Astronomy & Celestial Mechanics
the sky, computed
Series

Variable stars — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · stars
  2. The period spans 492× and the pulsation constant 2.1×. The pulsation constant Q = P√(ρ̄/ρ̄_⊙) for 7 radial pulsators, against their periods. A radial pulsation is a standing sound wave across a star and the sound speed in a self-gravitating body is set by its own gravity, so the period should go as (Gρ̄)^−1/2 and Q should be the same for every star pulsating in the same mode. Across a factor of 492 in period — from 1.3 hours to 27 days — Q varies by 2.07, with a mean of 0.0393 days. That is the whole reason a period–luminosity relation can exist: a period is a density, a density is a mass and a radius, and a radius with a temperature is a luminosity. The masses here are the weak link and the caption should say so — for the Cepheids they come from evolutionary models rather than from a dynamical measurement, and the drift of Q with period is where that assumption is showing.

    The valve that has to sit at the right depth

    A period–luminosity relation only exists because a period is a density in disguise. Which stars have a period at all is decided by whether a partial ionisation zone sits deep enough to have mass behind it and shallow enough that convection has not taken it over.

    part 2 · stars
  3. 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.

    part 3 · stars
  4. 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.

    part 4 · stars
  5. 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.

    part 5 · stars

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