A clock that slows, or a star that falls

A transiting planet is a clock. Every orbit it crosses its star at an instant that can be timed to a few tens of seconds, and over years the crossings line up on a straight line — transit number against time — whose slope is the period. Departures from that line are the subject of everything else in timing: a neighbour pulling the planet early and late, a star carried round a distant companion, each drawing its own shape in the residuals.

One shape is special, because it is the signature of the thing the hot Jupiters were long expected to do and were never seen doing: fall into their stars. A planet whose orbit is shrinking has a period that gets shorter, and its transit times curve away from any straight line as a parabola. The tides the planet raises on its star should be doing exactly that, for the closest planets, on timescales of millions of years; and by about 2020, after more than a decade of timing, one of them — WASP-12 b, on a 1.09-day orbit — was seen to be doing it, its period shrinking by about 29 milliseconds a year.

The difficulty, stated at the end of the essay on a star carried round its companion, is that another cause draws the same curve. This essay is about separating them.

One parabola, two causes #

A star with a distant companion is accelerating, slowly and steadily, towards or away from the observer. Over a few years that acceleration is constant, and the star’s distance changes as the square of the time. Each transit’s light then has a distance to travel that differs from the last by a steadily growing amount, and the arrival times curve away from a straight line — as a parabola.

The equivalence is exact, not approximate. A uniform line-of-sight acceleration $a$ changes the apparent period at a rate $\dot P/P = a/c$: each orbit’s light arrives earlier than the last by an extra $aP^2/c$, which is the definition of a shrinking apparent period. A real shrinking period and an apparent one are indistinguishable in the times of arrival, because the times of arrival are all a transit gives.

Two quite different physical statements are therefore made by one data set. In the first, the planet is losing orbital energy to tidal friction inside its star at a rate that measures how dissipative the star’s interior is. In the second, the planet is doing nothing at all, and a body far out in the system — a giant planet or a brown dwarf, perhaps unseen — is pulling the whole inner system along.

What the velocities can say #

The two hypotheses make different predictions for one other observable. A decaying orbit moves no star: the planet’s orbital period changes, but the star’s motion along the line of sight, averaged over the planet’s orbit, stays the same. An accelerating star shows the acceleration directly, as a linear trend in its radial velocity of $a$ per unit time.

So the radial velocities settle it — if the trend the acceleration would need is large enough to see.

For WASP-12 b the arithmetic is decisive. The velocities of its star have been measured for more than a decade, and they show no trend remotely as large as the ninety metres a second per year the acceleration hypothesis requires. The star is not falling; the planet is. That is what made the measurement the first convincing detection of tidal decay, rather than one more parabola with two explanations.

The scaling with period is what makes the case unusual. The apparent decay a given acceleration produces grows in proportion to the period — a longer clock accumulates more light-travel delay per tick — while the acceleration needed to fake a given decay falls as one over the period. A hot Jupiter on a four-day orbit decaying at the same rate as WASP-12 b could be faked by a velocity trend of about twenty-five metres a second per year. One on a ten-day orbit, by ten.

Where the companion wins #

The tides do not produce the same decay at every period. The tidal torque falls steeply with the planet’s distance from the star, and the decay rate falls with the period to about the minus ten-thirds power. The apparent decay from a companion rises in proportion to the period. Somewhere they cross.

The figure makes the selection plain. Measuring tidal decay requires a planet so close to its star that the tidal rate is large, and only for such a planet is the acceleration that would fake the decay large enough for the velocities to exclude it. For the great majority of hot Jupiters, on periods of three to five days, the expected tidal decay is a fraction of a millisecond a year, and an unseen Saturn ten astronomical units out would produce more apparent decay than that. Its velocity trend, half a metre a second per year, is at the limit of what the best spectrographs measure over a decade, and the star’s own activity adds noise of the same size.

The same comparison for a more ordinary system makes the point harder.

A parabola in the timing of an ordinary hot Jupiter is therefore more likely to be a companion than a decay, and claims of decay in such systems have to show that no companion is there — which, for a Saturn at ten astronomical units, the velocities can barely do. A slope in the velocities is itself a whole curve of possible companions, mass traded against distance, and a timing parabola is the same degeneracy seen through a different instrument.

A third mimic #

A parabola can also be the beginning of something that is not a parabola at all. A planet on a slightly eccentric orbit whose orientation turns — its periastron precessing under the star’s tidal bulge or general relativity — has transit times that oscillate with the precession period, years to decades long. Over a baseline shorter than that period, the oscillation’s curvature looks like a decay.

The discriminator is the occultation. Half an orbit after each transit, the planet passes behind its star, and the dip in the system’s infrared light — the planet’s own thermal glow disappearing — can be timed too, less precisely. A decaying orbit moves both events the same way, since the whole orbit is speeding up. A precessing eccentric orbit moves them oppositely, since the planet reaches the near side of its orbit earlier exactly when it reaches the far side later. For WASP-12 b, occultation times measured over the same years shifted with the transits rather than against them, which excluded precession as the explanation and left decay.

How the decay was found #

WASP-12 b was discovered in 2008 by a ground-based transit survey, and it was remarkable from the start: a giant planet on a 26-hour orbit, so close to its star that it is stretched into an egg shape by the tide and is losing gas, and so hot that its day side glows at more than two thousand degrees. It was an obvious candidate for tidal decay, and its transits were timed by many observers from the beginning.

The first claim of curvature in its timing came in 2016, after eight years of data, at modest significance and with apsidal precession as a viable alternative. Over the following four years the curvature grew as the baseline did, occultation times measured with space telescopes showed that transits and occultations were moving together, and the velocities ruled out an accelerating star. By 2020 the decay was established at about twenty-nine milliseconds a year, with a precision of a few per cent. That planet, sitting where one cannot form, is now the reference case for what tides do to a close planet — and the only case for a Sun-like star where the decay has been separated cleanly from every mimic.

The order of those steps is instructive. The parabola came first and was not believed until the other two explanations had been excluded by different measurements: the velocities for the acceleration, the occultations for the precession. Neither of those was part of the timing programme that found the curvature; each was a different instrument asking a different question of the same system.

Where the orbital energy goes #

A planet falling inward loses orbital energy, and the energy has to go somewhere. Tidal decay puts it into the star, as heat dissipated by the tidal flows in the star’s interior. For WASP-12 b the rate is easy to compute: the orbital energy, a few times $10^{37}$ joules, falls by the fractional rate of change of the semi-major axis — two-thirds of the fractional rate of change of the period, about $10^{-14}$ per second — which is about $10^{23}$ watts. That is a ten-thousandth of the star’s own luminosity, delivered deep in its interior, and invisible.

So the star cannot be seen to heat up, and the decay cannot be confirmed by its thermal consequences. How much heat a tide deposits depends on a dissipation factor nobody can compute from first principles, and here the dissipation factor is measured from the orbit and the heat is inferred from it — the reverse of how tidal heating is usually argued for moons, where the heat is seen and the dissipation inferred.

The spin of the star is a different matter. The angular momentum the planet loses goes into the star’s rotation, spinning it up. For WASP-12 the effect on the rotation over the planet’s remaining lifetime is small but not negligible, and in systems where a hot Jupiter has already been swallowed, a star spinning faster than its age allows is one of the proposed fingerprints.

How long the clock must run #

Detecting a decay at all requires time, and more of it than most programmes have.

That steep dependence on baseline is why the decay of WASP-12 b emerged slowly: the first hints, after a few years of timing, were consistent with zero, and the curvature became unambiguous only after about a decade. It is also why older transit times are disproportionately valuable, and why a forecast made from a straight-line ephemeris fails on a schedule: for a decaying planet the predicted transit times drift from the true ones as the square of the time since the ephemeris was fitted, by minutes within a decade for WASP-12 b — enough to miss the start of an observation booked on a space telescope. A single transit timed fifteen years ago extends the baseline more than a dozen timed this year, and archival observations, made for other purposes, are routinely searched for transits that can be timed.

What the decay measures #

A tidal decay rate measures the star’s tidal quality factor — a dimensionless number that expresses how much of the tidal energy is dissipated per cycle, and the one quantity in the problem that cannot be computed from first principles. For WASP-12 b the decay implies a value of about two hundred thousand, which means the star dissipates efficiently — more efficiently than the values of order a million or more that most theories give for Sun-like stars.

That efficiency may itself be a clue. WASP-12 is somewhat more massive than the Sun and may be beginning to evolve off the main sequence, and evolved stars are expected to dissipate tidal energy far better, because waves excited by the tide can break in their changed interiors. If so, WASP-12 b is decaying fast because its star has just begun to change, and the measured quality factor applies to a short phase of stellar evolution rather than to stars in general. A second decaying planet, around a star that has already left the main sequence, has since been reported, consistent with that picture.

The decay also bears on how hot Jupiters arrived where they are. A planet decaying at WASP-12 b’s rate will reach its star within a few million years. If such planets are found in significant numbers at the shortest periods, they must be continually supplied; if the supply is migration from further out, the decay rate and the numbers at each period together constrain how fast the supply arrives.

Where the model stops #

A single quality factor. The tidal decay curves use a constant quality factor at every period, which theory does not support: the dissipation depends on the tidal frequency, on the star’s structure, and on whether internal waves are excited and where they break. The curves show the scale of the effect, and the crossing periods with the mimics move with the assumed value.

Companions on wide circular orbits. The mimicking accelerations are drawn as constant, which holds when the companion’s orbit is much longer than the timing baseline. A companion with a period comparable to the baseline produces a cubic or higher curvature, which a long enough series begins to separate from a decay.

Timing errors that are only noise. The detectability curves assume independent Gaussian timing errors. Real transit times have systematic errors from different instruments and analyses, and offsets between data sets can mimic or mask curvature; homogeneous reanalysis of all the transits of a system is part of every credible decay claim.

Precession as the only other mimic. A third body on a nearby orbit produces its own timing variations, usually periodic and recognisable, but over a short baseline its signal can contain curvature too. The occultation test separates precession; it does not separate everything.

A test across many stars #

One planet’s decay is a measurement of one star’s dissipation. The question theorists most want answered is how dissipation depends on the star’s mass, age and structure, and that needs many decay rates — which, for ordinary hot Jupiters, are below what timing can reach for any single planet.

There is a way round that the mimics themselves suggest. A companion’s acceleration of the star can point towards the observer or away, and across many systems the signs are random: the apparent period derivatives it produces scatter symmetrically about zero. Tidal decay always shortens the period. A large sample of hot Jupiters, each timed for a decade, would show a distribution of fitted period derivatives with a symmetric spread from companions and noise and an offset towards negative values from decay. The offset is a population-average decay rate, measurable even when no individual planet’s decay is.

The sample needed is hundreds of planets, timed homogeneously over many years, which the space surveys now provide for the first time: the same hot Jupiters observed by successive missions a decade apart, with ground-based timing filling the gaps. The first such analyses have found most hot Jupiters consistent with no decay and a handful with tentative curvature, each of which then has to go through the same tests WASP-12 b did — the velocities for acceleration, the occultations for precession — before it can be counted.

The surprising connection #

The light-travel effect that mimics decay is how Ole Rømer measured the speed of light in 1676: the eclipses of Jupiter’s moon Io came progressively later as the Earth moved away from Jupiter and earlier as it approached, by up to sixteen minutes across the Earth’s orbit. Io’s clock was perfect; the delays were in the distance. A hot Jupiter’s transits are the same experiment run backwards. There the speed of light is known and the question is whether the delays are in the distance — an accelerating star — or in the clock itself. Three and a half centuries after Rømer read a distance from a delay, the first measurement of a planet falling into its star required showing that a delay was not one.

Still open: whether ordinary hot Jupiters are falling #

WASP-12 b is the extreme: the closest giant planet round a Sun-like star whose decay can be measured cleanly, around a star that may be unusually dissipative. For the typical hot Jupiter, on a three-day orbit round a main-sequence star, the expected decay is a fraction of a millisecond a year, below what decades of timing can reach and below what an unseen distant companion can fake. Whether those planets are decaying at the rates the standard tidal theories predict — which would tie the theories down — or barely at all, as some recent calculations suggest for stars with radiative cores, may be answered only statistically: by timing hundreds of hot Jupiters for decades and looking for a population-wide curvature that companions, distributed at random in their signs, would average away and decay would not.