A polarisation that a round star cancels

Light scattered by a free electron is polarised. An electron set oscillating by an incoming wave radiates most strongly perpendicular to its motion and not at all along it, so light scattered through a right angle is completely polarised, perpendicular to the plane containing the incoming and outgoing directions. That is the direction a photon count throws away, and the polarisation of scattered light — from the blue sky, from dust round a young star, from the electrons in a hot plasma — is how astronomers recover it.

In the atmosphere of a hot star, hotter than about twenty thousand kelvin, the dominant source of opacity in the outer layers is exactly this: scattering by free electrons, which absorb nothing and redirect everything. Light diffusing outward through such an atmosphere is scattered many times before it escapes, and in 1946 Subrahmanyan Chandrasekhar solved the problem of how the polarisation of that light depends on the direction in which it finally emerges. The answer was clean, surprising, and for seventy years almost impossible to see.

A star polarised only at its edge #

Light leaving the atmosphere straight up — towards an observer looking at the centre of the star’s disc — has been scattered from all horizontal directions about equally, and its polarisation averages to zero. Light leaving at a grazing angle — towards an observer looking at the very edge of the disc — has last been scattered mostly from directions in the plane of the atmosphere, and its polarisation does not average out.

Chandrasekhar’s number for the limb, 11.713 per cent, has become one of the classic results of radiative transfer. The polarisation is parallel to the limb, and it is concentrated in a thin ring at the edge of the disc: a quarter of a per cent halfway out, two per cent at nine tenths of the radius, rising steeply over the last few per cent of the radius to its maximum. That outer ring is also where the disc is faintest, because the same atmosphere is darker towards the limb, at about a third of the central brightness. The most polarised light is the least of it.

The result depends only on the physics of scattering, not on the star. Any atmosphere where electron scattering dominates gives the same curve, which made it a firm prediction. What made it hard to test is the geometry of the whole star.

A pattern that adds up to nothing #

A star seen from interstellar distances is a point. A polarimeter measures the polarisation of the star’s light as a whole, the sum of the light from every part of the disc.

The polarisation pattern is circularly symmetric: at every point it is tangential to the limb, with a degree that depends only on the distance from the centre. A tangential pattern on a circle has no preferred direction. For every patch of the limb polarised north–south there is a patch a quarter of the way round polarised east–west, of equal brightness and equal degree, and in the sum the two cancel. The net polarisation of a round, uniformly bright star is exactly zero, however strongly polarised its edge.

Chandrasekhar saw this at once and pointed to where the cancellation might fail. An eclipsing binary, in which one star passes in front of the other, covers part of the eclipsed star’s disc and removes part of the pattern; whatever is left need not cancel. A star that is not round — distorted by a companion’s tide or flattened by fast rotation — has a pattern that is not symmetric to begin with. Each of these leaves a net polarisation, small because only a thin limb carries much of it, but not zero.

An eclipse uncovers it #

The eclipse is the cleanest case. An opaque companion crossing a star’s disc blocks the light behind it, polarised and unpolarised alike.

While the companion covers the centre of the disc it blocks almost unpolarised light, and the remaining pattern stays symmetric about the line of the eclipse. Near the beginning and end of the eclipse it covers a piece of the polarised limb, and what is left over — the rest of the limb — is no longer balanced. The net polarisation rises to a sixth of a per cent, perpendicular to the direction the companion is moving, and falls again as the companion moves inward. The curve has a characteristic double-humped shape, peaking at ingress and egress, and its size depends on how much of the limb the companion covers relative to the light of the whole disc.

A sixth of a per cent is easily measurable for a bright star — an instrument more polarised than the sky is the main obstacle, and it can be calibrated — but it comes from an idealised star whose atmosphere is pure electron scattering. Real hot stars have other opacity too, from the bound–free and free–free absorption of hydrogen and helium, and the more of the opacity that absorbs rather than scatters, the less polarised the limb. Measurements during eclipses of hot binaries, beginning with claims in the eclipsing binary Algol in the 1980s, have found signals of the expected shape and roughly the expected size, but smaller, as the extra opacity predicts, and at a level where the companion’s own reflected light and the interstellar polarisation have to be removed with care.

How the signal scales with the eclipser #

The size of the signal depends on how much of the polarised ring an occulter can cover relative to the light of the whole disc, which makes it a steep function of the occulter’s size.

Halving the occulter roughly quarters the area it blocks, but it can still lie almost entirely on the polarised ring at ingress, so the peak falls by less than the area and moves outward towards the limb. That is what makes small occulters interesting: the polarisation at ingress is dominated by the outermost few per cent of the radius, the region where the atmosphere’s structure is least well known and where the plane-parallel approximation is weakest. A sequence of eclipses by bodies of different sizes, or a single transit sampled finely in time, maps the polarisation profile across the limb — the curve that the first figure draws from theory — and so tests the theory where it is most uncertain.

A planet’s transit, a million times fainter #

A planet crossing a hot star does the same thing on a smaller scale.

A planet a tenth of the star’s radius, the size of Jupiter against a star like the Sun, blocks a hundredth of the disc. When it covers a piece of the limb, the uncovered pattern is unbalanced by the polarisation of that piece, and the net reaches about a hundred and seventy parts per million for a pure scattering atmosphere — less for a real one, and less again for a cooler star. For a Sun-like star the limb polarisation in the visible is a few parts in a million at most, because the Sun’s continuum opacity is not electron scattering but the absorption of the negative hydrogen ion, which polarises nothing; the Sun’s limb carries the weak, line-dominated polarisation of the second solar spectrum instead. Transit polarimetry is therefore a measurement for hot stars, where it would give a planet’s orbit geometry — which limb it crossed and in which direction — independently of the light curve.

The polarisation signal also carries information the light curve does not. A transit light curve is symmetric whether the planet crosses the star’s equator in one direction or the other; the polarisation rotates with the direction of motion across the disc, so its angle records the orientation of the planet’s path on the sky. That is the same quantity the Rossiter–McLaughlin effect measures from spectral lines, read instead from the continuum.

A star spun out of round #

The most remarkable case is a single star with no companion at all, polarised because it is not round. A star rotating close to the speed at which its equator would fly off is flattened, its equatorial radius larger than its polar by up to half, and — by von Zeipel’s theorem — darker at the equator than at the poles, because the effective gravity there is weaker.

A flattened outline carries more limb along its long sides than its short ends, and the polarisation of the long sides wins. The crude model in the figure, which ignores the change in the atmosphere’s structure across the surface, gives hundreds to more than a thousand parts per million for the flattenings of the fastest rotators. Proper models, which include the gravity darkening and the real mix of scattering and absorption, give much less: the brighter poles push the other way, and the absorption dilutes the scattering. The effect was predicted in 1968 for rapidly rotating hot stars and was detected only in 2017, in the bright B star Regulus, which spins at about ninety-six per cent of its break-up rate. Its polarisation, a few tens of parts per million, varies with wavelength in the way the models predict, and fitting it gave the star’s inclination, rotation rate and gravity darkening, the last of these measured for the first time from polarisation rather than from interferometry.

The measurement needed a polarimeter sensitive to a few parts per million, which became possible only when instruments were designed to measure polarisation faster than the air changes and to calibrate their own contribution to the same level. Seventy years separate the prediction from the detection, and almost all of that time was spent building the instrument.

Why the Sun shows almost none of it #

The Sun is the one star whose limb can be studied point by point, and its continuum is almost unpolarised at the limb — a few parts per million in the visible, rising only in the ultraviolet. The difference from Chandrasekhar’s result is not geometry but opacity. In the Sun’s photosphere the continuum is made by one atom in ten thousand: the negative hydrogen ion, which absorbs and re-emits light thermally and has no preferred direction. Electron scattering contributes a tiny fraction of the opacity at the Sun’s temperature and density, and its polarising effect is diluted accordingly. The limb darkening the Sun shows is that of an absorbing atmosphere, in which the light from the limb comes from higher, cooler layers because a sight line stops at an optical depth of about two-thirds and a grazing line reaches that depth sooner.

In a hot star the balance shifts. Above about twenty thousand kelvin hydrogen is almost completely ionised in the outer atmosphere, the negative ion is gone, and the free electrons it has released dominate the opacity at the depths where the continuum forms. Chandrasekhar’s atmosphere is therefore an idealisation of the hottest stars, approached more closely the hotter the star, and never reached: even in O stars the bound–free absorption of helium and hydrogen contributes enough opacity to cut the limb polarisation to a fraction of 11.7 per cent. The idealisation is still the right starting point, because it fixes the pattern — tangential, concentrated at the limb — that every real atmosphere scales down.

The same physics polarises the light of other hot, electron-rich places: the discs of material round Be stars, whose scattered light is polarised perpendicular to the disc’s plane and reveals its orientation on the sky; the winds of massive stars; and the hot gas round accreting black holes. In each, the net polarisation of an unresolved source measures its departure from spherical symmetry — the falling polarisation and the angle that says why is a version of the same reading, made on scattering by dust rather than by electrons.

Why the cancellation is the whole story #

The pattern this essay describes is a case of a general principle: a measurement of an unresolved source sees only what survives averaging, and symmetry decides what survives. Chandrasekhar’s polarisation is strong at every point on the limb and cancels completely in the whole. Its detection in Regulus did not measure the polarisation of the limb; it measured the asymmetry of the star’s shape, through the small fraction of the limb’s polarisation that the asymmetry leaves uncancelled. The same is true of the gravity darkening a fast rotator shows: a spherical star’s brightness is the same from every direction, and only its deformation lets the inclination show through.

It also means the measurement is most sensitive to what the light curve and the spectrum are least sensitive to. A rotating star seen pole-on looks round and gives no polarisation; seen equator-on it gives the most. A transit crossing the centre of the disc gives almost no polarisation in mid-transit and the most at the limbs, where the light curve’s ingress and egress are hardest to model. Polarimetry fills in the geometry that photometry averages away.

The angle as well as the degree #

Every figure here plots one number, the degree of polarisation along a fixed axis. A polarimeter measures two, the Stokes parameters $Q$ and $U$, and together they give the angle of polarisation as well as its degree. For an eclipse or a transit the angle carries as much information as the degree. When an occulter covers a piece of limb, the uncovered pattern is polarised perpendicular to the radius through the covered piece; as the occulter moves along its chord the covered piece moves round the limb, and the angle of the net polarisation turns with it. A central crossing keeps the angle fixed; a grazing one swings it through a large angle between ingress and egress; and the direction of the swing says which way round the star the occulter is going.

That turns a polarimetric eclipse into a measurement of the occulter’s path across the face of the star, independent of the light curve’s shape and of the star’s limb darkening, which the light curve can only fit with a model. For an eclipsing binary it gives the orientation of the orbit on the sky; for a transiting planet, the angle between the planet’s path and the star’s rotation axis if the star’s own polarisation, from its flattening, gives the latter. In each case the polarisation measures a direction on the sky that photometry cannot, which is the reason astronomers have spent seventy years trying to see an effect that, for a round and undisturbed star, is exactly zero.

An ideal scattering atmosphere and a caricature of a flattened star #

The figures use Chandrasekhar’s results for a semi-infinite, plane-parallel atmosphere of pure electron scattering, tabulated at a dozen values of the emergent angle, and his law of limb darkening for the same atmosphere. Real atmospheres are not pure scatterers — the ratio of scattering to total opacity sets how much of the limb polarisation survives — and are curved, which matters at the extreme limb where the polarisation is greatest. The occultations treat the occulter as a sharp, opaque disc with no light of its own, and the flattened star is a geometric caricature that turns the polarisation without changing the atmosphere. The occultation numbers are therefore upper limits for a given geometry, and the flattened-star numbers are an order-of-magnitude guide that the detailed models revise downward by factors of several.

Still open: whether polarisation can weigh the gravity darkening of many stars #

Regulus was the first and remains one of very few single stars whose rotational polarisation has been measured. Its detection needed a bright star, a nearly edge-on view and a rotation close to break-up. The question is whether the method can be extended to the many rapidly rotating hot stars that are fainter, more inclined, or slower, where the signal falls to a few parts per million. If it can, polarimetry becomes a way to measure the inclinations and gravity darkening of stars too far away to resolve with interferometers, and to test the theory of how rotation redistributes a star’s flux between its poles and equator — a theory that, as interferometric images have shown, does not fit every star. If it cannot, Chandrasekhar’s 11.7 per cent will remain a number seen whole only in the eclipsed and the spun-out few.