Secular perion changes and fundamental parameters of long--period Cepheids

Hydrodynamic computations of nonlinear Cepheid pulsation models with periods from 20 to 100 day on the evolutionary stage of core helium burning were carried out. Equations of radiation hydrodynamics and time--dependent convection were solved with initial conditions obtained from selected models of evolutionary sequences of population I stars with initial masses from $8M_\odot$ to $12.5M_\odot$. For each crossing of the instability strip the pulsation period $\Pi$ and the rate of period change $\dot\Pi$ were derived as a function of evolutionary time. Comparing results of our computations with observational estimates of $\Pi$ and $\dot\Pi$ we determined fundamental parameters (the age, the mass, the luminosity and the radius) of seven long--period Cepheids. Theoretical estimates of the stellar radius are shown to agree with radius measurements by the Baade--Wesselink technique within 3\% for RS~Pup and GY~Sge whereas for SV~Vul the disagreement between theory and observations does not exceed 10\%.