A misalignment only cool stars forget

A transiting planet crossing a rotating star hides part of the approaching limb and then part of the receding one, and the resulting distortion of the star’s line profile is a measurement of the angle between the orbit’s plane and the star’s equator. The measurement is now made routinely, and the result was one of the surprises of the subject: hot Jupiters are frequently misaligned, and some of them orbit retrograde.

That was taken, for a while, as a straightforward statement about how such planets arrive. It is not, because half the sample has been reprocessed since arrival.

The observation #

The transit itself supplies four contact points and the geometry they fix; the spectroscopy during it supplies an angle. The measurement returns a projected angle rather than the true obliquity: the transit chord’s orientation on the sky relative to the star’s projected rotation axis. Recovering the three-dimensional angle needs the star’s inclination as well, which comes from combining a projected rotation speed with a rotation period and a radius.

Four ways to measure the same angle #

The velocity anomaly is the original method and it is now one of several, and the others matter because they fail differently — a result established by four techniques with unrelated systematics is a different kind of result from one established by four applications of the same technique.

Doppler tomography uses the same transit and the same spectra, and instead of collapsing each spectrum to a single fitted velocity it tracks the actual bump the planet leaves in the line profile as it crosses. That is a stronger use of the same data: the bump’s trajectory across the profile maps directly onto the transit chord’s path across the rotating disc, and it does not depend on how well a single velocity summarises a distorted line. It works best on rapid rotators, whose lines are broad enough for the bump to be resolved, which is to say on the hot stars.

Gravity darkening works on the photometry alone. A rapidly rotating star is oblate, its poles are hotter than its equator, and a planet transiting such a star crosses regions of different brightness — so the transit light curve is asymmetric in a way that encodes the chord’s orientation relative to the rotation axis. It needs no spectroscopy at all, and it needs a star distorted enough to be measurably darkened, which again means a hot rapid rotator.

Spot crossings work on cool stars instead. A planet passing over a starspot produces a brief brightening in the transit, and if the same spot is crossed on successive transits its apparent motion says whether the planet’s path is aligned with the star’s rotation. It requires a spotted star, which is a cool active one, and it delivers the true obliquity rather than a projected one.

Asteroseismology supplies the missing stellar inclination directly. The relative amplitudes of the components of a split oscillation mode depend on the angle between the rotation axis and the line of sight, so a star with measurable oscillations reports its own inclination, and combining that with a projected obliquity gives the three-dimensional angle.

The four are complementary in exactly the awkward way: the first two work on hot stars and the last two on cool ones. The temperature boundary the essay is about is also a boundary between the techniques used on either side of it, which is the kind of coincidence that has to be checked rather than assumed harmless — and the checking has been done on the handful of systems where two methods both apply.

The boundary #

Sort the measured obliquities by the host star’s effective temperature and the distribution splits.

Below roughly 6,250 kelvin, the projected obliquities cluster near zero with a scatter of a few degrees. Above it, they are scattered across the whole range, with a substantial fraction beyond ninety degrees.

Nothing about a planet knows the temperature of its host. Planets form in discs, and a disc’s plane is set by the angular momentum of the material that made the system, which does not consult the star’s photosphere. So the boundary is a statement about what happened to the orbits afterwards.

It could in principle be a statement about what happened before, since hot stars are more massive and more massive stars have more massive, shorter-lived discs. That alternative has been examined and it does not work: the disc’s plane is set at the very beginning, by the angular momentum of the collapsing core, and there is no mechanism by which a star’s envelope structure — a property of its outer few per cent, established after it reaches the main sequence — could reach back and tilt a disc that has long since dispersed. The temperature boundary is at the wrong place in the causal chain to be about formation.

There is one loophole worth naming, because it is taken seriously. If a star’s own outer envelope can be tilted relative to its interior, then a “misalignment” is a statement about the star rather than about the orbit, and a convective envelope is precisely what could be tilted and then be dragged back. That would produce the same temperature boundary from the opposite direction, and distinguishing it requires knowing whether the star’s interior is aligned with its surface — which is not currently measurable for any planet host.

Why dissipation is the variable #

Tidal realignment is the same calculation as tidal circularisation, run on a different element of the orbit. The rate goes as the sixth power of the ratio of the star’s radius to the orbital separation, times an efficiency that says how much of the tidal flow is converted to heat.

The sixth power is geometry and is the same for every star. The efficiency is not. In a star with a convective envelope, turbulence in the convection damps the tidal flow, and the effective viscosity is large. In a star with a radiative envelope there is no turbulence to do it, and the equilibrium tide is dissipated far more weakly — by orders of magnitude, though how many is exactly the quantity in dispute.

Take the contrast as four decades. The sixth root of ten thousand is 4.6, so the two realignment boundaries differ by a factor of 4.6 in separation. Hot Jupiters sit between them. A cool star realigns its planet within its lifetime and a hot star does not, at the same orbital distance.

What the numbers look like #

Some arithmetic makes the filter concrete.

A Jupiter-mass planet at eight stellar radii — a three-day orbit around a solar-type star — realigns in about a billion years if the star has a convective envelope. Move it to twelve radii and the sixth power multiplies that by eleven, giving eleven billion years, which is longer than the star has existed. Move it back to five radii and the same power divides it by seventeen, giving sixty million years, which is short compared with anything.

So for cool stars the realignment boundary is sharp in orbital distance as well as in temperature, and it lands right in the middle of the hot Jupiter population. That is the awkward part: the population being used to measure a migration history straddles the boundary of the process that erases it.

For a hot star the same three-day orbit takes ten thousand billion years. Nothing happens at all, and the obliquity that is measured is the obliquity the planet arrived with.

What the filter conceals #

If the observed distribution is an arrival distribution that has been partly erased, then the interesting quantity — how planets arrive — has to be read from the part that survives.

The hot stars are the clean sample. Their obliquities are essentially untouched, and they are broadly distributed, with a significant retrograde fraction. That is not what smooth migration through a disc produces: a planet spiralling inwards under a torque that nearly cancels stays in the disc’s plane, and the disc’s plane is close to the star’s equator.

It is what a violent history produces. Two mechanisms are on offer and both involve a third body.

The second is planet–planet scattering: a system of several giants becomes unstable, one is thrown outwards or ejected, and another is left on a highly eccentric, inclined orbit which tides then shrink. Both mechanisms deliver a hot Jupiter on a misaligned orbit; both leave the remaining planets, if any, disturbed; and the two are distinguished by whether a distant companion is present.

The population the filter predicts #

A filter makes predictions about what should be found where, and two of them have been checked.

The first is about age. Realignment takes time, so within the cool hosts the young systems should be less aligned than the old ones. Ages for field stars are poor, but the systems around cool stars that are demonstrably young do include misaligned cases, and the oldest cool hosts are uniformly aligned.

The second is about the shape of the aligned distribution. If cool hosts were aligned because their planets arrived aligned, the residual scatter would be whatever the disc’s own warp supplies — a few degrees at most. If they are aligned because a tide dragged them there, the scatter should be a decaying remnant of a broad distribution, and should be wider. The measured scatter for cool hosts is a few degrees, which is closer to the first prediction than to the second, and is the strongest argument that some hot Jupiters do arrive aligned and are not merely repaired.

That is the reason the field has converged on a mixture rather than on a single migration channel: disc migration delivers aligned planets, high-eccentricity migration delivers a broad distribution, and the observed pattern needs both. Estimating the proportions is what the temperature split is actually used for.

There is a third prediction that has not gone as well, and it is worth recording because it is the one the filter argument most obviously owes. Realignment by a tide should depend on how much tide there is, and the tide is a steep function of separation — the twelfth or so power in the relevant limit — so within the cool hosts the alignment should switch on over a narrow range of scaled orbital distance. It does, roughly, and the boundary sits where the four-decade dissipation contrast puts it. But the same tide that realigns should also destroy: a planet close enough to be realigned is close enough to be spiralling in, and the two timescales differ by less than an order of magnitude across the sample. Every aligned cool host is therefore a system caught in the interval between having been repaired and having been consumed, and the width of that interval is a prediction the population size can be checked against. The check is not clean, and the count of aligned cool hosts is somewhat larger than the simplest version allows.

Not scattered, but perpendicular #

The description above treats the misaligned population as broadly distributed, which is how it was read for a decade. Larger samples have sharpened that into something more specific and harder to explain.

When the projected obliquities are converted to true obliquities — using stellar inclinations from rotation periods or from seismology — the distribution of the misaligned systems is not uniform. It piles up near ninety degrees. A substantial fraction of the misaligned hot Jupiters are on orbits nearly perpendicular to their star’s equator, rather than being scattered evenly across every angle between aligned and retrograde.

A pile-up is a much stronger constraint than a scatter. Scattering mechanisms — a planet thrown by a neighbour, a chaotic history — produce broad distributions with no preferred angle, and would have to be tuned to prefer one. Something that ends at ninety degrees is a mechanism with an attractor there.

The candidate is a secular resonance. If a system’s orbital plane precesses at a rate that at some point matches the precession rate of the star’s own spin axis, the two lock, and the locked state can drive the mutual angle towards a fixed value. Because the rates depend on the orbital distance and on the star’s rotation, and both evolve — the orbit shrinking under tides, the star spinning down — a system can be swept into such a resonance during its history and left in it. Several versions of that mechanism produce a stable state near ninety degrees, and the ones that do also predict which systems should be caught, in terms of the planet’s mass and the star’s rotation.

The prediction is checkable and it is being checked, and the current position is that the pile-up is real and its significance is limited by sample size rather than by the measurements. What began as evidence for violence has turned into evidence for a particular kind of gentle process, which is the reverse of the usual direction of travel in this subject and is a reasonable warning about reading a broad distribution as a random one before it has been measured well enough to have a shape.

What is not settled #

Three difficulties keep this from being a closed argument.

The dissipation contrast is not measured. It is inferred from exactly the observations it is being used to interpret, which is uncomfortable — the four-decade figure above is a calibration to the boundary’s position rather than a prediction of it. An independent determination would come from binary star circularisation across the same temperature boundary, and the samples there are small.

The star’s own rotation complicates the accounting. Realigning an orbit means moving angular momentum between the orbit and the star’s spin, which is the transaction that lengthens the Earth’s day as the Moon recedes, and for a hot Jupiter around a solar-type star the orbit’s angular momentum is comparable to or smaller than the star’s. The tide can therefore realign the star rather than the orbit, or realign only the outer convective envelope while the interior stays put — in which case the measured photospheric alignment is not the whole system’s. And the sample is small and inhomogeneously assembled. Obliquities have been measured for a few hundred systems, chosen because they are bright, because their transits are deep — the depth is what fixes a density when combined with a mass —, and because their stars rotate fast enough for the anomaly to be detectable — the last of which correlates with temperature and is therefore not innocent, and none of which is helped by the metre per second that is not the star. The amplitude of the anomaly is proportional to the star’s projected rotation speed, so a slowly rotating cool star gives a small signal and is measured less often and less precisely than a rapidly rotating hot one. A selection that favours hot stars in a comparison between hot and cool stars is not fatal, because the aligned cool cases that are measured are measured well, but it does mean the two halves of the sample are not equally sampled and the fraction of misaligned cool hosts is an upper limit rather than a value.

One more calibration shows how much of the conclusion depends on the assumed realignment timescale.

Where the ladder goes #

The most useful extension is to systems that are not hot Jupiters. If the misalignments are produced by violent migration, then planets that did not migrate violently — compact multi-planet systems, or single planets on wide orbits — should be aligned regardless of host temperature. The measurements are harder, because the signal scales with the planet’s area, and the early results are consistent with alignment.

The other direction runs back into the star. The realignment tide deposits its energy in the stellar envelope, and a hot Jupiter close enough to be realigning its star is also spinning it up — which makes the host look younger by the rotation clock than it is. Several planet hosts are anomalously rapid rotators for their apparent age, and the tidal explanation makes that a measurement of the same dissipation the alignment argument needs.