Cepheid Variable Star: Pulsation, Period-Luminosity Law and the Distance Ladder

Presets

Live readouts

Graphs

Light curve: brightness change in magnitudes (orange, up = brighter) and radius change x5 (blue) versus phase

Period-luminosity diagram: -M_V versus log10 P with a synthetic LMC sample (grey) and the chosen star (white)

Apparent V magnitude versus distance for this star; the orange line marks V = 26 (about the HST limit)

Numerical checks

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How to use

Pick a famous star from the presets or set the period yourself. Watch the star swell and cool, then shrink and heat up, while the brightness curve (orange) lags the radius (blue). Try: (1) raise the period from 3 to 40 days and see the star's size and luminosity grow along the PL line; (2) LMC Cepheid: the white dot sits on the cloud of grey LMC stars; (3) push distance up to the point where the apparent V crosses the orange 26 mag limit, then Hubble in M31; (4) add dust and see the apparent magnitude fade but the Wesenheit index stay put; (5) Polaris shows what a feeble pulsator looks like. Space pauses.

The science

What a Cepheid is. A classical Cepheid is a yellow supergiant of 4 to 10 solar masses that swells and shrinks by roughly 10 percent in radius every 2 to 80 days, changing temperature and brightness with the same period. Henrietta Leavitt (1912) noticed that in the Small Magellanic Cloud, where all stars are at about the same distance, the longer the period the brighter the star. That period-luminosity (PL) relation turned Cepheids into standard candles.

Why it pulsates (kappa mechanism). In the envelope there is a layer where helium is partly ionised (He II to He III). Compressing the gas there raises the opacity instead of lowering it, so the layer absorbs heat during compression and releases it during expansion, like a heat engine pushing a piston. Only stars whose He II zone sits at the right depth, a narrow band in the Hertzsprung-Russell diagram called the instability strip, are driven; others damp the motion out.

Equations used on this page.

Period-luminosity: M_V = -2.43 (log10 P - 1) - 4.05 + gamma [Fe/H], gamma = -0.2 mag/dex Period-radius: log10 (R / Rsun) = 1.075 + 0.75 log10 P Stefan-Boltzmann: L = 4 pi R^2 sigma T^4 (T_eff follows from L and R) Pulsation constant: P sqrt(rho / rho_sun) = Q, Q = 0.040 d => M / Msun = (R / Rsun)^3 (Q / P)^2 Distance modulus: m - M = 5 log10 (d / 10 pc) + A_V Wesenheit index: W = V - 2.5 (V - I) (independent of dust, because A_V = 2.5 E(V - I) for the standard law)

The period-radius relation and Q give a mass of about 4.8 solar masses for a 10-day Cepheid, a typical value. The relations are the observational ones of Benedict et al. (2007), Gieren et al. (1999) and Leavitt & Pickering (1912); the zero point is anchored by Gaia parallaxes and the eclipsing-binary distance to the Large Magellanic Cloud (49.6 kpc, mu = 18.477; Pietrzynski et al. 2019).

The distance ladder. Parallax calibrates nearby Cepheids; Cepheids in galaxies up to about 40 Mpc then calibrate Type Ia supernovae, which reach into the Hubble flow. Edwin Hubble found a 31-day Cepheid in Andromeda in 1923, deduced a distance of roughly 300 kpc by the same method (the modern value is 780 kpc, because the PL zero point and the dust and the existence of two Cepheid populations were not yet understood) and so showed that the nebulae are other galaxies.

Checks on this page. (1) A synthetic LMC sample of 400 Cepheids with 0.1 mag scatter and a known distance modulus is fitted by least squares and must return the input slope and the modulus 18.477; (2) the Wesenheit index must not change when 0 to 2 magnitudes of dust are added; (3) the Stefan-Boltzmann luminosity from R and T must equal the luminosity from the PL law; (4) the distance recovered from period and apparent magnitude must equal the distance set; (5) the cycle-mean radius must equal the mean radius.

Honest simplifications. The light and radius curves are a phenomenological three-term Fourier shape with a temperature lag of 0.28 cycles, not a hydrodynamic model; amplitudes are tied to period by an approximate power law. The PL relation has an intrinsic width of about 0.1 to 0.2 mag that the single star here does not show, and real Cepheids sit inside a strip of temperature rather than on a single line. Mass loss, binarity, Cepheid blending and the real metallicity dependence (still debated, -0.2 to -0.5 mag/dex) are not modelled. The Sun is drawn to true scale in 2D and 3D; the star's size on screen is compressed for visibility.