Astronomy & Celestial Mechanics
the sky, computed
Concept

Cepheid variable — where it appears

A luminous pulsating star whose period of a few days to a few months is tied to its luminosity. Measuring the period gives the luminosity, and comparing that with the apparent brightness gives the distance, which made Cepheids the first step of the extragalactic distance scale.

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

A Cepheid's velocity, radius and angular size through two pulsation cycles. Three curves for a model Cepheid with δ Cephei's period of 5.37 days and a mean radius of 44 solar radii, at 272 parsecs, through two cycles. Top: the radial velocity of its spectral lines, in km/s, oscillating about the star's own velocity of -16.8 km/s — negative is towards the observer, so the surface is expanding when the curve is lowest. Middle: the change in radius, from integrating that velocity over time and multiplying by the projection factor p = 1.3: a swing of 4.9 solar radii, 11 per cent of the star. Bottom: the angular diameter the same star presents, in milliarcseconds, swinging by 168 microarcseconds about 1.505. The middle and bottom curves have the same shape because they are the same motion — one in kilometres, one in angle — and their ratio is the distance.

A distance that rests on one unmeasured number

A pulsating star's swelling can be measured twice — in kilometres from the Doppler shift of its lines, and in milliarcseconds by an interferometer — and the ratio is its distance, with no calibration against any other star. Except one: the factor that turns a line's velocity into the speed of the surface, which geometry puts at 1.33 to 1.5, measurement puts at 1.2 to 1.35, and every distance carries in proportion.

stars · Variable stars The end of the red giant branch, found as an edge. The luminosity function of a synthetic galaxy halo — the number of red giants against apparent magnitude in the I band, smoothed — and its derivative, which is what the edge detector computes, searched over the magnitudes brighter than the branch's crowded lower part, as every application of the method does. 1582 stars are drawn: red giants rising in number towards fainter magnitudes as ten to the 0.3 times the magnitude, ending at the tip at magnitude 24, with a sixth as many asymptotic giants thinning out above it, all measured with errors of 0.08 magnitudes. The derivative peaks sharply where the counts jump, here at 24.09 — 0.09 from the true tip. The red giant branch ends where helium ignites in a degenerate core of almost fixed mass, so the tip's absolute magnitude is close to −4.05 for old, metal-poor stars, and the edge's apparent magnitude is a distance.

An edge that needs no pulsation

The local measurement of the expansion rate rests on a calibration step, and for decades that step was the Cepheid variables. The tip of the red giant branch replaces it with an edge — the brightness at which old stars ignite helium, nearly the same in every galaxy — found by counting stars rather than modelling them. It gives a slightly lower expansion rate, and whether the difference is a flaw in one calibration or in both is the local half of the Hubble tension.

cosmology · Hubble constant How deep a hole the Hubble tension needs, and how deep holes are. The typical departure of the matter density from the mean, averaged over a sphere — one standard deviation, σ, from the linear power spectrum with σ₈ = 0.81 — against the sphere's radius, both on logarithmic axes. It falls from 0.55 at 20 Mpc to 0.022 at 300 Mpc. A local expansion rate 8.3 per cent above the global one, the size of the Hubble tension, needs the sphere the supernovae sample to be underdense by 0.47 (dashed), with a growth rate of 0.53. Across the radii the supernovae span (shaded, 100–600 Mpc) that is 4 to 67 standard deviations.

A hole too deep for the sky to have made

If the Milky Way sat inside a vast underdense region, everything around it would be expanding faster than the universe as a whole, and the locally measured Hubble constant would come out high — which is what Cepheids and supernovae find. The idea needs no new physics. It needs a void about half empty across the six hundred megaparsecs the supernovae sample, and the same cosmology that predicts the microwave background prices such a void at dozens of standard deviations.

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

Named alongside it

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

Hubble constantA type Ia supernovaAngular diameterBaade wesselink methodColour magnitude diagramCosmic varianceCosmic voidDegeneracy pressureDistance scaleGrowth rateHelium flashHubble tension

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