What good is asteroseismology? Well, the curse of astronomy is that as experimentalists, we are unable to perform manipulations on the objects we like to study so we are reduced to looking at the light we see from these objects, and observing only that which nature chooses to show us. Of course we only see the photospheres, the thin outer skins. Asteroseismology gives us a way to look inside.

How do we do asteroseismology? Through an approach we are all very familiar with as physicists: normal mode analysis. There is an obvious analogy with seismology of the Earth, from which the ideas for asteroseismology have been borrowed. Perhaps a more familiar analogy is with atomic structure. We see from the mathematics why the problems are so analogous. Both are simple spherical-potential problems. The pulsation modes in the star are described in the same way as the energy levels of the atom.

In order to understand the oscillations that we actually observe we need to consider the general class of non-radial spheroidal oscillations. We arrive at the form of the equations we actually solve by perturbing the fluid equations, and keeping only the terms of lowest order. Non-linear effects are clearly important, and of increasing interest, but they are well beyond the scope of our present discussion. We assume a static, spherical equilibrium structure which is given by a theoretical evolutionary model. Because the surface gravity is high, $\log g \sim 8$, in cgs units, and rotation rates are typically of order days, the spherical approximation is quite good, and so we can expand our solutions in terms of spherical harmonics, ( $Y_{\ell m}$). Further, we seek periodic solutions of the form,

\begin{displaymath}\xi({\bf r},t)=\xi({\bf r})e^{i\sigma t}. \end{displaymath}

The most relevant parameters which describe the equilibrium model are the Brunt-Väisälä frequency, which in the absence of chemical potential gradients is given by


\begin{displaymath}N^2 \equiv -Ag=-g\left[\frac{d\ell n\rho}{dr}-\frac{1}{\Gamma_1}\frac{d\ell nP}{dr}\right], \end{displaymath}

which is just the difference between the actual and the adiabatic density gradients, and the Lamb, or acoustic, frequency given by


\begin{displaymath}S^2_\ell \equiv \frac{\ell(\ell+1)}{r^2}\ \frac{\Gamma_1P}{\rho}=\frac{\ell(\ell+1)}{r^2}v_s^2, \end{displaymath}

where vs is the local sound speed, and all quantities have their usual meanings.

We can gain a great deal of physical insight into the oscillations, and how they sample the star through a sort of local analysis (see for example, Unno et al. 1989). If we assume a radial dependence of the form, eik, and wavelengths short compared to the relevant scale-heights for the physical quantities, we arrive at a local dispersion relation (LDR) of the form,


\begin{displaymath}k^2_r=\frac{k^2_h}{\sigma^2 S^2_\ell}(\sigma^2-N^2)(\sigma^2-S_\ell^2) \end{displaymath}

where we have defined a horizontal wavenumber,

\begin{displaymath}k_h^2 \equiv \frac{\ell(\ell+1)}{r^2}\ \frac{S^2_\ell}{v_s^2} \end{displaymath}

such that the total wavenumber is

\begin{displaymath}k^2 \equiv k^2_k+k^2_r. \end{displaymath}

The LDR allows us to see how the two characteristic frequencies determine the nonradial pulsation properties of the star. In order for a given mode to be locally propagating kr2 must be positive, so from the above expression we see this occurs only when the oscillation frequency is greater than both N and the $S_\ell$, or is less than both.

Taking the limits of large and small frequencies, LDR yields two physically distinct kinds of solutions that represent the two principle classes of non-radial spheroidal modes,

1. $\sigma\gg N^2,S_\ell^2$

\begin{displaymath}\sigma^2_p\approx\frac{k^2}{k^2_h}\ S_\ell^2=(k^2_r+k^2_t)v_s^2 \end{displaymath}

2. $\sigma^2\ll N^2,S_\ell^2$

\begin{displaymath}\sigma^2_g\approx\frac{k^2_h}{k_r^2+k_h^2}\ N^2. \end{displaymath}

The first class of solutions represent the p-modes, so called because pressure is the principle restoring force. Radial displacements are dominant, and for white dwarf stars these have timescales of seconds, too short to be the periods of the observed oscillations. Also, one would not expect to observe large radial displacements on such a high-gravity object as a white dwarf. The second class of modes are the g-modes, where gravity is the dominant restoring force. These have timescales of hundreds of seconds and longer, just like the observed oscillations in the white dwarf stars. Also, the motions are predominantly horizontal, along gravitational equipotential surfaces. These expressions also indicate that the frequencies of the g-modes decrease with increasing radial overtone number (shorter wavelength) with an accumulation point at zero frequency. We can construct an expression for g-mode frequencies from this by an integration over the star and arrive at an expression,


\begin{displaymath}\sigma_{k,\ell,m}\approx\langle\frac{N^2\ell(\ell+1)}{k^2r^2}\rangle^{1/2}+ \left(1-\frac{C_k}{\ell(\ell+1)}\right)m\Omega. \end{displaymath}

I have included the second term on the right hand side of the expression to indicate the effects of slow-rotation on the frequencies--note that there is a somewhat similar effect due to magnetic fields which we do not explicitly include here. The effect is to break the spherical symmetry and make the azimuthal quantum numbers, m, non-degenerate. Here $\Omega$ is the rotation frequency and the constant Ck is a quantity depending on the eigenfunction but approximately equal to one in most cases. This last term produces the potential for fine-structure, splitting each radial overtone into $2\ell+1$ components. Observing the spacing between these components then allows us to measure the rotation rate of the star.

When the first term in the above expression is dominant (the slow rotation, small magnetic field limit) we can also see another important feature of the g-mode frequencies: in a compositionally homogeneous star, we could expect the spacing of consecutive radial overtone g-modes, for a given spherical harmonic degree, $\ell$, to be approximately uniform in period. Further, the spacing depends on the integrated average of the N2, and so is set primarily by the total mass of the star. Deviations from this uniform spacing gives information about compositional stratification and allows us to measure the masses of the different layers.

We can use plots of the run of N2 and $S^2_\ell$ through the star to determine where modes of given frequencies will propagate. Such diagnostic plots are known as propagation diagrams, and contain essentially all the information we can obtain through asteroseismological analysis of the star. In a sense the ultimate of asteroseismological analysis is to use the distribution of observed frequencies to determine, empirically, the propagation diagram for the star. It is important to note that if this is known, all of the constitutive physics can be de-convolved--at least in principle. This kind of analysis is called seismological inversion, and has been used successfully so far on only one star, the Sun. This is because of the more limited number of modes observed in other stars, but there is hope for the future of this technique as applied to white dwarf stars. For now, we use mostly the forward technique which consists of matching the observed frequencies to models and trying to find the particular theoretical model which best fits the observed periods.

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