An angle hidden by the tilt of the star

A transiting planet crossing a rotating star hides a slice of the stellar disc at one velocity, and the distortion that slice leaves in the star’s spectral lines — measured as an anomaly in the radial velocity, or followed as a moving bump across the line profile — gives an angle. The angle is $\lambda$, the angle between the planet’s orbital axis and the star’s spin axis as both are projected onto the plane of the sky. More than two hundred systems have one. The distribution of those angles is what showed that hot Jupiters around hot stars are often wildly misaligned and those around cool stars are not.

The angle the physics is about is a different one. The true obliquity $\psi$ is the angle between the two axes in three dimensions, and a projection onto the sky loses one of the three. The transit fixes the orbit’s inclination to the line of sight at nearly ninety degrees — that is what a transit is — but it says nothing about the inclination $i_\star$ of the star’s own spin axis. With both,

$$\cos\psi = \cos i_\star \cos i_{\rm orb} + \sin i_\star \sin i_{\rm orb}\cos\lambda,$$

and with $i_{\rm orb} = 90^\circ$ this becomes simply $\cos\psi = \sin i_\star\cos\lambda$.

This essay is about the missing angle: what can be said without it, how it is measured, and why every method of measuring it fails in the same place.

A bound, and one exact angle #

Two consequences of the formula hold whatever $i_\star$ is, and they are the reason projected angles were useful long before true ones could be measured.

The first is that the projection can only hide misalignment. Since $\sin i_\star \le 1$, $\cos\psi \le \cos\lambda$ for an aligned-looking system, so $\psi \ge \lambda$: a star tipped towards the observer turns any projected angle into a larger true one, and never a smaller. For a retrograde-looking system the inequality reverses and $\psi$ lies between $\lambda$ and 90°. Either way the true angle sits between the projected one and a right angle, and a star seen pole-on sends every projected angle to exactly 90°, because from above the pole every orbit crosses the equator at right angles whatever its tilt.

The second is that a right angle is exact. If $\lambda = 90^\circ$ then $\cos\lambda = 0$, so $\cos\psi = 0$ and $\psi = 90^\circ$ for any tilt at all. A planet whose projected orbit crosses its star’s projected equator at a right angle is on a polar orbit, and needs no further measurement to say so. This is why the most striking recent finding about obliquities — that the misaligned planets are not scattered evenly over all angles but pile up near perpendicular — could be seen in the projected angles before the true ones were in hand: a cluster of $\lambda$ near 90° is a cluster of $\psi$ near 90°, with no inclination in the argument.

The Solar System is a useful check on both. Its planets orbit about seven degrees from the Sun’s equatorial plane. An observer somewhere in the plane of the Earth’s orbit, watching the Earth transit, would see the Sun’s spin axis tipped towards or away from them by an amount that depends on where around the orbit they stood, and would measure a projected angle anywhere between zero and seven degrees — zero from the two directions in which the tilt points along the line of sight, seven from the two at right angles to them. Every one of those observers would correctly report the Solar System as well aligned, and all but a few would quote an angle smaller than the true one.

Between those two statements lies everything else, and for most systems it is a lot.

An aligned system that may not be #

The case that matters most is the commonest: a planet with $\lambda \approx 0$, published as aligned. With $\lambda = 0$ the formula gives $\psi = 90^\circ - i_\star$ exactly, so the true obliquity is simply how far the star is tipped from equator-on. If nothing is known about the tilt, the true obliquity is as uncertain as a random orientation.

A random orientation in space is not uniform in angle. Spin axes are equally likely to point anywhere on a sphere, and there is more sphere near the equator than near the poles, so $\cos i_\star$ rather than $i_\star$ is uniformly distributed. With $\lambda = 0$ that makes the true obliquity’s distribution proportional to $\cos\psi$, and its median is exactly thirty degrees: a system with a perfectly measured projected angle of zero and nothing else known has even odds of being misaligned by more than thirty degrees, and about one chance in eight of being misaligned by more than sixty. For $\lambda = 45^\circ$ the median true angle is 52°; for 70°, 73°.

Those are the answers for complete ignorance of the star, and complete ignorance is not the state of the population. If most orbits really are aligned, then most transiting systems are aligned systems seen from within their orbital planes, and those stars are seen nearly equator-on; a small $\lambda$ then usually means a small $\psi$, and the distributions in the figure overstate the misalignment. The difficulty is that this is an assumption about the population used to interpret a member of it. A statement that the obliquity distribution is sharply peaked at zero cannot be supported by measuring $\lambda$ and assuming the stars are equator-on because obliquities are small. For individual systems — and for the question of whether a particular planet migrated gently or violently — the tilt of the star has to be measured.

A tilt measured through a sine #

The oldest method needs three numbers and no new instrument. A star’s lines are broadened by its rotation to a width set by $v\sin i_\star$, the equatorial speed times the sine of the tilt. The equatorial speed itself is $2\pi R_\star/P_{\rm rot}$, from the star’s radius and its rotation period, and a cool star’s rotation period can be read from the brightness modulation as starspots rotate into and out of view. Dividing gives $\sin i_\star$.

Dividing is where it goes wrong, and near 90° it goes wrong badly.

The sine is steep at small angles and flat near 90°. A ratio of 0.5 known to $\pm 0.08$ fixes the tilt to within ten degrees. A ratio of 0.95 known equally well allows anything from 60° to 90°, a range three times wider, and a ratio consistent with one says almost nothing except that the star is not nearly pole-on. The reason is a symmetry rather than a property of the sine. A star seen from ten degrees above its equator and from ten degrees below presents exactly the same appearance: the same projected speed, the same spot modulation, the same everything. Any observable of a spin axis is therefore symmetric about $i_\star = 90^\circ$, and any smooth symmetric function has zero slope at its centre. No measurement of this kind can pin down a star that is nearly equator-on, and an aligned transiting system is exactly such a star.

A second error is subtler and was only corrected in 2020. The projected speed and the equatorial speed are not two independent numbers whose ratio carries a propagated error: the unknown true equatorial speed appears in both. The correct treatment asks, for each possible tilt and each possible true speed, how probable both measurements are, and integrates over the true speed.

For a star whose projected speed is well below its equatorial speed the two treatments agree. For the aligned-looking case, where the two speeds are nearly equal, they do not: the proper distribution is broad and piled against 90°, allowing anything from about 57°, where the naive ratio returns a single number with an error bar that does not describe it. Applying the ratio method to a sample of stars also biases the inferred tilts systematically. The errors scatter the ratio above one as often as below it; every ratio above one is truncated to 90°, and every ratio below one is read as a genuine tilt, so a sample of stars that are all equator-on appears to contain a spread of moderately tilted ones — an artefact of the arithmetic rather than a property of the stars.

A tilt read from which notes are heard #

The other route to the tilt does not use a rotation speed at all. A Sun-like star oscillates in thousands of sound modes at once, its interior written in a comb of frequencies, and rotation splits each mode of angular degree $\ell$ into $2\ell + 1$ components of different azimuthal order $m$, separated in frequency by roughly the rotation rate. Which of those components can be seen depends on the direction from which the star is viewed.

For a dipole mode the central component’s visibility is $\cos^2 i_\star$ and each outer one’s is $\tfrac12\sin^2 i_\star$. A star seen pole-on shows only the central peak, one seen equator-on shows only the two outer ones, and the ratio of their heights in a power spectrum is the tilt. The quadrupole modes add a pair whose visibility, $\tfrac38\sin^2 2i_\star$, peaks at 45°. Fitting the heights and the splitting together measures both the rotation rate and the tilt, with no radius, no line width and no spot modulation.

The method has its own failure: it needs the splitting to be wider than the width of each peak, which requires the star to rotate fast enough and the oscillation to be long-lived enough, and it works best for stars somewhat more evolved than the Sun, whose modes are sharper. And it shares the one limitation every method shares, since its curves too are symmetric about 90° and flat there. What it adds is independence. A tilt measured from which notes are heard does not share a single systematic with a tilt measured from line widths and spots, and where the two have been compared on the same stars they have generally agreed within their errors.

A population tilted all at once #

The symmetry that defeats the tilt of one star can be turned round for a population, and it is how obliquities have been measured for planets far too small for any line-profile measurement at all.

The idea is that a transiting planet selects its star’s orientation only if the planet is aligned. If orbits and spins are aligned, every transiting system is seen from within the star’s equatorial plane, so every host star is equator-on, and $\sin i_\star$ is close to one for all of them. If orbits and spins are unrelated, the hosts’ spin axes point anywhere, and $\sin i_\star$ has the distribution of a random orientation, whose mean is $\pi/4 \approx 0.79$. The two populations differ by twenty per cent in the average of a quantity each star gives only poorly — and the difficulty near 90° that ruins an individual measurement barely matters for an average over hundreds, because the question is no longer where one star sits but whether the population is piled against the top of the sine or spread down it.

Two versions of the test have been run on the stars of the Kepler field, whose planets were found by the light they removed and are mostly Neptune-sized or smaller. One compares the projected rotation speeds of planet hosts with those of stars without known planets but with the same temperature and rotation period: the hosts’ speeds are systematically higher, as they should be if the hosts are seen equator-on and the comparison stars at random. The other uses the spots. A spotted star seen equator-on carries its spots into and out of view every rotation and varies by more than the same star seen near pole-on, whose spots stay in sight; so if planet hosts are equator-on, their rotational brightness variations should have larger amplitudes than those of comparison stars. They do, for cool stars.

Both tests divide at the same temperature the line-profile measurements found for hot Jupiters. Below about 6,000 kelvin the hosts of small Kepler planets are consistent with alignment; above it their orientations look closer to random. That result reaches planets whose own Rossiter–McLaughlin signal would be a few centimetres per second, and it says the temperature boundary is not a property of hot Jupiters but of the stars — a statement about the stars’ envelopes and how they respond to whatever tilts them, extended to planets that never migrated violently at all.

What the population method cannot do is say which system is which. It returns the fraction of hosts that are equator-on and an estimate of how widely the rest are spread; it cannot identify the misaligned members, and it depends on the comparison stars being truly comparable in every property that affects the rotation speed or the spot amplitude. It is an obliquity distribution with no obliquities in it, which is the complement of the line-profile method — a precise angle for a few hundred bright systems, and no population.

What the true angles have shown #

The first system in which seismology changed the answer was a star with two transiting planets on nearly coplanar orbits. Multi-planet systems had been assumed aligned, on the reasoning that whatever tilts one planet’s orbit should disrupt a system of several, and a projected angle could not test that because none had been measured. The star’s split modes put its spin axis at about 47° to the line of sight while both planets transit, so the planets’ shared orbital plane is misaligned with the star’s equator by at least 37°. A whole planetary system can be tilted together, which means the tilt was given to the star or to the disc rather than to one planet — the direction a distant companion’s slow torque on the disc would take.

For hot stars the tilt comes from a third route. A rapidly rotating hot star is hotter and brighter at its poles than at its equator, and a planet crossing that uneven disc produces an asymmetric transit whose shape fixes both the projected and the true angle at once. The same effect makes brightness-limited samples preferentially contain pole-on stars, because a hot fast rotator seen pole-on looks brighter than the same star seen equator-on — a selection that acts on exactly the variable at issue.

With a few dozen true obliquities now measured, the population has a shape the projected angles only hinted at. The aligned systems are aligned in three dimensions, not merely in projection. The misaligned ones are concentrated between about 80° and 125°, an excess of nearly polar orbits that no process which scatters orbits randomly produces. The projected angles had already suggested it, through the exact result at 90°; the true angles confirmed that the excess is real and not an accident of projection.

What the figures do not include #

The geometry is drawn for an orbit seen exactly edge-on. Real transiting orbits are inclined by a degree or two from that, which adds a small term to the formula and matters only for nearly pole-on stars. The distributions of the true obliquity assume an isotropic distribution of spin axes and nothing else; in a real analysis the prior is the population’s own obliquity distribution, which is what the analysis is trying to measure, and it has to be fitted jointly rather than assumed. The inference of the tilt treats the errors on the two speeds as Gaussian and ignores two systematics that matter in practice: differential rotation, which makes the equatorial speed from a spot at mid-latitude too small, and the difficulty of separating rotational broadening from the turbulent broadening of the lines in a slowly rotating star, which makes $v\sin i_\star$ itself uncertain at a level comparable to the signal. The seismic visibilities assume that energy is shared equally among the components of each mode, which holds well for stochastically excited oscillations and is the assumption every seismic tilt rests on.

Still open: how many aligned systems are aligned #

The projected angles established that misalignment exists and depends on the star; the true angles have established that it favours right angles. What neither has yet established is how well aligned the aligned systems are. A projected angle of zero leaves a thirty-degree median true obliquity for a star of unknown tilt, the rotation-period method cannot distinguish 70° from 90° for such a star, and the seismic method needs stars that oscillate cleanly and rotate fast enough to split their modes. The answer matters because a distribution sharply peaked at zero says that most planetary systems formed and stayed in their star’s equatorial plane, while a distribution a few tens of degrees wide says that stars and their discs are routinely tilted against each other before any planet migrates. The Sun’s own equator is tilted by about seven degrees to the plane of its planets, and it is not known whether that is typical, generous or unusually tidy.