• Rudolf Kippenhahn
• Alfred Weigert
Chapter
Part of the Astronomy and Astrophysics Library book series (AAL)

## Abstract

The functions P0(m), r0(m), and ϱ0(m) are supposed to belong to a solution of the stellar-structure equations (9.1–4) for the case of complete equilibrium. Let us assume that we perturb the hydrostatic equilibrium, say by compressing the star slightly and releasing it again suddenly. It will expand and owing to inertia overshoot the equilibrium state: the star starts to oscillate. The analogy to the oscillating piston model (see §6.6) is obvious. More precisely we assume the initial displacement of the mass elements to be only radially directed ( = = 0) and of constant absolute value on concentric spheres. This leads to purely radial oscillations (or radial pulsations) during which the star remains spherically symmetric all time. For the perturbed variables at time t we write
$$\begin{gathered} P(m,t) = {P_0}(m) + {P_1}(m,t) = {P_0}(m)\left[ {1 + p(m){e^{{i\omega t}}}} \right], \hfill \\ r(m,t) = {r_0}(m) + {r_1}(m,t) = {r_0}(m)\left[ {1 + x(m){e^{{i\omega t}}}} \right], \hfill \\ \varrho (m,t) = {\varrho_0}(m) + {\varrho_1}(m,t) = {\varrho_0}(m)\left[ {1 + d(m){e^{{i\omega t}}}} \right] \hfill \\ \end{gathered}$$
(38.1)
where the subscript 1 indicates the perturbations for which we have made a separation ansatz with an exponential time dependence [as in (25.17)]. The relative perturbations p, x, d are assumed to be ≪ 1.

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