Astronomy & Celestial Mechanics
the sky, computed
Concept

Starspot — where it appears

A region of a stellar surface darkened by a magnetic field strong enough to suppress convection there. As the star rotates the spot distorts the disc-integrated line profile, producing an apparent velocity signal at the rotation period from no orbital motion at all.

Named by 7 essays across 3 fields — each of them below, with the objects they name alongside it.

A 28.7 km/s correction, and a 12.5 m/s planet underneath it. Two years of radial velocities of a star at ecliptic latitude 12°, orbited by a companion whose reflex semi-amplitude is 12.5 m/s — the Sun's own, from Jupiter. The upper panel is what the spectrograph measures: the Earth's motion about the barycentre of the solar system, amplitude 28.72 km/s, which is V⊕ cos β to a fraction of a per cent. The planet is in that curve and is 2,297 times smaller than it, which is a line thinner than the stroke it is drawn with. The lower panel is the same data after the correction, and the correction is not a fit: it is computed from an ephemeris, the observatory's position on a rotating deformable Earth, and the star's own coordinates and proper motion. To leave a centimetre a second it has to be right to one part in 2.9·10⁶ — the light-travel time across the Earth's orbit, the relativistic terms, and the fact that the star moves are all inside that budget. What remains is the planet, at 4333 days, and a scatter of 1.2 m/s that is the star rather than the instrument.

The metre per second that is not the star

A line shift is a speedometer, and the speed it reads is mostly the observer's. Getting to a metre a second means removing thirty kilometres of the Earth's own motion to a part in three million, and then confronting a floor that is the star's own surface rather than the instrument.

starlight · The Doppler effect 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 Two models through the same points, one with a planet. 60 simulated velocities over 200 days of a star rotating every 25 days, with spot activity drawn from a quasi-periodic process of 3 m/s amplitude and a 40-day decay, white noise of 1 m/s, and a planet of 2.5 m/s semi-amplitude on a 27.0-day orbit — 1.08 times the rotation period. Two models are fitted, each with its activity amplitude and decay time chosen by maximum likelihood: activity plus the planet (solid), whose planet comes out at 3.3 m/s, and activity alone (dashed), which has no planet in it. Both pass through the points; the likelihoods differ by 3.6 in the logarithm, a ratio of 34.9, which is not decisive. Near the rotation period a flexible model of the star can carry the planet's signal by itself.

A model of the star that can carry the planet

Starspots are modelled as correlated noise with the star's rotation period built in, and the model is good enough to remove signals that would otherwise be announced as planets. It is also good enough to remove planets. Near the rotation period no amount of data in one season separates the two, and away from it a flexible noise model quietly takes most of the evidence a planet would have had.

exoplanets · Reflex velocity 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 Larger and cooler, along a line of constant light. The fractional change in effective temperature against the fractional change in radius for a star whose luminosity is unchanged, and for luminosities −10% and +10%, from L = 4πR²σT⁴. Holding the luminosity, a radius 10 per cent larger needs a temperature 4.7 per cent lower: about half the radius change, with the sign reversed. The shaded box is the range reported for red dwarfs in eclipsing binaries against standard models — radii 5 to 15 per cent too large, temperatures 3 to 5 per cent too low — and it lies along the line of constant luminosity.

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

Red dwarfs in eclipsing binaries, whose radii are measured directly, come out several per cent larger than stellar models predict for their masses and a few per cent cooler. They are not brighter or fainter. The light is decided deep inside, by mass and opacity, and whatever disturbs the convecting outer layers can change only the radius and temperature through which that light leaves.

stars · Energy transport 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 Water that is on the star and not on the planet. Excess transit depth, in parts per million, against wavelength, for a planet of depth 7000 ppm around a star of 2600 K. The upper curve is a bare rock with no atmosphere on a star whose disc is 10% covered by 2300 K spots that hold more water than the rest of the photosphere: it shows a broad slope and a band near 1.4 μm, 36 ppm from its shoulder. The lower curve is the signal of an Earth-like air column with water on an Earth-sized planet, about 93 ppm at the band. The band in the spot spectrum is an imposed illustration rather than a model atmosphere, but its sign is measured: it adds where a planet's own water would add.

Water that is on the star and not the planet

A transit depth is the planet's shadow divided by the brightness of the whole star, and the shadow falls on only one strip of it. If the strip is cleaner than the rest of the disc, the depth comes out too large, by an amount that changes with colour. The star's own spots hold water, so the error has water's fingerprint. On the smallest red dwarfs, whose rocky planets are the best targets there are, that false signal is larger than the real atmosphere being looked for.

exoplanets · Exoplanet atmospheres

Named alongside it

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

Radial-velocityConvective blueshiftFalse-positiveRotation periodStellar activityLine bisectorM dwarfStellar jitterActivity indicatorBarycentric correctionBlackbodyConvection

All concepts