is skewed: the star moves fast near periastron and slowly near apastron, as the second law requires, so one half of the cycle is compressed and the other stretched. The skew is how an observer measures the shape of an orbit nobody can see.

It is also exactly what a second planet would do. A skewed periodic curve is a sine wave plus harmonics — components at twice the frequency, three times, and so on — and the largest of them, at twice the frequency, is indistinguishable from the signal of another planet orbiting at half the period on a circular orbit. For the eccentricities most planets have, one reading of the data is a single planet on a moderately eccentric orbit, and the other is two planets on circular orbits with periods in the ratio two to one. The two are the same measurement.

The expansion that makes them the same #

The reason is a line of algebra. The star’s velocity along the line of sight is

$$v = K\left[\cos(\nu + \omega) + e\cos\omega\right],$$

where ν is the true anomaly — the angle of the planet from periastron — ω is the orientation of the orbit, and K the semi-amplitude. The true anomaly does not advance uniformly in time; it runs ahead near periastron. Expanded in the mean anomaly M, which does advance uniformly, it is

$$\nu = M + 2e\sin M + O(e^2).$$

Put that into the velocity, expand the cosine, and keep terms to first order in e. The constant term cancels against the $e\cos\omega$, and what is left is

$$v \approx K\left[\cos(M+\omega) + e\cos(2M+\omega)\right].$$

That is a sinusoid at the orbital frequency with amplitude K plus a sinusoid at twice the frequency with amplitude eK. And a circular orbit at half the period has a velocity curve that is exactly a sinusoid at twice the frequency, with any amplitude and any phase. So an eccentric orbit of eccentricity e, to first order, is a circular orbit plus a second circular orbit at half the period with an amplitude e times as large.

There is no phase constraint that saves the distinction. The second harmonic’s phase is locked to the fundamental’s through ω, but ω is a free parameter of the orbit. Any relative phase the two-planet model wants can be supplied by choosing ω; any ω the eccentric model has can be imitated by choosing the phase of the inner planet. The mapping between the two models is exact at first order, and it runs in both directions.

A difference that is second order #

The two models part company at the next order in e. The true velocity curve contains a third harmonic of amplitude about $e^2 K$, a fourth of amplitude about $e^3 K$, and so on; the two-planet model has none of these. The difference between the two curves is therefore of order $e^2 K$, and how visible it is depends on how that compares with the noise.

For a planet with a semi-amplitude of 30 metres a second observed with an instrument delivering one metre a second, the noise is about three per cent of K, and eccentricities below about 0.2 are ambiguous from any single set of data of modest size. That covers most of the eccentricities in the known population. The distinction can still be made with enough data — noise averages down as the square root of the number of observations and the residual does not — but the amount of data required rises steeply as the eccentricity falls, and a published orbit fitted from a few dozen velocities has usually not reached it.

This is not the same effect as the preference every detection method has for circular orbits, though it is related to it. That effect is about which planets are found: a periodogram built from sine waves loses the power in an eccentric orbit’s harmonics, so an eccentric planet is slightly harder to detect than a circular one of the same mass. The degeneracy here is about what is concluded once a planet has been found: the power that went into the harmonics is still in the data, and it can be attributed to the planet’s orbit or to another planet.

The planet an eccentricity can hide #

Read in one direction, the degeneracy says an eccentric orbit might really be two planets. Read in the other, it says a real second planet might be absorbed into the first planet’s orbit and never reported.

So a real inner planet of a fifth of the outer planet’s mass, on a 2:1 orbit, produces the same velocities as a single planet with an eccentricity of about a quarter. Fitted as a single planet, the inner one disappears from the catalogue, and the outer one is recorded as more eccentric than it is. Both errors propagate. The catalogue’s count of planets in 2:1 resonance is too low, and its distribution of eccentricities is too high, by amounts that depend on how many such pairs there are — which is the quantity that cannot be measured without resolving the degeneracy.

That matters because the two readings imply different histories. A pair of planets in a 2:1 resonance is the characteristic product of convergent migration in a gas disc: two planets drifting inward at different rates meet at a resonance and are captured into it, and chains of such captures are the strongest evidence that planets migrate. A single eccentric planet is the characteristic product of the opposite kind of history: a violent one, in which planets scattered each other or a distant companion pumped the eccentricity. The same velocity curve supports a quiet history or a violent one, and the demographics of planetary systems depend on which is chosen.

There is one constraint the two-planet reading has to satisfy that the single-planet reading does not, and occasionally it decides the matter. Two planets at a period ratio of two sit at orbital radii in the ratio $2^{2/3} \approx 1.59$, and a pair that close is stable only if the planets are not too massive and their orbits not too eccentric. For a pair of small planets around a Sun-like star this is no restriction at all. For the fits that come out with an inner planet of a Jupiter mass or more, and with eccentricities of their own, it can be: a numerical integration of the fitted system over a few million orbits sometimes shows it tearing itself apart in a few thousand, and a solution that could not have survived to be observed is excluded whatever its likelihood. The opposite does not follow — stability says a pair could exist, not that it does — and a resonance can protect a pair that would otherwise be unstable, which is exactly the configuration convergent migration produces.

How the second planet gets manufactured #

The degeneracy would be only a curiosity if the analysis always fitted both models and compared them. In practice the analysis is often sequential, and the sequence decides the answer.

The standard procedure for finding multiple planets in velocity data is exactly the one in the figure: compute a periodogram, fit the strongest signal, subtract it, and compute the periodogram of the residuals. If the first fit is a circular orbit, the residuals contain the second harmonic, and the second harmonic looks like a planet. If the first fit is a Keplerian with free eccentricity, the second harmonic is absorbed and the residuals are clean.

Neither choice is neutral. A circular first fit manufactures an inner planet whenever the real orbit is eccentric; an eccentric first fit absorbs an inner planet whenever the real system is a 2:1 pair. The residual periodogram does not say which happened. A peak in a periodogram is a statement about the model subtracted before it was computed, and a peak at exactly half the period of a known planet is the least informative peak there is.

That is not a hypothetical concern. Re-analyses of published single-planet systems have found that a substantial fraction of the moderately eccentric ones are fitted as well, or better, by a pair of planets near a 2:1 period ratio, and several systems first reported as a single eccentric planet have later been reported as two. The reverse has also happened: signals at half a known planet’s period have been withdrawn when the first planet’s orbit was refitted with a small eccentricity.

When the data cannot decide, the prior does #

Faced with two models that fit equally well, a careful analysis compares them formally, and the comparison is less objective than it sounds.

A single eccentric orbit has five parameters: period, semi-amplitude, eccentricity, orientation and a reference time. Two circular orbits have six: a period, amplitude and phase each. The eccentric model is therefore the simpler of the two, and any criterion that penalises extra parameters prefers it when the fits are equally good. But the fits are not equally good in general; the two-planet model can always absorb some noise the eccentric one cannot, and the penalty for one extra parameter is small. The outcome of the comparison depends on how the penalty is set.

In a Bayesian comparison the penalty is replaced by priors, and the priors are where the answer hides. A prior on eccentricity that favours small values — as the observed distribution does, with most planets below 0.3 — pushes the eccentric model towards zero and makes the two-planet reading more attractive. A prior on multiplicity that says 2:1 pairs are rare pushes the other way. Both priors are taken from the catalogue, and the catalogue is built from fits that faced this very choice. The circle closes: the eccentricity distribution is inferred from orbits whose eccentricities were partly decided by an assumed eccentricity distribution.

None of that makes the analysis wrong. It makes the eccentricity of an individual planet, below about 0.2 and without a transit or a long baseline, a quantity whose value depends on what else is believed about planetary systems. That is a different kind of uncertainty from a measurement error, and it does not shrink with better calibration.

Why noise alone makes orbits look eccentric #

There is an older version of the same problem, and it pushes in the same direction. An eccentricity is a positive quantity: it is the length of a vector whose two components are $e\cos\omega$ and $e\sin\omega$. If the true orbit is circular, both components are zero, but a fit to noisy data returns small non-zero values for each, and the length of the resulting vector is always positive. Noise therefore never makes a circular orbit look circular. It makes it look slightly eccentric, with an eccentricity of order the noise divided by the semi-amplitude.

For spectroscopic binary stars the effect was recognised in 1971, and the standard response is a significance test: an eccentricity is quoted only if it is significantly larger than the value noise alone would produce for a circular orbit, and set to zero otherwise. The same bias afflicts planets, and it compounds with the degeneracy rather than cancelling it. A circular planet with a real inner companion at half its period is fitted, as a single planet, with an eccentricity inflated both by the companion’s harmonic and by the noise — two effects of the same sign, neither of which can be seen in the fit.

The combined result is that the lower end of the eccentricity distribution is the least secure part of it, and it is precisely the part that distinguishes a system that migrated quietly from one that did not.

The harmonic that only an ellipse makes #

The way out is the third harmonic, and it is worth being precise about what it can decide.

An eccentric orbit predicts a third harmonic with a definite amplitude — about $\tfrac{9}{8}e^2K$ — and a definite phase relative to the first two. The two-circular-planet model predicts none. Detecting a third harmonic of the right size and phase is therefore evidence for eccentricity, and failing to detect one where the eccentric model says it should be large enough is evidence against it. The required precision is set by $e^2 K$, which at an eccentricity of 0.1 is a hundredth of the semi-amplitude.

Two complications prevent this from being a clean test. First, two real planets in or near a 2:1 resonance perturb each other, and the perturbation produces signals of its own at combinations of the two frequencies, including the third harmonic of the outer one. A resonant pair does not look like the sum of two independent circular orbits once the data span long enough for the interaction to matter. Second, the two planets need not be on circular orbits either: a pair can have small eccentricities of its own, with harmonics of its own. The comparison is therefore not between one eccentric orbit and two circular ones, but between one eccentric orbit and two interacting, slightly eccentric ones — a model with more parameters, which can always fit a little better.

What the long baseline buys is exactly the interaction. A single eccentric orbit is strictly periodic. A resonant pair is not: its orbits precess and exchange energy on a timescale of years to decades, and its velocity curve changes shape slowly. A data set spanning several of those cycles can tell the two apart by the drift, whatever the third harmonic does — which is why the ambiguous systems are resolved by time rather than by precision.

Transits break it, when they exist #

For a planet that also transits, the degeneracy has an independent resolution, because a transit time is a direct statement about where the planet is.

A planet on an eccentric orbit transits at a phase that depends on ω, and its transit duration differs from that of a circular orbit by a factor depending on e and ω together. An inner planet at half the period would transit too, if its orbit were aligned closely enough with the outer one’s, and it would be seen. And a resonant pair of transiting planets shifts each other’s transit times by minutes to hours, in a pattern that has no counterpart for a single planet. Any one of those three is enough to decide.

Most planets found by velocities do not transit, and for those the degeneracy is resolved only by more velocities. That is why the eccentricity distribution of velocity-detected planets — the distribution every theory of planetary dynamics is ultimately tested against — carries an uncertainty that is not in any individual orbit’s error bar.

What the picture assumes #

The figures draw the velocities as coming only from the planet. A real star adds its own signals: rotation, spots and convection, many of them periodic, and the rotation signal of a spotted star is itself strongly non-sinusoidal, with harmonics of its own. A star rotating in forty days with a spot pattern that persists for several rotations produces power at forty days and at twenty, just like the eccentric planet in the figures. A signal at half a known period may be the orbit’s harmonic, a second planet, or the star’s rotation harmonic, and the three have to be separated by other means — activity indicators, line shapes and the persistence of the phase.

The figures also assume full phase coverage when they compute the imitation. With gaps, the best two-planet fit is not simply the first two Fourier terms, and the sampling’s own periodicities can make the degeneracy worse — a gap pattern with power at the orbital period mixes the harmonics with each other.

Still open: how many 2:1 pairs are catalogued as eccentric planets #

The fraction is not known, and it is a fraction that matters. If a substantial part of the moderately eccentric planets found by velocities are really unresolved 2:1 pairs, then resonant pairs are more common than the catalogues say and the eccentricity distribution is less extreme than it looks — which would move the balance between the quiet, migration-driven histories and the violent, scattering-driven ones. The test is data: long, dense velocity series on the ambiguous systems, analysed with both models fitted from the start rather than in sequence. Those series also run into the floor that the star’s own surface sets on every velocity measurement, which at the precision a third harmonic needs is no longer the instrument’s to set.