Astronomy & Celestial Mechanics
the sky, computed
Series

Reflex velocity — the series

7 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The wobble of a star with a circular companion. The star's velocity along the line of sight, over two orbits, computed from the companion's orbit. A circular orbit gives a sine wave; an eccentric one gives a skewed curve whose shape encodes the eccentricity.

    The star moves, and the mass is a lower bound

    A planet is found by watching its star fall towards it. The wobble gives a mass multiplied by the sine of an angle nobody has measured — and the shortfall is not a rounding error.

    part 1 · exoplanets
  2. 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metres a second at the rotation period, from no planet at all — and two of the most celebrated nearby planets were withdrawn on exactly this evidence.

    part 2 · exoplanets
  3. One eccentric planet, drawn as two circular ones. Top: the velocity of a star with one companion on an orbit of eccentricity 0.25 (argument of periastron 0.9 radians), in units of its semi-amplitude K, over two orbits — and on top of it the sum of two circular orbits, one at the same period and one at half of it, fitted by taking the first two terms of the exact curve's Fourier series. The second orbit's amplitude comes out at 0.246 of the first, which is the eccentricity to first order. Bottom: the difference between the two, on a scale 12 times magnified; its largest excursion is 0.0889 K and its rms 0.0473 K, second order in the eccentricity. For a planet with K = 30 m/s that is 1.42 m/s rms — below the noise of most instruments — so an eccentric single planet and a pair of planets in a 2:1 period ratio are the same measurement unless the data are good enough to see a term of order e².

    One eccentric planet, or two circular ones

    The velocity curve of a star with one planet on an eccentric orbit is, to first order in the eccentricity, exactly the curve of two planets on circular orbits with periods in the ratio two to one. The difference between the two readings is a term in the eccentricity squared, and for the modest eccentricities most planets have, it is smaller than the noise.

    part 3 · exoplanets
  4. An exposure time chosen by the star. The fraction of a star's p-mode velocity signal that survives averaging over an exposure of the length on the horizontal axis, for an envelope of modes centred on νₘₐₓ with the stated width. The dashed curve is a single oscillation at νₘₐₓ = 3.1 mHz, a period of 5.4 minutes: it averages to exactly zero after one period and at every multiple. The solid curve is the real case, a band of modes 1.2 mHz wide, which cannot all be nulled at once: its first minimum is at 5.8 minutes, where 15% of the rms survives, and beyond it the residual falls only as one over the exposure time. Short exposures on bright stars sample the oscillations rather than averaging them, which is why a bright star observed quickly can be noisier than a faint one observed slowly.

    An exposure time chosen by the star

    A Sun-like star's surface rings at five minutes and boils on timescales from ten minutes to a day, and each of those motions shifts its spectral lines by more than the nine centimetres a second an Earth would. The ringing can be averaged away by an exposure tuned to the star; the slowest boiling cannot be averaged away within a night at all, and it sets the number of nights an Earth twin costs.

    part 4 · exoplanets
  5. 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.

    part 5 · exoplanets
  6. 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.

    part 6 · exoplanets
  7. Curvature turns a curve into an island. The companions — separation across, mass up, on edge-on circular orbits — that fit simulated velocities of a star with a 12 Jupiter-mass companion at 22 AU (period 103 years), within a Δχ² of 6.18 of the best fit, for observing spans of 6, 12, 24 years with 60 velocities of 2 m/s noise each. Over 6 years the data show a slope and the allowed region is a long band along M ∝ a², spanning 3.3 decades in mass. As the span grows the curvature of the orbit appears, and the band contracts — to 1.0 decades at 24 years — towards the true companion. Inclined orbits, which the fit leaves out, would extend every region to higher masses.

    A slope that is a whole curve of companions

    A companion on an orbit longer than the observations shows in a star's velocity only as a steady drift. A drift is one number, an acceleration, and an acceleration is a mass divided by the square of a distance — so a planet at ten astronomical units and a brown dwarf at fifty produce the same straight line. Turning the line into a companion needs the orbit to bend, a picture, or the other two components of the acceleration.

    part 7 · exoplanets

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