Astronomy & Celestial Mechanics
the sky, computed
Concept

Selection effect — where it appears

A pattern in a sample produced by how the sample was gathered rather than by the population. Every survey has one, and the difference between an occurrence rate and a raw count is entirely the work of undoing it.

Named by 8 essays across 4 fields — each of them below, with the objects they name alongside it.

Two radial-velocity curves, and one mass ratio. The line-of-sight velocity of each star through one orbit of AI Phoenicis. Both curves are computed from the two masses and the period; what a spectrograph delivers is the reverse. The ratio of the amplitudes is the inverse ratio of the masses — 48.2 to 50.3 kilometres a second, so the heavier star moves more slowly — and the sum of the amplitudes with the period gives the mass sum, 2.437 solar masses, once the inclination is known from the eclipses.

The only stars whose masses are known

A star's mass cannot be measured by looking at it. It can be measured by watching two stars pull on each other, and if the pair also eclipses, the same observations give both radii as well — with no stellar model anywhere in the chain. A few hundred such systems calibrate everything else.

stars · Binary stars What each method can see. Planet mass against orbital distance, both logarithmic, with the detection threshold of each method drawn as the boundary it actually is. Radial velocity at 1 m/s needs mass rising as √a; astrometry at 20 µas needs it falling as 1/a, which is the only method that gets easier further out; a 100 ppm transit is a threshold on radius and so a horizontal line at about 1.4 Earth masses, cut off at 1.21 AU by the need for three transits in 4 years; direct imaging begins outside the diffraction limit, 0.6 AU at 10 parsecs for a 39 m aperture at 10 µm. The solar system is drawn on top: for two decades every one of its planets except Jupiter lay outside every region, which is the whole of what the early census was measuring.

Every survey draws a different sky

The first exoplanets found were enormous and impossibly close to their stars. That was not a discovery about planets. It was a measurement of what a 10 m/s spectrograph watching for three years is able to see.

exoplanets · Detection bias What distance does, and does not, do to a galaxy. Three quantities against distance, each relative to its value at 5 Mpc, on logarithmic axes so that a power law is a straight line and its exponent is the slope. Flux falls with slope −2 and angular size with slope −1, both of which are ordinary. Their ratio has slope zero: a galaxy of surface brightness 23.5 magnitudes per square arcsecond has that surface brightness at every distance, and a sky of 22 is brighter than it at every distance too. The contrast against the sky — the quantity that decides whether the thing is detectable at all — is -1.5 magnitudes wherever it is put.

The brightness distance cannot touch

Flux falls as the inverse square of distance and so does solid angle, so their ratio does not fall at all. A galaxy's surface brightness is the same number wherever it is put, which means whole populations can be undetectable at any distance whatever.

galaxies · Surface brightness The same orbit, realigned by one star and not by the other. The time an equilibrium tide takes to bring a planet's orbit into the plane of its star's equator, against orbital separation in stellar radii, for a planet of 1e-3 stellar masses. Both axes are logarithmic. The two curves differ only in how efficiently the star dissipates the tide, by a factor of 10⁴ — the contrast between a star with a convective envelope, where turbulence turns the tidal flow into heat, and one hotter than about 6,250 K, which has almost none. The lower curve is calibrated so that a Jupiter at 8 stellar radii realigns a cool star's orbit in 1 billion years, which is what the aligned systems require; the tidal quality factor of a star is not known from first principles and this is the honest way to say so. Everything else follows from the sixth power of the separation, which is measured off the drawn curve as 6.000. The two curves cross a Hubble time at 12.4 and 2.7 stellar radii, and their ratio is the sixth root of the dissipation contrast. Hot Jupiters sit between those two numbers. So the same arrival distribution of orbital tilts is erased around cool stars and preserved around hot ones, and a survey that finds cool hosts aligned and hot hosts scattered has measured the filter rather than the arrivals.

A misalignment only cool stars forget

A third of hot Jupiters orbit at a large angle to their star's equator, and some go round backwards. Sort the same planets by the temperature of their host and the picture changes — below about 6,250 kelvin almost all are aligned, and above it almost none are. The boundary is not about the planets.

exoplanets · Spin–orbit alignment Residuals of 112 per cent nearby and 2.2 far out. Deviations from a pure Hubble flow, in per cent, against distance. Each galaxy carries a peculiar velocity of a few hundred kilometres a second — part a coherent bulk flow shared with its neighbours and part a random dispersion — and that velocity is added to its recession. Since the recession grows with distance and the peculiar velocity does not, the fractional error falls as one over the distance: it is 112 per cent at 5 megaparsecs and 2.2 at 250. The practical consequence is a lower cut-off on any Hubble-constant measurement: below about 40 megaparsecs the motions dominate, and the coherent part does not average away over a sample because neighbouring galaxies share it. Choosing that cut-off is one of the analysis decisions a local expansion rate depends on.

A residual that is somebody else's velocity

A redshift is not a distance until the galaxy's own motion has been removed, and galaxies move at a few hundred kilometres a second. Nearby that is comparable to the expansion itself, so the local Hubble diagram's scatter is motions rather than measurement — and the motions are shared between neighbours, so they do not average away.

cosmology · Dark energy What comes back is a ramp, not a threshold. Detection efficiency against signal-to-noise: the fraction of synthetic transits injected into real photometry that the pipeline afterwards finds. The measured curve is a gamma cumulative distribution of shape 4.65 and scale 0.98 beginning at 4.1, which is the form a survey's own injection tests are fitted with; the dashed line is the step at 7.1 that a threshold calculation assumes instead. Half the injections are recovered at 8.33, 1.2 units above the nominal threshold — the ramp is a property of the search and the cut is a separate decision, so the two need not meet anywhere in particular. The rest of the disagreement is the area between the curves. The pipeline does not reach 99 per cent efficiency until 14.9, four units above the threshold, and it recovers 45 per cent one unit above it. Over a population whose signal-to-noise falls as s^-2 — which is what a planet population looks like, because there are far more small planets than large ones — the step function counts 1.21 times as many detections as the ramp does. That factor is not an error bar. It multiplies every occurrence rate computed without it, and it is larger for the small planets than for the large ones, because the small ones live where the ramp is.

The threshold that is not a threshold

A survey's detection limit is quoted as a number — seven point one — and a pipeline does not behave that way. Half the injected signals come back at the threshold, and full efficiency arrives four units above it.

exoplanets · Detection bias Three biases against eccentricity, and they do not agree. Four quantities against orbital eccentricity, each relative to a circular orbit of the same semi-major axis, averaged over the argument of periastron. The transit probability rises as (1 − e²)⁻¹, because an eccentric planet spends part of its orbit inside its own semi-major axis: at e = 0.5 a transit is 1.33 times as likely. The transit duration falls as √(1 − e²), so the event carries less signal-to-noise, and the two together — probability times the square root of the time in transit — come to 1.24 at the same eccentricity. They very nearly cancel, and that is the surprise: a transit survey has almost no eccentricity bias at all. The radial-velocity curve is the one that does. A Keplerian of eccentricity e puts less of its variance in the fundamental and more into harmonics no sinusoidal search is looking at — 68 per cent remains at e = 0.6 and 47 per cent at e = 0.8 — so a velocity survey loses amplitude exactly where a transit survey does not. What no figure here can show is which of these the measured eccentricity distribution is made of, because the correction depends on a detection pipeline rather than on geometry, and the two surveys have to be corrected separately before their answers can be compared.

Every method prefers a circle, and not for the same reason

A transit is more likely on an eccentric orbit and shorter when it happens, and the two very nearly cancel. A velocity curve loses amplitude to harmonics no sinusoidal search is looking at, and that one does not cancel at all.

exoplanets · Detection bias How many planets a star has is the hardest thing a catalogue measures. The multiplicity distribution a transit catalogue would contain, for systems that all truly hold 5 planets, at four mutual inclination dispersions. 40,000 systems are drawn per dispersion with an isotropic viewing direction and Rayleigh-distributed inclinations about a common plane, at semi-major axes of 12, 16, 21, 27, 34 stellar radii; the bars are conditioned on at least one planet transiting, which is what makes a system appear in a catalogue at all. At 0.5° of dispersion 33 per cent of the detected systems show all 5 planets and the mean apparent multiplicity is 3.13; at 10° it is 1.39, with 68 per cent of them showing exactly one. Every one of those systems has 5 planets. The entire difference between a catalogue of singles and a catalogue of compact multiples is one number that nothing in the light curve measures. And the two effects run in opposite directions: the fraction of stars showing any planet RISES with the dispersion — 8%, 9%, 12%, 19% across the four — because scattering the orbits gives more of them a chance to cross the line of sight, while the number seen per detected star falls by a factor of 2.2. A survey that scatters its systems finds more stars with planets and fewer planets per star, and neither number on its own says which has happened. What no figure here can show is the true dispersion, because the observable is the ratio of those two and a system with fewer planets and a tighter plane reproduces it exactly.

How many planets a star has is not a measurement

Draw five thousand identical five-planet systems, scatter their orbital planes by half a degree, and a third of the detections show all five. Scatter them by ten degrees and two thirds show exactly one. Every system has five.

exoplanets · Detection bias

Named alongside it

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

Detection limitSurvey completenessDetection thresholdRadial velocitySignal-to-noiseHot jupiterMalmquist biasOccurrence rateTransit probabilityArgument of periastronBulk flowConvective envelope

All concepts