3.4: Variational Form of the Surface Terms

In deriving the virial theorem, we noted earlier that the use of the divergence theorem yields some surface integrals which are generally ignored. Formally they may be ignored by taking the bounding surface of the configuration to be at infinity. However, in reality, this generally proves to be inconvenient for stars as they usually have a reasonably well-defined surface or boundary. For stars possessing general magnetic fields which extend beyond the surface, these surface contributions should be included. They are usually wished away by assuming they are small compared to the total magnetic energy arising from the volume integration. Although this may be true for simple fields in stars, it is unlikely to be true for other gaseous configurations such as flares and in any event a numerical estimate of their importance is far more re-assuring than an intuitive feeling. For this reason, let us consider the way in which these surface terms affect the variational formalism of the previous section. To facilitate the calculations, we will assume the star is nearly spherical and the pulsations are radial. If the magnetic field is strong this will clearly not be the case and the full tensor virial theorem must be used. However, the simplicity generated by the use of the scalar virial theorem justifies the approach for purposes of illustration. Let us begin by sketching the origin of the virial theorem as rigorously presented by Chandrasekhar⁶. The Equations of motion for a gas with zero resistivity are

\[\rho \frac{\mathrm{d} \mathbf{u}}{\mathrm{dt}}=-\nabla \rho+\rho \nabla \Phi+\frac{1}{4 \pi}(\nabla \times \mathbf{H}) \times \mathbf{H}.\label{3.4.1}\]

Employing the identity \((\nabla \times \mathbf{H}) \times \mathbf{H}=(\mathbf{H} \cdot \nabla) \mathbf{H}-\nabla(\mathbf{H} \cdot \mathbf{H}) / 2\) and taking the scalar product of Equation \ref{3.4.1} with the position vector r then integrating over all space enclosed by the bounding surface, we get

\[\int_{\mathrm{v}} \rho \mathbf{r} \cdot \frac{\mathrm{du}}{\mathrm{dt}} \mathrm{dV}=-\int_{\mathrm{v}} \mathbf{r} \cdot \nabla \mathrm{PdV}+\int_{\mathrm{v}} \rho \mathbf{r} \cdot \nabla \Phi \mathrm{dV}+\frac{1}{4 \pi} \int_{\mathrm{V}} \mathbf{r} \cdot (\mathbf{H} \cdot \nabla) \mathbf{H d V}-\frac{1}{8 \pi} \int_{\mathrm{v}} \mathbf{r} \cdot \nabla\left(\mathrm{H}^2\right) \mathrm{dV}.\label{3.4.2}\]

As in section 3, this becomes

\[\frac{1}{2} \frac{\mathrm{d}^2 \mathrm{I}}{\mathrm{dt}^2}-2 \mathrm{T}=3(\Gamma-1) \mathcal{U}+\Omega+\mathscr{M}-\int_{\mathrm{s}} \mathrm{P}_0 \mathbf{r} \cdot \mathrm{d} \mathbf{S}-\frac{1}{8 \pi} \int_{\mathrm{s}} \mathrm{H}_0^2 \mathbf{r} \cdot \mathrm{d} \mathbf{S}+\frac{1}{4 \pi} \int_{\mathrm{s}}\left(\mathbf{r}_0 \cdot \mathbf{H}_0\right)\left(\mathbf{H}_0 \cdot \mathrm{d} \mathbf{S}\right),\label{3.4.3}\]

where \(\mathrm{P_o}\) and \(\mathbf{H}_\mathrm{o}\) are the gas pressure and magnetic field present at the surface \(\mathbf{r}_\mathrm{o}\). It is the behavior of the three integrals in Equation \ref{3.4.3} that will interest us as hopefully the remaining terms are by now familiar. Consider first the effect of a pulsation on the surface term arising from the pressure by taking the variations of the surface pressure integral.

\[\delta \int_{\mathrm{s}} \mathrm{P}_0 \mathbf{r}_0 \cdot \mathrm{d} \mathbf{S}=\int_{\mathrm{s}} \delta \mathrm{P}_0 \mathbf{r}_0 \cdot \mathrm{d} \mathbf{S}+\int_{\mathrm{s}} \mathrm{P}_0 \delta \mathbf{r}_0 \cdot \mathrm{d} \mathbf{S}+\int_{\mathrm{s}} \mathrm{P}_0 \mathbf{r}_0 \cdot \mathrm{d}(\delta \mathbf{S}).\label{3.4.4}\]

For radial variations only

\[\mathrm{d}(\delta \mathbf{S})=2 \mathrm{r}_0 \delta \mathrm{r}_0 \sin \theta \mathrm{d} \theta \mathrm{d} \phi=2 \xi \mathrm{r}_0^2 \sin \theta \mathrm{d} \theta \mathrm{d} \phi=2 \xi \mathrm{d}\mathrm{S},\label{3.4.5}\]

where, as in section 2, \(\xi=\delta \mathrm{r} / \mathrm{r}\). In Chapter III, section 2 [Equations (N3.2.13) and Equation \ref{3.2.14}], we already have shown that for adiabatic pulsations

\[\frac{\delta \mathrm{P}}{\mathrm{P}}=\gamma \frac{\delta \rho}{\rho}=-\gamma\left(3 \xi+\mathrm{r} \frac{\mathrm{d} \xi}{\mathrm{dr}}\right).\label{3.4.6}\]

Combining Equation \ref{3.4.5} and Equation \ref{3.4.6} with Equation \ref{3.4.4}, we get

\[\delta \int_{\mathrm{s}} \mathrm{P}_0 \mathbf{r}_0 \cdot \mathrm{dS}=3(\gamma-1) \int_{\mathrm{s}} \xi \mathrm{P}_0 \mathbf{r}_0 \cdot \mathrm{dS}-\gamma \int_{\mathrm{s}}\left(\left.\mathrm{r}_0 \frac{\mathrm{d} \xi}{\mathrm{dr}}\right|_{\mathrm{r}_0}\right) \mathbf{r}_0 \cdot \mathrm{dS}.\label{3.4.7}\]

Earlier we assumed that \(\xi\) was constant throughout the star and hence its derivative vanished. Here, we only require the derivative to vanish at the surface in order to simplify Equation \ref{3.4.7} to get

\[\delta \int_{\mathrm{s}} \mathrm{P}_0 \mathbf{r}_0 \cdot \mathrm{dS}=3(\gamma-1) \xi \int_{\mathrm{s}} \mathrm{P}_0 \mathrm{r}_0 \mathrm{dS}.\label{3.4.8}\]

Now consider the variation of the two magnetic integrals in Equation \ref{3.4.3}.

\[\begin{gathered}
\delta\left[\frac{1}{8 \pi} \int_{\mathrm{s}} 2\left(\mathbf{r}_0 \cdot \mathbf{H}_0\right)\left(\mathbf{H}_0 \cdot \mathrm{d} \mathbf{S}\right)-\frac{1}{8 \pi} \int_{\mathrm{s}} \mathrm{H}_0^2 \mathbf{r}_0 \cdot \mathrm{d} \mathbf{S}\right]=\frac{1}{8 \pi} \int_{\mathrm{s}} 2\left(\delta \mathbf{r}_0 \cdot \mathbf{H}_0\right)\left(\mathbf{H}_0 \cdot \mathrm{d} \mathbf{S}\right)+\frac{1}{8 \pi} \int_{\mathrm{s}} 2\left(\mathbf{r}_0 \cdot \delta \mathbf{H}_0\right)\left(\mathbf{H}_0 \cdot \mathrm{d} \mathbf{S}\right) \\[4pt]
+\frac{1}{8 \pi} \int_{\mathrm{s}} 2\left(\mathbf{r}_0 \cdot \mathbf{H}_0\right)\left(\delta \mathbf{H}_0 \cdot \mathrm{d} \mathbf{S}\right)+\frac{1}{8 \pi} \int_{\mathrm{s}} 2\left(\mathbf{r}_0 \cdot \mathbf{H}_0\right)\left(\mathbf{H}_0 \cdot \mathrm{d} \delta \mathbf{S}\right)+\frac{1}{8 \pi} \int_{\mathrm{s}} 2\left(\mathbf{H}_0 \cdot \delta \mathbf{H}_0\right)\left(\mathbf{r}_0 \cdot \mathrm{d} \mathbf{S}\right) \\[4pt]
-\frac{1}{8 \pi} \int_{\mathrm{s}} \mathrm{H}_0^2 \delta \mathbf{r}_0 \cdot \mathrm{d} \mathbf{S}-\frac{1}{8 \pi} \int_{\mathrm{s}} \mathrm{H}_0^2 \mathbf{r}_0 \cdot \mathrm{d} \delta \mathbf{S}
\end{gathered}\label{3.4.9}\]

This truly horrendous expression does indeed simplify^3.7 by using

\[\delta \mathbf{H}=-2 \xi \mathbf{H}_0 .\label{3.4.10}\]

Using this result, Equation \ref{3.4.5} and the definition of \(\xi\), Equation \ref{3.4.9} becomes:

\[\delta \mathrm{Q}_{\mathrm{m}}=-\frac{\xi}{8 \pi}\left[\int_{\mathrm{s}} 2\left(\mathbf{r}_0 \cdot \mathbf{H}_0\right)\left(\mathbf{H}_0 \cdot \mathrm{d} \mathbf{S}\right]-\int_{\mathrm{s}} \mathrm{H}_0^2 \mathbf{r}_0 \cdot \mathrm{d} \mathbf{S}\right..\label{3.4.11}\]

where \(\mathrm{Q_m}\) stands for the original magnetic surface term that appears in Equation \ref{3.4.3}. Thus, the variation of the surface terms can be represented as

\[\left.\begin{array}{l}
\delta \mathrm{Q}_{\mathrm{m}}=-\xi \mathrm{Q}_{\mathrm{m}} \\[4pt]
\delta \mathrm{Q}_{\mathrm{P}}=3(\gamma-1) \xi \mathrm{Q}_{\mathrm{P}}
\end{array}\right\}.\label{3.4.12}\]

If we assume a linear or homologous pulsation, as was done in the previous two sections, then the expression for the pulsational frequency [Equation \ref{3.3.45}], becomes

\[\sigma^2=-\frac{(3 \gamma-4)\left(\Omega_0+\mathscr{M}_0\right)-(5-3 \gamma) \omega_0 \mathscr{L}_0+(3 \gamma-1) \mathrm{Q}_{\mathrm{P}}-\mathrm{Q}_{\mathrm{m}}}{\mathrm{I}_0}.\label{3.4.13}\]

Since \(\gamma>4/3\), the contribution of the surface pressure term is such as to increase \(\sigma^2\) and thereby improve the stability of the system. Basically, this results because an unstable system will have to do work against the surface pressures either in expanding or contracting the surface. This energy is thus not available to feed the instability. The situation is not as obvious for the magnetic contribution \(\mathrm{Q_m}\), since \(\mathrm{Q_m}\) is the difference between two positive quantities. Thus, the result depends entirely on the geometry of the field. The effect of the field geometry can be made somewhat clearer by considering a spherical star so that the radius vector is parallel to the surface normal. Under these conditions \(\mathrm{Q_m}\) becomes

\[\mathrm{Q}_{\mathrm{m}}=\frac{1}{8 \pi} \int_{\mathrm{s}}\left[2\left(\hat{\mathbf{H}}_0 \cdot \hat{\mathbf{r}}\right)^2-1\right] \mathrm{H}_0^2 \mathrm{rdS}=\frac{1}{8 \pi} \int_{\mathrm{s}} \cos \beta \mathrm{H}_0^2 \mathrm{rdS},\label{3.4.14}\]

where β is the local angle between the field and the radius vector. Thus, the average of cos2β weighted by \(\mathrm{H_0^2}\) over the surface will determine the sign of \(\mathrm{Q_m}\). In any event, it is clear that

\[\left|\mathrm{Q}_{\mathrm{m}}\right|<\frac{1}{8 \pi} \int_{\mathrm{s}} \mathrm{H}_0^2 \mathrm{r}_0 \mathrm{dS}=\frac{1}{2} \mathrm{R}_0^3 \overline{\mathrm{H}_0^2}.\label{3.4.15}\]

It is worth noting that in the case where the magnetic field increases slowly with depth, this term can be of the same order of magnitude as the internal magnetic field energy and must be included. Furthermore, whether or not the local contribution to \(\mathrm{Q_m}\) is positive or negative depends on whether or not the local value of β is greater or less than \(\pi/4\). Since a positive value of \(\mathrm{Q_m}\) increases the value of \(\sigma^2\), fields exhibiting a local angle to the radius vector greater than \(\pi/4\) tend to stabilize the object, whereas more radial fields enhance the instability. This simply results from the fact that a radial motion will tend to compress fields more nearly tangential to the motion than 45^o, thereby removing energy from the motion. Conversely, more nearly radial fields will tend to feed the perturbation leading to a decrease in stability.

One thing becomes immediately clear from this discussion. If \(\mathrm{Q_m}\) is an important term in Equation \ref{3.4.13}, radial pulsation will not occur. Since a magnetic field cannot exhibit spherical symmetry, the departures from symmetry will yield a variable "restoring force" over the surface inferring that non-radial modes will be excited. In this case the tensor virial theorem must be used and the field geometry known. Lastly, for purposes of simplicity, we have assumed no coupling between the gas pressures and magnetic pressures. Unless the system is rather bizarre, the gas will be locally relaxed on a time scale less than the pulsation period and hence the two cannot be treated independently. This assumption was made merely for the sake of simplicity and doesn't affect the illustrative aspects of the effects. However, unless \(\mathrm{Q_m}\) and \(\mathrm{Q_p}\) are comparable the coupling between the two will be weak and we may expect Equation \ref{3.4.13} to give good quantitative results.