A planet radius is a stellar radius

A planet is measured by the light it removes, and what the light curve actually delivers is a fraction rather than a size.

A transit light curve is a dip. Its depth is the fraction of the star’s disc the planet covers, which is the square of the ratio of the two radii, and that ratio is measured superbly — to a fraction of a per cent for a well-observed system.

What is wanted is the planet’s radius in kilometres. That is the ratio times the star’s radius, and the star’s radius is not in the light curve. It arrives from spectroscopy, or from a model, or from a parallax and a temperature, and whatever its error is, the planet inherits it in full.

A gap in a histogram says how planets are built, and this essay is about how that histogram spent a decade being too blurred to show one — not because the planets were poorly observed, but because the stars were.

The transit measures nothing absolute #

It is worth being precise about what is and is not in the light curve, because the division is unusually clean.

In the light curve: the ratio of radii, the ratio of the orbital distance to the stellar radius, the impact parameter, the period, and the limb-darkening coefficients if the photometry is good enough. All of these are dimensionless or are times.

Not in the light curve: the stellar radius, the stellar mass, and therefore the planet’s radius, the orbital distance, and the planet’s insolation.

That division is not a defect. A measurement of a ratio is often better than a measurement of an absolute — it is why differential photometry reaches precisions absolute photometry cannot — and the transit’s ratio is one of the best-determined quantities in exoplanet science. The difficulty is that the questions people want to ask are about the absolute.

There is a second absolute quantity in the same position and it is worth naming because it is the one most arguments actually need. The insolation a planet receives is the star’s luminosity divided by the square of the orbital distance, and the transit gives the distance only as a multiple of the stellar radius. So insolation, like radius, is a ratio times a stellar property, and every statement about atmospheric loss — which depends on the flux the planet receives — carries the stellar uncertainty twice, once through the luminosity and once through the distance. The improvement in stellar parameters therefore sharpened the insolation axis as well as the radius axis, and the valley’s dependence on insolation is one of the features that emerged.

It is also worth noticing which questions the ratio alone can answer, because they are more than one might expect. Whether a planet’s radius changes with wavelength — the transmission spectrum — is a ratio of ratios and needs no stellar radius. Whether two planets in one system have the same size is a comparison of two ratios around one star. Whether a planet’s radius varies between transits, which would indicate a variable atmosphere, is a time series of ratios. All of those are immune to everything in this essay, and they are the measurements that were possible before the parallaxes and are unaffected by them.

Why the smearing hides rather than moves #

The distribution of planet radii is a histogram of ratios multiplied by stellar radii. If the stellar radii carry a fractional error with no bias, then in the logarithm the error is an additive Gaussian of constant width — so the observed distribution is the true one convolved with a fixed kernel.

Convolution has two properties that decide everything here. It preserves the mean in the logarithm, so a symmetric error does not shift a feature’s position. And it suppresses structure on scales narrower than the kernel, so a gap narrower than the smearing disappears.

The radius valley is about 0.1 in the logarithm — some twenty-five per cent in radius — so a stellar radius error of twenty-five per cent erases it and one of five per cent barely touches it. That is the whole of what happened when parallaxes arrived: the same planets, in the same catalogue, with the same transit depths, plotted against a better set of stellar radii.

Be explicit about the arithmetic, because it explains why the effect is so abrupt. Convolving a Gaussian gap of width $w$ with a kernel of width $\sigma$ leaves a gap of width $\sqrt{w^2 + \sigma^2}$ and reduces its depth by roughly the ratio $w/\sqrt{w^2 + \sigma^2}$. For $\sigma$ well below $w$ almost nothing happens; for $\sigma$ comparable to $w$ the depth halves; for $\sigma$ twice $w$ the feature is gone. The transition covers a factor of about three in the error, which is exactly the factor by which stellar radii improved — so the feature went from undetectable to obvious in one step, with no intermediate stage in which it was marginal and being argued about.

One further property of the convolution deserves a sentence because it is the diagnostic. A symmetric smearing preserves not only the mean but every odd moment about it, so a feature’s position, its skewness and the asymmetry of the surrounding distribution all survive. What it destroys is the even structure — the depth and the sharpness. So an observed distribution’s shape away from the feature is trustworthy even when the feature itself is not, and comparing the wings of a distribution before and after a reference improves is a check that the improvement was a smearing rather than a shift.

What the parallaxes did #

The transiting planet catalogue was assembled over four years of continuous photometry. The stellar radii used to interpret it came from broad-band colours and low-resolution spectra, giving errors of twenty to forty per cent for typical hosts — dominated by the difficulty of telling a dwarf from a slightly evolved subgiant, which have similar colours and radii differing by a factor.

A parallax removes that difficulty entirely. With a distance, an apparent magnitude gives a luminosity; with a temperature, the luminosity gives a radius through the Stefan–Boltzmann law. The precision is a few per cent, and the dwarf-versus-subgiant question answers itself, because the two have very different luminosities at the same colour.

The reanalysis that followed reduced the median stellar radius error from about twenty-five per cent to about four, and the median planet radius error with it. Nothing about the planets was reobserved.

There is a detail of the improvement that is worth having because it explains why the gain was so large. The dominant error before was not a random one at all: it was a classification error, between dwarfs and subgiants of the same colour. A subgiant has a radius twice a dwarf’s at the same temperature, so a misclassification is a factor-of-two error rather than a twenty-five per cent one — and the misclassified fraction was of order a quarter. So the twenty-five per cent “error” was really a mixture of a small error for most stars and an enormous one for a minority, and the parallax fixed the minority by making the luminosity directly visible. That is a common structure for a catalogue systematic: a quoted uncertainty that is the average of two quite different populations.

What the stellar radius is checked against #

A parallax radius is not a direct measurement either. It is a luminosity and a temperature run through the Stefan–Boltzmann law, and the temperature comes from a spectrum or a colour. So the reference has its own reference, and it is worth knowing what sits at the bottom of that chain because the whole planet catalogue rests on it.

Three measurements give a stellar radius without a temperature scale, and all three are expensive.

An interferometric angular diameter. Resolving a star’s disc with a long-baseline optical interferometer gives an angle; with a parallax that is a radius, directly. It works on a few hundred stars — the nearest and largest — and it is the only route that is purely geometric. Those stars then define the temperature scale that everything else uses, because a radius and a bolometric flux give an effective temperature by definition.

An eclipsing binary. Two stars eclipsing each other give both radii as fractions of the orbit, and the orbit’s size follows from the two velocity curves. That yields radii to a per cent with no distance and no temperature involved, and it works for a few hundred systems.

Asteroseismology. The frequencies of a star’s oscillations scale with its mean density and its surface gravity, so a light curve long enough to resolve them gives a radius to a few per cent. It applies to bright stars observed continuously, which for the transiting sample is a subset of a few thousand.

Each of the three is a small, precise sample used to calibrate a large, convenient one — which is the same architecture as every distance ladder, arriving inside stellar astrophysics. The parallax radii agree with all three at the level of a couple of per cent, and the residual disagreements are what the current systematic budget is made of.

The consequence for reading a planet radius is a hierarchy of trust. A planet around a seismic host has a radius good to two or three per cent; around an ordinary parallax host, four or five; around a faint star with only colours, fifteen or more. Those are the same catalogue and the same measurement technique, and they are not interchangeable — a population study that mixes them without weighting is convolving its distribution with three different kernels at once, which is not the same as convolving with their average.

That last point is more than a technicality. A mixture of kernels produces a distribution with broader wings than any single Gaussian, so a feature can be partly visible and partly smeared in one histogram, and the natural response — cut to the best-measured subsample — trades precision against a sample that is no longer representative. Every analysis of the radius valley has to choose where on that trade to sit, and the choices are not the same between papers.

What was actually measured #

Three of them, and the third is the one that matters most for the interpretation.

The gap’s depth. Before the parallaxes, the deficit at the valley was a factor of about 1.3 below the surrounding distribution and was consistent with no feature at the two-sigma level. Afterwards it was a factor of two or more and unambiguous.

The gap’s position, unchanged. The valley sits at about 1.8 Earth radii in both analyses. That is the signature of a symmetric error: it hid the feature without moving it, exactly as convolution requires. Had the position moved, the explanation would have had to be a bias rather than a smearing, and the interpretation would be quite different.

The gap’s slope with period. The valley’s centre moves to smaller radii at longer periods, with a power-law slope of about $-0.1$. That slope is the discriminating measurement between formation theories — photoevaporation and core-powered mass loss predict slightly different values — and it was completely inaccessible while the feature itself was marginal.

Where the picture stops #

There are three, and the second is the one that will bite next.

Not every error is symmetric. The parallax-based radii inherit whatever bias the parallaxes have, and a parallax zero point is a systematic that does not average away. A seventeen-microarcsecond offset at a kiloparsec is nearly two per cent in distance, one per cent in radius, and one per cent in every planet radius derived from it — small, and no longer negligible against a four per cent error.

The temperature is now the limiting term. With a parallax the radius comes from the luminosity and the temperature, and the temperature enters as its square. A spectroscopic temperature good to a hundred kelvin is 1.7 per cent, which is 3.4 per cent in radius — larger than the parallax’s contribution and the reason the median radius error has stopped falling.

And the ratio itself has systematics. The transit depth depends on the assumed limb darkening, on contaminating light from unresolved companions, and on how the light curve was detrended. Each contributes a per cent or so and each is correlated across a survey rather than random, so they do not smear the distribution — they shift it.

One more belongs on that list, and it concerns the sample rather than the measurement. The improved stellar radii also improved the detection efficiency calculation, because whether a transit is detectable depends on the depth relative to the noise, and the depth depends on the stellar radius. So the occurrence rates — planets per star — changed at the same time as the radii, and by an amount that is not simply related. What a survey could have seen has to be recomputed when the stars change, and a comparison between a pre-parallax and a post-parallax occurrence rate is a comparison of two different quantities.

Why an improvement to one catalogue moved another #

The general point deserves separating from the instance because it is the pattern rather than the instance.

When a measurement is a ratio, the population it belongs to inherits the population of references. Every planet radius is a stellar radius; every stellar radius from a parallax is a distance; every distance is a parallax zero point. The chain is short here and it is a chain, and a feature in the planet distribution can be created or destroyed by a change three links away.

That has a practical consequence for reading such results. A claimed feature in a distribution of derived quantities is only as sharp as the reference catalogue used, and the first question about a new feature is what changed in the references. The radius valley is a case where the answer is reassuring — the feature appeared when a reference improved, in the direction improvement predicts, without moving. The opposite pattern, a feature that appears when a reference changes and sits at a new place, is what a reference artefact looks like.

The episode is also a good argument for a practice that is not universal: publishing the ratio alongside the derived quantity. A catalogue that lists $R_p/R_\star$ as well as $R_p$ can be re-derived when the stars improve, and one that lists only the planet radius cannot. The same discipline applies to any quantity derived through a calibration, and it is what made the reanalysis a matter of months rather than a new survey.

It is worth recording one more consequence, because it changed how such catalogues are built. The reanalysis demonstrated that a large, homogeneous, well-characterised stellar catalogue is worth more to exoplanet science than additional planet observations — and the follow-up programmes that resulted are spectroscopic surveys of planet hosts rather than photometric surveys for new planets. That is an unusual conclusion for an observational field to reach about itself, and it followed directly from the arithmetic in this essay: when a derived quantity is a product of a well-measured ratio and a poorly measured reference, the return on improving the reference is the whole of the available improvement.

The same logic is now being applied one step further out. The stellar radii from parallaxes are limited by the temperatures, the temperatures are limited by the systematics of spectroscopic analysis, and the effort has moved there. Each generation of the problem is the same shape: find the term that is limiting, notice that it belongs to somebody else’s subject, and go and improve it.

Where the ladder goes next #

The immediate next rung is the composition inference: what a radius and a mass together say about a planet’s interior, and why the relation is one-to-many. The rung beyond it is the insolation — the other absolute quantity a transit does not measure, which is the orbital distance times the stellar luminosity and which is what any argument about atmospheric loss actually depends on.