A moon read from how late and how long

The Solar System holds more than four hundred known moons, and its eight planets hold them all but for the two innermost. Thousands of planets around other stars are known. No moon has been confirmed around any of them, and the reason is not any shortage of moons. A moon is small, and a large one has a transit depth of perhaps one part in ten thousand against a star’s noise. Its direct signal is out of reach for most targets.

But a moon pushes its planet. The planet and moon go round their common centre of mass, so the planet is not on the smooth Keplerian track that its orbital period suggests. It is displaced along the track by a small and regular amount, and transits are timed to tens of seconds. That is the idea behind reading companions from a transit running late, and it applies to a moon with the same logic and a different set of numbers.

The trouble is that a timing signal is one number for each transit, and a moon has two unknowns: how massive it is and how far out it sits. A second number was needed, and in 2009 David Kipping pointed out that the planet gives it away itself. It is displaced, and it is also moving. The first changes when a transit happens. The second changes how long it lasts.

Two shifts from one small orbit #

Let a planet of mass $M_p$ carry a moon of mass $M_m$ at distance $a_m$. The planet orbits the common centre at $(M_m/M_p)\,a_m$, in the time $P_m$ of the moon’s month. At the moment of a transit, with the moon at phase $\phi$, the planet’s position along the track is offset by

$$x = \frac{M_m}{M_p}a_m\sin\phi,$$

and its speed along the track is changed by

$$v = \frac{M_m}{M_p}\frac{2\pi a_m}{P_m}\cos\phi .$$

The planet’s orbital speed $v_{\rm orb}$ turns an offset into a time, $\delta t=x/v_{\rm orb}$, the transit timing variation. The transit lasts $T=2R_\star/v_{\rm orb}$ and a speed change alters that by a fraction $v/v_{\rm orb}$: the transit duration variation. Those two have the same origin, but since one comes from position and the other from velocity, they are a quarter-cycle apart.

The quadrature is a signature that nothing else readily produces. A perturbing planet changes when a transit happens through a slow change of orbital period and phase; its effect on the speed at the moment of transit is tiny, because it changes the period and the speed by the same fraction of a very small amount. A starspot crossing the chord changes the apparent centre of a transit and its shape together. A moon alone has to do both, at comparable sizes, a quarter-cycle apart.

Two lines that cross once #

Since the moon’s period depends on its distance through Kepler’s law, $P_m\propto a_m^{3/2}$ with the planet’s mass fixed, the two amplitudes come out as

$$\delta t \propto M_m a_m, \qquad \delta T \propto \frac{M_m}{a_m^{1/2}} .$$

Moving the moon outward makes the planet’s small orbit larger but slower. The timing therefore grows, and the duration falls.

One amplitude alone gives a line in the plane of mass and distance. A timing shift could equally be a heavy moon nearby or a light one far out, which is the familiar difficulty of inferring two numbers from one and the same difficulty that dogs the masses read from transit timing. The duration gives a second line, of a different slope, and two lines cross once.

That dependence on the planet’s mass is a real restriction. The transit gives the planet’s radius, and a mass has to come from radial velocities or from a model. For a giant planet a guess at the mass may be good to tens of per cent, and that feeds straight through. But compared with the position before 2009, where the timing signal gave a single product and little more, it changes the problem from one with no solution to one with an uncertain solution, which is what a measurement is.

The moon goes round many times between looks #

The most awkward feature of this method is that a planet transits only once an orbit. A planet in a 287-day orbit is seen once in 287 days, and if its moon takes thirteen days to go round, the moon makes over twenty circuits between each pair of measurements. The moon’s period is far too short for the sampling to follow.

The problem is aliasing, the same effect that makes a wagon wheel appear to turn backward on film. All frequencies that differ from the true one by a whole number per planetary orbit fit the samples equally, and the apparent period is only the remainder. The timing series can be fitted perfectly by a wrong moon on a wrong orbit. Its amplitude is unaffected by the aliasing, which is why the amplitude remains useful, but the period cannot be read.

The duration gives it. Take the ratio of the two amplitudes:

$$\frac{\delta T}{\delta t} = \frac{2\pi T}{P_m} .$$

All of the moon’s mass and its distance have cancelled. What remains is the moon’s period, written in units of the known transit duration. The ratio contains no aliasing, since it compares amplitudes rather than following a phase through time. A pair of amplitudes measured over a long enough series is enough.

There is a satisfying symmetry to the outcome. The timing series knows the moon’s phase but cannot distinguish its period. The ratio knows its period and has no phase in it. And the two are sampled at the same transits, so there is nothing extra to be observed for it, provided the durations can be measured well enough.

How small the second signal is #

“Provided” is the operative word, and a realistic calculation is sobering. Timing a transit to a precision $\sigma_t$ means fitting a light curve with a depth and an ingress. A rule of thumb from the shape of a trapezoid is that the mid-transit time and the duration are nearly independent in the fit, and the duration’s error is larger by about $\sqrt2$.

For the example in the figures the signals are enormous: ±42 minutes in timing, ±10 in duration, for a Neptune-like moon. Such an object would be conspicuous with data of ordinary quality, and it is also likely to be a body that nothing is known to form. For a body like Earth’s Moon around a Jupiter the signals would be some thousand times smaller. For one Earth mass around a Saturn analogue in a year-long orbit, at a tenth of the distance of the stability limit, the amplitude in timing is of order a minute and the amplitude in duration ten or twenty seconds. The precision achieved for a typical faint Kepler target was no better than a few minutes per transit.

Long-period planets are favoured in principle, because moons can survive at wider separations where the planet’s Hill sphere is large, and the region is quite restrictive close to the star. A longer orbit also gives a longer and hence better-timed transit. However, a long-period planet gives few transits per mission and a small transit probability. The compromise leaves relatively few systems where a moon as massive as Ganymede could be seen. The Hunt for Exomoons with Kepler project ran through several hundred candidates over the decade after 2012, and set upper limits on most of them, typically around an Earth mass at best.

The candidate that shows the whole method #

The most publicised candidate, Kepler-1625 b, tests every assumption in this account. In 2018 Alex Teachey and David Kipping reported that a transit of this Jupiter-sized planet, measured with the Hubble Space Telescope, arrived about 78 minutes earlier than predicted, and also that the light curve after it showed a shallow second dip of the kind a moon would make. They read this as a Neptune-sized moon, a large and strange body that no standard formation theory would have predicted.

The claim drew an independent reanalysis of the same data. Laura Kreidberg and colleagues, using a separate reduction, found that the second dip weakened and could be made to go away, and that its significance depended on the treatment of systematic trends in the telescope’s detector. Heller and others then showed that a planet-only fit, with a little extra stellar activity and an orbital perturber, could account for the early transit by itself. A further candidate, Kepler-1708 b-i, was reported in 2022, and has been disputed along the same lines.

The whole dispute is a fair example of what the method needs. The 78-minute early transit is a single large timing point. A single point cannot have a quadrature and it cannot have a ratio; there is no duration series, because only four transits of a long-period planet had been seen. A moon was a fair hypothesis and so was a perturbing planet. The test described above needed perhaps a dozen more transits, which the planet’s 287-day orbit will take a decade to deliver.

What a moon on an alias can and cannot be #

There is a further limit, one that the figures can only hint at. In a real analysis the amplitudes are not given; they are fitted, with a phase that drifts through the series at the aliased rate. A planet observed only for six or eight transits does not sample a full aliased cycle, so the amplitude is correlated with the phase. And the noise that remains in a timing series, from stellar spots and granulation, is correlated from one transit to the next, so a small sample can look coherent when it is not.

The most effective way to deal with these is to model the moon directly. The photometric model includes a moon as a transiting body with its own orbit, and all the light curves are fitted at once, using the shape of the transit and any small second dip it produces. This does two things that the simple timing argument cannot do. It uses the moon’s shadow if it is large enough. And it gives a direct estimate of the moon’s radius, in addition to its mass from timing, and hence a density that would be a first measure of what a moon around another star is made of.

But the model-based method has a cost of its own, in that many models fit the same data, and a high-dimensional search will find apparent moons in noise. A false positive rate has to be calibrated by searching in fake data, and doing so has been a major part of the published effort. There may be an instructive parallel in the way that the same multiplicity problem affects a planet validated without confirmation.

Why a moon may be unusual, not absent #

What would be surprising about a moon? There are several reasons to think that the best candidates could be strange by the standards of the Solar System.

Moons in the Solar System amount to about one part in ten thousand of their planet’s mass, for the giant planets. That regularity has an explanation in how a satellite forms in a circumplanetary disc, where accretion and inward migration balance. On that basis a Jupiter-sized planet would not carry a moon of more than a few Mars masses, and the Earth-mass and Neptune-mass moons being searched for could not be there at all. Capture events and giant impacts can make larger satellites, which is how the Earth got the Moon and Neptune got Triton. But those are unlikely, on the giant planets, to deliver a body as big as the signals can detect.

Timing therefore searches in a place where the expected objects are rare, for the reason that the common ones cannot be seen. The non-detections say so far that very massive moons are not common. They do not say very much about Ganymede-sized bodies, and the real population, if it follows the Solar System’s rule, lies beyond what has been observed. There is another way of stating that: the transit timing method, used in this way, has not yet measured the thing most worth knowing.

A test that needs no moon #

The argument has a useful reverse form which comes free. Every planet searched that shows no signal sets a limit on a combination of moon mass and distance, for a given time of observation, without any assumption about how moons form. This has been done for a few hundred planets, and the upper limits, even when individually weak, constrain the fraction of planets carrying moons above a given size.

The same data have also yielded something the search was not aiming at. A timing series is a sensitive test of whether the planet’s orbit has anything perturbing it. In long-period giants, timing excesses appear more often than moons do, and these are largely due to outer companions, including some on eccentric orbits that would have remained unknown otherwise. They are found by excluding the moon: an excess timing signal with no quadrature in its duration is not a moon. This is how a search that has found no moons has found other planets instead.

What an impostor would have to do #

The real strength of the method is seen in what else could imitate it, and the list is short but not empty. Consider the alternatives one at a time, and what each does to the two signals.

A second planet. Near a mean-motion resonance, a companion drives a sinusoidal timing signal with a long period, and at first sight it is an excellent mimic of an aliased moon. Its effect on duration, however, is small, being a change in the planet’s speed by the small fractional change in its period, plus a slow drift from any inclination change. A duration signal with the moon’s expected size cannot be produced that way. The inverse problem, of which planet could draw a given timing curve, is degenerate already, and a moon adds another branch to that. A good upper limit on any duration signal removes it.

A spinning spotted star. A planet crossing a starspot produces a bump in the light curve that shifts the fitted mid-time, by tens of seconds for a large spot. It may also change the fitted duration. These are correlated with the star’s rotation and not with anything of the moon’s, and they will depend on the colour observed. A moon’s timing is the same in every colour. That is an effective discriminator wherever multiple bands exist. It is the same test used to separate a planet’s velocity from a stellar one.

A distant massive companion of the star. The star’s own orbit about a companion shifts every transit by the light travel time, which appears as a slow curve across many years. It would show no periodicity on the scale of the aliased moon and has no duration effect. It is removed as a long-term trend.

The fit itself. With a small number of transits and several parameters per transit, the fitted mid-times and durations have correlated errors, which are usually smaller for times than durations. Correlated noise can then produce a spurious quadrature for a small number of transits, because any two-parameter pattern can be fitted to a few points. That is why a moon cannot be claimed from a handful of transits, however good each may be. The required number of transits depends on how many cycles of the alias they can sample: about three would be a minimum, and for a one-year planet that is a decade of observation.

So a convincing detection has to pass four separate tests, and each needs more transits than almost any long-period planet has so far been given.

Still open: the first moon with a confirmed quadrature #

No moon has yet been shown to have both effects a quarter-cycle apart with the amplitudes in the predicted ratio. That single demonstration would be stronger than any amount of depth fitting, because it is a signature rooted in geometry, and a telescope large enough to do it exists. The timing and duration of one transit are measurable by the James Webb Space Telescope to a precision of several seconds for bright stars. It is the number of transits that is the constraint, because a long-period planet gives few.

The planets observed by the next wide-field missions will have longer time baselines and brighter stars. Whether those, followed up with the largest telescopes, reach down to a moon of a few Mars masses is what will decide this. If they do, the first confirmed exomoon may be found not as a dip in a light curve at all but as a small, regular disagreement between how late a planet was and how long it took.