A period read from a single dip

Every search for transiting planets described so far relies on folding. A light curve is cut at a trial period and the pieces are stacked, so that a dip too shallow to see in any one transit adds up across many; the period at which the stack is deepest is the planet’s. The box-shaped search that finds most transiting planets is a machine for doing exactly that, and it needs at least two transits and preferably dozens.

A planet whose orbit is longer than the observations transits once. A space telescope that stares at one patch of sky for twenty-seven days catches the transit of a planet with a year of three hundred days only if the planet happens to cross in those twenty-seven, and then it sees it once. There is no second dip to fold on and no spacing to measure. The planets this happens to are the ones most worth finding — long-period planets, cool enough to be like the Earth or Jupiter rather than like the scorched hot planets that dominate the catalogues — and for years the single transits that surveys recorded were set aside, because a planet without a period cannot be followed up or characterised.

The period is not lost. A transit has a duration, and the duration is set by how fast the planet moves across the face of its star, which is set by the orbit’s size through Kepler’s third law. This essay follows that inference: how a duration and a stellar density give a period, why the answer is a distribution rather than a number, what narrows it and what widens it, and what happens when a second single transit arrives years later and replaces one broad distribution with a comb of sharp ones.

A duration that grows as the cube root of the year #

A planet on a circular orbit of radius $a$ crosses a star of radius $R_\star$ along a chord whose length depends on how far from the centre it passes. That offset, in units of the stellar radius, is the impact parameter $b$, and the chord is $2R_\star\sqrt{1-b^2}$. The planet covers it at its orbital speed $2\pi a/P$, so the duration is

$$T = \frac{P}{\pi}\frac{R_\star}{a}\sqrt{1-b^2}.$$

Kepler’s third law ties $a$ to $P$ and the star’s mass, and dividing by the radius turns the mass into a density: $R_\star/a = (3\pi/G\rho_\star P^2)^{1/3}$.

For a central crossing the duration grows only as the cube root of the period — the planet moves more slowly on a larger orbit, but it also has less far to go relative to the orbit’s size, and the two nearly cancel. A planet on a ten-day orbit crosses the Sun in four hours, an Earth in thirteen, a planet with a ten-year orbit in twenty-eight. A grazing crossing is shorter by the factor $\sqrt{1-b^2}$, and that is where the inference becomes difficult: a grazing chord of a long orbit can last exactly as long as a central chord of a short one.

One duration, a family of periods #

Inverting the relation gives the period from the duration:

$$P = \frac{\pi^2 G \rho_\star T^3}{3\,(1-b^2)^{3/2}}.$$

The period goes as the cube of the duration and as the star’s density, and both can be measured: the duration from the light curve, the density from the star’s spectrum and luminosity, or from the shape of the transit itself. What is not measured from a single dip is the impact parameter, and the period depends on it through an inverse power of three-halves.

The central crossing gives a floor — the shortest period consistent with the duration — and every other crossing gives a longer one. Halfway out the period is half as large again; at eight-tenths of the radius it is four and a half times the floor; at the limb it runs away. A twelve-hour transit of a Sun-like star belongs to a planet with a year of at least 289 days, and possibly of decades.

The relation is the same one that lets a transit measure an eccentricity when the period is known, run in the other direction. There the period and the density are given, the expected duration follows, and a mismatch is read as the orbit’s speed at transit. Here the duration and the density are given and the period is the unknown, so the same mismatch — an eccentric orbit crossing faster or slower than a circular one — appears as an error in the period. One equation, three quantities, and a single transit can solve for one of them only by assuming the others.

What one transit says #

A distribution of periods follows from a distribution of impact parameters, and for transiting planets the impact parameter is uniform: every chord across the disc is equally likely for orbits whose orientations are random. Mapping a uniform $b$ through the period relation, and multiplying by what is known in advance about which periods are likely, gives the probability of each period given the one transit.

Two pieces of that prior knowledge matter. A planet on a longer orbit is less likely to transit at all, as $P^{-2/3}$, because its orbit is larger compared with its star. And a planet seen to transit exactly once in a window of length $B$ must have a period longer than about half the window; above the window, the chance that its single transit fell inside is $B/P$. Both favour shorter periods.

With the impact parameter unconstrained, the distribution rises sharply at the floor of 289 days and falls away in a tail towards longer periods. The prior pulls the probability towards the floor, so the median sits only a few per cent above it, and the central two-thirds of the probability span less than a factor of 1.4. That is far better than the bare geometry suggested, and the reason is the steepness of the prior: grazing orbits that would give long periods are individually unlikely to transit and to be caught once, and there is only a narrow range of $b$ near the limb that produces them.

The shape of the transit narrows it further. Ingress and egress — the minutes during which the planet’s disc is only partly over the star — last longer for a grazing chord than for a central one, because the planet crosses the limb obliquely; their fractional length is roughly $k/(1-b^2)$ for a planet of radius ratio $k$. Measuring them pins the impact parameter, and with $b = 0.30 \pm 0.08$ the period’s central range closes to about fifteen per cent. That is a period good enough to predict the next transit to within a few weeks.

The eccentricity nobody measured #

The calculation so far assumed a circular orbit, and long-period planets are not usually on circular orbits. An eccentric planet crosses its star at a speed that depends on where in its orbit the transit happens: faster near periastron, slower near apastron, by the factor $(1 + e\sin\omega)/\sqrt{1-e^2}$. The period inferred from the duration goes as the cube of that factor.

A modest eccentricity of the kind long-period planets typically have — a Rayleigh distribution of width two-tenths — changes the duration by tens of per cent either way, and the cube turns that into a factor of two or three in the period. The distribution spreads out, and the prior, which favours short periods, drags much of it below the circular floor: an eccentric planet caught near periastron moves fast enough to imitate a much shorter orbit. The eccentricity, which a single transit cannot see at all, becomes the largest term in the uncertainty.

That is not a defect of the method so much as a statement of what a single transit contains. The duration measures the planet’s speed across the star. The speed is set by the orbit’s size only if the orbit is circular, and a period derived from a speed carries every uncertainty in the shape of the orbit cubed. Radial-velocity measurements of the star, if the planet is massive enough, measure the eccentricity directly, and a second planet in the same system whose period is known supplies an independent density that can separate the two.

A second dip, years later #

The survey that found the single transit usually returns. A space telescope that observes a field for a month and comes back to it two years later may catch the same planet again, once, in the second visit. Two single transits do not give a period either — the planet may have gone round once between them, or twice, or a hundred times, with every intermediate transit falling in the gap when nobody was looking. What they give is a comb: the period must be the gap divided by a whole number, $P = \Delta/n$.

The comb is exactly the structure that a period found in the gaps produces in a periodogram of sparsely sampled data: aliases, all fitting the observed transits equally well, separated by the frequency of the gap between visits. In a periodogram they are a nuisance to be broken by more data. Here the single transit’s duration breaks them partially before any more data are taken. Each tooth of the comb is a definite period, and the duration-based distribution says how probable each one is. The teeth far from where the duration points are suppressed; those near it carry the probability.

For a six-hour transit of a Sun-like star the surviving teeth cluster round twenty-five days, with the eccentricity’s spread letting a few dozen of them share the probability. Without the duration every one of the hundred and twenty teeth up to the gap would be equally likely, and the period would be a guess.

How many nights it costs #

The practical question a comb poses is how many further observations it takes to find the true tooth. Each allowed period predicts a transit at a specific time; a telescope watching at that time either sees the dip or does not, and a negative result eliminates the tooth.

Observing the teeth in order of probability, half the probability is covered by thirteen predicted transits and ninety per cent by thirty. Without the duration’s ranking, ninety per cent would take a hundred and eight. Several predicted transits can sometimes be covered in a single night, and the most probable teeth are usually checked first, so a candidate that would have cost a hundred nights becomes a programme of a few weeks — the difference between a planet that is confirmed and one that is left in a list of candidates.

Two caveats limit the ranking’s power. It is only as good as the stellar density and the ingress measurement; an error of ten per cent in the density moves the whole distribution by ten per cent and can shift probability between neighbouring teeth. And it assumes the two dips belong to the same planet. A star with two long-period planets can show one transit of each, and two dips of similar depth and duration years apart are evidence for one planet, not proof.

The Earth, seen once from somewhere else #

The clearest way to see what the method can and cannot do is to turn it round. Suppose a survey telescope on a planet orbiting another star, somewhere near the plane of the Earth’s orbit, watched the Sun for twenty-seven days. The chance that its line of sight lies close enough to the ecliptic for the Earth to transit at all is the Sun’s radius over the Earth’s orbit, about half a per cent. Given that it does, the chance that the one transit a year falls inside a twenty-seven-day window is twenty-seven in three hundred and sixty-five, about seven per cent. One survey in about three thousand, looking in random directions, would catch the Earth, and it would catch it once.

What it would record is a dip of eighty-four parts per million lasting up to thirteen hours. If it crossed the middle of the Sun, the duration and a solar density would give a floor of about 365 days — the right answer, because a central crossing is what the floor assumes. If it crossed at an impact parameter of a half, the duration would be eleven hours and a central-crossing assumption would give a floor of about 240 days, with the true period inside the distribution’s tail. The ingress, lasting seven minutes for a central crossing and longer for an oblique one, would be at the edge of what such a shallow transit lets anyone measure, so the observers would be left with a distribution spanning a factor of one and a half or two, centred somewhere near a year. They would know they had found a temperate planet. They would not know its year to better than a few months until their survey came back.

And it would come back. A second visit to the same field two years later would see the Earth again with the same seven per cent chance — so most such pairs of visits would find nothing, and a few would produce a comb with teeth at 730, 365, 243, 183 days and so on. The duration would rank them, and the tooth at 365 days would be among the most probable but not the only one. The Earth, observed the way a long-period planet is observed today, would be a candidate with a ranked list of periods and a follow-up programme attached, not a planet with a known year.

That is the honest scale of the problem. The planets most like the Earth are, by their nature, the ones least likely to be seen more than once by any survey of practical length, and the period of each will be an inference from a duration for years before it is a measurement from a spacing. Kepler’s third law, used here to get a period from a speed rather than a mass from a period, is the bridge across those years.

What a single transit is worth #

The method turns a class of detections that used to be discarded into planets with periods. That matters disproportionately for the census, because single transits are where the long periods are. The odds that a signal is a planet depend on how common planets of that size and period are thought to be, and for the long periods those numbers come mostly from radial-velocity surveys and microlensing, which measure different populations; transiting long-period planets with measured radii fill a gap between them.

The same inference connects the transit to the star in a way that is easy to miss. A duration is a time, and turning a time into a period requires a density. The density comes from stellar models, from the spectrum and the luminosity, or from asteroseismology; its uncertainty enters the period linearly. A planet’s radius is already a stellar radius times a ratio, and a single-transit planet’s period is a stellar density times a cubed duration: both are measurements of the star first and the planet second.

What the distributions assume #

The orbits are circular in all but one figure, and in that one the eccentricity distribution is a single Rayleigh form with orientations uniform; long-period planets show a range of eccentricities that depends on the planet’s mass and multiplicity, and transits preferentially catch eccentric planets near periastron, which a full calculation weights in. The star is a uniform disc for the purpose of the chord, without limb darkening, which affects how precisely the ingress can be measured but not the geometry. The prior on period is flat in its logarithm, which is roughly what surveys find for long periods but is not established for the planets whose periods are being estimated.

The window function is a single uninterrupted stretch. Real space photometry has gaps every orbit of the spacecraft and at each data download, and a transit that would have fallen in a gap is a transit not seen, which reshapes the prior on period in detail. And the comb uses the duration-based distribution as each tooth’s weight with no further information; in practice the two transits’ depths, durations and shapes are fitted together, which tightens the impact parameter more than either alone.

Still open: a census built from single transits #

Single transits from space surveys now number in the hundreds, and two-transit combs from return visits in the dozens. Each comb can be resolved by a few weeks of ground-based or space-based follow-up, and the resolved planets have periods of tens to hundreds of days around stars bright enough for their masses and atmospheres to be measured. That is the regime of temperate planets, and it has very few well-characterised members.

What is not yet clear is whether the single-transit sample can be used for occurrence rates as well as for individual planets — whether the probability of detecting a planet once, the probability of its period being recovered and the probability of its comb being resolved can be modelled well enough to turn a list of planets into a count. Each step depends on the stellar densities, on the eccentricity distribution that the method has to assume and on the window functions of surveys that were not designed for long periods. If they can, the tallest peak a long-period planet ever makes in any periodogram will be no peak at all — just one dip, and a duration read through Kepler’s third law.