A polarisation capped by the light the planet catches

Venus’s clouds were identified as droplets of sulphuric acid by their polarisation, and Venus is brightest as a crescent for reasons its scattering particles also explain. The fraction of Venus’s reflected light that is polarised, and the way it changes as Venus goes from full to crescent, depends on the size and refractive index of the particles doing the scattering, and two clouds that shine alike can be told apart that way when their brightness alone cannot. The same measurement on a planet round another star would be the most direct probe of what its atmosphere is made of — and, because the polarisation angle turns as the planet goes round, of how its orbit is oriented on the sky.

It is also a measurement at the edge of possibility, and the first claim to have made it shows why. In 2008 a team reported polarisation varying with the orbital period of the hot Jupiter HD 189733 b, with an amplitude of about a part in ten thousand of the starlight, and interpreted it as Rayleigh scattering in a bright, hazy atmosphere. Later measurements with more sensitive instruments did not reproduce it, setting upper limits several times lower.

That figure shows that the claimed signal was not just unconfirmed but impossible. It draws the polarised part of the planet’s reflected light, as a fraction of the star’s, across one orbit. A bright atmosphere of Rayleigh-scattering haze, with a geometric albedo of 0.4 and a peak polarisation of a half, gives at most twenty-four parts per million. A ceiling — the largest albedo any scattering atmosphere can have, 0.75, with every scattered photon polarised — gives ninety-one. The claimed amplitude of a hundred lies above the ceiling.

The ceiling #

The arithmetic behind the ceiling is the arithmetic of how a planet is seen at all. A planet of radius $R$ at distance $a$ from its star intercepts a fraction $(R/2a)^2$ of the star’s light. Of what it intercepts it sends some fraction back toward the observer, which at full phase defines its geometric albedo, $A_g$, and at other phases is reduced by a phase function $\Phi(\alpha)$ that depends on the phase angle — the angle at the planet between the star and the observer. The reflected light as a fraction of the starlight is therefore

$$\frac{F_p}{F_\star} = A_g\left(\frac{R}{a}\right)^2\Phi(\alpha),$$

and the polarised part is that times the degree of polarisation $P(\alpha)$, which cannot exceed one.

Every factor but the first is geometry. For HD 189733 b, a planet of 1.14 Jupiter radii orbiting at 0.031 astronomical units, $(R/a)^2$ is $3\times10^{-4}$. A Lambert phase function at quadrature — the planet half lit, ninety degrees from full — is $1/\pi$. The geometric albedo has a hard upper limit: a semi-infinite atmosphere of perfectly white, conservatively scattering molecules reflects at most about three-quarters of the light back toward the star, and anything that absorbs reflects less. So at quadrature the polarised light is at most $0.75\times3\times10^{-4}/\pi\approx72$ parts per million, and allowing the phase function and polarisation their most favourable combination over the orbit raises the peak to about ninety.

A signal larger than that would need the planet to send back more light than falls on it. That is not a statement about atmospheric models; it is energy conservation applied to a disc of known size at a known distance. The ceiling is the first number to compute before any polarimetric detection is believed.

Across the hot Jupiters the ceilings span a factor of ten, and only WASP-12 b — inflated to nearly twice Jupiter’s radius and orbiting at two hundredths of an astronomical unit — exceeds a hundred parts per million. The realistic signals, for atmospheres bright and polarising enough to be worth detecting, are a few to a few tens of parts per million. Most hot Jupiters are in fact dark: their measured geometric albedos in visible light are mostly below 0.1, because alkali metals and their own heat absorb the light. For those the polarised signal is a few parts per million at most.

What a real atmosphere would send #

The degree of polarisation depends on how the light is scattered. Single scattering by molecules much smaller than the wavelength — Rayleigh scattering — polarises light completely at a scattering angle of ninety degrees, which for a planet is quadrature. Multiple scattering scrambles the polarisation: a photon scattered several times before leaving the atmosphere carries a random mixture of polarisation states. A bright, optically thick Rayleigh atmosphere has a high albedo precisely because photons scatter many times without being absorbed, and the same multiple scattering lowers its peak polarisation to about a third or a half. Brightness and polarisation trade against each other.

That trade is why the solid curve in the first figure, with an albedo of 0.4 and a peak polarisation of 0.5, sits at a quarter of the ceiling. It is a generous model — brighter than most hot Jupiters are measured to be — and it gives twenty-four parts per million at the peaks, which fall a little before and after quadrature because the phase function rises toward full phase while the polarisation falls.

The orbit in the polarisation angle #

The polarisation carries a second piece of information, independent of the atmosphere, and in some ways more useful.

Light scattered once is polarised perpendicular to the plane of scattering — the plane containing the star, the planet and the observer. Seen from the Earth, that plane contains the line from the star to the planet on the sky, so the polarisation direction is perpendicular to the star–planet line. As the planet orbits, the line turns, and the polarisation direction turns with it. The two linear Stokes parameters, $q$ and $u$, which describe the polarisation’s strength along two pairs of axes, therefore cycle twice per orbit, with shapes set by how the orbit is tilted and oriented on the sky.

For a nearly edge-on orbit the star–planet line on the sky barely turns — the planet moves back and forth along a line — so the polarisation angle stays nearly fixed, and the signal varies mainly in strength. For an orbit tilted toward face-on the line sweeps round, the polarisation angle turns through a full half-circle each orbit, and $q$ and $u$ trade off. The shapes of the two curves together give the inclination and the orientation of the line of nodes, which is the same information an astrometric orbit gives from the star’s wobble on the sky, obtained from the planet’s light instead.

The polarisation angle is defined only to within half a turn — a polarisation along one direction is indistinguishable from one along the opposite direction — so the planet at a given point of its orbit and the planet diametrically opposite produce the same angle. The measurement fixes the orientation of the orbit’s line of nodes only up to which end is which, the same ambiguity an astrometric orbit has, and it is broken the same way: a radial-velocity curve says which node the planet passes while receding. Combined with the velocities, the polarisation gives the full orientation; alone, it gives everything but that one bit.

The sense of the planet’s motion on the sky — clockwise or anticlockwise — is not ambiguous, because $q$ and $u$ follow each other in a definite order as the angle turns, and reversing the motion reverses the order. That piece of geometry, which velocities cannot give at all, is free.

That is the measurement’s strongest case. Most planets found by radial velocities do not transit, and their orbital inclinations, and therefore their masses, are unknown; the velocity curve gives only the mass times the sine of the inclination. A polarimetric detection of such a planet would give the inclination directly, and a nearly face-on orbit — the one hardest to find by velocity, because its signal is diluted by the small sine — gives the largest polarisation-angle swing. The method works best exactly where the others are weakest.

Photons are cheap #

If the signal is tens of parts per million and the star is bright, photon noise is not the obstacle.

A bright star delivers billions of photons a second to a large telescope, and a few times $10^{10}$ of them suffice to measure a fraction to a part in a hundred thousand. For the nearby hot-Jupiter hosts, the photon budget is a matter of minutes to hours. The difficulty is everything else.

A telescope polarises the light passing through it. Every oblique reflection off a mirror, every coated surface, every lens at an angle introduces a few parts in ten thousand to a few per cent of instrumental polarisation, which changes as the telescope tracks and the mirrors’ angles change. The atmosphere’s flicker turns a polarimeter’s sequential measurements into false polarisation unless the instrument modulates faster than the air changes. The star’s light already carries polarisation from interstellar dust along the line of sight, constant but unknown, and from its own surface — starspots and magnetic regions polarise the light slightly and rotate with the star, at a period that may be commensurate with the planet’s orbit.

The star’s own contribution is the hardest to separate, because it can mimic the planet. A spotted, magnetically active star like HD 189733, which is a young K dwarf with large starspots, polarises its light by up to a few parts per million through the asymmetry the spots introduce on its disc and through magnetic effects in its spectral lines, and that polarisation varies with the star’s rotation period of about twelve days. The planet’s orbit is 2.2 days. The two periods are distinct, but a measurement that samples a few orbits over a fortnight samples about one rotation too, and the two signals have to be separated by their periods alone.

Detecting a planet at twenty parts per million means calibrating all of these to about a part in a million, ten times below the signal, and keeping them calibrated over the many nights needed to sample an orbit. The instruments built for it — designed with symmetric optics, fast modulation and beam exchange — reach that level for the brightest stars on good nights. The claimed 2008 signal, at ten times what a bright atmosphere could produce, is consistent with an uncorrected instrumental drift of a few parts in a hundred thousand: small by the standards of most polarimeters, enormous by the standard this measurement requires.

The same light, without the star #

The ceiling is so low because the polarised light is compared with the star’s total light, which is unpolarised and a hundred thousand times brighter. Take the star away and the comparison changes completely. A planet imaged directly, with the starlight suppressed by a coronagraph and subtracted using the star’s own images, is measured against its own light rather than its star’s, and its polarisation is then tens of per cent of the signal rather than tens of parts per million of the total.

That is why polarimetry has become a tool of direct imaging rather than an alternative to it. The residual starlight that defeats a coronagraph — speckles scattered by imperfections in the optics — is mostly unpolarised, while light scattered by a planet’s atmosphere or a debris disc’s grains is strongly polarised near ninety degrees of scattering angle. Imaging in two orthogonal polarisations and subtracting one from the other removes most of the speckles and keeps most of the planet. The instruments that image debris discs and young planets around nearby stars routinely do this, and the polarisation of a directly imaged object’s light is measured to a few per cent.

The cost is that direct imaging reaches planets far from their stars, where they are faint in reflected light; the hot Jupiters whose polarisation the 2008 claim concerned are a few milliarcseconds from their stars and cannot be resolved by any telescope. For them the comparison with the star’s total light, and the ceiling it imposes, cannot be avoided. The two approaches sample different planets: polarimetry of the combined light reaches the close-in, bright ones at parts per million; polarimetric imaging reaches the distant ones at per cent levels, once they can be separated from a star nine orders of magnitude brighter.

The planet whose polarisation is known #

There is one planet whose disc-integrated polarisation, seen from far away, has been measured over its full range of phases: the Earth. Its light reflected from the Moon’s dark side, the earthshine, carries the Earth’s polarisation, and measurements of earthshine through a polarimeter have shown the degree of polarisation rising to a few tens of per cent near quadrature, falling toward full phase, and changing with the fraction of the Earth’s visible face covered by ocean and cloud. Rayleigh scattering by the Earth’s air dominates the blue; the ocean’s glint and the clouds’ droplets shape the red.

That record is the calibration for every model of a terrestrial exoplanet’s polarisation, and it has a sobering scale. The Earth seen from ten parsecs is about ten billion times fainter than the Sun, so its polarised light is a few parts in a hundred billion of the total. No combined-light polarimeter will ever measure that; only direct imaging, with the Sun’s light suppressed by ten orders of magnitude, could. The hot Jupiters are the only planets for which the ceiling leaves room above the instrumental floor, and even there the room is a factor of a few.

A record of claims at the ceiling #

The episode is a useful pattern. A claimed exoplanet signal can be checked against a ceiling computed from quantities already known — the planet’s radius from its transit, its orbit from its period and its star’s mass, the physical limits on albedo and polarisation — before any atmospheric model is consulted. When a claim exceeds the ceiling, something other than the planet is producing it.

The same check applies to reflected-light phase curves and to secondary eclipses: a planet cannot reflect more than it intercepts, and the measured eclipse depths of hot Jupiters in visible light, mostly tens of parts per million and below, imply low albedos consistent with the ceiling and with the polarimetric upper limits. Polarisation adds the factor $P \le 1$, which lowers the ceiling further and makes polarimetric claims especially easy to test against it.

What the figures leave out #

The phase function is Lambert’s and the polarisation Rayleigh’s single-scattering shape, scaled. A real atmosphere has a phase function peaked by forward or backward scattering from haze or cloud particles, and a polarisation curve that depends on their size; for cloud droplets the polarisation near quadrature can even change sign. The ceiling holds whatever the shapes, because both factors are bounded.

The planet is a uniform disc. Hot Jupiters have day–night contrasts and possibly inhomogeneous clouds, which make the reflected light and its polarisation asymmetric about the orbit. That shifts the peaks; it does not raise them above the ceiling.

Non-transiting planets’ radii are assumed. For τ Boötis b, HD 179949 b and 51 Pegasi b the ceiling scales as the square of the assumed radius; a planet inflated like WASP-12 b would have a ceiling three times larger.

The photon budget is for a single broad visible band. Narrower bands, as needed to separate Rayleigh scattering’s steep colour dependence from cloud scattering, cost photons in proportion; the conclusion that photons are not the limit holds to bands a few hundred ångströms wide for these stars.

Still open: a detection below the ceiling #

Polarimetric measurements of hot Jupiters since 2008 have produced upper limits of a few to a few tens of parts per million and, for HD 189733 b in blue light, a reported signal of a few tens of parts per million that is below its ceiling and consistent with a bright Rayleigh-scattering haze — the kind of atmosphere its transmission spectrum, which slopes steeply toward the blue, also suggests. Whether that signal is real, and whether it shows the twice-per-orbit rotation of the polarisation angle that would tie it unambiguously to the planet rather than to the star or the instrument, is not settled.

The next generation of polarimeters, on large telescopes with optics designed for symmetry and with calibration at the part-per-million level, is aimed at the brightest hot-Jupiter hosts first and at non-transiting planets after. For the latter the prize is not an atmosphere but an orbit: the inclination and orientation on the sky, and therefore the true mass, of a planet known until then only as a minimum mass from its star’s velocity. Every one of those measurements will be checked first against a ceiling the planet’s size and distance set, and the check will cost one line of arithmetic.