Cepheid Pulsation & the Period–Luminosity Relation

Star Properties

Mass 7.0 M☉

Pulsation amplitude 12%

Playback speed 3.0 d/s

Live Readouts

Pulsation period

— d

Phase

—

Radius

— R☉

Surface temp.

— K

Luminosity

— L☉

Mean density

— ρ☉

Light Curve

Apparent luminosity vs. time (last 2 cycles)

How it works

The κ-mechanism. In a normal star, compressing a gas layer makes it more transparent, so extra heat escapes immediately and the layer settles back down — a stable star. In the partially-ionized helium layer of a Cepheid, the opposite happens: compression ionizes more He⁺→He²⁺, which increases opacity κ. Heat gets trapped, builds up pressure, and overshoots the layer outward; as it expands and cools, helium recombines, opacity drops, the trapped heat escapes, and gravity pulls the layer back in. This opacity "valve" pumps energy into the oscillation each cycle faster than pulsation damping removes it, so the star self-sustains a radial pulsation instead of decaying to rest.

Period–mean-density relation (Ritter's relation). The pulsation period is set by the star's own free-fall/sound-crossing timescale:

P·√(ρ̄ / ρ̄☉) ≈ Q        (Q ≈ 0.05 d, fundamental mode)
ρ̄ ∝ M / R³
⇒  P ≈ Q · √(R³ / M)     (R, M in solar units)

Bigger, puffier stars pulsate more slowly — this is why the mass slider changes the period.

Radius, temperature and the light curve. This model drives the radius and effective temperature as R(θ)=R₀(1+A·sinθ) and T(θ)=T₀(1+B·cosθ), a quarter-cycle out of phase — maximum outward velocity (θ=0) occurs about ¼ period after minimum radius, exactly as observed. Luminosity then follows the Stefan–Boltzmann law self-consistently:

L = 4πR²σT⁴  ⇒  L(θ)/L₀ = (R(θ)/R₀)² · (T(θ)/T₀)⁴

Because T⁴ dominates, brightness rises sharply near minimum radius and fades slowly during expansion — the characteristic asymmetric "sawtooth" Cepheid light curve, visible in the strip chart.

Leavitt's Law — the period-luminosity relation. Because P depends only on ρ̄ (mass and radius) while L depends on R and T, and Cepheids occupy a narrow instability strip in temperature, longer-period Cepheids are systematically more luminous. The approximate bolometric calibration is:

log₁₀(L/L☉) ≈ 1.15·log₁₀(P/days) + 2.35

Because period is easy to measure from a light curve alone, this relation turns Cepheids into "standard candles": measuring P gives L, and comparing L to the observed apparent brightness gives distance — the method Henrietta Leavitt discovered in 1908 and Hubble used to first measure the distance to Andromeda.

Photosphere (κ-mechanism) Convective granulation

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