A mass that is only a mass once the eccentricity is known

The first rung of this anchor ended on a promise: the amplitude of a transit-timing residual is proportional to the perturbing planet’s mass, so fitting the curve weighs a planet that may never cross its star. That is true of a circular pair, and no pair is circular.

The amplitude of the slow, near-resonant part of the signal is proportional not to the perturbing mass but to the perturbing mass multiplied by a bracket, and the bracket contains the pair’s free eccentricity divided by the fractional distance from exact resonance. That distance is small, by construction — the signal is only large because it is small — so an eccentricity of a hundredth is not a small correction. It is a factor of two.

An observer handed one of those curves and no other information cannot say which system produced it. That is not a limitation of the fitting; it is a statement about what the data contain.

Why the slow part is a sinusoid at all #

Before the eccentricity, the shape. A pair near a $j\!:\!(j-1)$ commensurability has a resonant angle that would be stationary if the resonance were exact and instead drifts slowly, at the rate

$$\frac{1}{P_{\rm super}} = \left| \frac{j}{P'} - \frac{j-1}{P} \right|$$

which is small precisely because the two terms nearly cancel. The perturbation delivered at each conjunction therefore points in nearly the same direction for many orbits, accumulates, and then slowly reverses as the angle circulates — and the accumulated displacement, sampled at each transit, is a sinusoid at that beat.

The amplitude of that sinusoid is the accumulated kick, so it is the per-conjunction kick multiplied by the number of conjunctions over which the accumulation is coherent. The first factor is proportional to the perturbing mass; the second is proportional to the super-period, and therefore to $1/\Delta$. That is where the $\mu'/\Delta$ in the expression below comes from, and it explains a fact that would otherwise look perverse: a signal can be made arbitrarily large by moving the pair closer to resonance without changing a single mass.

Where the eccentricity gets in #

The transiting planet’s timing residual near a first-order resonance is, to first order in the masses,

$$\delta t \;\propto\; \frac{\mu'}{\Delta}\left| f + \frac{3}{2}\frac{Z^*_{\rm free}}{\Delta} \right|$$

where $\mu'$ is the perturber’s mass in stellar masses, $\Delta$ is the fractional distance from exact commensurability, $f$ is a coefficient of order unity belonging to the particular resonance, and $Z_{\rm free}$ is a complex combination of the two planets’ free eccentricities.

Two things about that expression matter and neither is the constant. The first is that the eccentricity appears divided by $\Delta$ where the mass does not, so the eccentric term grows faster than the resonant term as a pair is placed nearer to resonance. The second is that $Z_{\rm free}$ is complex — it has a phase — so the correction can add or subtract. A pair with the right relative apsidal alignment produces a smaller signal than a circular pair of the same masses, which means the degeneracy runs in both directions and a fitted mass can be too small as easily as too large.

That last sentence is the difficulty in one line. The quantity that doubles the answer is one the same data are almost blind to.

It is worth being precise about which eccentricity. A near-resonant pair has two of them and they decompose into a forced part, which the resonance itself imposes and which is a known function of the masses, and a free part, which is whatever the system was left with by its formation and its subsequent damping. Only the free part is a nuisance. The forced part is not an extra unknown at all — it is determined by the same masses being solved for, and a fit that included it as a free parameter would be double-counting. This is why the expression above carries $Z_{\rm free}$ rather than the eccentricities themselves, and it is why the degeneracy is not simply “the fit has more parameters than data”.

The exchange rate is set by the distance from resonance #

Since the eccentric term carries an extra factor of $1/\Delta$, how much eccentricity it takes to imitate a given change in mass depends on how close the pair sits to the resonance — and it takes less, the closer it is.

This is worth stating as a rule because it inverts the usual relation between signal and information. The systems with the largest transit-timing signals are the ones whose masses are least determined by them, because both the amplitude and the degeneracy scale as inverse powers of the same small number. A pair that sits a long way from resonance has a small, clean, nearly eccentricity-free signal that is often too small to measure at all.

There is a second reason the eccentric term is hard to bound from outside the fit. Its phase matters, and the phase is the relative orientation of the two orbits’ apsides. Two systems with identical eccentricities and opposite alignments produce residuals differing by a factor of several, so even a perfect external measurement of the two eccentricity magnitudes would not remove the ambiguity. What is needed is the eccentricity vector difference, which nothing else observes.

What broke, historically #

The degeneracy was not an anticipated difficulty that was carefully managed. It showed up as a discrepancy in a catalogue.

By around 2014 there were enough planets with both a transit-timing mass and a radial-velocity mass to compare, and the transit-timing masses were systematically the smaller — by tens of per cent on average, and by factors of two on individual objects. Both techniques were mature; both had internally consistent error bars; and the error bars did not overlap.

The comparison sample was small and the objects in it were not typical. A planet with both kinds of mass is a planet in a compact multi-planet system — because that is what produces timing variations — that also orbits a star bright enough and quiet enough for a velocity measurement at the metre-per-second level. Those are competing requirements, and the handful of systems meeting both carried a great deal of weight.

Several explanations were offered and the one that survived is the least dramatic. A fit that holds the eccentricities at zero, or at small values with a prior that pulls them to zero, absorbs the eccentric part of the bracket into the mass. Which way it is absorbed depends on the sign of the eccentric term, and the sign is not random across a real sample: pairs that arrived at their resonances by convergent migration and were subsequently damped tend to share an apsidal alignment, and that alignment biases the fits one way. The effect was a property of the priors and of the systems’ common history, not of either instrument.

The signal that is not degenerate #

There is a second component in every one of these curves, and it does not carry the eccentricity.

The near-resonant term is a slow sinusoid because it is a nearly resonant argument slipping slowly out of phase. Superimposed on it is the direct effect of each conjunction: a discrete kick delivered whenever the two planets pass one another, at the synodic period. This is the chopping signal, it comes from the geometry of a close approach rather than from a resonant argument, and to leading order it is proportional to the perturbing mass and to nothing else that is unknown.

The physical reason it is clean is worth a sentence. A conjunction is a close approach, and the impulse it delivers depends on how close the two planets come and how fast they pass — a geometry fixed by the two semi-major axes, which are known to many digits from the periods. What the eccentricity changes is where along the orbit the conjunction happens, and that enters the chopping amplitude only at second order, because the kick is dominated by the separation at closest approach rather than by the phase at which it occurs. The resonant term has the opposite character: it is an accumulation over many conjunctions, and an accumulation is exactly what a slowly precessing apsidal line interferes with.

The chopping term is what turned transit-timing masses from ratios into measurements, and it is a demanding thing to measure. It is a sixth of the amplitude of the signal it has to be pulled out of; it lives at the synodic period, which for a near-resonant pair is a few times the transiting planet’s own period, so it is sampled barely above the rate at which it can be followed; and its own amplitude is a fraction of a minute where the timing precision of a good ground-based light curve is a minute.

Space photometry is what made that possible. Transit times good to tens of seconds, over four years, for hundreds of consecutive transits, is what the chopping signal needs, and it is why the technique’s masses became credible in the same years that its data did. It is also why the technique does not travel well: a survey with a shorter baseline sees fewer super-periods and fewer conjunctions, and both losses fall on the chopping term first, because it is the smaller signal and the one that needs the denser sampling.

The two systems where it worked first are worth naming, because they bracket the difficulty. Kepler-36, a pair at 13.8 and 16.2 days in a 7:6 period ratio, produces timing variations of tens of minutes — so large, and with such a strong chopping component from conjunctions every 97 days, that both masses came out to a few per cent. The result was the pair of densities, 7.5 and 0.9 grams per cubic centimetre for planets 10 per cent apart in orbital distance, that became the sharpest single piece of evidence for atmospheric stripping in the census. Kepler-18, by contrast, has a pair near the 2:1 whose timing gives a mass ratio cleanly and whose absolute masses needed the radial velocities as well — the two techniques used together on one system rather than checked against each other.

What is actually measured #

Nothing above is what a fit does. It is worth setting the idealisation beside the practice.

An observer has a list of transit times with error bars, one per transit, for one or more planets. The model is an N-body integration whose parameters are the masses, the periods, the eccentricity vectors and the epochs of every planet in the system — fourteen parameters for a two-planet fit — sampled with a Markov chain. There is no fitting of a super-period and no subtraction of a sinusoid; the chopping term is in the integration automatically, and it does its work by tightening the posterior rather than by being measured separately.

The integration is not optional and the reason is worth stating, since a closed form exists for the case this essay has been describing. The analytic expressions are expansions in the eccentricity and in the distance from resonance, and they are excellent where they apply. They do not apply to the interesting cases — a deep resonance, an eccentricity of a tenth, or three planets interacting at once — and those are the systems whose masses are worth having. Every published mass from a compact chain comes from fitting a numerical integration to the times.

That means the degeneracy shows up not as a wrong answer but as a shape in the posterior: a long, curved ridge in the plane of mass against free eccentricity, along which the likelihood barely changes. Whether the reported mass is meaningful depends entirely on whether the data are good enough to close that ridge, and the honest reporting of such a fit is the two-dimensional contour rather than a mass with a symmetric error bar. Many published masses from the first years of the technique were quoted as the latter.

There is a further complication the two-planet picture hides. Most systems with measurable timing variations have more than two planets, and a third body contributes its own near-resonant terms at its own super-periods. Untangling several superposed slow sinusoids from a few hundred transit times is not a harder version of the same problem; it is a different one, because two of those periods can beat against each other over the length of the dataset and mimic a single term of a third period. The three-body problem has no closed-form solution, so there is no expression to appeal to, and the chain systems where the technique is most powerful are exactly the ones where this is worst.

The other thing the practice supplies is a prior. Eccentricities in compact multi-planet systems really are small — tidal damping and the pairs’ own dynamics see to that — so a prior that concentrates near zero is defensible. It is also exactly the assumption that produced the systematic offset against radial velocities, which is a reminder that a defensible prior and a harmless one are different things.

The generalisation #

The structure worth extracting is that a degeneracy is a statement about a combination, and the way out is never a better fit to the same combination.

Here the observable is $\mu'(f + \frac{3}{2}Z^*/\Delta)$, and no amount of timing precision on that quantity separates its factors. What separates them is a second observable with a different dependence — the chopping term, which carries $\mu'$ alone — and it is smaller and harder by an order of magnitude than the one that was degenerate. That relation is not a coincidence. The strong signal is strong because it is resonantly enhanced, and the resonant enhancement is precisely what brings the eccentricity in with it.

The same shape recurs throughout the subject. A radial velocity gives $m\sin i$ and no improvement in the velocities gives the mass; a transit does, because it constrains the inclination separately. A transit depth is a radius ratio and better photometry never yields a radius; a stellar model does. In each case the second measurement is of a different kind, and in each case it is the weaker one.

There is a positive version of the same observation, and it is what makes the technique worth its difficulties. Two measurements of one planet with different degeneracies are together stronger than either is alone, and much stronger than two of the same kind. A transit and a radial velocity give a density because one supplies a radius and the other a mass; a transit and a timing signal give a mass without a velocity at all, which is the only route to the masses of planets around stars too faint for a spectrograph. Most of the planets in the census are around such stars.

The corollary is a rule for reading a catalogue. When two techniques disagree about the same objects, the parameter that differs is usually not the one either technique measures. Transit timing and radial velocity both measure a planet’s mass in the sense that they report one; what they actually measure is a resonant amplitude and a velocity semi-amplitude, and the disagreement lived entirely in what had to be assumed to convert one of them.

Where the ladder goes next #

The next rung takes the same integration and asks what happens when the timing is used to predict rather than to measure. A near-resonant pair’s transit times wander on the super-period, so a linear ephemeris fitted during one phase of that wander mispredicts the next season by tens of minutes — which is a scheduling problem for a follow-up campaign and, more interestingly, a way of detecting a third planet from the failure of a two-planet solution.

Further rungs on this anchor: transit duration variations, which measure the mutual inclination that the timing alone cannot; the use of timing to establish that a candidate is a planet rather than a blend, since a blended eclipsing binary produces no such signal; the systems with only one transiting planet, where the perturber’s period is itself a fitted parameter and the solutions are multiply valued; and the Trojan configuration, which produces a timing signal with no accompanying transit at any period.