The derivation of the Leavitt law treated the instability strip as vertical

PHYSICS · Multi-part · Fermi University Physics · Astrophysics & Cosmology · Observational foundations and instrumentation · Stellar structure, energy generation and evolution

Stars from the Inside, Ch Pulsating stars: the kappa mechanism, the Leavitt law and asteroseismology

Problem

The derivation of the Leavitt law treated the instability strip as vertical, every Cepheid at the same

T_{\rm eff}

. Suppose instead that the strip's centre line tilts towards cooler temperatures at higher luminosity,

T_{\rm eff} \propto L^{-b}

with

b

a small positive constant. Keep

\Pi \propto R^{3/2}M^{-1/2}

L \propto R^2T_{\rm eff}^4

and

L \propto M^a

. ((a)) Show that

\Pi \propto L^{3/4 + 3b - 1/2a}

. ((b)) For

a = 3.5

and

b = 0.05

, find the exponent and the slope

\mathrm{d}\log L/\mathrm{d}\log\Pi

, and compare the slope with the vertical strip's

1.6

and with the calibrated

0.97

. ((c)) With

a = 3.5

, what value of

b

would the tilt alone need to bring the slope down to

0.97

? By what factor would

T_{\rm eff}

then fall across the two decades of luminosity that Cepheids span, and is that compatible with a strip that runs from about

6000

7500\,\mathrm{K}

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