Astronomy & Celestial Mechanics
the sky, computed
Series

Distance ladder — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The fractional distance error, rung by rung. Cumulative fractional uncertainty in a measured distance against the distance itself, both on logarithmic axes. Each rung of the ladder is calibrated against the one below it, so its scatter adds in quadrature to everything already inherited, and the total can only rise: radar to the planets 10⁻⁵%, parallax 1.0%, main-sequence fitting 5.1%, Cepheids 6.5%, Type Ia supernovae 8.2%.

    Every distance is measured with the last one

    No single method reaches from a planet to a distant galaxy. The ladder is built rung by rung, each calibrated on the one below, and the errors multiply all the way up.

    part 1 · starlight
  2. Four distances to the same galaxy. Four quantities all called "the distance", against redshift, at a Hubble constant of 67.36 km/s/Mpc, a matter density parameter of 0.3153 and a dark-energy parameter of 0.6847. They agree below z ≈ 0.1 and then part company completely. Comoving distance is the separation now, and it is what a map of the universe is drawn in. Luminosity distance is what a brightness gives, and it is larger by (1+z) because the photons arrive both redshifted and spread out in time. Light-travel distance is the age difference times c, and it is bounded by the age of the universe. Angular-diameter distance is what an angle gives, and it is the odd one: it rises, turns over at z = 1.59 where it reaches 1.79 Gpc, and falls thereafter. Past that redshift a galaxy of fixed size looks bigger the further away it is, because the universe it is being seen across was smaller when the light left it. At z = 10 the four differ by a factor of 121 between the largest and the smallest, so a sentence quoting a cosmological distance without saying which one has not given a number.

    Four distances to the same galaxy

    Inside the Local Group the word "distance" has one meaning. Past a redshift of about a tenth it has four, they disagree by factors of a hundred by the time the light is old, and one of them stops increasing and starts coming back.

    part 2 · cosmology
  3. 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.

    part 3 · starlight
  4. The ratio of two distances, for a universe with dark energy and one with dust. The distance-duality ratio η — the luminosity distance divided by (1 + z)² times the angular-diameter distance — against redshift. In a universe with a cosmological constant, or any universe in which light travels on the geometry and is conserved, it is exactly one (solid). In a universe of matter alone, decelerating, with grey intergalactic dust dimming its supernovae just enough to make their distances match the accelerating one's, it rises with redshift: 1.19 at redshift 0.5, 1.30 at 1, because the dust makes supernovae look farther than the geometry puts them while leaving the acoustic-scale distances untouched. The shaded band is η = 1 ± 0.03z, about what supernova and acoustic-scale distances together allow. The dusty universe's η rises at the start with a slope of about 0.45 per unit redshift, more than ten times the band's, and leaves it at once.

    Two distances that must agree if light is conserved

    A supernova's distance comes from how faint it is; an acoustic ruler's distance comes from how small it looks. In any universe whose light follows the geometry and is neither created nor destroyed, the two are tied by an exact factor of (1 + z)², whatever the universe contains. Dust that dimmed distant supernovae without reddening them could have faked the accelerating expansion — and would have broken that tie by a fifth by redshift a half. The tie holds to a few per cent.

    part 4 · starlight

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