Astronomy & Celestial Mechanics
the sky, computed
Concept

Period luminosity relation — where it appears

The tight line between how slowly a variable star pulsates and how bright it intrinsically is, which turns a clock into a ruler. It exists because a pulsation period is a mean density, and a mean density with a temperature is a luminosity.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

Leavitt's law, through the Milky Way calibrators. Mean absolute magnitude against the logarithm of the pulsation period, for classical Cepheids with independently known distances, and RR Lyrae itself for comparison. The line is M = −2.81 log P − 1.43: a tenfold longer period is 2.81 magnitudes brighter, a factor of 13 in luminosity. The vertical axis runs the astronomers' way, with brighter upward.

A star that tells its distance by how slowly it blinks

Some stars pulsate, and the slow ones are the bright ones. That single correlation turns a clock into a ruler, and it is how the size of the universe was first measured.

stars · Variable stars H₀: nine determinations in two families. Published determinations of H₀, each with its quoted one-sigma interval, sorted into two families — measured locally, in the late universe, against inferred from z ≈ 1100 through a model. The shaded band behind each family is that family's inverse-variance weighted mean: 72.66 ± 0.75 across 5 of them, against 67.40 ± 0.41 across 4. The difference is 5.26 ± 0.85 km/s/Mpc, which is 6.2 standard deviations, computed here from the quoted errors alone. That number is an upper bound on the significance rather than the significance: the determinations within each family share calibrations, samples and in two cases the same supernovae, so they are not independent, and a correlated pair combines to something wider than the formula used here gives. What the figure does establish is that the split is not one discrepant measurement against a consensus — it is two internally consistent groups, and the grouping is by method rather than by result.

The same constant, measured twice, five sigma apart

Cepheids and supernovae give an expansion rate of about 73 kilometres per second per megaparsec. The microwave background gives 67.4. Both quote errors near one per cent, both have been rebuilt from scratch by rival teams, and the gap between them has grown as the measurements have improved.

cosmology · Hubble constant One population, two period–luminosity relations. The period–luminosity relation of 400 synthetic Cepheids spread across the instability strip, each behind a random amount of dust up to E(B − V) = 0.6: absolute magnitude in V (upper scatter) and in the Wesenheit index W = V − 2.44(V − I) (lower). In V the scatter about the fitted line is 0.57 magnitudes — 0.20 from the strip's width alone, the rest from the dust. In W it is 0.07: the dust is gone by construction, and most of the strip's width has gone with it, because a star that is hotter at a given period is bluer by about as much as dust would make it redder. That leftover 0.07 magnitudes is 3 per cent in distance per star.

A scatter that points the way dust does

A Cepheid's period fixes its luminosity only to within the width of the instability strip, and at a given period the hotter stars are the brighter ones. Dust moves stars in the colour–magnitude diagram along a line too, and the two lines point nearly the same way. A magnitude built to cancel the dust — the Wesenheit index — therefore cancels most of the strip's width as well, which is why the tightest period–luminosity relations are the ones that ignore the extinction law's own advice.

stars · Variable stars How far a wrong metallicity term moves the Hubble constant. The fractional change in the Hubble constant against the error in the adopted metallicity term, in magnitudes per dex, for supernova hosts whose Cepheids are 0.1, 0.2, 0.4 dex richer in metals than the geometric calibrators'. A term too weak (too close to zero) by 0.2 mag/dex with hosts 0.2 dex richer moves H₀ by 1.9%. The shaded band is the ±8% disagreement between the local measurement and the microwave background's prediction: reaching it with a 0.2-dex difference needs the term wrong by 0.9 mag/dex.

A metallicity term too small to close a gap

Cepheids of the same period are slightly brighter when they are richer in metals, by about a fifth of a magnitude per factor of ten in metal content. The galaxies whose supernovae calibrate the Hubble constant hold Cepheids a little richer than the galaxies whose distances are known geometrically, so the term enters the answer. How much it can move it is simple to compute: a few per cent at most for any plausible error — and the eight per cent separating the local Hubble constant from the early universe's would need the term to be wrong by four times its own size.

stars · Variable stars

Named alongside it

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

Cepheid variableDistance ladderHubble constantHubble tensionInstability stripAbsolute magnitudeColour magnitude diagramEarly dark energyInterstellar extinctionInverse distance ladderIonisationLarge magellanic cloud

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