Astronomy & Celestial Mechanics
the sky, computed
Concept

Stellar activity — where it appears

Magnetic phenomena on a star's surface — spots, plage, flares — that vary its brightness and distort its spectral lines. It is the limiting systematic in radial-velocity planet searches and in any transit measurement at the parts-per-million level.

Named by 7 essays across 2 fields — each of them below, with the objects they name alongside it.

A 6.2 m s⁻¹ signal at the rotation period, made by no planet at all. Above: the apparent radial velocity of a star carrying one dark spot over 0.4 per cent of its disc, rotating with an equatorial velocity of 3.2 km s⁻¹, over 3 rotations — and beside it a circular-orbit planet of the same period fitted to the same amplitude, 6.2 metres a second. That amplitude is several times the precision of a modern spectrograph and squarely inside the range in which warm sub-Neptunes are claimed, so the two are competing on equal terms. The spot signal is not a sinusoid: the spot is in view for half the rotation and hidden for the other half, so the curve is truncated, and its power at half the period is 1.16 of its power at the period. A Keplerian orbit at the same period has none there at all. Below: the diagnostic that actually settles it. A planet moves the whole spectrum bodily, so every line keeps its shape and the bisector — the locus of midpoints up a line profile — does not move; a spot removes light from one side of the profile, so the bisector tilts in step with the velocity and against it. The two clouds correlate at -0.92 and 0.14. What the picture cannot show is why this took so long to become routine: measuring a bisector to a few metres a second needs a signal-to-noise ratio of several hundred per spectrum, so for two decades the diagnostic existed and could not be applied to the faint stars the interesting claims were about.

A planet that was the star's own rotation

A dark spot rotating across a star removes light from the approaching limb and then the receding one, and the line centroid moves. That is several metres a second at the rotation period, from no planet at all — and two of the most celebrated nearby planets were withdrawn on exactly this evidence.

exoplanets · Reflex velocity The same wind strips a mini-Neptune inside 0.06 au and leaves a hot Jupiter intact. The fraction of a planet's hydrogen envelope removed in 5 billion years by an energy-limited wind, against orbital distance, at a heating efficiency of 0.15 and an extreme-ultraviolet fluence integrated over the star's own history — saturated at L_XUV/L_bol = 3.2·10⁻⁴ for the first 100 million years and declining as t to the power −1.23 after, which comes to 4.9·10¹⁵ J m⁻² at one astronomical unit and is some 7 times what today's flux would give over the same span. This is a different mechanism from the tail of a Maxwellian, not a correction to it. Close to a star the upper atmosphere absorbs more extreme ultraviolet than it can radiate away, expands, and flows off as a wind whose rate is set by the energy arriving — Ṁ = ηπR³F/GM — so the exponential in the Jeans parameter vanishes entirely and what remains is a ratio of radius cubed to mass, which is one over the density. That is why the three curves are ordered as they are. The hot Jupiter is dense enough to lose 0.0087 of itself even at 0.047 au, where HD 209458 b sits and where its escaping hydrogen makes a transit fifteen per cent deep in Lyman α against one and a half per cent in the optical — an exosphere filling and overflowing the Roche lobe, and still costing the planet almost nothing. The mini-Neptune loses its whole envelope anywhere inside 0.06 au, and what is left when it does is a bare core about 1.5 Earth radii across. That is one of the two standard accounts of the gap in the radius histogram, and this figure is what it looks like before any of the observations are brought in.

A planet ten times larger in one colour

A hot Neptune that blocks one and a half per cent of its star's light in the optical blocks fifteen per cent of it in the ultraviolet line of hydrogen. No bound atmosphere can be that large — the material is well outside the planet's Roche lobe — so the observation is not a measurement of an atmosphere but of one leaving.

exoplanets · Atmospheric escape The evidence for a period, one trial period at a time. The natural logarithm of the Bayes factor for a sinusoid at each trial period against no signal, for 120 radial velocities spread over 359 days with 2.0 m/s white noise per point and a planet of semi-amplitude 1.0 m/s at 11.3 days. The amplitude prior is Gaussian with scale 3.0 m/s in each component. At the planet's period the evidence is 14.4 — odds of about 1.8·10⁶ to one — and almost everywhere else it is near zero or negative, because a sinusoid that fits nothing is penalised for the amplitude range it wasted. The quantity that answers the question "is there a planet at some period" is not the peak but the average of the Bayes factor over the prior on the period, here uniform in the logarithm of the period from 1.5 to 1,000 days, and that average is ln B = 7.7: 6.7 lower than the peak, the Bayesian form of the look-elsewhere penalty, paid because the data had to find the period as well as the amplitude.

Odds for a planet that depend on a guess about its size

A false-alarm probability asks how often noise alone would produce a peak this tall. A Bayes factor asks something better — how much more probable the data are with a planet than without one — and pays for it with two numbers nobody measured. The width allowed for the period barely matters. The width allowed for the amplitude can turn strong evidence into none, from the same velocities.

starlight · Periodograms The red velocity against the blue one. Simulated velocities of two stars, each measured simultaneously in a blue channel at 0.45 micrometres (across) and a red one at 0.8 (up), with 0.6 m/s of noise in each. One star (filled) has a 4 m/s planet on a 9.3-day orbit: its points lie along the diagonal, slope 0.97, because a Doppler shift is the same velocity in every colour. The other (open) has no planet and a spot 900 K cooler than its 5100 K photosphere, rotating every 12 days and evolving: its points lie along a shallower line, slope 0.64, which is the ratio of the spot's contrasts in the two channels. Neither star's blue velocities alone say which it is. The flux-deficit spot model is the simplest case; a spot whose signal is mostly suppressed blueshift lies closer to the diagonal.

A planet has one velocity in every colour

A Doppler shift moves every line in a spectrum by the same velocity, because it is the whole star that moves. A starspot does not move anything; it removes light, and how much light it removes depends on the colour it is seen in. Measuring the velocity in the blue and the red at once separates the two in a single night — until the star's magnetic field, which grows more visible towards the red, takes part of the separation back.

exoplanets · Reflex velocity The width of an emission core against a star's luminosity. Absolute visual magnitude against the logarithm of the width of the calcium K emission core in km/s, for the Wilson–Bappu relation Mᵥ = 27.59 − 14.94 log W₀ (line), with 120 illustrative stars scattered about it by 0.5 magnitudes, the relation's typical scatter, and the Sun marked at about 33 km/s and Mᵥ = 4.83. Over a range of fifteen magnitudes in luminosity — from dwarfs fainter than the Sun to supergiants a million times brighter — the logarithm of one width predicts the magnitude to half a magnitude, whatever the star's temperature. A factor of two in width is four and a half magnitudes in brightness, sixty times in luminosity.

A width that knows how bright the star is

In the dark trough of a cool star's calcium K line sits a small bright emission core made in its chromosphere. In 1957 Olin Wilson and Vainu Bappu found that the core's width predicts the star's absolute magnitude over fifteen magnitudes — from dwarfs to supergiants, whatever the star's temperature and however active its chromosphere — and the simplest physical account of why reproduces the relation's existence and misses its slope by half.

starlight · Spectra A spot's mark on a line, and how much of it a shift explains. Above: the disc-averaged profile of one absorption line from a star rotating at 5 km/s, observed at a resolution of 100,000, without (dashed) and with (solid) a dark spot 8° in radius at latitude 20°, 70 per cent darker than the surrounding surface, on the receding side of the disc. Below, magnified 185 times: the change the spot makes (blue), the part of it a pure Doppler shift of the whole line would make (orange, a shift of −36.3 m/s), and what is left over (green). The spot removes light from the velocity at which its own surface is moving, putting a bump into the line where it was darkest. Most of the change, 92 per cent of its square, looks like a shift; 8 per cent does not, and that remainder is the only part of the spot's signal a spectrum could tell apart from a planet.

A spot a slow star hides inside its velocity

A dark spot on a rotating star removes light from one part of each spectral line, and the line's centre seems to move: a velocity of metres per second that no planet caused. In principle the spot betrays itself, because it changes the line's shape and a planet does not. In practice the betrayal is a few per cent of the signal on a star that rotates as slowly as the Sun, and seeing it takes a signal-to-noise of hundreds per pixel across thousands of lines.

starlight · The Doppler effect The same worlds, placed by the light that strips. The bodies of the shoreline plane, each moved from its present instellation to the cumulative X-ray and extreme-ultraviolet energy it has received over its star's life, both relative to the Earth's (arrows). The Sun's planets do not move, by construction. A planet of a small red dwarf moves up by a factor of 7 to 16: its star stayed at its saturated activity level — a thousandth of its light in X-rays and extreme ultraviolet — for 2.3 billion years for TRAPPIST-1 against 0.1 for the Sun, and its planets have absorbed that for longer. TRAPPIST-1 e, which receives 0.65 of the Earth's sunlight, has received 10 times the Earth's lifetime XUV and moves from well inside the line to above it. Every planet measured as bare, TRAPPIST-1 b, TRAPPIST-1 c, LHS 3844 b, now lies above the line. The saturation times are a scaling with stellar mass rather than a fit to any one star.

A shoreline measured in light already spent

In the solar system one line of slope four, in the plane of escape velocity against sunlight, separates every world with air from every world without. Drawn against the light a planet receives today it puts the temperate planets of a small red dwarf safely on the side with air. Drawn against the stripping light their star has already spent, it moves them an order of magnitude toward the airless side — and leaves most of them inside a band that the line's unknown height cannot resolve.

exoplanets · Exoplanet atmospheres

Named alongside it

The objects these essays reach for when they reach for this one.

Radial-velocityConvective blueshiftStarspotEnergy-limited escapeFalse-positiveRotation periodAbsolute magnitudeActivity indicatorBayes factorBlackbodyCharge exchangeChromosphere

All concepts