8.2: Linear Adiabatic Radial Oscillations

Just as the pitch or oscillation frequency of an orchestral chime conveys information about its length, so the oscillation period of a pulsating variable should indicate its size. The low-pitch, or long- period, chimes or bells are characteristically larger than higher-pitch ones. So it is with the most common of the pulsating stars. The RR Lyrae stars having pulsational periods of less than a day are both hotter and less luminous, and therefore smaller, than the longer-period Cepheid variables. The Cepheid variables in turn are significantly hotter and smaller than the even longer period Mira variables. Only for the cataclysmic variables, where the pulsations are probably mediated by an outside source, does this size-period relationship fail. This is as it should be, for the pulsation of a star, being a hydrodynamic phenomenon, should take place on the hydrodynamic time scale roughly equal to the sound crossing time. Let us now consider how we may quantify this notion.

Begin by assuming that the pulsations take place in an adiabatic manner. By this we mean that the energy associated with the motion of the pulsation does no net work on the gas of the star and is therefore conserved from one oscillation to the next. Thus an adiabatic oscillation can proceed forever once it is established. This lack of sharing of energy with the star means that driving and damping terms that must be present in any real situation are assumed to be negligible. Thus we cannot hope to determine anything about the evolutionary history of such oscillations in real stars. The origin or fundamental cause of such oscillations will not be found by such analysis. However, we can learn something about the eigenfrequencies for those oscillations and the stellar parameters upon which they depend.

The history of the theory of stellar pulsation can be traced to Eddington^{1, 2}, but a conceptually simple picture was introduced by Ledoux³ and is nicely described by Ledoux and Walraven⁴. This approach involves the Virial theorem and so focuses on the global properties of the stars and their relation to the periods of pulsation. This approach was developed by Chandrasekhar⁵ to deal with some extremely complicated problems and is summarized by Collins⁶.

a. Stellar Oscillations and the Variational Virial theorem

Consider a spherical star in equilibrium. Now allow the moment of inertia, the internal energy, and the gravitational potential energy to vary in such a manner that the Virial theorem always holds. Then the Virial theorem as given by equation \ref{1.2.34} can be written as \[\frac{1}{2} \frac{d^2(\delta I)}{d t^2}=2 \delta U+\delta \Omega\label{8.1.1}\]

The conservation of mass requires that the mass interior to some radial distance r remain constant throughout the oscillation, so that \[M(r)=M(r+\delta r)=M\left(r_0\right)=\int_0^{r_0} 4 \pi r_0^2 \rho d r\label{8.1.2}\]

This is virtually a definition of what is meant by M(r). The variations of the moment of inertia and gravitational potential energy can simply be calculated from their definitions as \[\delta I=2 \int_0^{I_0} \frac{\delta r}{r_0} d I_0 \quad \delta \Omega=-\int_0^{\Omega_0} \frac{\delta r}{r_0} d \Omega_0\label{8.1.3}\]

To calculate the effect of the oscillations on the total internal energy, we have to consider how the gas responds to the radial motion of material. In terms of the state variables, the total internal energy is \[2 U=3 \int_V P d V=3 \int_0^M \frac{P}{\rho} d M(r)\label{8.1.4}\]

For adiabatic pulsations \[\frac{\delta P}{P_0}=\gamma \frac{\delta \rho}{\rho_0}\label{8.1.5}\]

Now let us assume that the pulsations can be characterized by \[\xi \equiv \frac{\delta r}{r_0}\label{8.1.6}\]

and make the sensible requirement that \(\xi\) remain finite at the origin.

The conservation of mass [equation \ref{8.1.2}] requires that \[\mathrm{d}[\mathrm{M}(\mathrm{r})]=0=\int_0^{\mathrm{r}} 8 \pi \mathrm{r}_0 \rho \delta \mathrm{rdr}+\int_0^{\mathrm{r}} 4 \pi \mathrm{r}_0^2 \delta \rho_0 \mathrm{dr}+\int_0^{\mathrm{r}} 4 \pi \mathrm{r}_0^2 \rho_0 \mathrm{~d}(\delta \mathrm{r})\label{8.1.7}\]

while the definition of \(\xi\) requires \[d \xi=\frac{d(\delta r)}{r_0}-\frac{\delta r d r}{r_0^2}\label{8.1.8}\]

Assuming the variations are small and by keeping only the first-order terms, we get \[\int_0^r 4 \pi r_0^2 \delta \rho d r=-\int_0^r 4 \pi \rho_0 r_0^2\left(3 \xi+r_0 \frac{d \xi}{d r_0}\right) d r\label{8.1.9}\]

Since this integral must hold for all values of r, the integrands must be equal, or \[\frac{\delta \rho}{\rho_0}=-\left(3 \xi+r_0 \frac{d \xi}{d r_0}\right)\label{8.1.10}\]

When this is combined with equation \ref{8.1.5}, the variation of the internal energy as given by equation \ref{8.1.4} becomes \[2 \delta U=-3 \int_0^M \frac{P_0}{\rho_0}(\gamma-1)\left(3 \xi+r_0 \frac{d \xi}{d r_0}\right) d M(r)\label{8.1.11}\]

This may be combined with equations \ref{8.1.1} and \ref{8.1.3} to give the constraint on radial oscillations implied by the Virial theorem. However, to arrive at some sensible result, we have to make some further assumption about the nature of the oscillation. Let us assume that the amplitude variation is linear with position, so that the pulsation varies homologously within the star and is simply periodic in time and \[\xi=\xi_0 e^{i \sigma t}\label{8.1.12}\]

Substitution of this form of the variation into the variational forms of the moment of inertia, internal and potential energies, and subsequently into the variational Virial theorem [equation \ref{8.1.1}] gives \[\sigma^2=-\frac{\langle 3 \gamma-4\rangle \Omega_0}{I_0}\label{8.1.13}\]

where \(<\gamma>\) is an average throughout the star weighted by the gravitational potential. For a homogeneous uniform star that behaves as a perfect gas throughout, this becomes \[\sigma^2=\frac{9 G M}{4 R_0^3}=3 \pi G \bar{\rho}_0\label{8.1.14}\]

If we compare this to the free-fall time given by equation \ref{3.2.6}, we see that the pulsation period, which is just \(2\pi/\sigma\), is of the same order. Specifically \[P_p(\mathrm{sec} .)=\frac{8 \sqrt{2} \tau_f}{3}=\left(\frac{4 \pi}{3 G \bar{\rho}}\right)^{1 / 2} \approx\left(8 \times 10^3\right)(\bar{\rho})^{-1 / 2} \mathrm{~g} / \mathrm{cm}^3\label{8.1.15}\]

Since pulsation is a dynamic phenomenon, we should not be surprised that it takes place on a dynamical time scale and its period is slightly longer than the free-fall time. For Cepheid variables, observationally determined values for the mean density (that is, \(10^{-3} \mathrm{gm} / \mathrm{cm}^3>\mathrm{p}>10^{-6} \mathrm{gm} / \mathrm{cm}^3\)) imply that the characteristic pulsation periods should lie in the interval \(\mathrm{0.3 d<P_p<90 d}\) which, conveniently, is observed for these stars. Thus these stars can be understood as undergoing radial oscillations.

b. Effect of Magnetic Fields and Rotation on Radial Oscillations

The impact of rotational motion or the presence of a strong magnetic field can be significant to the characteristic periods of pulsation for a star. Detailed predictions are difficult, for they depend on knowledge of the internal angular momentum and magnetic field distribution. However, the Virial theorem gives us some insight into the nature of such effects. We need only calculate the variational behavior for the magnetic energy density and angular momentum (see Collins⁶) to find that the pulsational frequency, under the assumptions made in obtaining equation \ref{8.1.13}, is \[\sigma^2=-\left[<3 \gamma-4>\left(\Omega_0-\mathrm{M}_0\right)+<5-3 \gamma>\omega_0 \quad \mathbf{L}_0\right] / \mathrm{I}_0\label{8.1.16}\]

Here M0 is the total internal magnetic energy of the equilibrium configuration while \(\mathbf{L}_0\) is the total angular momentum. It is clear that the effect of rotation on stars where \(4 / 3<\gamma<5 / 3\) will cause a decrease in the pulsational period, while the presence of a strong magnetic field will cause the period to increase.

c. Stability and the Variational Virial Theorem

If we eliminate the total internal energy in favor of the total energy E, the Virial theorem for static stars as given by equation \ref{1.2.34} can be written as \[\frac{1}{2} \frac{d^2 I}{d t^2}=2 E-\Omega>2 E\label{8.1.17}\]

Thus, if the total energy of any configuration is greater than zero, the moment of inertia will increase without bound and the system can be said to be dynamically unstable. This condition is sometimes called Jacobi's stability criterion and as stated is a sufficient condition for a system to be dynamically unstable. We may extend this condition to the variational Virial theorem by noting that if \(\sigma^2<0\), then the perturbations have the form \[\xi=\xi_0 e^{ \pm 2 \pi t / t_0}\label{8.1.18}\]

Since we may expect the full spectrum of perturbations to be present in any configuration, the ∀ sign does not matter, for some perturbation will grow exponentially without bound and the object will be unstable. A quick inspection of equations \ref{8.1.13} and \ref{8.1.16} shows that they represent a set of sufficient conditions for a star to be dynamically unstable. In the absence of magnetic fields and rotation, equation \ref{8.1.13} shows that a necessary condition for a star to be stable is that \(\gamma>4/3\), which is consistent with what we learned in Chapter 6 about the relativistic polytrope. In the absence of rotation, equation \ref{8.1.16} implies that the magnitude of the gravitational energy must exceed that of the magnetic energy if the star is to remain stable. In chapter 7 we saw that the presence of a magnetic field would tend to distort the star and this would seem to be a destabilizing process. This is not necessarily the case for rotation since equation \ref{8.1.16} indicates that, for \(4/3<\gamma<5/3\), rotation actually seems to help stabilize the star. This occurs because as an oscillation takes place in a rotating star, it is necessary for the pulsating material to conserve angular momentum. For reasonable values of \(\gamma\), this removes energy from the gas, thereby enhancing the stability of the motion of the perturbation.

The oscillations described so far represent only the fundamental or lowest frequency of oscillation that one could expect. It is this fundamental mode that is limited by the global characteristics of the star. However, it is possible for the star to oscillate at higher frequencies. Under these conditions, the oscillations will take the form of standing waves, with nodes at the surface and the center of the star and possibly elsewhere. However, to show this, it is necessary to consider the internal structure in greater detail than that afforded by the Virial theorem.

d. Linear Adiabatic Wave Equation

To find higher-order modes of oscillation, it is necessary to track the motion of the gas within the star. This can be done by considering the equations of motion for the gas. In Chapter 1 we developed the Euler-Lagrange equations of hydrodynamic flow [equation \ref{1.2.27}]. These can be use to develop equations of motion for small-amplitude oscillations. Remember that \[\frac{\partial \vec{u}}{\partial t}+(\vec{u} \cdot \nabla) \vec{u}=-\nabla \Omega+\frac{\nabla P}{\rho}\label{8.1.19}\]

Since \(\vec{\mathrm{u}}\) represents the motion of the gas during the pulsation, we can assume it to be small. Under these conditions, the second term of equation \ref{8.1.19} will be second-order, and we may write the equations of motion as \[r_0 \frac{d^2 \xi}{d t^2}=-\nabla \Omega+\frac{\nabla P}{\rho}\label{8.1.20}\]

where \[\begin{array}{ll}
\vec{u}=\frac{d(r \xi)}{d t}=r_0 \frac{d \xi}{d t} & P=P_0+\delta P \\
\Omega=\Omega_0+\delta \Omega & \rho=\rho_0+\delta \rho
\end{array}\label{8.1.21}\]

The subscript ₀ refers to the equilibrium configuration, so hydrostatic equilibrium requires that \(\rho \nabla \Omega_0=-\nabla P_0\). The variational form of the equations of motion becomes \[r_0 \frac{d^2 \xi}{d t^2}=-\nabla \delta \Omega+\frac{\nabla \delta P}{\rho_0}-\frac{\delta \rho \nabla P}{\rho_0^2}\label{8.1.22}\]

Assuming that the variation has the form given by equation \ref{8.1.12} and that the variation of the density is given by equation \ref{8.1.10}, we can use the fact that \[\nabla \delta \Omega=\xi \nabla \Omega_0=-\frac{\xi \nabla P_0}{\rho_0}\label{8.1.23}\]

and some algebra to write \[\frac{d}{d r}\left(\gamma_0 r^4 \frac{d \xi_0}{d r}\right)+\xi_0\left\{\sigma^2 \rho r^4+r^3 \frac{d}{d r}\left[(3 \gamma-4) P_0\right]\right\}=0\label{8.1.24}\]

This is known as the linear adiabatic wave equation for \(\xi(\mathrm{r})\) (see Cox⁷); since all the coefficients are real, \(\sigma^2\) must also be real, and pure standing waves are possible. Clearly this is a linear homogeneous second-order differential equation in the displacement \(\xi_0\), so we can expect some ambiguity in the solution. This ambiguity takes the form of an amplitude that can be scaled. That is, if \(\xi_0(\mathrm{r})\) represents a solution to the wave equation, so does \(\mathrm{A\xi_0(r)}\). Since the spatial variation must vanish at the origin, the boundary condition at the center is \[\left.\frac{d \xi_0}{d r}\right|_{r=0}=0\label{8.1.25}\]

The appropriate boundary condition for the surface is rather more difficult to obtain and is given by Cox⁷ (pp. 77-80) as \[\begin{aligned}
\left.\frac{\delta P}{P_0}\right|_{r=R} & =-\left(\frac{\sigma^2 R^3}{G M}+4\right) \xi_0=-\gamma\left(3 \xi_0+\left.R_0 \frac{d \xi_0}{d r}\right|_{r=R}\right) \\
& =-\left.\gamma \frac{\delta \rho}{\rho_0}\right|_{r=R}
\end{aligned}\label{8.1.26}\]

These conditions provide the two constants required for the integration of the wave equation, but the actual solution takes the form of a two-point boundary-value problem. Actually with the equilibrium structure known from models, the problem is to find the eigenvalue \(\sigma^2\) which satisfies the wave equation subject to the boundary conditions. Thus, in principle, we can find the entire spectrum of allowed adiabatic oscillations for a particular star. Since the solutions are pure standing waves, the state variables oscillate locally about the equilibrium values passing through them twice each cycle. However, to understand the origin of these oscillations, we must describe the nonadiabatic processes which drive and damp them.