Brightest as a crescent, and not as a disc

An inner planet’s cycle as seen from the Earth runs between two configurations that pull its brightness in opposite directions. At inferior conjunction it is between the Earth and the Sun: as close as it ever comes, and turned almost entirely dark side towards the observer. At superior conjunction it is beyond the Sun: fully lit, and as far away as it ever gets. Galileo’s phases of Venus were decisive precisely because the lit fraction and the apparent size varied together in the proportions a Sun-centred arrangement demands.

Brightness is the product of the two. The lit fraction rises from nothing to one as the planet moves from inferior to superior conjunction; the inverse square of the distance falls, for Venus, by a factor of forty. Where the product peaks is not obvious, and for Venus the answer is well known from observation: the planet is brightest about 36 days before and after inferior conjunction, at an elongation of about 39° from the Sun, when roughly a quarter of its disc is lit. At that moment it can be seen in daylight, when it is measured against a sky thousands of times brighter than the night sky it is usually seen in.

That observation contains two separate things. One is geometry, which says that a peak between the extremes exists at all. The other is a property of the planet’s surface, which says where exactly the peak falls — and the geometry alone does not put it at 39°.

The contest, written down #

Put the Earth on a circle of radius one and the planet on a circle of radius $a$, and measure the planet’s position by ψ, the angle at the Sun from inferior conjunction. The distance from the Earth is

$$\Delta^2 = 1 + a^2 - 2a\cos\psi$$

and the phase angle α — the angle at the planet between the Sun and the Earth — satisfies $\cos\alpha = (a - \cos\psi)/\Delta$. At inferior conjunction α is 180°, the planet is new; at superior conjunction α is zero, the planet is full. The lit fraction of the disc is $(1 + \cos\alpha)/2$.

How bright a partly lit disc is depends on more than its lit fraction, because a surface lit obliquely sends less light towards the observer than one lit face-on. For a perfectly matte surface — one that scatters equally in every direction from every point, a Lambert surface — the brightness of a sphere at phase angle α is proportional to

$$\Phi(\alpha) = \frac{\sin\alpha + (\pi - \alpha)\cos\alpha}{\pi}$$

which is one at full, $1/\pi$ at half, and falls to zero as the crescent thins. The planet’s flux at the Earth goes as $\Phi(\alpha)/(a\Delta)^2$, and everything in the figures follows.

For Venus the contest has a winner in the middle. The distance falls fast enough near inferior conjunction that approaching the Earth more than compensates for losing the lit face, down to a phase of a little over a third; beyond that the crescent thins faster than the distance shrinks. The model’s peak is at 44.6° elongation. Venus’s real peak is at 39°.

Mercury loses the contest #

Mercury behaves the other way. Its orbit is so small that its distance from the Earth varies only between 0.61 and 1.39 AU, a factor of 2.3, and a factor of 2.3 squared is 5.2 — far less than the fall in the phase function as the planet goes from full to crescent. The disc wins everywhere. Mercury is brightest at full phase, close to superior conjunction, where it is also closest to the Sun in the sky and hardest to see; observed, it reaches its brightest magnitude near superior conjunction and is a faint crescent near inferior conjunction.

That is why Mercury is almost always seen as a gibbous or half-lit object low in twilight, and Venus as a brilliant crescent high in a dark sky. The difference is not the planets’ surfaces or sizes. It is the ratio of their orbits to the Earth’s.

Where the winner appears #

The transition is abrupt. At an orbital radius a little below 0.464 AU the brightness curve through the cycle has a single maximum at superior conjunction; as the radius grows, a second hump appears in the curve, grows, and at 0.464 AU overtakes the first. The elongation of greatest brightness therefore jumps from zero to twenty degrees or more rather than growing smoothly — the signature of a maximum that changes from one branch to another, the same kind of discontinuity as a boiling point.

Just above the transition the brightest moment is not a crescent at all. A planet on a 0.5 AU orbit is brightest 51 days from inferior conjunction with 73 per cent of its disc lit — on the far side of greatest elongation, heading towards superior conjunction. Its brightness through most of the cycle barely changes: only four hundredths of a magnitude separate the gibbous peak from full phase. Such a planet would have no conspicuous “greatest brilliancy” at all.

As the orbit grows further, the brightness peak moves inward, towards inferior conjunction, and becomes a thinner and thinner crescent, because the distance near inferior conjunction — $1 - a$ — shrinks towards nothing.

A magnitude and a half is the whole range #

The magnitude scale compresses all of this into small numbers, and it is worth reading them in the scale’s own terms. A magnitude runs backwards and multiplies: five magnitudes are a factor of a hundred in flux, one magnitude a factor of 2.512. On the Lambert model Venus at superior conjunction is 0.82 magnitudes fainter than at its brightest, a factor of 2.1 in flux; the real planet ranges from about −3.9 near superior conjunction to about −4.6 at greatest brilliancy, a factor of about 1.9.

So the whole contest between a factor-of-forty change in the inverse square of the distance and a change from a full disc to a thin crescent resolves into a brightness that varies by only a factor of two over most of the cycle. The two effects nearly cancel, which is exactly why where they fail to cancel is sensitive to the details of the surface. A product of two strongly varying quantities that nearly cancel is a sensitive instrument for anything that shifts the balance, and the scattering law is such a thing.

Mercury’s range is larger in magnitudes, from about −2.5 near superior conjunction to fainter than +5 as a thin crescent, because there the two factors do not come close to cancelling: the phase function falls much further than the distance factor rises. The contrast between the two inner planets is the contrast between a balanced product and an unbalanced one.

The Earth, seen as an inner planet #

The same geometry applies from any planet looking inward, and the answer depends only on the ratio of the two orbits. From Mars, the Earth is an inner planet on an orbit 0.656 times the size of the observer’s. That is well above the transition, so a Lambert Earth seen from Mars is brightest as a partly lit disc at about 41° from the Sun, with 45 per cent of it lit — a gibbous-to-half phase rather than a thin crescent, because the ratio of 0.656 is smaller than Venus’s 0.723. From Jupiter, the Earth’s orbit is 0.19 of the observer’s: far below the transition, and the Earth seen from Jupiter would be brightest full, beside the Sun, and never conspicuous at all.

The loop a planet traces depends only on the ratio of the two orbital radii, and so does this. A civilisation on Mars would have a brilliant blue evening star with a pronounced greatest brilliancy; one on Jupiter would have a faint, Mercury-like companion to the Sun. The Earth’s actual surface, with its oceans’ glint and its clouds, is neither matte nor uniform, and its real phase curve from Mars would depend on which hemisphere and which weather were turned towards the observer.

A peak that recurs on an eight-year calendar #

Because greatest brilliancy is tied to the configuration of Venus relative to the Sun and the Earth, it recurs with the synodic period of 584 days — and so it inherits the near-commensurability that makes Venus’s conjunctions trace a pentagram. Five synodic periods are eight years less two and a half days, so every eight years Venus reaches greatest brilliancy on almost the same calendar date, in almost the same part of the sky, as the evening or morning star.

That regularity is why the planet’s appearances could be tabulated long before its phases could be seen. Venus’s first and last visibility as the morning and evening star, its greatest elongations and its brightest nights all come back on the same eight-year schedule, drifting by two or three days per cycle. A calendar that tracks Venus has the same structure as a calendar that tracks the Sun: a near-fit between two periods, corrected when the drift accumulates.

What Venus’s clouds do #

The model puts Venus’s greatest brilliancy at 44.6° elongation and 38 per cent lit. The observed greatest brilliancy is at about 39° and about a quarter lit. The difference is several weeks in time and it is in one direction: the real planet stays bright to a thinner crescent than a matte sphere would.

A peak further towards inferior conjunction needs a phase function that falls off more slowly at large phase angles than the Lambert law does — a surface that sends proportionally more light forward, towards an observer who sees it nearly back-lit. Venus is covered completely by clouds of sulfuric acid droplets about a micrometre across, and droplets of that size scatter light strongly forward. Measurements of Venus’s brightness across the full range of phase angles, including observations made within a few degrees of inferior conjunction, confirm that its phase curve departs from the Lambert form in exactly that sense, and rises again at the largest phase angles as forward scattering takes over.

So the position of greatest brilliancy is a measurement of Venus’s clouds. It was available to naked-eye observers for millennia, sitting in the record as a regularity of the planet’s appearances, and what it says about the scattering properties of the cloud particles could only be read once the geometry was separated out. The same separation is how the size and composition of the droplets were first constrained from the ground: the degree of polarisation of Venus’s light as a function of phase angle picked out spherical droplets of a specific size and refractive index, and sulfuric acid was identified as the only plausible match before any spacecraft arrived.

The Moon, which is not matte either #

The Moon makes a useful contrast, because its surface departs from the Lambert law in the opposite direction. A Lambert sphere at half phase has a third of its full-phase brightness. A quarter moon has less than a tenth of the full moon’s.

A phase is a matter of geometry, not of shadow, and the geometry says the quarter moon is half lit. What the geometry does not say is how that half is lit. The lunar surface is rough on every scale from craters to grains of dust, and at any phase other than full, each rough element shades part of its neighbours. At exactly full phase the shadows vanish behind the objects casting them, and the surface brightens sharply — the opposition surge — and fine-grained regolith adds a coherent backscattering enhancement on top. The Moon is a strongly backscattering surface; Venus’s clouds are forward-scattering. A matte sphere is neither, and both real objects are measured against it.

If the Moon orbited the Sun as an inner planet, its backscattering would move its greatest brilliancy the other way, towards superior conjunction, and an inner Moon at Venus’s distance might have no crescent peak at all.

What the figures leave out #

The orbits are circles. Venus’s eccentricity is small enough not to matter, but Mercury’s is 0.21, and its distance from the Earth at a given configuration can differ by twenty per cent between one apparition and the next — enough to change how bright a given phase is and, near the transition, whether a crescent can briefly compete with the full disc. The figures also treat the planets as seen from the Earth’s centre in a sky with no Sun: near either conjunction the planet is lost in daylight, and the curves there describe light nobody can measure.

The model also ends at the limb. Very close to inferior conjunction, when the lit crescent of a matte sphere would shrink to nothing, Venus’s thick atmosphere refracts and scatters sunlight round the planet, so the cusps of the crescent extend beyond a semicircle and, within a few degrees of conjunction, join into a thin complete ring. The effect was seen during the transit of 1761, when a luminous arc outlined the part of Venus’s disc still off the Sun, and it was among the first evidence that Venus has an atmosphere at all. A Lambert sphere has no atmosphere and no ring; its phase function goes to zero at α = 180°, where Venus’s does not.

And a magnitude is a disc-integrated brightness. When Venus is at greatest brilliancy its crescent is about 40 arcseconds long, resolved in binoculars, and the eye’s impression of brilliance depends on surface brightness as well as on total flux. The contest drawn here is the total; a sharper crescent concentrated into fewer square arcseconds can look more striking than its magnitude alone suggests.

The same contest round other stars #

For a planet orbiting another star, observed from far away, the distance factor disappears: every point of the orbit is at the same distance from the observer, to one part in millions. What remains is the phase function alone. A planet’s reflected light rises and falls through its orbit as its lit face turns towards and away, and a light curve of that rise and fall — a phase curve — is a direct measurement of Φ(α), with nothing to separate out.

That is how the reflectivity and the scattering of hot exoplanet atmospheres are measured: from a phase curve that is Venus’s cycle with the geometry removed, and from the moment the planet disappears behind its star, which fixes its brightness at full phase. A phase curve that peaks before or after the geometric full phase measures how the atmosphere scatters and where it is brightest, exactly as Venus’s greatest brilliancy measures its clouds.

The maximum of a product sits where its factors’ slopes balance #

A brightness is a product, and the position of a product’s maximum depends on how steeply each factor changes, not on either factor’s extremes. Two factors with extremes at opposite ends of a cycle — lit fraction and distance — can produce a maximum at one end, at the other, or between, and which one is set by their relative steepness. For an inner planet that steepness is fixed by the ratio of the orbits and by the scattering law, so the location of greatest brilliancy measures a combination of geometry and surface.

The same structure runs through the conjunctions that trace a slowly turning triangle: an observed regularity of the sky, known for millennia, whose position encodes a physical quantity nobody could have measured directly when the regularity was first recorded. In both cases what made the regularity readable was separating the part set by geometry from the part set by the objects.

Still open: what a phase curve cannot separate #

A phase curve of an exoplanet measures its phase function, but a phase function is itself a product — of how much light the surface reflects and of how that light is distributed in angle. A bright, strongly forward-scattering atmosphere and a darker, more isotropic one can produce similar curves over the range of phase angles a transiting planet’s orbit presents, which rarely reaches the thin-crescent angles where forward scattering dominates. The polarisation of the reflected light breaks that degeneracy for Venus, and whether it can be measured for a planet a hundred million times fainter than its star is the open question.