Too large and too cool by just enough to keep the light

A star’s radius is not normally measured. It is inferred from a temperature and a luminosity through a model. Detached eclipsing binaries are the exception: the light curve and the two velocity curves give both stars’ masses and radii without any model of a stellar interior. For stars like the Sun the measured radii and the models agree to a per cent or two, which is how the mixing length came to be trusted.

For red dwarfs they do not. Eclipsing pairs of M dwarfs, from about 0.2 to 0.7 solar masses, have radii several per cent larger than standard models give for their masses, and up to fifteen per cent larger in the worst cases. Their effective temperatures come out lower than the models by a few per cent. The discrepancy has been confirmed in system after system since the first well-measured pairs, and it has three candidate explanations. All three act through the convecting outer layers.

This essay is about a feature of the discrepancy that each explanation has to reproduce, and why the explanations reproduce it so easily. The stars are larger and cooler than the models, by amounts that leave their luminosities about where the models put them.

A line of constant light #

The luminosity, radius and effective temperature of a star are tied by definition:

$$L = 4\pi R^2\sigma T_{\rm eff}^4.$$

If the luminosity is held fixed and the radius grows by a small fraction $r$, the temperature must fall by $r/2$. A radius ten per cent larger needs a temperature 4.7 per cent lower. The relation is exact on a line that, plotted as percentage changes, runs through the origin with a slope close to $-1/2$.

The shaded box marks the range of radius excesses and temperature deficits reported for red dwarfs against standard models. It sits on the line of unchanged luminosity. A star five per cent too large is about two and a half per cent too cool; one twelve per cent too large is about five per cent too cool. If the inflated stars were also brighter, the box would lie above the line. If they were fainter, it would lie below. Within errors it does neither.

That is a constraint, and a strong one. Whatever makes these stars larger must do it without changing how much light they emit. It must reach the surface and not the core.

The light is decided in the interior #

For a star with a radiative interior, the reason the light stays fixed has a familiar derivation. The luminosity carried by radiative diffusion through the bulk of the star scales, in the homology argument, as

$$L \propto \frac{\mu^4 M^3}{\kappa},$$

with the mean molecular weight $\mu$, the mass $M$ and the opacity $\kappa$. With electron scattering, whose opacity is constant, the radius has dropped out entirely. With Kramers’ opacity, which depends on density and temperature, it comes back only weakly, as $L \propto M^{5.5} R^{-1/2}$. The interior’s capacity to carry energy outward is set by its mass, composition and opacity, and almost not at all by the size of the envelope around it.

So a star with a radiative core cannot easily change its luminosity by changing its envelope. The convective envelope of a red dwarf above about 0.35 solar masses is an outer layer that takes whatever energy the core delivers and passes it to the surface. If something makes that layer less efficient, the energy still has to get out. The envelope adjusts until it carries the same flux: it swells, and the photosphere sits further out and cooler. The luminosity, set below, is unchanged.

That is the reason all three candidate explanations land on the same line. Each makes the envelope worse at carrying heat. Each therefore inflates the star at nearly constant luminosity, and the figure alone cannot tell them apart.

Three ways to make an envelope worse at its job #

A magnetic field that inhibits convection. A field threaded through the convecting gas resists the motions that carry heat, because rising and sinking parcels have to bend field lines. In a one-dimensional model the effect can be represented by a stricter criterion for convection or by a shorter mixing length: the gas must be more superadiabatic before it moves, or moves less far before it gives up its heat. Both make the envelope less efficient and the star larger. Models built this way inflate red dwarfs by the observed amounts with surface fields of a few kilogauss, and those strengths are measured on active M dwarfs.

Starspots. A spot is a region where a strong field has locally suppressed convection, so less heat arrives and the surface is cooler. Spots covering a fraction of the surface block part of the flux. Over a short time the blocked heat is stored in the envelope. Over the envelope’s thermal timescale the star re-establishes its luminosity, and the only way to do so with part of its surface dimmed is to expand.

The model itself. The mixing length was calibrated on the Sun, whose convection zone is a thin shell, and applied unchanged to stars that are convective through most or all of their mass. The equation of state and the molecular opacities in the cool envelope of an M dwarf are harder to compute than the Sun’s. Any of these could leave standard models too compact. Unlike the first two, this explanation predicts that the discrepancy is a property of all red dwarfs of a given mass, not of the active ones.

How many spots it would take #

The spot explanation can be made quantitative with almost no model. Take a fraction $f$ of the surface covered by spots at a temperature $x$ times that of the unspotted photosphere. The star’s effective temperature, defined from its total flux, satisfies

$$T_{\rm eff}^4 = T_{\rm ph}^4\big[(1-f) + f x^4\big].$$

To lower the effective temperature by three to five per cent with dark spots at seven-tenths of the photospheric temperature, spots must cover between fifteen and twenty-five per cent of the star. Warmer spots, at nine-tenths, must cover a third to a half. Those are enormous by the Sun’s standard: even at maximum activity spots cover a few tenths of a per cent of its disc. For the most active red dwarfs they are not unreasonable. Doppler imaging and the depths of molecular bands formed in cool gas suggest that active M dwarfs can be covered by spots over a large fraction of their surface.

The eclipsing binaries that show the inflation are mostly short-period pairs. Their tides have locked them into rotating every day or two, much faster than a single red dwarf of the same age, and fast rotation makes red dwarfs magnetically active. The inflated stars are therefore, for the most part, the active ones. Both the magnetic explanations predict exactly that. Single, slowly rotating red dwarfs, whose radii come from interferometry and distances, show discrepancies too, but with more scatter and on the whole smaller ones than the tidally spun-up binaries. That points partly at activity, and leaves room for a smaller model error underneath.

A radius with a long memory #

A star cannot swell overnight. Inflating the envelope means doing work against gravity and storing heat in a larger volume, and the time over which a star can do that is set by its thermal timescale: the gravitational binding energy divided by the luminosity, $GM^2/RL$. For a red dwarf of 0.4 solar masses and 0.4 solar radii, shining at a fortieth of the Sun’s luminosity, that is about five hundred million years. A star of 0.6 solar masses takes about half as long. The convective envelope, which holds only part of the mass, responds faster, but still over tens of millions of years or more.

Starspots change on much shorter timescales. They form and decay over weeks to months, and a star’s activity cycle, if it has one, runs over years. A burst of spot coverage blocks flux at once, so the star dims a little, and the blocked heat is stored in the envelope just under the spots. Most of it leaks out again as the spots decay, long before the star has had time to expand. Only coverage sustained over the thermal timescale changes the radius.

So the radius of a red dwarf records its activity averaged over a very long time, and not its activity on the night it was observed. That has a practical consequence for the test. A star’s present spot coverage, measured from its light curve or its molecular bands, is a snapshot of something that varies. Its radius is the integral. A correlation between inflation and current activity should be real but noisy, and the scatter is not evidence against the explanation. The quantity that should correlate cleanly with inflation is the long-term rotation rate, which sets the long-term activity. In the eclipsing binaries, locked by tides to their orbital periods for their whole lives, it has been fixed and fast throughout.

How well “the same luminosity” has been tested #

The constant-luminosity line is easy to state. How tightly the data constrain it is less obvious. The radius of an eclipsing binary’s component can be measured to a per cent or better. The luminosity is not measured directly. It is computed from that radius and an effective temperature, and for red dwarfs the temperature is the weak link. Their spectra are dense with molecular bands, the continuum is hard to find, and temperatures from different methods disagree by tens of kelvin at best and a hundred or more at worst: two to three per cent.

The fourth power makes this matter. A temperature uncertain by two and a half per cent makes the luminosity uncertain by ten. A ten-per-cent inflation at fixed temperature would raise the luminosity by twenty-one per cent, so the data can exclude that, but a few per cent either way is not excluded. “The inflated stars have the luminosities the models predict” is true to about a tenth. Within that tenth there is room for the different explanations to differ.

Two more precise routes exist. One is the temperature ratio of the two stars, which the eclipse depths give far better than either temperature, so the inflation of one component relative to the other is known more precisely than either absolute value. The other is the bolometric flux, measured directly by integrating the spectrum from the ultraviolet to the infrared. Combined with a parallax it gives the luminosity without any temperature. Gaia’s parallaxes have made the second route available for nearby systems, and it is the test that would measure how constant the luminosity really is.

A star that is too large moves its planets #

Red dwarfs are where small planets are easiest to find, because a small planet blocks a larger fraction of a small star’s light. A transit measures the ratio of the planet’s radius to the star’s, so the planet’s radius is only as good as the star’s. If the star’s radius came from a model that left it too small, every planet around it is too small by the same fraction.

The consequences multiply. A planet whose mass comes from radial velocities has a density proportional to the inverse cube of its radius, so a ten-per-cent stellar radius error makes the planet a third denser than it is. That is enough to move it from a water-rich composition to a rocky one in the composition diagrams. The location of the radius valley around small stars, a quantity used to test how planets lose their atmospheres, shifts with every planet’s radius at once.

What does not move is the insolation, and that is the constant-luminosity line again. The light a planet receives is $L/4\pi a^2$; the orbit’s size comes from the period and the star’s mass, and the luminosity is the one quantity the model gets right. The boundaries of the habitable zone, which are set by insolation, are unaffected. The inflation corrupts what a planet is made of and leaves alone how much light reaches it.

Modern transit studies of red dwarfs avoid most of this by not using model radii at all. They take the star’s radius from empirical relations between absolute magnitude in the K band and radius, calibrated on interferometric diameters and eclipsing binaries. Those relations include the inflation, because they are built from the stars that show it. The cost is that they describe the average star, and the inflation is not average. An active, rapidly rotating host may be larger than the relation says, and a quiet one smaller.

The mass that survives the inflation #

The same argument decides which of a single red dwarf’s properties can be trusted. Most red dwarfs are not in eclipsing binaries, and their masses have to be estimated from something observable. For a star near the Sun, two things are observable with high precision: its distance from a parallax, and therefore its absolute magnitude, and its colour. The mass can be read from either, through a calibration.

The calibrations that work are the ones built on luminosity. An empirical relation between the absolute magnitude in the infrared K band and the mass, calibrated on visual and eclipsing binaries with dynamical masses, gives single red dwarfs’ masses to two or three per cent. It works because the infrared luminosity is nearly the quantity the interior fixes. Mass decides the luminosity, and inflation, which moves radius and temperature against each other, leaves the luminosity almost where it was. An active red dwarf and a quiet one of the same mass sit at nearly the same absolute magnitude, so the relation reads the same mass for both.

A relation built on temperature or colour would not work as well. The inflated star is cooler, so it would be read as a lower-mass star; along the lower main sequence the temperature changes only slowly with mass, by about a fifth of a per cent for each per cent of mass, so a five-per-cent temperature deficit near 3,500 kelvin would be read as a mass a fifth or more too low. A relation built on radius would be worse still, since the radius is the quantity that moves most. The hierarchy of trust among a red dwarf’s properties follows the constant-luminosity line: luminosity best, temperature worse, radius worst.

What the figures leave out #

The constant-luminosity argument was made for a star with a radiative core, and red dwarfs below about 0.35 solar masses have none: they are convective throughout. For them the luminosity is set by the nuclear burning at the centre and by the outer boundary condition together, and there is no radiative interior to pin it independently of the envelope. Models that inflate fully convective dwarfs by inhibiting convection or adding spots find the same trade, a larger radius and a lower temperature at only slightly changed luminosity. The reason is less transparent, and the strong interior fields some of those models need are hard to sustain. The well-measured fully convective pairs, such as CM Draconis, remain the hardest systems to fit.

The homology scaling is a scaling, not a model. It says how the luminosity responds to a change in radius at fixed mass and composition. It does not say the response is zero, and for Kramers opacity it is not. The spot model takes the photosphere outside the spots as unchanged, while a real spotted star also brightens slightly around its spots and redistributes some blocked flux. Both refinements change the numbers by a fraction of the effect without changing its direction.

Still open: activity, or the models underneath #

The two magnetic explanations predict that inflation follows activity: rapid rotators, strong fields, deep molecular bands. The model explanation predicts that it follows mass, the same for every star of a given mass whether or not it is active. The binaries cannot easily separate these, because almost all of them are active, spun up by their companions. The separation needs inactive red dwarfs with radii measured to a per cent. Interferometry provides those for a few dozen nearby single stars, and long-period eclipsing binaries, wide enough that tides have not spun them up, provide them in principle and are rare.

If the quiet stars agree with the models, the models are right and red dwarfs are swollen by their own magnetism. If the quiet stars are also inflated, by a smaller amount, the convection theory that fixed the Sun’s radius is missing something for stars whose convection reaches almost to the centre, and activity adds on top. Either way the luminosity stays where it is, because it is decided below the layer where the disagreement is.