A velocity measured from a shape

A star’s spectral lines are broadened by its rotation. One limb is approaching and the other receding, so the light from the disc is a superposition of Doppler shifts spanning $\pm v\sin i$, and the resulting profile is the rotation kernel — an ellipse-like shape with finite support and hard edges, quite unlike a thermal Gaussian.

Now put an opaque disc in front of part of it.

The planet hides light from one particular strip of the stellar surface, moving at one particular line-of-sight velocity. That velocity is missing from the profile. The profile is no longer symmetric, its fitted centre moves, and a spectrograph measuring radial velocities reports that the star has a velocity of a few tens of metres a second which it does not have.

The arithmetic of a missing slice #

Suppose the star’s disc has total flux $F$ and the light from it has a flux-weighted mean line-of-sight velocity of zero — which it does, by symmetry, for a star rotating rigidly and observed as a whole.

Block a fraction $f$ of the flux, all of it coming from material moving at velocity $v$. The remaining light has mean velocity

$$\bar v = \frac{0 - f v}{1 - f} \approx -f v \quad \text{for small } f.$$

So the apparent shift is the product of the transit depth and the velocity of the hidden strip. For a Jupiter-sized planet in front of a Sun-like star, $f \approx 0.01$; for a star rotating at $v\sin i = 4.5$ km/s the hidden velocity reaches a few km/s; the product is a few tens of metres a second, which is comfortably above the precision of a modern spectrograph.

Both factors matter and they scale differently. The amplitude goes roughly as $(R_p/R_\star)^2\,v\sin i$ — so a rapidly rotating star gives a large anomaly with a small planet, and a slowly rotating one gives almost nothing with a large planet. That is the opposite of the ordinary reflex-velocity signal, which is largest for slowly rotating stars because those are the ones whose lines are sharp enough to measure precisely.

Where the asymmetry comes from #

If the transit chord is perpendicular to the projected rotation axis — the aligned case — the planet spends equal time over the approaching half and the receding half, entering on one and leaving on the other. The anomaly is a positive excursion followed by a negative one of the same size, antisymmetric about mid-transit.

Tilt the chord and that stops being true. A planet whose orbit is inclined to the star’s equator crosses the two halves for unequal times, or crosses only one of them, and the anomaly becomes lopsided in a way that reads off directly as an angle.

What was actually measured #

The observation is a time series of radial velocities taken during a transit, at a cadence of a few minutes, with a precision of a few metres per second. That is it — the same spectrograph, the same reduction, the same cross-correlation against a template — the same shift in a line that is a speedometer — that produces an ordinary reflex velocity curve, applied over a window of a few hours instead of a few years.

What makes it a measurement of an angle rather than of a velocity is that the shape of that few-hour excursion is fitted, against a model containing $\lambda$, $v\sin i$, the impact parameter, the radius ratio and a limb-darkening law.

The quantity that comes out is not the obliquity. It is $\lambda$, the angle between the two vectors as projected onto the sky, and the true angle $\psi$ between the orbit normal and the stellar spin axis is related to it by

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

which needs $i_\star$, the inclination of the star’s own rotation axis to the line of sight. That is not available from the transit at all. It comes, when it comes, from asteroseismology — the relative amplitudes of the split rotational multiplets in a star’s oscillation spectrum depend on the viewing angle — or from combining $v\sin i$ with a measured rotation period and a radius.

So a published “obliquity” is usually a projected obliquity, and a system reported as aligned at $\lambda = 0$ could still have a large true misalignment if the star happens to be seen pole-on. The distinction matters for individual systems and washes out statistically.

The first measurement is older than the effect’s use. Richard Rossiter and Dean McLaughlin, working independently on eclipsing binaries in 1924, both found that the radial-velocity curve of an eclipsing system shows an excursion during eclipse that no orbital motion could produce. It was recognised immediately as a rotation effect and used for seventy-five years as a way of measuring stellar rotation in binaries. Its first application to a planet was HD 209458 b in 2000, and the first misaligned system, XO-3 b, was found in 2008 — which is when the technique stopped being a curiosity — and it was found in a system detected by the light its planet removes, because nothing without a transit can be measured this way at all.

The result nobody wanted #

A hot Jupiter cannot have formed where it is. The temperatures at 0.05 AU are far too high for the ices that a giant planet’s core needs, so these objects formed beyond the snow line and moved inward. The question is how.

Two families of answer, and they make opposite predictions about obliquity.

Disc migration moves a planet inward by exchanging angular momentum with the protoplanetary disc it formed in. The disc is a flat rotating structure, very nearly coplanar with the star’s equator, and a planet migrating through it stays in that plane. Prediction: aligned, always.

High-eccentricity migration throws a planet onto a highly eccentric orbit by a scattering encounter with another planet or a distant stellar companion, after which tidal friction at each close perihelion passage shrinks and circularises the orbit. The scattering has no memory of the disc plane. Prediction: obliquities distributed over a wide range, including retrograde.

The measurements said the second, and loudly. Of the roughly 200 systems with measured $\lambda$, about a third are misaligned by more than 20°, and a dozen or more are retrograde. Compare the resonant chain of TRAPPIST-1, whose planets could only have been assembled by a smooth migration through a disc: the two populations record two entirely different histories. That is not a tail on an aligned distribution; it is a population.

The measurement is of a distortion, not of a shift #

There is a distinction here that the units hide, and it is worth drawing out because it decides what the technique can and cannot do.

An ordinary radial velocity is a shift: every line in the spectrum moves by the same fraction of its wavelength, and cross-correlating the observed spectrum against a template recovers that fraction with a precision set by how many lines there are and how sharp they are. Nothing about the line’s shape matters, which is why the method works on a spectrum too noisy to show any individual line.

The Rossiter anomaly is a distortion. No line has moved; a piece has been removed from each of them, at a wavelength that depends on where the planet is. The cross-correlation reports a shift because that is the only thing it is built to report, and the number it returns is a projection of the distortion onto the one degree of freedom the algorithm has.

That projection loses information, and the loss is not small. The full distortion is a function of wavelength across each line — a bite whose position moves through the profile as the transit proceeds — and reducing it to a single centroid throws away everything except the first moment. Doppler tomography does not: it subtracts the out-of-transit profile from each in-transit one and shows the residual directly, as a dark trace crossing the line from one wing to the other. The trace’s slope is $v\sin i$, its intercept is the obliquity, and the two are separated by geometry rather than by a fit.

The consequence is practical. Doppler tomography works on rapidly rotating stars, where the lines are broad and shallow and the centroid is poorly determined — which is exactly the population the centroid method struggles with and exactly where the interesting obliquities are, since rapid rotators are hot stars and hot stars are the ones tides have not realigned.

The anomaly as a contaminant #

Before it was a measurement it was a nuisance, and it is still one for anybody fitting an orbit rather than an obliquity.

A radial-velocity campaign to measure a planet’s mass samples the star at whatever times the telescope is available, and some of those times fall during a transit. A measurement taken then contains the anomaly — tens of metres a second, in a direction that depends on where in the transit it was taken — superimposed on the orbital curve the fit is trying to recover.

For a hot Jupiter that contamination is small compared with the orbital amplitude and is still enough to bias the fitted eccentricity, because a spurious excursion at one particular orbital phase looks exactly like the asymmetry an eccentric orbit produces. Several early reports of small eccentricities in hot Jupiter orbits were traced to in-transit points that had not been excluded.

The remedy is trivial once recognised — discard any measurement taken within the transit window — and it costs nothing, because the transit occupies a per cent or two of the orbit. What it illustrates is a general hazard: a signal that is a measurement in one analysis is a systematic in another, and the two analyses use the same numbers.

The reverse contamination also exists and is harder. An obliquity fit has to remove the orbital curve, and the orbital curve is known only from the out-of-transit measurements — so an error in the orbital solution propagates into the anomaly’s baseline, and a baseline error is degenerate with the impact parameter. Systems where both are measured well are systems with a great deal of data on either side of the transit as well as during it.

Where the model stops #

The star is not a rigid rotator. Differential rotation — the equator turning faster than the poles, as in the Sun by about 20 per cent — changes the velocity field the planet is crossing, and it is degenerate with obliquity in the fit for some geometries. So is convective blueshift, whose suppression under the planet produces a small additional anomaly of its own with a different shape.

Tides realign the star too, eventually. The same tidal friction that circularises the orbit also drags the star’s spin towards the orbit normal, on a timescale that depends steeply on the star’s internal structure. Cool stars below about 6,250 K have thick convective envelopes and dissipate tides efficiently; hotter stars have radiative envelopes and do not. The observed obliquity distribution splits at almost exactly that temperature — cool hosts are mostly aligned, hot hosts are all over the sky — which is strong evidence that the aligned systems were realigned rather than born aligned, and that the primordial distribution is the hot stars’ one.

A small planet gives nothing. The anomaly scales as the transit depth, so an Earth-sized planet in front of a Sun-like star produces about 0.5 m/s, which is at or below the state of the art and buried in stellar activity noise. Essentially all measured obliquities are of giant planets, so the sample is shaped by the same detection bias every survey has, and the sample is therefore about the migration of giants rather than about planetary systems in general.

And the modelling is degenerate. $\lambda$, $v\sin i$ and the impact parameter shape the same curve, and for a nearly central transit of a slowly rotating star the three are strongly correlated. Published uncertainties on $\lambda$ are frequently dominated by that correlation rather than by the photon noise.

Reading the profile instead of its centroid #

The distinction between a shift and a distortion has produced a technique that abandons the velocity entirely, and it is now the preferred one where the data allow.

Instead of cross-correlating each in-transit spectrum to a single number, take the average of the out-of-transit profiles and subtract it from each in-transit one. What remains is the light the planet blocked: a small bright residual — bright because it is what was removed from an absorption profile — at the wavelength corresponding to the velocity of the strip the planet was covering.

That residual is the local line profile of one piece of the stellar surface, isolated. Its position gives the rotational velocity of that strip directly, so a sequence of them through the transit maps the velocity across the chord without any model of the whole disc.

Two things follow that the centroid method cannot deliver. The obliquity comes from the trajectory of the residual across the profile, which is a geometric measurement rather than a fit to an amplitude — so it is insensitive to the degeneracies between obliquity, projected rotation and impact parameter that afflict the velocity fit.

And the shape of the residual is the local line profile, which carries the local temperature, the local convective blueshift and any local magnetic field. That turns a transit into a scan across a stellar surface: the planet is an aperture, and what it reveals is the spectrum of the region behind it, one strip at a time.

A method built to measure a planet’s orbit has become a way of resolving a star, and the resolution achieved — the width of the planet’s own shadow — is far finer than anything an image of that star could reach.

The cost is signal-to-noise. Subtracting one profile from another doubles the noise and leaves a residual of a per cent or so of the line depth, so the technique needs a bright host and a large telescope where the centroid method needs neither.

What the picture cannot show #

The third angle. Every figure here is a projection onto the sky, and the quantity drawn is $\lambda$. The true obliquity needs the stellar inclination, which is not in any of these pictures and is not obtainable from the data they represent.

The line profile. The anomaly is drawn as a velocity, which is what a cross-correlation reports. What is physically happening is a distortion of a line shape, and the modern technique — Doppler tomography — fits the residual profile directly rather than its centroid, recovering more information and being much harder to plot.

The star’s surface. The disc is drawn as a smooth velocity gradient. A real star has spots, plage and granulation, all of which move with the rotation and all of which produce their own centroid shifts of comparable size. For an active star those are the dominant systematic, and the drawing of a clean rotating disc is the assumption most likely to be wrong. The anomaly’s shape is a function of one angle, and the chord it is measured across is a function of another, so both are worth drawing at an intermediate value.

Where the ladder goes next #

Later rungs on this anchor: Doppler tomography and the reconstruction of the planet’s shadow directly in the line profile. The stellar-inclination measurement from asteroseismic mode splitting, and the handful of systems where the true $\psi$ is known. The obliquity distribution split at 6,250 K, and what it says about tidal quality factors. Obliquity in multi-planet systems, where the whole plane can be tilted together. The Kozai–Lidov mechanism, which is the specific way a distant companion produces the eccentricity that high-eccentricity migration needs. And the McLaughlin effect on eclipsing binaries, where it began, and which is still the best way to measure rotation in a close pair.

The measurement is a curve of a quantity that is not what it is called, produced by a star that is not moving, fitted for an angle that is not the angle of interest — and it is nevertheless the sharpest observational discriminant anyone has between two theories of how planetary systems rearrange themselves.