3.2: Radial Pulsations for Self-Gravitating Systems- Stars

In this section we shall use the virial theorem, to obtain an expression for the frequency of radial pulsations in a gas sphere. The approach will be to apply a small variation to the virial theorem and by making use of several conservation laws, obtain expressions for the variation of the moment of inertia, kinetic energy, and potential energy as a function of time.

Remember from the earlier section that Lagrange's identity is

\[\frac{1}{2} \frac{\mathrm{d}^2 \mathrm{I}}{\mathrm{dt}^2}=2 \mathrm{T}+\Omega.\label{3.2.1}\]

1. In this form, no time averaging has been carried out and the equation must apply to a dynamic system at any point in time. Now, consider a star with radius \(R\). Let \(r\) be the distance from the center of symmetry to any point in the configuration and or be the displacement of a point mass from the equilibrium position \(r_o\). Conservation of mass requires that for a spherical shell of radius

\[\mathrm{m}\left(\mathrm{r}_0+\delta \mathrm{r}\right)=\mathrm{m}\left(\mathrm{r}_0\right).\label{3.2.2}\]

We wish to find the variations \(δI\), \(δT\), \(δΩ\) of the quantities \(I\), \(T\), \(Ω\), from the equilibrium values \(I_0\), \(T_0\), and \(Ω_0\). The variational form of the virial theorem then becomes

\[\frac{1}{2} \frac{\mathrm{d}^2(\delta \mathrm{I})}{\mathrm{dt}^2}=2 \delta \mathrm{T}+\delta \Omega .\label{3.2.3}\]

Since I was defined as the moment of inertia about the center of the coordinate system, we have by definition that,

\[\mathrm{I}=\int_0^{\mathrm{M}} \mathrm{r}^2 \mathrm{dm}(\mathrm{r}).\label{3.2.4}\]

Thus, we have

\[\delta \mathrm{I}=\int_0^{\mathrm{M}} 2 \mathrm{r} \delta \mathrm{rdm}(\mathrm{r})+\int_0^{\mathrm{M}} \mathrm{r}^2 \delta[\mathrm{dm}(\mathrm{r})].\label{3.2.5}\]

Since, by the conservation of mass [Equation \ref{3.2.2}], δm(r) = 0 for all r, d(δm(r) ) = δ(dm(r) ) =0 for all r and the second integral of Equation \ref{3.2.5} vanishes leaving

\[\delta \mathrm{I}=\int_0^{\mathrm{M}} 2 \mathrm{r} \delta \mathrm{rdm}(\mathrm{r}).\label{3.2.6}\]

Now from Equation \ref{3.2.4}, we have

\[\frac{\mathrm{dI}}{\operatorname{dm}(\mathrm{r})}=\mathrm{r}^2.\label{3.2.7}\]

Since this must always be true, it is true at the equilibrium point r_o. Therefore,

\[\mathrm{dI}_0=\mathrm{r}_0^2 \operatorname{dm}(\mathrm{r}).\label{3.2.8}\]

So, to first order accuracy in r, we may re-write Equation \ref{3.2.6} as

\[\delta \mathrm{I}=2 \int_0^{\mathrm{I}_0} \frac{\delta \mathrm{r}}{\mathrm{r}_0} \mathrm{dI}_0.\label{3.2.9}\]

In a similar manner we may evaluate the variation of the gravitational potential energy

with respect to small variation in r ^3.1 and obtain

\[\delta \Omega=2 \int_0^{\Omega_0} \frac{\delta \mathrm{r}}{\mathrm{r}_0} \mathrm{~d} \Omega_0.\label{3.2.10}\]

All that now remains to be determined in Equation \ref{3.2.3} is the variation of the total kinetic energy T. To first order only the variation of the thermal kinetic energy will contribute to Equation \ref{3.2.3}.^3.2

\[2 \delta \mathrm{~T} \cong 3 \int_0^{\mathrm{M}} \frac{\mathrm{P}_0}{\rho_0}(\gamma-1) \frac{\delta \rho_0}{\rho_0} \mathrm{dm}(\mathrm{r}).\label{3.2.11}\]

In order to facilitate obtaining an expression for \(\delta \rho / \rho_0\) we shall now specify a time dependence for the pulsation about r_o. For simplicity, let us assume the motion is simply periodic. Thus, defining a quantity ξ as

\[\xi=\frac{\delta \mathrm{r}}{\mathrm{r}_0}=\xi_0 \mathrm{e}^{\mathrm{i} \sigma \mathrm{t}},\label{3.2.12}\]

where \(2π/σ\) is the period of oscillation, we may re-write the variations of \(I\) and \(Ω\) as follows:

\[\left.\begin{array}{l}
\delta \mathrm{I}=2 \mathrm{e}^{\mathrm{i} \sigma \mathrm{t}} \int_0^{\mathrm{I}_0} \xi_0 \mathrm{dI}_0 \\[4pt]
\delta \Omega=-\mathrm{e}^{\mathrm{i} \sigma \mathrm{t}} \int_0^{\Omega_0} \xi_0 \mathrm{~d} \Omega_0
\end{array}\right\} .\label{3.2.13}\]

Conservation of mass requires that^3.3

\[\mathrm{\frac{\delta \rho}{\rho_0}=-\left(3 \xi_0+r_0 \frac{d \xi_0}{d r_0}\right) e^{i \sigma t}}.\label{3.2.14}\]

Substitution of this back into the expression for the variation of the kinetic energy yields

\[2 \delta \mathrm{T}=-3 \int_0^{\mathrm{M}} \frac{\mathrm{P}_0}{\rho_0}(\gamma-1)\left(3 \xi_0+\mathrm{r}_0 \frac{\mathrm{~d} \xi_0}{\mathrm{dr}_0}\right) \mathrm{e}^{\mathrm{i} \sigma \mathrm{t}} \mathrm{dm}\left(\mathrm{r}_0\right).\label{3.2.15}\]

Equation \ref{3.2.15} may be simplified to yield^3.4

\[2 \delta \mathrm{T}=-3 \mathrm{e}^{\mathrm{i} \sigma \mathrm{t}} \int_0^{\mathrm{M}} \frac{\mathrm{P}_0 \xi_0}{\rho_0} \frac{\mathrm{~d}}{\mathrm{dr}_0} \mathrm{dm}\left(\mathrm{r}_0\right)+3 \mathrm{e}^{\mathrm{i} \sigma \mathrm{t}} \int_0^{\Omega_0} \xi_0(\gamma-1) \mathrm{d} \Omega_0.\label{3.2.16}\]

We now have all the material necessary to evaluate the variational form of the virial theorem to first order accuracy. Substituting Equations \ref{3.2.13} and \ref{3.2.16} into Equation \ref{3.2.3}, we obtain

\[-\sigma^2 \mathrm{e}^{\mathrm{i} \sigma \mathrm{t}} \int_0^{\mathrm{I}_0} \xi_0 \mathrm{dI}_0=3 \mathrm{e}^{\mathrm{i} \sigma \mathrm{t}} \int_0^{\mathrm{M}} \frac{\mathrm{P}_0 \xi_0 \mathrm{r}_0}{\rho_0} \frac{\mathrm{~d} \gamma}{\mathrm{dr}} \mathrm{dm}(\mathrm{r})+3 \mathrm{e}^{\mathrm{i} \sigma \mathrm{t}} \int_0^{\Omega_0} \xi_0(\gamma-1) \mathrm{d} \Omega_0-\mathrm{e}^{\mathrm{i} \sigma \mathrm{t}} \int_0^{\Omega_0} \xi \mathrm{~d} \Omega_0 .\label{3.2.17}\]

Solving for σ², which is related to the pulsation period, we have

\[\mathrm{\sigma^2=\frac{-\int_0^{\Omega_0}(3 \gamma-4) \xi_0 d \Omega_0+3 \int_0^M \frac{P_0 \xi_0 r_0}{\rho_0} \frac{d \gamma}{d r_0} d m(r)}{\int_0^{I_0} \xi_0 d I_0}}.\label{3.2.18}\]

For a model of known equilibrium structure, the integrals in Equation \ref{3.2.18} may be evaluated and the frequency for which it is stable to radial pulsations may be computed. However, for purposes of examining the behavior of a pulsating star we may assume the star is sufficiently homogeneous so that γ is constant. Also, let us assume the pulsation increases radially outward in a linear manner. Under these admittedly ad hoc assumptions, Equation \ref{3.2.18} reduces to the extremely simple form

\[\sigma^2=-\frac{(3 \gamma-4) \Omega_0}{\mathrm{I}_0}.\label{3.2.19}\]

In order to obtain a feeling for the formula we have developed, we shall attempt to estimate some approximation pulsation frequencies. For a sphere of uniform density

\[\Omega_0=\frac{3}{5} \frac{\mathrm{GM}^2}{\mathrm{R}_0}.\label{3.2.20}\]

The moment of inertia for a sphere about an axis is equal to 3/2 the moment of inertia about its center and is given by

\[\mathrm{I}_{\mathrm{z}}=\frac{2}{5} \mathrm{MR}_0^2=\frac{3}{2} \mathrm{I}_0 .\label{3.2.21}\]

Therefore,

\[\mathrm{I}_0=\frac{4 \mathrm{MR}_0^2}{15}.\label{3.2.22}\]

Prior theory concerning stellar structure implies that \(\gamma>4 / 3\). If we take \(\gamma=5 / 3\), appropriate for a fully convective star, we obtain

\[\begin{aligned}
\sigma^2 & =\frac{9}{4} \frac{\mathrm{GM}}{\mathrm{R}_0^3}, \\[4pt]
\text{or}\quad\sigma^2 & =3 \pi \mathrm{G} \bar{\rho} .
\end{aligned}\label{3.2.23}\]

Remembering that the period T is just 2π/σ we have

\[T=\left(\sqrt{\frac{4 \pi}{3 G}}\right) \rho^{-\frac{1}{2}}.\label{3.2.24}\]

Thus, we see that the theory does produce a period which is inversely proportional to the square root of the mean density. This law has been found to be experimentally correct in the case of the Classical Cepheids. It should be noted that this property will be preserved even for the integral form Equation \ref{3.2.18}, only the constant of proportionality will change. If we evaluate the constant of proportionality from Equation \ref{3.2.24}, we have

\[T \cong 7.92 \times 10^3 \bar{\rho}^{-\frac{1}{2}} \mathrm{sec}.\label{3.2.25}\]

where \(\overline{\rho}\) is given in (g/cc).

Taking an observed value for the mean density of a Cepheid variable to be between 10^-3 and 10^-6 cm/cc (eg. Ledoux and Walraven)⁴, we arrive at the following estimate for the periods of these stars.

\[0.3 \text { days }<T<90 \text { days }\label{3.2.26}\]

It is freely admitted that this estimate is arrived at in the crudest way, however, and it is comforting that the result nicely brackets the observed periods for Cepheid variables. It should also be noted that for most stars the expression arrived at in Equation \ref{3.2.23} for σ² is a lower limit. As the mass becomes more centrally concentrated the magnitude of the gravitational energy will increase while the moment of inertia will decrease. Even for reasonable density distributions the value arrived at in Equation \ref{3.2.23} will not differ by more than an order of magnitude. This would imply that a value for the period calculated in this manner should be correct within a factor of 2 or 3. Thus, without solving the force Equations, an estimate for a very important parameter in describing the pulsation of a gas sphere may be obtained which is the period for which that sphere is stable to radial pulsation.