Astronomy & Celestial Mechanics
the sky, computed
Concept

Distance indicator — where it appears

An observable property of an object that fixes its luminosity or size, so that comparing it with the observed brightness or angle gives a distance. Cepheid periods, galaxy rotation speeds and supernova light curves are the main ones.

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

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 A velocity error that grows with distance. The uncertainty in one galaxy's peculiar velocity, v = cz − H₀d, against its distance in Mpc/h, when the distance comes from an indicator with a fractional error of 5 per cent, 20 per cent, 25 per cent (type Ia supernovae, 5%; Tully–Fisher, 20%; fundamental plane, 25%). The redshift is exact to a few kilometres a second, so the whole error is the distance error times H₀, and it grows in proportion to the distance. The horizontal line is the rms velocity of a galaxy along one direction in linear theory for the model drawn, 302 km/s. Each galaxy's own velocity is measured to better than its own size only inside 60 Mpc/h for 5 per cent, 15 Mpc/h for 20 per cent, 12 Mpc/h for 25 per cent; beyond that a single velocity is mostly noise, and a velocity survey is useful only as an average over many galaxies.

Velocities that see past the edge of the map

A redshift survey infers the motions of galaxies from the distortions of its own map. A peculiar-velocity survey measures them directly, one galaxy at a time, by subtracting a distance from a redshift — and every one of those measurements is dominated by its own error beyond a few tens of megaparsecs. What makes the survey worth doing is that a velocity answers for structure far larger than the volume it is measured in.

cosmology · Large-scale structure 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

Named alongside it

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

Malmquist biasTully fisher relationAbsolute magnitudeThe baryonic Tully–Fisher relationBulk flowChromosphereCosmic varianceDamping wingsEmission lineFreeman's lawGalaxy biasGrowth rate

All concepts