A helium line that weighs the star first

For a decade the only way to watch a planet lose its atmosphere was the Lyman-α line of hydrogen, seen from space, in the few systems near enough that the interstellar gas had not already swallowed the line’s core. In 2018 a second line arrived. Helium atoms in a long-lived excited state — the 2³S level, nineteen electronvolts above the ground state and unable to decay by any fast route — absorb at 1083 nanometres in the near infrared, where the interstellar medium is transparent and where ground-based spectrographs work. Within a year the line had been seen around half a dozen planets, and it is now the workhorse of the subject.

The essay that introduced it noted, in a paragraph, that the line has a selection effect of its own: the level it comes from is filled and emptied by the star’s light, so a non-detection is a statement about the star as much as about the planet. This essay works the paragraph out. It follows one planet’s outflow through the level’s balance to a transit depth, around four kinds of star, and asks what a measured depth says about the rate at which the planet is losing gas — which is the number the whole theory of escape is built to predict.

From a rate to a depth #

The calculation has three steps, and each has a simplification worth stating.

The first is the outflow. Gas heated by the star’s extreme ultraviolet to several thousand kelvin flows away from the planet as a wind, accelerating through a sonic point, much as the Sun’s corona flows away as the solar wind. The simplest description is the one Parker wrote down for the Sun’s own corona, an isothermal wind: a single temperature, a single mass-loss rate $\dot M$, and a velocity that follows from the balance of pressure and gravity. The density at each radius is then $\dot M / (4\pi r^2 v)$. Real outflows are not isothermal — they are heated at the base and cool as they expand — but the isothermal wind has the right structure, and its temperature is the one parameter that stands in for everything the heating does.

The second is the level’s population. Helium in the wind is ionised by photons shorter than 504 ångströms. When an ion recombines, about three-quarters of the time it lands in the triplet system, and every triplet cascade ends in 2³S, where the atom is stuck: the transition back to the ground state is forbidden twice over, and the radiative lifetime is two and a quarter hours. It leaves the level in three ways. A photon with wavelength shorter than 2593 ångströms — the mid-ultraviolet — can ionise it. A collision with an electron can knock it across to the singlet system, from which it decays at once. Or, if neither happens for hours, it decays. Balancing the filling against the three emptyings at each radius gives the fraction of helium in the level.

The third is the geometry. During transit — the same geometry that turns a colour-dependent radius into an atmosphere — the starlight passing through the wind at each impact parameter outside the planet is absorbed in proportion to the column of metastable helium along that chord, each atom absorbing over a line broadened by its thermal motion and shifted by the wind’s speed along the line of sight. Summing over the annuli and dividing by the star’s disc gives the excess depth at the line’s centre — what a spectrograph records as a dip in the line, deeper than the dip the planet’s body makes in the continuum.

The same wind around four stars #

The hero figure holds the planet and the wind fixed and changes only the star. The planet is modelled on HAT-P-11 b, a Neptune-sized world on a five-day orbit around a K dwarf, one of the first planets in which the line was seen; the stars are four representative spectra, an F star, a Sun-like star, an active K dwarf and an active M dwarf, each reduced to three photon fluxes — below 912 ångströms, which ionises hydrogen; below 504, which ionises helium; and between 504 and 2593, the band that empties the metastable level.

The differences are large. At a mass-loss rate of $10^{10}$ grams a second, a typical estimate for a Neptune this close to its star, the same wind absorbs 0.06 per cent of the light at line centre around the F star and 0.40 per cent around the M dwarf, a factor of seven. Read in reverse, as an observer must, a measured excess of one per cent means a mass-loss rate of $1.1 \times 10^{11}$ grams a second if the host is the F star and $2.9 \times 10^{10}$ if it is the M dwarf, a factor of four that comes from the star alone.

The pattern matches what the surveys have found. Detections cluster around K dwarfs — HAT-P-11 b, WASP-107 b, HAT-P-18 b — and some of the planets expected to be losing gas fastest, hot Jupiters around F and early G stars, show nothing. A survey that ranked planets by their helium depths would rank their stars as much as their atmospheres.

A balance set by two bands of light #

The level holds a few millionths of the helium at most, which is why the absorption is measurable only because helium is abundant and the line is strong. What the figure shows is how differently that fraction is distributed through the wind.

Near the planet the gas is dense enough that collisions with electrons empty the level far faster than any photon does, and the fraction is set by the ratio of recombination to collisional transfer — a property of the gas, nearly the same around every star. Further out the density falls as the square of the radius times the inverse of the speed, collisions become rare, and the star takes over. Around the F star, whose photosphere is hot enough to pour out mid-ultraviolet light — 150 times as much as the K dwarf’s in the band that matters — the level is photoionised as fast as it fills, and the fraction falls by two orders of magnitude between the base of the wind and the Hill radius. Around the K dwarf it falls by a factor of three. Around the M dwarf, with thirty times less mid-ultraviolet again, it hardly falls at all.

The absorption is mostly made in the outer parts of the wind, because that is where the area is: an annulus at five planet radii covers twenty-five times the area of one at the planet’s surface. So the stars that empty the outer wind lose most of the signal, and the peak fractions, which differ by a factor of three, understate a difference in depth that is a factor of seven.

The two bands do not pull with equal strength. At a fixed mass-loss rate the depth answers to the mid-ultraviolet with roughly the square-root power of its inverse, and to the extreme ultraviolet with roughly the quarter power. The asymmetry has a simple cause. Most of the helium in the wind is already ionised; more ionising light ionises a little more, adds a few electrons, and slightly raises the recombination that fills the level. But every extra mid-ultraviolet photon acts directly on the atoms in the level.

The figure holds the mass-loss rate fixed, and in reality it is not fixed. The extreme ultraviolet is what heats the wind and drives it, and a planet receiving ten times more of it loses gas several times faster — in proportion, up to a point, and as a square root beyond it. So the extreme ultraviolet acts on the line twice: weakly through the level’s balance and strongly through the rate. The mid-ultraviolet acts only on the level, and for a cool star it is the one band whose influence on the depth has nothing to do with escape at all.

A depth is a line, not a point #

The mass-loss rate is not the only property of the wind the depth depends on. The temperature sets the wind’s speed — the sound speed rises as the square root of the temperature — and a faster wind of the same rate is thinner at every radius.

So a measured depth fixes a curve in the plane of temperature and rate rather than a point in it. A cool wind at four thousand kelvin needs fifty times less gas than a hot one at fourteen thousand to absorb the same one per cent: the cool wind is slow and dense, it lingers near the planet, and its electrons fill the metastable level by recombination quickly; the hot wind is fast and thin and carries its helium out to where the star’s light empties the level.

The temperatures that matter are not exotic. Models of the heating, which balance the energy absorbed against the energy radiated and carried off, give outflow temperatures between five and ten thousand kelvin for Neptunes on close orbits, with hotter winds around stars with more extreme ultraviolet and cooler ones where radiative cooling by Lyman-α is efficient — the thermostat that caps the heating in the photon-limited regime. Even across that narrower range the rate a depth implies changes by a factor of five. Every published helium mass-loss rate quotes a temperature, and the temperature is usually assumed rather than measured.

The width that breaks the degeneracy #

A temperature from outside the depth is needed, and the line itself can supply one, if it is resolved.

Two winds on the curve, with the same central depth, make lines of very different shapes. The cool wind’s helium sits close to the planet, moving slowly, with a thermal width of a few kilometres a second; its line is narrow and peaked. The hot wind’s helium is spread across the whole Hill sphere and moving outwards at tens of kilometres a second, half of it towards the observer and half away; its line is three times wider, with the same depth at the centre and far more absorption in its wings. The equivalent width — the area of the dip — differs by the same factor.

That is the measurement that turns a depth into a rate, and it has requirements that are worth stating plainly, because they decide which observations can do it. Twelve kilometres a second is 0.04 nanometres at 1083, so the line has to be resolved by a spectrograph of resolving power near a hundred thousand; most of the first detections were made with lower resolution, which gives a depth and an equivalent width but blurs the shape. The profile also has to be free of other broadening: the planet’s own orbital motion changes its velocity by several kilometres a second during a transit, winds around the planet blow the gas from the day side to the night, and gas beyond the Hill radius, swept into a tail by the star’s radiation and wind, adds absorption offset to one side — the same tail that made the Lyman-α transit so deep. Each of these has been seen. The fits that extract a temperature from a resolved helium line are fitting all of them at once, and the temperature they return is the one the model’s broadening needs, not a measurement of the gas’s kinetic temperature alone. A line’s width is rarely one thing.

A spectrum that cannot be observed #

The last step of the inference is the one most often skipped. To turn a depth into a rate, the model needs the star’s flux in the three bands, at the planet. None of them is measured for most host stars.

The extreme ultraviolet is absorbed by the neutral hydrogen between the stars within a few parsecs at most wavelengths, so for every planet host it is reconstructed — from the star’s X-ray luminosity through an empirical scaling, from the core of its Lyman-α line reconstructed through the interstellar absorption, or from models of a chromosphere and corona fitted to what can be seen. The methods disagree with one another by factors of two to five for the same star. The mid-ultraviolet is easier, since it is not absorbed, but it needs a space telescope, and the ones that can measure it are few and old; for most hosts it comes from a model photosphere, which for active stars misses the chromospheric emission that dominates the band below about 2,000 ångströms.

The errors propagate partly. Around a Sun-like star, where the mid-ultraviolet controls the level in the outer wind, an error of a factor of three in that band becomes an error of about one and a half in the rate: 38 per cent of the error, in the logarithm, survives the inversion. Around an active K dwarf, where electrons empty most of the level that matters, the same error costs 17 per cent. The extreme ultraviolet, at fixed temperature, costs least — but at fixed temperature only. Through the heating, it also sets the temperature, which the previous figures showed to be worth a factor of fifty.

So the honest uncertainty on a helium mass-loss rate is the product of three things: the depth’s measurement error, usually small; the temperature, wide unless the line is resolved and modelled; and the star’s spectrum, a factor of a few on its own and amplified or damped according to which process controls the level. Published rates often carry error bars that reflect only the first.

What the line is good for anyway #

None of this makes the line less valuable; it changes what kind of measurement it is. Three things survive the uncertainties.

The detection itself. An excess absorption at 1083 nanometres extending above the planet’s body, lasting longer than the transit, means gas outside the planet in a state that requires it to be escaping or at least extended far beyond any bound atmosphere. That is a yes-or-no answer to whether a planet is losing its envelope now, and it is robust to every factor above.

The comparison of one planet with itself. A planet observed at two epochs, around one star whose spectrum is unknown but unchanged, gives a ratio of depths that cancels the stellar uncertainty — and a star’s activity cycle, which changes its extreme ultraviolet by factors of several, is then a controlled experiment on the wind. The mid-ultraviolet of a cool star varies far less over a cycle than its extreme ultraviolet does, because it comes mostly from the photosphere rather than the chromosphere and corona, so a depth that rises and falls with the star’s activity is the wind responding to its driver, with the level’s balance nearly held still. Following one planet through its star’s cycle, or through a single flare, is the closest thing the subject has to turning the driver up and down and watching the outflow answer.

And the comparison of planets around the same kind of star. Within a sample of K dwarfs the conversion from depth to rate differs much less than across spectral types, and the trend of depth with a planet’s gravity or irradiation, inside that sample, is a trend in escape rather than in stars. The radius valley, where the stripping of envelopes is supposed to have sculpted the planet population, is best tested this way — young planets near the valley around stars of one type, observed in the one line that the ground can reach.

What the calculation assumes #

The wind is isothermal, spherically symmetric, and extends no further than the Hill radius. A real outflow is heated at its base and cools as it expands, which puts more of its helium at the temperatures near the base; it is shaped by the star’s tide and by the stellar wind it runs into, which confine it on the day side and stretch it into a tail on the night side; and the gas beyond the Hill radius absorbs too.

The level populations are in local balance. In a fast wind, gas is carried outwards faster than the level can come to equilibrium at large radii, and the metastable fraction there remembers conditions further in. Full calculations that include the transport find fractions differing from the local ones by tens of per cent in the outer wind, in the direction of more absorption for fast winds.

The photon fluxes are attenuated by the neutral columns between each radius and the star, with cross-sections averaged over each band. A calculation resolved in wavelength attenuates the softest photons first and hardens the radiation reaching the inner wind. The four stellar spectra are representative rather than measured, and real stars of each type differ from them by factors of several in the extreme ultraviolet.

The helium is assumed to flow at its cosmic proportion to hydrogen, one atom in ten. It need not. Helium is four times heavier than hydrogen, and it leaves the planet only because the escaping hydrogen drags it along; a wind whose hydrogen flux falls below a crossover set by the two species’ masses and their mutual collision rate lets the heavier atoms fall behind. For the winds drawn here the hydrogen flux is far above helium’s crossover and the fractionation is small. For the weakest winds, around old, quiet stars, it is not, and a missing helium line there could mean that the helium was left in the atmosphere while the hydrogen escaped — a third reading of a non-detection, after a weak wind and an unfavourable star.

And the line is the blend of the two strongest components of the triplet. The third, 1.2 ångströms to the blue with an eighth of their combined strength, is sometimes used as a check on the optical depth: in an optically thin wind the ratio of the two features is fixed by atomic physics, and a departure from it says the line is saturated — which, at these depths, it begins to be in the cool, dense winds.

Still open: the star behind the planet #

The helium line moved the measurement of atmospheric escape from space to the ground, and from one planet to dozens. What it measures is a column of atoms whose number is set as much by the host star’s ultraviolet spectrum as by the planet’s mass loss, and whose interpretation needs a temperature the depth alone cannot supply.

The two improvements in view address the two halves. Spectrographs resolving the line at a hundred thousand and better, on telescopes large enough to reach the faint K and M dwarfs that host most small planets, give profiles from which a temperature can be fitted, at the cost of modelling every other source of broadening. Ultraviolet spectroscopy of the hosts — the mid-ultraviolet directly, and the extreme ultraviolet through better-calibrated proxies — gives the level’s balance. Until both are in hand for a planet, its helium depth is a robust detection and an uncertain rate, and the comparison that matters most, between the planets that kept their gas and those that lost it, is safest when it is made between planets whose stars are alike.