Anharmonic vibrations in pulsating stars

Abstract

We have studied the anharmonic vibrations in pulsating stars using the Hamiltonian formulation of Newtonian dynamics. We have considered the spherical and homogeneous stars as simple case which can however put a constraint on the model in a generalized sense. Utilizing the equation of motion, it has been found that the theoretical model investigated in this study could produce the anharmonic nature of pulsations in radial velocity, luminosity and apparent magnitude with respect to time. We could improve the understanding of such anharmonic pulsation at theoretical front in comparison to the already existing models. However, there is still a large discrepancy in between the observed data and calculated values for the known anharmonically pulsating star (viz., δ Cepheid).

Appendix

The mass of the star is given by $M = \frac{4}{3}\pi R^{3} \rho$ where R is the radius and ρ is the density. From the mass conservation, we obtain,

$$ \frac{dM}{dt} = \frac{d}{dt}\left( {\frac{4}{3}\pi R^{3} \rho } \right) = \frac{4\pi }{3}\left( {3\rho R^{2} \dot{R} + R^{3} \dot{\rho }} \right) = 0 $$

The above equation implies,

$$ \dot{\rho } + 3\frac{{\dot{R}}}{R}\rho = 0 $$

Inserting this equation into the continuity equation $div(\rho v) + \dot{\rho } = 0$, leads to,

$$ dE_{\text{kin}} = \frac{1}{2}dmv^{2} = \frac{1}{2}\rho 4\pi r^{2} dr\frac{{\dot{R}}}{R}r^{2} $$

which in turn results

$$ E_{\text{kin}} = \int\limits_{0}^{R} {dE_{\text{kin}} = 2\pi \rho \frac{{\dot{R}}}{R}} \int\limits_{0}^{R} {r^{4} dr} = \frac{2}{5}\pi \rho \dot{R}R = \frac{3}{10}M\dot{R}^{2} $$

$$ E_{\text{kin}} = \frac{3}{10}M\dot{R}^{2} $$