A peak and an amplitude that should be one number

A planet on a two-day orbit shows its star the same face for ever, so one hemisphere is in permanent noon and the other in permanent midnight. As the planet goes round, an observer sees first its night side, at transit, then more and more of its day side, until the planet disappears behind its star. Infrared photometry of the star and planet together rises and falls through this cycle by a part or two in a thousand, and that slow wave is the phase curve. It is the nearest thing there is to a map of an unresolved world.

Two numbers come off the wave at once. The first is how large it is: the difference between the brightest part of the curve and the faintest, as a share of the maximum. The second is where its top falls. The top is expected at the eclipse, when the observer sees the point under the star head on, and it was the great surprise of the first such curve, published for HD 189733b in 2007, that it comes early. The hottest air is not under the star. It is downwind, displaced to the east by an equatorial jet.

Those two numbers are generally reported together and discussed separately, the first as a measure of heat redistribution and the second as a measure of winds. In the simplest model they are not two numbers. A single ratio generates both, and this leaves a clean prediction: the measurements should fall on one curve. Testing that prediction asks whether the planet’s atmosphere can be described by a wind and a cooling time at all, a question that a table of offsets and a table of amplitudes do not ask.

A parcel going round #

Follow a parcel of gas round the equator at a steady speed. It is heated by the star when on the day side, and it radiates all the time. Nicolas Cowan and Eric Agol put that into the form of a single equation, with temperature in units of the no-wind value at the substellar point and longitude $\Phi$ measured east from that point:

$$\frac{d\tilde{T}}{d\Phi} = \frac{1}{\varepsilon}\left[\max(\cos\Phi, 0) - \tilde{T}^4\right].$$

Everything about the atmosphere is in $\varepsilon$, the product of the radiative time and the wind’s angular speed, which says how many radians a parcel travels in the time it takes to cool. For a small $\varepsilon$ the parcel sheds its heat instantly, so the temperature follows the sunlight locally. For a large one it carries its heat all the way round and the temperature hardly changes.

The picture has a simple structure. The parcel’s temperature peaks not where the heating is greatest but where heating and cooling balance, at the point where $\cos\Phi = \tilde{T}^4$. Downstream of noon, the heating has fallen and the parcel has not yet caught up. That crossing moves further east as the parcel’s memory grows. The same memory keeps the night side warm, since a parcel that took a long time to heat also takes a long time to cool. So the offset of the hottest point and the warmth of the night are the same physical fact.

The observer does not see a temperature profile, and the curve the telescope records is an average of the emission over the visible hemisphere, weighted towards the middle of the disc.

The averaging is why the curve’s peak is further from eclipse than the hottest air’s longitude alone would suggest, and why it is smooth even when the planet is not. It is also why a phase curve’s offset is not simply a longitude. At the largest values of the ratio, the peak-to-trough amplitude has fallen so low that a measurement of its position is hard, and offset approaches a quarter of an orbit only when there is almost nothing left to measure.

One line in the plane of two measurements #

Run the ratio through its whole range and plot the results against each other. Each planet in this model should lie somewhere along the line, and its position gives its ratio.

Some planets sit on the line, and they are worth noting first. WASP-18b, a ten-Jupiter-mass planet close to its star and more than 3,000 kelvin on its day side, shows essentially no offset and a nearly undetectable night side, just what a very small ratio predicts. Its air is so hot that it radiates almost as soon as it is heated. In that limit nothing about the atmosphere’s motion can be read from the curve at all, and what is seen is close to the local re-radiation of bare rock. So the upper left of the figure is where the model has the least to explain, and it is also where the planets differ least from each other.

The rest of the diagram is more awkward. HD 189733b at 8 microns has a modest offset and a small amplitude, and it falls well below the line. For a peak displaced by sixteen degrees of orbit, the model expects a nearly full-amplitude curve, and the real one has only a third of that height. Translating band flux into bolometric flux closes part of this gap, since a band-limited measurement sees only a piece of the blackbody curve, whose contrast is smaller than the total flux would be. It does not close all of it. The night side of this planet is much warmer than its peak offset allows, in a one-parameter description. The gas that crosses the night side must have kept heat that the gas near the hot spot did not.

The figure also carries a warning about how easily a planet can land on the line for the wrong reason. 55 Cancri e is a rocky world, or something near that, with a lava day side and possibly a thin atmosphere of its own, and its point falls on the curve with no help. A one-parameter locus can catch something very different by accident.

What a single ratio leaves out #

A real atmosphere has several reasons for failing the model, and they fail in distinct ways. The most obvious is that it has more than one depth. Each band sees emission from a different height, because the opacity varies with wavelength, and different pressure levels have different cooling times. A planet measured in two bands is really two parcels at two heights, going round with two ratios. A single-band point on this plane is therefore a point on one of many curves, and a spectroscopic phase curve from a space telescope with broad coverage is what makes that visible. Measurements of WASP-43b and other planets, at many wavelengths, show the offset and amplitude changing with the band in ways that have to be read as a temperature structure.

Another reason is the wind itself. The model has a uniform jet. A real atmosphere has a broad equatorial jet with slower or reverse flow near the poles, and at high altitudes flow from the day side to the night side over both limbs. The averages taken across the disc weight these in a way that has no counterpart in one equation.

The third reason appears to be the most consequential: nightside clouds. Compilations of hot-Jupiter phase curves published around 2019 found that the nightside brightness temperatures of planets with day sides from 1,000 kelvin to 2,500 kelvin clustered in a surprisingly narrow range around 1,100 kelvin. A one-parameter model cannot produce that, since a hotter day side should raise the night along with it. A simple explanation is that the air on the night side condenses, and a deck of cloud sets the temperature at the level from which light escapes: the observer sees only the cloud tops, regardless of what lies beneath. If so, the amplitude is partly a statement about clouds, and the offset partly a statement about dynamics, and the two have been decoupled. The clouds would break the very link this argument began with.

Why hot air has less time #

Within a family of similar planets, one quantity changes predictably: temperature. The time for a layer of gas to radiate its heat falls steeply as the temperature rises:

$$\tau_{\rm rad} \approx \frac{c_P P}{4 g \sigma T^3}.$$

The energy to be radiated is proportional to the temperature, and the radiated power to its fourth power, and so the time falls as the cube.

This predicts a trend, and the data show one, loosely: the hottest planets have the smallest offsets and the largest amplitudes, at least in the broad sense that planets above about 2,500 kelvin look much like a dayside glow with little transport. The trend is not smooth, and the scatter around it is of the same order as the trend itself, partly because measurement errors on an offset are large and partly because a hot-spot model with a fixed wind is unreasonable. Faster jets are expected on hotter planets, and ionised air on the hottest ones is dragged by the planet’s magnetic field, which could slow the jet down, and this is a further reason the offset falls.

A wind and a cooling time alone also leave the surprising fact that some planets offset the other way. A wind model cannot produce a peak after eclipse. Reflected light can.

A cloud deck that reverses the sign #

The Kepler telescope worked in visible light, and for a planet that is not too hot, most of what it saw in a phase curve was reflected rather than emitted. In 2013, Brice-Olivier Demory and colleagues published the phase curve of Kepler-7b, a planet of very low density and a peak that came after eclipse. That is a peak on the western side. Infrared observations with Spitzer showed the thermal emission to be too weak to contribute, so this had to be reflection, and the reflectors could not be uniform.

The energy-balance picture accounts for this naturally. The air arriving at the morning side from the night side is the coldest on the day hemisphere, because it has just spent half a rotation in darkness. If clouds condense there and evaporate as the parcel warms up on its way past noon, then the western side is bright and the eastern side dark, in visible light.

So a single wind in a single direction produces peaks on both sides of the eclipse, depending on which light is being recorded, and the two together make a stronger case for the flow than either alone. The difference between a planet’s visible and infrared peak offsets is a measure of how the clouds are laid out relative to its temperature. Since polarisation has its own dependence on what reflected the light, the ceiling polarimetry can reach would be another test of cloud location, were it not so low.

Here the model is being used to explain something it was not constructed for, and the agreement is qualitative. Its threshold temperature is a free choice, it has been picked so the effect is visible, and the real clouds will have a composition, a size distribution and a height. That the sign comes out right from physics this simple is suggestive, and it is not a measurement of Kepler-7b’s clouds.

The same effect in the eclipse’s timing #

A hot spot has another consequence, one that links this question to the timing of the orbit. A bright patch displaced from the centre of the disc is also a displacement of the planet’s centre of light. During an eclipse, the star’s limb covers the disc from one side, and the time at which half the light is gone is then not the time at which half the disc is gone. The planet appears to enter and leave eclipse a little early, by the time the planet takes to cover the displacement of its own centroid.

That is how Eric Agol and his colleagues in 2010 detected a shift of this kind in repeated Spitzer eclipses of HD 189733b, consistent with the hot spot already found from its phase curve. It is a strikingly small effect, a minute or so in a long observation, and it was found by stacking many eclipses. It is also a trap for a different measurement. The usual way of constraining an orbit’s eccentricity is by the time of eclipse relative to the midpoint between transits: a displacement proportional to $e\cos\omega$. The density-based constraints on eccentricity rest on different physics, but they cannot reach this level for a hot Jupiter. So an inferred eccentricity of about a thousandth, found from eclipse timing alone, might be weather.

The way out comes from what each signal does to the shape. An eccentricity shifts the whole eclipse. A hot spot changes the light curve’s shape during ingress and egress, which take about half an hour here, because different longitudes of the planet are covered at different moments. The residual can be inverted into a brightness map of the disc, a method known as eclipse mapping, which gives a two-dimensional view of a planet no telescope can resolve. To do this well, the signal-to-noise of the largest telescope in space has been needed. And it adds a third measurement to the plane this essay began with: a direct measure of where the bright spot is, with no appeal to an orbit-long curve.

The connection across two parts of the subject #

The surprise that comes out of this is that a tool from orbital mechanics turns out to be contaminated by meteorology, and an effect discovered by atmospheric scientists can be misread as dynamics. Astronomers doing the timing and those doing the climate use one signal for different purposes. In the end the timing gives a limit on the winds and the winds give a limit on the orbit’s shape, and only an analysis fitting both, as with any two explanations of a single residual, can say how much belongs to each.

The other surprise is in the first figure’s lesson about what a phase curve’s amplitude means. It has long been described as the efficiency of heat transport: a flat curve means efficient redistribution. Viewed through the model, a flat curve means a long radiative time, not necessarily a strong wind. With nightside clouds in the picture, it may not even mean that. The same number can mean a fast jet, a deep emitting layer, or a cloud. That is why the plane of two numbers is worth more than either alone, even though it was the failure to fall on a line that carried the information, not the line.

What the picture cannot show #

The model tracks a single band of air at the equator in a steady state. It has no vertical structure, no poles, no time dependence, no weather. All the phase curves in the figures are bolometric and noiseless, whereas each real one comes through a filter and a detector, whose systematic drifts over a two-day stare can be comparable to the signal. Removing those drifts has at times changed a planet’s published offset by tens of degrees, as was the case for early Spitzer results, so some of the points in the figure may yet move.

The measured points are placed without error bars and without a correction for the band, because a correct translation would need a spectrum for each planet at each longitude. The fixed wind in the temperature trend is an assumption with an unmeasured value, although Doppler shifts of the planet’s own absorption lines during transit have given a few wind speeds, of several kilometres a second, from a transit’s spectrum. Nothing here shows an atmosphere’s composition, on which the opacity depends and therefore the depth seen. And since a tidally locked planet’s rotation has been assumed, a world with a residual eccentricity or an asynchronous spin would receive its heating in a pattern this model cannot represent.

Still open: whether a night side is a temperature or a cloud top #

If hot Jupiters’ nightsides are a cloud-deck temperature rather than the temperature of the deep air, then much of what the amplitude has been taken to mean has to be reworked. The direct test is spectroscopic: clouds flatten the spectrum, so a night side’s emission should lose its molecular features if the cloud is high and thick, and it should keep them if it is clear. The largest space telescope has been measuring full phase curves at those wavelengths for a growing list of planets. Whether the features fade by longitude as the cloud model requires, and at which temperature the condensation sets in for each species, would say whether nightside uniformity is a property of chemistry or a coincidence in a small sample. If the answer is chemistry, the offset and the amplitude will truly have become two numbers, and then the single ratio this argument began with describes only the planets too hot to have clouds at all.