A shadow crossing a rotating line

A rotating star’s absorption lines are broadened because different parts of its disc move at different velocities: the approaching limb contributes to the blue side of the line and the receding limb to the red. Every velocity across the line corresponds to a strip of the disc.

A transiting planet covers one of those strips. It removes the light that would have come from that velocity, so the line profile acquires a deficit at one position — and as the planet crosses, the deficit travels from one side of the line to the other.

Tracking that bump is a measurement of a path rather than of a number, and it is the reason the alignment between planetary orbits and stellar equators is known for the systems where it matters most.

The technique is called Doppler tomography, by analogy with the medical reconstruction: a sequence of one-dimensional projections is used to build a two-dimensional picture, and here the projections are line profiles and the picture is the path across the disc.

The same physics, two ways of reading it #

The classical version of this measurement takes the centroid of the distorted line and reports it as a velocity. The star appears to shift because part of its line has been removed asymmetrically, and the apparent shift as a function of time is the Rossiter–McLaughlin effect — a velocity measured from a shape rather than from a motion.

Tomography takes the same distortion and does not collapse it to a number. It keeps the whole residual profile, at every epoch, and fits the path of the deficit through it.

The two use the same photons and they are not equally informative. Collapsing a profile to a centroid discards everything about the deficit except its first moment, and the first moment is a weighted average that depends on the line’s shape, on the limb darkening, and on the star’s own macroturbulence. Keeping the profile keeps the deficit’s position and width, which are the physical quantities.

There is a further and less obvious advantage. The velocity anomaly is a small perturbation on a much larger quantity — the star’s own orbital reflex velocity — so it has to be separated from the Keplerian signal, and any error in the orbit propagates into the anomaly. The tomographic residual is a difference between two line profiles taken hours apart, so the orbital motion contributes a shift of a few metres a second that is negligible against a line hundreds of times wider. The measurement is nearly independent of the orbit, which is a useful decoupling when the orbit itself is being fitted from the same data.

Where the advantage is largest #

The gain is not uniform. It is largest for rapidly rotating stars, and that is where the interesting planets are.

The velocity anomaly’s amplitude is roughly the projected rotation velocity times the transit depth, so it grows with rotation. Its significance does not, because a rapidly rotating star has broad, shallow lines and its velocities are correspondingly imprecise — the two effects nearly cancel, and the classical technique works poorly above about thirty kilometres a second.

Tomography does not suffer that cancellation. A faster rotator has a wider line, so the deficit is better resolved and its position better determined. The technique gets better as the rotation increases, up to the point where the line becomes so shallow that the residual is lost in the noise.

It is worth being explicit about what “better resolved” means, because it is the crux. The deficit’s width in velocity is set by the planet’s size relative to the star — a planet covering a tenth of the stellar radius produces a deficit a tenth of the line’s full width. So on a line of thirty kilometres a second the deficit is three wide, and on a line of a hundred it is ten. The position of a feature is measured to roughly its width divided by the signal-to-noise, so a wider line gives a better-determined position in absolute terms and the same in relative terms — and since the quantity wanted is the position as a fraction of the line width, the relative precision is what matters and it is unchanged. What improves with rotation is that the deficit becomes resolved at all: below about fifteen kilometres a second it is narrower than the instrumental profile and its shape is unmeasurable.

How big the deficit is, and what it costs to see #

The technique’s demands follow from one piece of arithmetic, and it is worth doing because it explains why the target list is short.

A planet covering a tenth of the stellar radius blocks one per cent of the light. That missing light is not spread across the line — it is removed from the strip of disc the planet covers, which occupies about a tenth of the line’s velocity range. So the residual bump has an area equal to one per cent of the line’s total absorption, concentrated into a tenth of its width, and its amplitude is therefore around a tenth of the line’s depth. For a mean line profile with a depth of forty per cent of the continuum, that is a bump of four per cent.

Four per cent sounds comfortable and is not, because it has to be measured in a single exposure short enough that the planet does not move during it. Seeing the bump at ten sigma requires a signal-to-noise of a few hundred per exposure per resolution element, and a few hundred in a few minutes restricts the technique to stars of eighth magnitude and brighter on a four-metre telescope, or eleventh on an eight-metre.

The exposure length is a hard constraint rather than a preference. A planet crosses the disc in two to four hours and its own shadow is a tenth of the way across, so the deficit moves by its own width in fifteen to twenty minutes. An exposure longer than that smears the bump, widening it and flattening it in exactly the way an unresolved deficit looks — so a long exposure does not merely blur the measurement, it biases the impact parameter, which is read from the track’s velocity range.

The residual is also not built from one line. A single photospheric line is too shallow and too noisy, so the profile that is actually analysed is a mean line: a cross-correlation against a mask of thousands of lines, or a least-squares deconvolution that solves for the common profile they share. Both assume every line has the same shape and differs only in depth, which is nearly true and is not exactly true — lines formed at different heights have different convective blueshifts and different centre-to-limb behaviour, so the mean profile is a weighted average over formation depths and the deficit’s position inherits that weighting. It is the same assumption, and the same small error, that averaging lines into one velocity makes.

What the track measures #

The path of the deficit across the line profile is a projection of the planet’s path across the disc, so the track determines two quantities.

The projected obliquity, from the track’s asymmetry. A chord parallel to the star’s equator produces a track symmetric about the line centre; one inclined to it does not, and the degree of asymmetry is the angle.

The impact parameter, from the range of velocities covered. A planet crossing near the equator sweeps the full range from one limb to the other; one crossing near a pole covers a narrower range at the centre of the line.

Both are also constrained by the photometric light curve, which determines the impact parameter independently, so a tomographic analysis is over-determined — and the consistency between the two is a check that the star is behaving as assumed.

There is a third quantity the track supplies almost for free, and it is a useful cross-check: the projected rotation velocity itself. The deficit reaches the line’s edge exactly when the planet reaches the stellar limb, so the velocity extent of the track is the projected rotation, measured from the planet rather than from the line’s width. Fitting a line’s width for a rotation velocity requires disentangling it from macroturbulence and from the instrumental profile, which is one of the several things that broaden a line for different reasons; reading it off a transit track does not.

What was actually measured #

The technique’s results are the reason the field’s picture of hot Jupiter formation changed.

Misalignment is common. Of the hot Jupiters with measured projected obliquities, a substantial fraction are misaligned, and several orbit retrograde. That is not what disc migration predicts — a planet that spiralled in through the disc it formed in should share the disc’s plane, which is the star’s equatorial plane.

Misalignment depends on the star. Planets around stars cooler than about 6,250 kelvin are predominantly aligned; those around hotter stars are not. The temperature is close to where stars acquire a convective envelope, and the interpretation is that cool stars tidally realign while hot ones do not — so the observed obliquity of a cool star’s planet is a record of tidal history and a hot star’s is a record of arrival.

And the rapid rotators were only accessible tomographically. The hot stars — which are the ones that preserve the primordial obliquity — rotate fast, and their obliquities were unmeasurable by the classical technique. The whole of the temperature dependence above was established using the method in this essay.

A fourth result is worth recording because it demonstrates the technique doing something the classical one cannot do at all: detecting a planet. A transiting planet’s deficit is a coherent feature moving through the line profile at a predictable rate, and searching a time series of profiles for such a feature is a detection method in its own right — one that works for planets around stars far too rapidly rotating for any radial-velocity search. Several hot Jupiters around early-type stars were confirmed this way when their velocities were useless, and the confirmation is a direct image of the planet’s shadow rather than an inference from a wobble.

One consequence of the over-determination is worth stating, because it is how the analysis fails safely. The photometric light curve gives the impact parameter and the planet-to-star radius ratio; the tomographic track gives the impact parameter again, plus the projected rotation velocity and the obliquity. If the two impact parameters disagree, something in the stellar model is wrong — a spot on the chord, an unaccounted-for companion diluting the light, a rotation profile that is not solid-body. The disagreement is visible before any obliquity is quoted, which is more than can be said for the classical velocity anomaly, where the same errors move the fitted angle and leave no residual to notice.

Where the picture stops #

Three limits stand out, and the first is the one shared with every version of this measurement.

It measures a projection. The angle determined is the one between the orbit’s projection on the sky and the star’s projected rotation axis. The true three-dimensional obliquity requires knowing the inclination of the star’s rotation axis to the line of sight, which needs an independent measurement — a rotation period combined with a radius, or asteroseismology. Without it, a projected alignment can hide a large true misalignment.

The star’s own line profile has to be modelled. The residual is the difference between the observed profile and the profile the star would have had, and the latter is built from out-of-transit exposures. Any change in the star between them — an evolving spot, a pulsation, a change in the instrument — enters the residual and looks like a deficit.

And convection contaminates the deficit’s position. The strip the planet covers has its own convective blueshift, which depends on the position on the disc through the centre-to-limb variation, so the deficit is not exactly at the rotational velocity of that strip. The correction is at the level of a few hundred metres a second — small against a rapid rotator’s line width and not small against a slow one’s, which is another reason the technique prefers fast rotators. It is the same centre-to-limb convective effect that limits ordinary radial velocities, arriving as a spatial rather than a temporal systematic.

There is a fourth, because it is what the field is now working around. The technique needs a transit, and a transit requires the orbit to pass in front — so the sample is restricted to systems whose orbital inclination is near ninety degrees, and every measured obliquity is for such a system. Whether the misalignments seen are representative of planets in general or peculiar to the ones that happen to transit is not answerable from this data alone, and it matters because a planet where one cannot form has to have arrived somehow, and the obliquity distribution is the main evidence about how.

Why keeping the profile is worth the trouble #

The general lesson is one this collection has met from several directions, and this is one of its cleanest instances.

A summary statistic discards the information that distinguishes hypotheses. A radial velocity is the first moment of a line profile; a line profile is a function. Collapsing the second to the first is lossless only if the distortion is a pure translation, and a transiting planet’s distortion is not a translation at all — it is a hole. The centroid of a profile with a hole in it moves, and the movement is a lossy summary of where the hole is.

That is exactly the structure of the line-by-line velocities that separate a planet from stellar activity: the averaged velocity is a number and the individual velocities are a pattern, and the pattern distinguishes causes the number cannot. It is also the structure of an eclipse-timing residual, where three causes are separated by their shapes rather than by their amplitudes.

The practical prescription that follows is short and is not always followed: keep the data at the level the physics acts on. A planet occults a region of a star, so the physics acts on the spatial distribution of the star’s surface, which is encoded in the line profile. Any statistic computed before that encoding is used has thrown away the thing the measurement is about.

End on what the obliquity distribution has actually established, because the technique’s value is entirely in that result. Before these measurements, the default assumption was that hot Jupiters arrived by migrating inward through their protoplanetary discs, which predicts alignment. The measurements found a substantial misaligned and retrograde population around hot stars, which disc migration cannot produce — those orbits require a violent history: a scattering encounter with another planet, or an inclination cycle driven by a distant companion followed by tidal capture. So the obliquity is a fossil of the arrival mechanism, and reading it required a technique that works on rapidly rotating stars, which required not collapsing a line profile to a number.

That is a satisfying chain: a decision about how to reduce a spectrum, made for reasons of information content, turned out to determine which stars could be measured, which determined which formation histories could be tested. The reduction choice was not a detail.

A last word on the archive. Because the measurement is a sequence of high-resolution spectra taken through a transit, any programme that observed a transit spectroscopically for another reason — a sodium detection, a velocity anomaly — recorded the data this technique needs. Several obliquities have been measured from archival observations taken years before anybody intended to make them, which is an unusually direct return on publishing raw spectra.

Where the ladder goes next #

The next rung is the true obliquity: how a projected angle is turned into a three-dimensional one, and what an asteroseismic or a rotation-period constraint on the stellar inclination adds. The rung beyond it is the same technique applied to a planet that does not transit at all — the deficit’s track exists for any occulting body, and a sufficiently large one can be detected in the line profile of a star whose planet passes only near the limb.