Two clouds that shine alike until the crescent

Venus is brightest as a crescent, weeks from inferior conjunction, and the moment of its greatest brilliancy falls closer to conjunction than any perfectly matte planet would allow. The difference is its clouds throwing light forward. That reading needed one thing the geometry supplied for free: Venus is an inner planet, so the Earth sees it from inside its orbit at every phase angle, from nearly full to a crescent a few degrees wide, and the part of its phase curve where forward scattering shows is in plain view several times a decade.

A planet round another star offers no such view. Its light arrives as one unresolved point — for a temperate planet, a hundred million or more times fainter than the star beside it — and the range of phase angles over which that light can be measured is set by the planet’s orbit and by the instrument, not by the observer’s choice. Within that range, a phase curve measures a product: how much light the cloud tops return, and how that light is distributed in angle. The question this essay asks is whether the two factors can be separated from the curve alone.

Over the middle of an orbit they cannot. A grey cloud that scatters light evenly in all directions and a whiter one that throws most of it forward return the same light, to one per cent, at every phase angle from 30° to 110°. They part only towards the crescent, where the forward-scattering cloud becomes six times brighter than its twin — and where a planet is faintest, closest to its star in projection, and hardest to observe. What separates them in the middle of the orbit is not the brightness at all but its polarisation.

Two numbers that describe a cloud #

A cloud deck seen from above is, for reflected light, two numbers to a first approximation. The single-scattering albedo, written $\varpi$, is the chance that a photon striking a cloud particle is scattered rather than absorbed; a value of 0.99 means one photon in a hundred is lost at each scattering. The asymmetry parameter, $g$, is the average cosine of the angle through which a scattered photon is deflected: zero for a particle that scatters evenly in all directions, approaching one for a particle that sends nearly everything straight on. Small particles — molecules, fine haze — scatter almost symmetrically. Droplets comparable with the wavelength of light scatter strongly forward, with $g$ around 0.7.

The light a planet returns towards an observer at phase angle $\alpha$ — the angle at the planet between the star and the observer, zero at full phase and 180° at new — depends on both. The figures here use a model with both kinds of light in it: the light scattered once, computed exactly with a Henyey–Greenstein phase function of asymmetry $g$, and the light scattered many times, computed for the evenly scattering cloud that is equivalent to it in the sense described below. The disc of the planet is integrated point by point, and the result is the planet’s geometric albedo times its phase function — the quantity a phase curve measures, once the star’s brightness, the planet’s radius and its distance from the star are divided out.

A grey cloud and a white one #

The evenly scattering cloud absorbs one photon in a hundred at each scattering. The forward-scattering cloud that matches it absorbs one in five thousand. That is a large physical difference — a factor of fifty in how much each particle absorbs — and over the whole middle of the orbit it does not show.

The reason is a balance between two effects that run in opposite directions. A cloud that scatters forward sends less light back towards a star-facing observer at each scattering, so at any phase angle short of the crescent it looks darker for a given albedo. A cloud whose particles absorb less lets light survive more scatterings, and after many scatterings the direction of the first one has been forgotten, so it looks brighter. Raise $g$ and raise $\varpi$ together in the right proportion and the two effects cancel. Over the range of angles where most of the returned light has been scattered many times, the curve the two clouds trace is the same curve.

The ratio shows where the information lives. It is flat across the whole window, and towards full phase it stays flat as well, because matching the level across the window fixes it there too. Only when the observer moves round behind the planet does the forward lobe of the scattering point at them, and then the forward-scattering cloud pulls ahead fast: by a factor of two and a half at 150°, six at 165°, more beyond. The information that separates albedo from asymmetry is concentrated at the phase angles where the planet shows a crescent.

A valley rather than a point #

Two models agreeing is a statement about two points. What a measurement constrains is better seen by asking, for every pair of $g$ and $\varpi$, how well that cloud reproduces a measured curve. With the curve taken every ten degrees across the window, each point uncertain by five per cent, and the planet’s radius known — as it is when the planet also transits its star — the answer is a curved valley across the whole diagram. Clouds anywhere along it fit nearly as well as the true one, and within the tightest contour the fit is acceptable in two separate places: the true evenly scattering cloud, and a family of forward-scattering clouds near $g = 0.65$ that are almost perfectly white.

The valley has a name. Hendrik van de Hulst showed, in his work on multiple scattering, that the light returned by a thick, many-times-scattering medium depends to a good approximation on one combination of the two numbers,

$$\varpi^\ast = \frac{\varpi\,(1-g)}{1-\varpi g},$$

so that a forward-scattering medium behaves like an evenly scattering one with albedo $\varpi^\ast$. The dashed curve in the figure holds that combination fixed. The valley runs the same way, more steeply: a forward-scattering cloud also sends less of its once-scattered light to these angles, and has to be whiter still to make up for it. The degeneracy is not an accident of one model but a property of multiple scattering, which erases most of what a single scattering knew about direction.

The same structure appears elsewhere in measuring planets, and in the same shape. A transmission spectrum cannot fix a planet’s radius and its atmosphere’s composition separately, because a missing reference level trades against an abundance; here the trade is between how much a particle absorbs and where it sends what it does not. In both cases the measured quantity is a product, and the product is constant along a line.

How far towards the crescent #

If the separation lies at the crescent, the obvious question is how much crescent is needed. The answer is less than the ratio figure might suggest, because the measurement errors are fractional and a ratio of two only has to be measured to a few per cent to be unmistakable. A curve that stops at 110° allows any asymmetry from zero to 0.7. One that reaches 120° allows almost none but the truth, and one that reaches 150° pins it. The information arrives over a span of a few tens of degrees, just past where the curves begin to diverge.

Whether those angles can be reached depends on how the planet is seen. For a planet on a circular orbit of radius $a$, the projected distance between the planet and its star is

$$\rho = a \sin\alpha,$$

because the phase angle is the angle at the planet between the star and the observer. At full phase and at new, the planet is directly behind or in front of its star and the separation is zero; it is greatest at $\alpha = 90°$. A coronagraph — the instrument that blocks the star to make the planet visible, against a contrast of nine orders of magnitude — has an inner working angle inside which nothing can be seen. If that angle is half the planet’s orbital radius on the sky, then every phase angle with $\sin\alpha < 1/2$ is lost: everything below 30° and everything above 150°.

The coronagraph hides both ends of the orbit, symmetrically, and the ends are where the information is. The crescent, where forward scattering shows, is lost for the same reason as full phase, and the orbit’s inclination narrows the range further: a planet whose orbit is seen tilted $i$ from face-on shows phase angles only between $90° - i$ and $90° + i$, so an orbit within twenty degrees of face-on never leaves 70°–110°. A transiting planet passes through every phase angle, but at the crescent it is a hundred or more times fainter than at full phase, its own heat dominates its night side for the hottest planets, and the reflected light near transit is a small signal measured against a much larger one.

A measurement with no albedo in it #

There is another property of the reflected light, and it does not depend on how much of it there is. A phase is geometry rather than shadow, and so, in large part, is polarisation: light scattered by a particle is partly linearly polarised, by an amount that depends on the scattering angle, the particle’s size compared with the wavelength, and its refractive index. The degree of polarisation is the polarised flux divided by the total flux. It is a ratio of two measurements of the same planet at the same moment, so the planet’s radius cancels, the star’s brightness cancels, and the albedo — the very number the phase curve could not separate — cancels with them.

The polarisation curves are computed exactly for spheres, with Mie theory, rather than from a parameterised phase function. Particles much smaller than the wavelength polarise the way molecules do: almost completely at 90°, where the scattering is at right angles, and less towards either end. That is why the sky is polarised most strongly at right angles to the Sun. Droplets comparable with the wavelength behave differently. They polarise weakly over the middle of the range, mostly in the opposite sense, and produce sharp features near full phase — the rainbow, the same optics as the bow in rain, displaced and blurred by the droplets’ size.

This is what the two islands of the photometric valley look like in polarisation. A cloud near $g = 0$ is made of particles small enough to scatter evenly, and particles that small polarise strongly in the middle of the window. A cloud near $g = 0.65$ is made of droplets, and droplets barely polarise there at all. Ninety per cent against a few per cent in the opposite sense, at 90°, is a difference no photometric error can hide, and it is available in the window where the phase curve was blind.

What the window still cannot say #

Polarisation in the window separates a haze from a droplet deck. It does much less to separate one droplet deck from another, and the reason shows in the asymmetry parameter itself. Once droplets are larger than a few tenths of a micrometre, $g$ hardly changes with size: droplets of 0.6, 1.05 and 2.5 μm all scatter forward by about the same amount. Their polarisation in the window is also similar, within a few points. Where they differ is at small phase angles, in the rainbow — seventy points apart at 12° for two sizes whose asymmetry parameters agree to 0.02.

So the polarisation splits the problem in two. The middle of the orbit says whether the cloud is made of small particles or droplets. The size and composition of the droplets are written near full phase, at the other end of the orbit from the crescent, and the coronagraph hides that end too.

A second way to brighten a crescent #

Forward-scattering cloud is not the only thing that makes a crescent bright. An ocean does it too, by a different route. Water is smooth, and at grazing angles it reflects sunlight specularly, the way a lake shows a dazzling streak at sunset. Seen from far away, a planet with oceans carries a glint — a bright spot where the sea is tilted to mirror the star towards the observer — and the glint strengthens steeply as the planet narrows to a crescent, because a mirror’s reflectivity rises towards grazing incidence. Calculations of the Earth seen as a distant planet show its crescent brightened by the glint well beyond what its clouds and land alone would give.

So a crescent brighter than an evenly scattering planet allows has at least two explanations, forward-scattering cloud and specular ocean, and the phase curve alone cannot say which. Polarisation can. Light reflected from water is polarised most strongly near Brewster’s angle, an angle of incidence of about 53° for water, which corresponds to a phase angle near 106°. An ocean’s glint is strongly polarised in the middle of the window, where a deck of droplets is barely polarised at all. The same measurement that tells a haze from a droplet cloud tells a cloud from a sea, and both are made at the phase angles a coronagraph can see.

That is a large part of why polarimetry is argued for in the search for planets like the Earth. An ocean is the feature such a search most wants to find, and a bright crescent is its most visible signature in brightness — and a bright crescent is exactly what a forward-scattering cloud deck would also produce, at the end of the orbit the instrument cannot reach.

How Venus’s droplets were identified #

For Venus all of this was measured, because an inner planet shows the Earth every phase angle in turn. Bernard Lyot measured the polarisation of Venus’s light across the whole range of phase angles in the 1920s, from the ground, with no knowledge of what the clouds were made of. In 1974 James Hansen and Joop Hovenier fitted those curves, and newer ones at several wavelengths, with full multiple-scattering calculations for clouds of spherical droplets. The fit was sharp enough to fix the droplets’ effective radius at 1.05 μm, their spread in size, and their refractive index at 1.44 — and the index, with its change across the spectrum, matched concentrated sulfuric acid and essentially nothing else. Sulfuric acid had been proposed a year or two earlier from the clouds’ refractive index; the polarisation settled it. The probes that had already descended through the clouds by then had not identified what they were made of, and the composition was established from light collected on the Earth.

The decisive features were the ones the window would have missed. The rainbow near 15°–20° phase angle fixed the refractive index and the size; the behaviour near the crescent fixed the forward scattering that moves Venus’s greatest brilliancy towards inferior conjunction. Venus could be read because its geometry gives the full curve. A planet round another star will usually give a slice through its middle.

What the curves assume #

The photometric model is a single thick cloud. A real atmosphere has gas above the cloud, which adds molecular scattering and absorption; cloud particles of mixed sizes and shapes; and patchy cover, which can shift a phase curve’s peak away from full phase, as it does for some hot Jupiters. Each of these adds parameters and, usually, more ways to trade one against another rather than fewer.

The multiple scattering is approximate. The once-scattered light is exact; the light scattered many times is computed for the evenly scattering cloud that van de Hulst’s relation makes equivalent, using an approximation to Chandrasekhar’s H-function good to a per cent or so. A full calculation would move the valley’s islands slightly and would not remove the valley, since the valley is the similarity relation.

The polarisation curves are single scattering. Light scattered many times is only weakly polarised, so in a real cloud the degree of polarisation is diluted several-fold. The features keep their positions, which is why Hansen and Hovenier could read Venus’s droplets from curves whose amplitudes were a few per cent rather than tens.

And the measurements are assumed, not simulated. Five per cent errors every ten degrees is optimistic for a planet a hundred million times fainter than its star, and measuring a degree of polarisation of a few per cent requires counting many times more photons than measuring the brightness. Every telescope also polarises the light passing through it, by more than most astronomical sources do, and that has to be calibrated away before a planet’s own polarisation can be read.

Still open: whether an exoplanet’s polarisation can be measured #

Reflected-light phase curves of exoplanets have been measured for hot Jupiters, the planets bright enough in reflected light to show one, mostly by space photometry over the whole orbit, with the moment the planet passes behind its star fixing the level at full phase. Their polarisation has not been measured convincingly. A claimed detection from a hot Jupiter in 2008 was not reproduced by later, more sensitive measurements, which set upper limits of several parts in a hundred thousand of the total light or found only marginal signals — consistent with what the planet’s reflected light, diluted by the star’s, could produce. The next generation of space coronagraphs is being designed to image temperate planets in reflected light, and whether they should carry polarimeters is among the questions their design has to settle. The case for one rests on the argument here: within the window a coronagraph allows, brightness alone leaves a cloud’s albedo and asymmetry tied together, and the polarisation is the one measurement in that window that can say whether the cloud is a haze or a deck of droplets, without needing to know how big the planet is.