Astronomy & Celestial Mechanics
the sky, computed
Concept

Malmquist bias — where it appears

The tendency of a flux-limited sample to contain only the brightest members at large distances, so that the mean luminosity of the sample rises with distance. Correcting for it requires knowing the luminosity function, which is what the sample was meant to measure.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

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 Tully–Fisher, from two assumptions and no fitting. 44 model galaxies spanning two and a half decades in luminosity, each built from a constant disc surface brightness of 380 L☉ per square parsec with 0.13 dex of scatter and a stellar mass-to-light ratio of 1.4 with 0.1 dex. Nothing about a luminosity–speed relation is put in: the scale length follows from the surface brightness, the mass from the light, and the speed from v² = GM/R. The construction makes L ∝ v⁴ exactly, so the true slope is −10.0 magnitudes per decade of speed. Least squares of magnitude on log speed returns -8.45, and the reverse regression -9.15: the scatter here is scatter in the speed at fixed luminosity, and error in the abscissa flattens a fitted slope, so neither fit returns the value the family was built with and the distance between them is the size of the effect. Residual scatter about the drawn line is 0.54 magnitudes. Observed slopes run from about −7.5 in blue light to −10 in the near infrared, where the mass-to-light ratio is steadiest — which is the same statement as the second assumption above.

A line width that is a distance

The width of a galaxy's hydrogen line depends on how fast it rotates, which depends on its mass, which is tied to its luminosity — so a quantity no distance enters gives an absolute brightness, and the distance follows from the brightness that is seen.

galaxies · Tully–Fisher 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 luminosity function has a knee, and the knee is the point. A Schechter function with a faint-end slope of -1.25, a characteristic magnitude of -20.9 and a normalisation of 0.0093 per cubic megaparsec per magnitude, plotted logarithmically. Fainter than the knee the curve is a straight line — a power law — and brighter than it the count falls off exponentially, which is why there is no such thing as a galaxy ten times brighter than the brightest. Integrated across the range drawn, galaxies brighter than the knee are 1.3 per cent of the number and 26 per cent of the light: almost every galaxy is a dwarf, and almost all the light is not in one.

A count with a knee in it

Count galaxies by luminosity and the answer is a power law at the faint end and an exponential cut-off at the bright one. Almost every galaxy is a dwarf; almost none of the light is in one; and the bend between those two statements is where galaxy formation stops being efficient.

galaxies · Luminosity function A sample that gets brighter with distance because the faint ones drop out. The mean absolute magnitude of a magnitude-limited sample, relative to the population it is drawn from, against distance. The population has a spread of 0.5 magnitudes about a mean of -4, and the survey stops at apparent magnitude 20. Nearby, everything is detected and the sample is unbiased. Beyond about 316228 parsecs the faint end of the distribution starts falling below the limit and the survivors are brighter than average; further out the bias deepens without limit, because eventually only the extreme tail is detectable. The horizontal line is the classical Malmquist value of 1.382 times the square of the spread, which is what the bias averages to over a magnitude-limited sample as a whole — it is a property of the sample rather than of any one object, and using it as a correction for an individual star is a common and specific mistake.

A sample brighter than the population it came from

Every survey stops at some apparent brightness. At any distance it therefore contains only the objects luminous enough to make the cut, so the average object in it is brighter than the average object in the universe — by an amount that grows with distance and that has been shortening every distance in astronomy since 1920.

starlight · Distance ladder Where the axes of rapid rotators point, in a sample chosen by brightness, at a darkening exponent of 0.19. The distribution of rotation-axis inclinations — 0° pole-on, 90° equator-on — for gravity-darkened stars whose axes point at random in space, surveyed down to a limit in apparent brightness, with the surface temperature following the local gravity to the power 0.19. The dashed curve is random orientation, which puts 13.4 per cent of stars within 30° of pole-on. At 0.9 of the critical rotation rate a star looks 1.093 times as luminous pole-on as equator-on, the survey reaches correspondingly further for the pole-on ones, and 14.4 per cent of the sample lies within 30° of pole-on; the nearly pole-on stars are 1.09 times as common as random orientation would make them. At 0.98 of the critical rotation rate a star looks 1.188 times as luminous pole-on as equator-on, the survey reaches correspondingly further for the pole-on ones, and 15.3 per cent of the sample lies within 30° of pole-on; the nearly pole-on stars are 1.19 times as common as random orientation would make them. Averaged over random orientations the apparent luminosity of each star equals its true luminosity to better than half a per cent, as it must; the tilt towards pole-on comes entirely from choosing stars by how bright they look.

The pole-on stars a brightness limit prefers

A rapidly rotating star looks brighter and hotter from its pole than from its equator, so a survey that picks stars by how bright they look should pick more of them pole-on. It does — and the surprise is how little. Averaged over random orientations, a gravity-darkened star's apparent luminosity is exactly its true one, because every photon goes somewhere; a brightness limit restores only a fraction of a per cent of bias, while the error in any single star is ten times larger and of either sign.

starlight · Gravity darkening

Named alongside it

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

CompletenessDetection limitFreeman's lawLuminosity functionSelection effectStandard candleVolume-limited sampleThe baryonic Tully–Fisher relationCharacteristic luminosityCritical rotationDetection thresholdDistance indicator

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