A transit that runs differently backwards

A transit light curve is usually symmetric in time. The planet enters the stellar disc, crosses it, and leaves, and if the star is a uniformly bright, limb-darkened circle the planet meets the same brightness at the same distance before and after the midpoint. Run the light curve backwards and nothing changes. That symmetry is so reliable that the time of mid-transit is defined by it.

It fails for one kind of star, and the failure is a measurement. A star rotating close to the rate at which its equator would fly off is not a sphere: it is flattened, and its equator is cooler than its poles, because the centrifugal effect weakens the effective gravity there and the emergent flux follows the gravity. A planet whose orbit crosses such a star at an angle to its equator passes over bright polar regions on one side of the transit and dim equatorial regions on the other. The light curve becomes lopsided in time, and the lopsidedness depends on the angle at which the orbit crosses the star and on how the star is tipped towards the observer — both of the angles that the line-profile method can only partly measure.

Why the poles are brighter #

The flattening and the darkening have one cause. In the frame rotating with the star, every surface element feels gravity pulling inwards and the centrifugal effect pushing outwards from the axis. At the poles the centrifugal term vanishes; at the equator it is largest and opposes gravity directly. The surface the star settles into is an equipotential of the two together — the Roche surface, which at the critical rate is exactly one and a half times as wide at the equator as it is tall at the poles — and the effective gravity on that surface falls from pole to equator.

Hugo von Zeipel showed in 1924 that for a star in radiative equilibrium the flux leaving each point of such a surface is proportional to the local effective gravity. The temperature then goes as the gravity to the power one quarter, and the intensity as the gravity itself. That exponent, written β, is the gravity-darkening exponent: $T \propto g^{\beta}$, with von Zeipel’s β = 0.25. At sixty per cent of critical rotation the gravity at the equator is about a quarter weaker than at the pole, and with an exponent of 0.2 the equator’s intensity is about four-fifths of the pole’s.

Interferometers have imaged this directly on a handful of nearby rapidly rotating stars — Altair, Vega, Regulus — and seen the bright poles and the dark equatorial band. The images fit an exponent somewhat smaller than von Zeipel’s, near 0.2, and that discrepancy is one of the uncertainties that carries through to everything below.

The part of the light curve that is not symmetric #

The effect on a transit is small next to the transit itself, and the cleanest way to see it is to separate the light curve into its symmetric and antisymmetric parts about the midpoint. The symmetric part is what any transit model describes: depth, duration, ingress shape, limb darkening. The antisymmetric part — half the difference between the flux at each moment and at the mirror moment — is zero for any star that is the same on both sides of the chord, and is the gravity darkening’s signature.

At λ = 0 the chord runs perpendicular to the star’s projected axis, and the star’s brightness pattern is mirror-symmetric about that axis: the planet meets the same brightness on the way in and on the way out, and the antisymmetric part is exactly zero. At any other angle the chord runs from one latitude to another, from brighter towards dimmer or the reverse, and the transit is deeper on the brighter side. The resulting antisymmetric curve is a wiggle: negative where the planet covers dim surface, positive where it covers bright, with a shape set by the path across the brightness map.

The amplitude, a few hundred parts per million on a transit six thousand deep, is a few per cent of the depth. For a bright star observed by a space photometer, folding several transits together reaches that precision easily. The wiggle is also unmistakable in its symmetry: no instrumental drift, stellar variability or detrending error produces a signal that is exactly antisymmetric about the transit midpoint, so it can be found even where its amplitude is comparable to the noise.

One detail had to be handled carefully, and it shows how the measurement is actually made. A flattened star crossed obliquely has its transit midpoint shifted slightly from the moment of closest approach to the star’s centre, because the chord across an ellipse is not centred on the point nearest the ellipse’s centre. That shift alone produces an antisymmetric signal about the geometric midpoint. A fit absorbs it into the transit time, so the figures here measure the asymmetry about the transit’s own best midpoint, and only what no shift of the time can remove is counted.

A transit is a scan along a line #

It helps to think of a transit as an instrument rather than an event. The planet is an opaque disc of known size moving at a steady speed along a straight line across the face of the star, and at each moment the light missing from the system is the brightness of the stellar surface behind it, summed over its area. The depth of a transit is not simply the planet’s area for exactly this reason: the star is brighter at its centre than at its limb, so a planet crossing the middle removes more than its share of the light and one grazing the edge removes less. The light curve of an ordinary transit is a one-dimensional scan of a limb-darkened disc, and its curvature during the flat part of the transit is how limb darkening is measured from photometry.

A gravity-darkened star adds a second pattern to the scan. The limb darkening is symmetric about the centre of the disc; the gravity darkening is symmetric about the star’s projected axis and, unless the star is seen equator-on, not about its equator. The planet’s path crosses both patterns at once, and the light curve is the sum of a symmetric scan of the limb darkening and an asymmetric scan of the latitude-dependent brightness. The separation into symmetric and antisymmetric parts used throughout this essay is simply the separation of those two patterns, and the reason it works is that the planet’s path is a straight line traversed at constant speed. The four contact points fix where the path enters and leaves the disc; everything between them is the brightness along the path.

The inversion has an attractive form. The latitude along the chord is a known function of position once the star’s orientation is given, and the brightness is a known function of latitude once the rotation and the exponent are given. A trial orientation predicts a brightness profile along the chord, and the light curve either agrees with it or does not. The fit is a search over the two orientation angles for the one that makes the scan match the map — which is why the shape of the wiggle, and not only its size, carries the information.

Two angles in one curve #

The antisymmetric curve depends on λ and on the star’s inclination together, and the dependence is not separable into a simple product.

That reversal is the reason the method gives the true obliquity rather than only the projected one. For a star seen equator-on, the brightness map is symmetric about the equator as well as about the axis, and the antisymmetric signal depends on λ in one way. For a star tipped towards the observer, one pole is prominent and the other hidden, the map loses its north–south symmetry, and the signal depends on λ in another. The shape of the wiggle — not only its size — distinguishes those cases, so a fit of the light curve constrains both angles, and through them the true obliquity $\psi$ from $\cos\psi = \sin i_\star\cos\lambda$.

The first system measured this way was the Kepler satellite’s KOI-13, a hot A-type star whose transiting companion produced a light curve so lopsided that its asymmetry was noticed before it was explained. Modelled as a gravity-darkened transit, it gave a true obliquity of about sixty degrees: a planet on an orbit tipped steeply to its star’s equator, measured without a single spectrum. Since then the same analysis has been applied to several hot Jupiters around rapidly rotating A and F stars, where it has found both aligned and strongly misaligned orbits.

The stars on which it works #

The method needs a star whose equator is substantially darker than its poles, and that means a fast rotator.

The square law makes the method a sharp function of rotation. A star turning at a quarter of its critical rate gives a signal near the limit of what folded transits of a bright star reach; one turning at half the critical rate gives a signal four times larger; one turning as slowly as the Sun, at well under one per cent of critical, gives nothing that will ever be measured. In practice this means hot stars. Below about 6,250 kelvin stars have convective envelopes, lose angular momentum to magnetised winds and spin down to slow rotation within their first billion years; above it the envelope is too thin for an effective wind, and stars keep the rapid rotation they were born with.

That boundary — the Kraft break — is the same temperature that divides aligned from misaligned hot Jupiters, and the coincidence matters. The hot stars are the ones whose planets most need an obliquity measurement, and they are the ones on which the line-profile method is hardest. A rapidly rotating star has broad, shallow spectral lines, few of them, and pulsations that distort them, so the radial-velocity anomaly of a transit is hard to extract and the shadow crossing the line profile has to be followed against a broadened, variable background. Photometry has none of those problems. Where the spectroscopic method weakens, the photometric one strengthens, and the two together cover the whole range of stellar temperatures.

A darkening law that is not known #

The weakness of the method is the exponent.

Von Zeipel’s derivation assumed a star in strict radiative equilibrium with no circulation, and his exponent is an idealisation. A star with a convective envelope transports energy differently, and Lucy’s estimate for that case is about a third of von Zeipel’s. Interferometric images of real rapid rotators fit intermediate values near 0.2, and more recent models that solve for the flux on the surface self-consistently find that the effective exponent depends on the rotation rate itself, starting at von Zeipel’s 0.25 for slow rotation and falling well below 0.2 as the rotation approaches critical. None of this is settled to better than a few hundredths.

The signal is nearly proportional to the exponent. An exponent wrong by a quarter is a gravity-darkening contrast wrong by a quarter, and a fit that takes the exponent as known has to explain the wrong contrast with something else.

The fit does not fail; it succeeds with the wrong answer. The asymmetric signal has enough freedom in its shape that a change in the inclination and the projected angle can compensate for a change in the darkening contrast to within a few parts per million — below what even many folded transits of a bright star reach. The obliquity that comes out carries an error of several degrees that no statistical uncertainty quoted from the light curve includes.

The defence is to measure the exponent somewhere else or to measure the geometry two ways. Interferometry fixes the exponent for the few stars bright and near enough to be imaged. Asteroseismology, for the hot stars that pulsate, constrains the inclination. And for stars bright enough for both methods, a line-profile measurement of λ breaks the degeneracy, because it measures the projected angle without involving the brightness map at all — leaving the photometry to fix the inclination.

The star’s own brightness depends on the same angle #

Measuring the inclination of a gravity-darkened star does more than complete a spin–orbit angle. It corrects the star.

A rapidly rotating star seen pole-on presents its hot, bright polar cap to the observer and hides its cool equator on the limb, where it is foreshortened anyway. Seen equator-on, the same star shows its dark equatorial band across the middle of the disc. The two views give different apparent temperatures and different apparent luminosities for one object, and the difference is tens of per cent for a star near critical rotation. A survey that selects stars by brightness prefers the pole-on ones, because the same star is brighter seen that way.

For a planet host this matters twice. The star’s radius, inferred from its temperature and luminosity, sets the planet’s radius through the transit depth; and the star’s luminosity sets the planet’s equilibrium temperature. Both are biased if the star’s inclination is unknown and the star is treated as a sphere of uniform temperature. A gravity-darkened transit fit returns the inclination along with the obliquity, and with it the correction from the apparent to the true stellar luminosity — so the same asymmetry that measures the orbit’s tilt also fixes the size and temperature of the planet in it.

There is a further consistency check in the same measurement. The rotation rate that the darkening contrast implies, together with the star’s radius and the inclination from the fit, predicts the projected equatorial rotation speed. That speed is measured independently from the width of the star’s spectral lines. A fit whose geometry predicts a line width the spectrum does not show has found a wrong solution, and the check is available for every star bright enough to have a spectrum at all.

A chord that moves #

A flattened star is not only darker at its equator; its gravity field departs from a point mass by a quadrupole moment, and an orbit tilted to the star’s equator precesses about the star’s spin axis. It is the same physics by which the Earth’s shape is read off a satellite’s node: the oblate body torques an inclined orbit, and the orbit’s node regresses at a rate set by the quadrupole moment, the orbital radius and the angle between orbit and equator.

For a transiting planet around a rapidly rotating star the precession is fast enough to see. As the orbit’s node moves, the transit chord slides across the stellar disc: the impact parameter changes, the transit’s duration changes with it, and over enough years the planet may stop transiting altogether. KOI-13’s transits have changed in duration and shape over the years they have been followed, as its chord migrates across the star. The rate of that drift depends on the cosine of the true obliquity, so a measured drift is a second, independent constraint on the same angle — one that does not involve the brightness map, and so does not care what the gravity-darkening exponent is.

The two measurements have complementary weaknesses. The asymmetry is measured in one season and depends on the darkening law; the drift takes years and depends on the star’s quadrupole moment, which in turn depends on how centrally concentrated the star is. A system with both measured is overdetermined, and the overdetermination is a test of the stellar model rather than of the planet’s orbit.

What the model leaves out #

The figures use a star whose surface is an exact Roche equipotential with a single rotation rate, a single exponent for the whole surface, and a linear limb darkening. Real rapid rotators may rotate differentially, faster at the equator or the poles, which reshapes the brightness map; their limb darkening varies with latitude because the temperature does; and a hot star’s atmosphere is not grey, so the brightness contrast in the instrument’s band is not the bolometric contrast computed here. A blue band sees a stronger contrast than a red one, which is itself a test: the asymmetry should grow towards shorter wavelengths in the proportion the temperature contrast predicts.

The planet is opaque, round and without an atmosphere, and its orbit is circular and fixed for the duration of the transit. Its slow precession, described above, is the one departure from a fixed orbit that matters, and it acts over years rather than hours.

A symmetry the star breaks for the observer #

There is a general point in this. Most ways of measuring a spin–orbit angle rely on the star’s rotation as a velocity: the line profile is distorted because the hidden part of the disc was moving towards or away from the observer. Gravity darkening relies on the rotation as a shape. The star’s spin does not need to be measured as a speed at all; it is visible in the brightness map it has printed on the surface, and the transit reads that map by covering it one strip at a time.

The two kinds of measurement fail in opposite places. The velocity method needs sharp lines and fails on fast rotators; the shape method needs fast rotation and fails on slow ones. Stars below the Kraft break are measured by their line profiles, stars above it by their light curves, and the boundary between the two methods falls at the same temperature as the boundary between the aligned and the misaligned planets. It is difficult to design a better arrangement deliberately.

Still open: whether the darkening law is the same star to star #

The method’s precision is set by the exponent, and the exponent has been measured directly for only a few stars. If it is the same for every rapidly rotating star of a given temperature and rotation, a calibration on the interferometric targets carries to every transiting system, and gravity-darkened transits become as reliable as line profiles for obliquities above the Kraft break. If it varies — with the depth of a thin convective layer near the surface, with the star’s age, with differential rotation — then each photometric obliquity carries an unknown of its own. The test is a sample of systems measured both ways: line-profile angles for the hot stars where they can be extracted, gravity-darkened light curves for the same stars, and the exponent left free. Where the two methods agree on the projected angle, the light curve’s remaining freedom is a measurement of the exponent star by star — and of whether von Zeipel’s law, a century old, describes the stars that turn the fastest.