The brightest instant of an occultation is its middle

An occultation by an atmosphere is usually described as a fade: the star does not switch off, it dims over several seconds as refraction spreads its light, and how fast it dims is a scale height.

That description is right for an observer anywhere except one place. At the exact centre of the shadow something else happens, and it is the opposite of a fade.

A spherically symmetric atmosphere bends every ray toward the axis. Rays passing on opposite sides of the body are bent toward each other, and there is one impact parameter for which the bending is exactly enough to bring the ray onto the axis. An observer standing there sees not an arc of the limb but the whole ring of it, and the flux rises above what an unocculted star would give.

The construction, in one equation #

The whole figure follows from a single relation and it is worth writing out because everything else is read off it.

For an isothermal atmosphere the refractivity falls exponentially with height, so the bending angle does too. A ray with impact parameter $r$ is deflected by $\omega(r)$, and by the time it reaches the observer at distance $D$ it has moved inward by $D\omega(r)$, landing in the shadow plane at

$$\rho(r) = r - D\,\omega(r),\qquad D\omega(r) = H\,u(r),\quad u(r) = e^{-(r-R)/H}.$$

Differentiating, $d\rho/dr = 1 + u$, so $dr/d\rho = 1/(1+u)$ — which is exactly the flux the ordinary isothermal light curve reports. The refraction compresses a range of impact parameters into a smaller range of ground positions, and the compression factor is the dimming.

The extra factor for the centre is geometric rather than refractive. A ray landing at $\rho$ came from a circle of radius $r$, and the light from that circle is being delivered to a circle of radius $\rho$ — so the surface brightness carries $|r/\rho|$ as well. Far from the centre $r\approx\rho$ and the factor is one. On the axis it diverges.

The total flux is therefore

$$\phi(\rho) = \frac{r_1}{|\rho|}\cdot\frac{1}{1+u(r_1)} \;+\; \frac{r_2}{|\rho|}\cdot\frac{1}{1+u(r_2)},$$

where $r_1$ is the ray landing at $+\rho$ and $r_2$ the one landing at $-\rho$ — the far limb’s ray, which has crossed the axis. Both terms are present everywhere; the second is negligible until close to the centre.

What stops it being infinite #

A point source focused by a perfect sphere gives an unbounded flux, which no observer has ever recorded. Three things cut it off and each is a measurement.

The star has a size. Every point of the stellar disc has its own focus, displaced by the same angle, so the focus is smeared over the star’s angular diameter projected to the shadow plane. For a star of a fraction of a milliarcsecond at twenty astronomical units that is a few kilometres — an angular diameter measured by how blurred an edge is, and it is the softening used in the figures here. It is also why occultation campaigns prefer small hot stars: a giant is a worse lens target for the same reason it is a worse ring probe.

The body is not a sphere. A rotating planet’s atmosphere is flattened, so the rays at different position angles focus at different places, and the point focus opens into a closed curve — a caustic. An observer crossing it sees several sharp spikes rather than one broad peak, and the positions of the spikes measure the oblateness of the atmosphere at the flash level.

The atmosphere is not clear. Haze absorbs, and the flash’s ray passes through the deepest layer of the event, so it is attenuated most. A flash weaker than the clear-atmosphere prediction is a measurement of the optical depth down there, and a flash entirely absent is a statement that the haze is thick.

The depth it reaches, and why that matters #

The single most valuable property of the flash is where its ray comes from.

An ordinary occultation light curve is dominated by the region near the half-light level, because that is where the flux is changing fastest. Everything deeper is already dimmed to nothing by the time it would contribute, so the thermal profile an occultation returns covers a few scale heights around one pressure level — typically a microbar or so.

The flash’s ray is different. It is the ray bent by exactly its own offset, and reaching that much bending requires going deep. In the hero figure the flash ray has an impact parameter 161 kilometres below the half-light radius, at a pressure nineteen times higher.

So an occultation light curve measures one level of an atmosphere and the flash measures another, several scale heights beneath it — from the same event, the same photometer, the same few seconds.

That is worth a great deal for the bodies concerned. Pluto, Triton and Titan are visited by a spacecraft once in a generation, and a flyby’s own tracking is the other way their interiors are read, and their atmospheres are otherwise known only from occultations. A technique that adds a second, deeper sounding for free is not a refinement.

What has been seen #

Central flashes have been recorded at several bodies and each one taught something different.

Titan, in 1989 and again in 2003, produced flashes observed from several stations. The 2003 event was observed with enough spatial sampling to map the caustic, and the positions of the spikes gave the atmosphere’s oblateness at the flash level — which turned out to be larger than the solid body’s, indicating a zonal wind field strong enough to flatten the atmosphere beyond what rotation alone does. A wind speed, measured from the shape of a bright spot on the ground.

Triton showed a flash in 1997 whose strength implied a clear atmosphere at the sounded level, contrary to expectations from the hazes seen by the flyby a decade earlier.

Pluto has produced flashes in several events since 2007, and their strength varies between events. The variation is read as changes in the haze layer, which is consistent with the seasonal volatile transport its atmosphere undergoes — its surface pressure has changed by a factor of three since its atmosphere was discovered.

Neptune and Uranus have produced flashes whose asymmetry constrains their atmospheric flattening, which for a rapidly rotating giant is substantial.

In each case the observation is the same: a photometer, a few seconds, and an observer who happened to be standing in the right place on the Earth.

Turning a light curve into a temperature profile #

The flash is one number in a procedure whose main output is a profile, and the procedure is worth describing because it explains what the flash is being compared against.

The forward problem is easy: given a temperature against height, compute the refractivity, integrate the bending along each ray, and get a light curve. The inverse problem — a light curve to a profile — is the one that has to be solved, and it is done by a direct inversion rather than by fitting.

The inversion works because the geometry is invertible. Each instant of the light curve corresponds to one ray, the observed flux gives the compression factor $1/(1+u)$ and therefore the bending gradient, and integrating the bending inward recovers the refractivity as a function of radius. The refractivity is proportional to the density, and hydrostatic equilibrium turns a density profile into a pressure and then into a temperature.

Two things have to be supplied from outside. The composition enters through the refractivity per molecule and the mean molecular weight, and for a nitrogen atmosphere both are known. And the upper boundary condition enters because the hydrostatic integration starts from a pressure at the top that the light curve does not determine — an assumed value that propagates downward and decays, so the profile is unreliable in the first scale height and good below it.

The profile from a single station typically covers three or four scale heights, and the flash extends the reach by another three. It is also the only part of the sounding that does not depend on the upper boundary, because the flash ray’s bending is measured against the axis rather than integrated from above.

Where the technique came from #

The first central flash was recorded before anybody was looking for one.

An occultation by Neptune in 1968 produced a light curve with an unexplained brightening near mid-event. The interpretation as a focus followed quickly, because the geometry is elementary once somebody has drawn it, and the subsequent theoretical work in the 1970s established the inversion and the caustic structure for a flattened body.

What made it a technique rather than a curiosity was the arrival of two things. Precise astrometry made the centreline predictable to within the caustic’s own size, and portable telescopes with fast photometers made it possible to put a dozen stations across it. The Titan campaign of 2003 is the model: stations across southern Africa, several of them inside the caustic, and a map of the flash structure that gave the atmosphere’s shape — the same kind of inference a set of chords makes about a silhouette.

The observation had been possible since the 1970s and the campaign was not, and the difference is entirely in knowing where to stand.

The prediction problem, and it is severe #

Standing in the right place is the difficulty, and it is worse than it is for an ordinary occultation.

An ordinary chord can be recorded from anywhere in the shadow, which for a body a thousand kilometres across is a band a thousand kilometres wide. The flash is confined to a region the size of the caustic — tens of kilometres — so catching it requires predicting the shadow’s centreline to a precision thirty times better.

That was impossible before precise astrometry and is now routine for the larger bodies. The current practice is to deploy portable telescopes in a line across the predicted centreline, spaced by a few kilometres, and accept that most of them will record an ordinary light curve while two or three catch the caustic.

The payoff for that effort is the spatial structure. A single station recording a flash gives its height; a line of stations gives a cut through the caustic, and the caustic’s shape is the atmosphere’s shape.

What a station has to be able to do #

The instrumental requirements are unusual and worth stating, because they explain why this is a technique for small telescopes.

The event lasts seconds and the caustic crossing lasts a fraction of one. A station has to sample at tens of hertz to resolve the spikes, which rules out a conventional imaging camera with a read-out time and rules in a photometer or a fast frame-transfer detector.

The timing has to be absolute, to milliseconds, which is a question about which clock is being read, because the whole measurement is where the station was when the flash arrived — and the station’s position relative to the shadow is derived from the timing. Satellite navigation supplies that for nothing, which it did not before the 1990s.

The aperture barely matters. The star is bright by the standards of photometry and the required signal-to-noise per sample is modest; a thirty-centimetre telescope records a usable curve. What matters is where the telescope is, and a telescope that can be driven to a point on a road is worth more than one that cannot be moved.

That inverts the usual economics of observational astronomy. The measurement is limited by spatial sampling rather than by photons, so twenty small telescopes spread across four hundred kilometres beat one large one on a mountain, and the community that does this work is accordingly a mixture of professional and amateur observers in a way few others are.

Where the model stops #

An isothermal atmosphere is an idealisation and the flash is where it fails. The construction above assumes a single scale height, and the flash’s ray samples a level several scale heights below where the light curve is measured — precisely the range over which a real temperature profile changes. Inverting a flash properly means inverting the whole light curve for a temperature profile and then computing the focus from that, which is what is actually done and which brings the profile’s own uncertainties with it.

Spherical symmetry is assumed and then removed by hand. The flattening in the figures is applied as a smear of the spherical answer. That gets the width right and the structure wrong: a real caustic is a closed curve with cusps, and the number and spacing of the spikes an observer records depend on where the track crosses it, which is the same dependence a light curve with a fold in it has on its own trajectory. Only a two-dimensional ray trace reproduces that, and the published analyses do one.

And the geometric factor assumes a smooth limb. Any structure in the atmosphere — a wave, a layer, a local temperature inversion — the fluctuating index that blurs every ground-based image is the same refractivity at a smaller scale — produces its own small focus, and these are seen as spikes away from the centre in the best-recorded events. They carry information about the layering and they are not in any model here.

Why the far limb is the interesting half #

There is a detail in the two-term flux expression that is easy to pass over and is where most of the information is.

Away from the centre, the near limb dominates completely: its ray is barely bent and carries nearly all the flux, while the far limb’s ray has been deflected across the whole body and is attenuated to nothing. The light curve an ordinary observer records is a near-limb measurement.

Near the centre the two terms become comparable, and then the far-limb term dominates — because the far limb’s ray has the larger geometric factor at small $\rho$. The flash is mostly light from the side of the body facing away from the observer, which has travelled through the atmosphere on that side.

That has a consequence for what the flash measures. An ordinary occultation samples the limb the star is setting behind — one longitude, one latitude, one local time. The flash samples both limbs at once, and a difference between them appears as an asymmetry in the flash rather than as anything in the light curve.

Titan’s 2003 flash was asymmetric, and the asymmetry was read as a difference in the atmosphere’s structure between the two limbs — a north–south contrast in the zonal wind. Two hemispheres compared, in one photometric time series, from a body whose disc is three-quarters of an arcsecond across.

The same construction elsewhere #

A focus produced by a smooth deflecting medium is a shape that recurs, and naming it connects this to two other parts of the collection.

Gravitational microlensing is the same geometry with gravity in place of refraction. A point mass deflects rays toward the axis, the bending falls with impact parameter, and an observer near the axis sees an amplified image. The difference is the sign of the gradient: gravity bends more for rays passing closer, so the image structure differs, but the divergence on the axis and its softening by the source’s finite size are identical — a caustic crossing in a microlensing light curve is the same object as the spikes in a flattened body’s flash.

A mirage is the same physics in the Earth’s atmosphere at a scale of metres, where a temperature gradient near the ground bends rays enough to produce multiple images and, occasionally, a bright line at the horizon.

What the three share is that a smooth medium with a monotone gradient focuses, that the focus is a caustic rather than a point once the symmetry is broken, and that the caustic’s shape carries more information than its brightness. The structure is worth more than the amplitude, which is the opposite of the usual situation in photometry and is the reason a line of small telescopes beats one large one here.

Still open: a flash that should be there and is not #

Several predicted flashes have failed to appear, and the non-detections are harder to interpret than the detections.

A missing flash means the deepest ray was absorbed, which means haze, and the optical depth implied is a real measurement. It can also mean the centreline prediction was wrong by tens of kilometres, which at the time was entirely possible and is now less so. And it can mean the atmosphere is flattened more than assumed, spreading the caustic beyond the deployed array.

Distinguishing the three requires observers on both sides of the predicted centreline, so that a miss can be localised rather than merely recorded. That is an argument for larger deployments, and larger deployments are how the field has gone — a well-covered event now has dozens of stations, many of them amateur, spread across a country.

What nobody yet has is a flash observed at two wavelengths simultaneously with enough signal to separate the haze’s colour dependence from the geometry’s. That measurement would turn a non-detection from an ambiguity into a constraint, and it needs a bright star and a lot of luck.