4 Propagation of acoustic-gravity waves

The stellar oscillations are now described by one or another phase equations (Eqs. (36) and (37)). These phase equations with the respective boundary conditions 44 and 47 or 45 and 48 form a boundary value problem with $\omega ^2$ as an eigenvalue. In this section we start by determining numerically the eigenfrequencies that are solutions of the eigenvalue problem presented. By the following, we will present a classification scheme base in the result obtained.

Let's start by observing that the phase function $\theta_i (\omega,l,r)$ (mod $\pi$), has the same number of zeros at the same location as the wave function $\Psi$, provided that the transformations are made as was indicated previously. However, the number and location of the zeros of the wave function, $\delta p$, (obtained from the equation of motion, Sect. 2), are the same as $\Psi$, only for pure acoustic or gravity modes, due to the fact that the discriminant fas given by Eq. (10) is strictly positive or negative in all the star. In the case of mixed modes, another zero occurs for the wave function, $\delta p$, at the radius of the star where f=0.

The main properties of the stellar oscillations, can be well defined in a linear phase diagram (see Fig. 4) that determines the phase dependence of a given eigenmode with the radius, for a given equilibrium structure. The construction of the linear phase diagram is made by introduction of an effective phase, $\phi (\omega,l;r)$. This effective phase for each eigenmode, of an eigenvalue $\omega$ and degree l, is given by

$$\phi (\omega,l;r) =\theta_i (\omega,l;r) -\theta_i (\omega,l;0)$$ (53)

where $\theta_i$, is the phase function. The computation of the linear phase diagram is made using the phase Eq. (36) in the case of acoustic modes, and the Eq. (37), with $\theta_2=\theta_1\pm \pi/2$, in the case of gravity modes. In that case the linear phase diagram for acoustic and gravity modes is constructed based in the phase $\theta_1$. This is justified by the simple relation between acoustic and gravity description when the function ${\cal T}$ is neglected in the phase equation. We remember that an eigenmode is the product of an amplitude function by a phase function, and the main properties of the eigenmode can be discussed using the phase equation.

Figure 3: Propagation diagram for a polytropic model with index $n_{\rm e}=3.0$, adiabatic index $\gamma =5/3$ and modes of degree l=2. The solid curves represent the critical frequencies $\omega _{\pm }$, delimiting the two propagation regions of the stellar oscillations as it is obtained from the first Cowling approximation motion equation (see text). The eigenstates corresponding to the eigenmodes with dimensionless square of the circular frequency, $\omega ^2$: 234, 196, 163, 132.4, 104.8, 80.55, 59.42, 41.47, 26.72, 15.26, 8.175, 4.915, 2.828, 1.822, 1.270, 0.9360, 0.7188.

The nonradial oscillations of a star can be interpreted as a superposition of stationary waves generated by the competition between the pressure and the gravity. These two driving mechanisms, define the two propagation regions illustrated in the Fig. 3 for a polytropic equilibrium structure of index $n_{\rm e}=3$. An eigenstate, can be interpreted heuristically as being the result of the competition between these two propagative regions, as it is determined by the phase Eq. (36) (or Eq. (37)). The eigenstates corresponding to the acoustic waves of high frequency, can be determined neglecting the contribution of the gravity propagation region (which corresponds to neglecting the second term of Eq. (36), which case is similar to the Sturm-Liouville problem. In this case, the modes form a well-ordered sequence, starting from a given fundamental state. Conversely, the eigenstates of gravity waves of very low frequency, can be determined by neglecting the contribution of the acoustic propagation. A particular state occurs in the star, even in a very simple equilibrium structure, caused by the fact that both propagative regions contribute in the same order of magnitude for this standing wave. This corresponds to the f-mode. Furthermore, this eigenmode presents a mixed nature and propagates mainly or uniquely in the evanescent region of the star. Cowling (1941) was the first to point out the fundamental difference of nature of the f-mode, relatively to the p-modes and g-modes. The f-mode corresponds to the fundamental oscillation of the system for which no nodes occur on the wavefunction. Furthermore, for stars where mixing of modes can occur, this f-mode, as it is defined here, cannot exist.

It is possible to prove that the modes form a well-ordered sequence when the star is simple, given that two different eigenmodes never cross. However, this can be guaranteed only for modes that propagate locally as an acoustic type wave or gravity type wave. The evanescent type mode f is the only exception. The fact that the phase of two eigenmodes never crosses, is reliable on the topological properties of the propagation diagram (see Figs. 1, 2 and 3) and the properties of the phase equations. For two waves with frequencies, $\omega_a$ and $\omega_b$, such as $\omega_b < \omega_a$, the propagative characteristics are fixed differently for an acoustic-gravity wave, in function of the type of propagation. For a wave that propagates acoustically, the higher frequency presents always a larger propagative region. Conversely, for a wave that propagates by gravity type, the higher frequency presents always a smaller propagative region. Furthermore, on the propagative region, we have that $f_{\rm p}(r;\omega_a) \ge f_{\rm p}(r;\omega_b) \ge f_{\rm g} (r;\omega_a) \ge f_{\rm g} (r;\omega_b) \ge 0$, for two waves that propagate acoustically, and $f_{\rm g}(r;\omega_a) \le f_{\rm g}(r;\omega_b) \le f_{\rm p} (r;\omega_a) \le f_{\rm p} (r;\omega_b) \le 0$, on the other cases. Using the phase equations corresponding to each one of these descriptions, we have that $\theta_{1,a} (r) \ge \theta_{1,b} (r)$for acoustic type propagation and $\theta_{2,a} (r) \le \theta_{2,b} (r)$for gravity type propagation, for each point inside the star. This determines that two eigenmodes can never cross each other, if they are acoustic waves, gravity waves or mixed waves.

Figure 4: Linear phase diagram for a polytropic equilibrium model of index, $n_{\rm e}=3$, and adiabatic index $\gamma =5/3$. The curves represent the phase function $\phi (\omega ,l,r)$ of eigenmodes of frequency $\omega$ and degree l=2. The thin-dashed straight lines represent the number $\pi$-cycle that a phase function has developed from the internal endpoint r=0, until the external endpoint r=R. The inner boundary condition at r=0 and the outer boundary condition at r=R for the different cycles of $\pi$ are represented by a black dot. In the propagative regions where the eigenmodes behave like an acoustic wave the phase increases (clockwise direction) and in propagative regions where the eigenmodes behave like a gravity wave the phase decreases (anticlockwise direction). The phase function, $\phi$ is calculated for a given eigenstate, from the higher to the lower eigenfrequencies. The radial order number, $n=\phi /\pi$, takes values from 10 to -10 (see text).

In Figs. 3 and 4 we represent the propagation diagram and the linear phase diagram for the eigenmodes of degree l, for a polytropic equilibrium structure of index $n_{\rm e}=3$ and adiabatic index $\gamma =5/3$. The phase function of acoustic waves and gravity waves, present important propagative topological differences between them. However, for the very low frequency acoustic waves and high frequency gravity waves, the propagation is similar to a f-mode.

The main asymptotic properties can be illustrated in the planar approximation (see Appendix B). In the case of acoustic modes, corresponding to the higher frequencies ( $\omega \rightarrow +\infty$), the phase Eq. (36) is dominated by the term $\omega/c$, through the discriminant $f_{\rm p}$ and $f_{\rm g}$. Then the phase, $\theta_1$, is given approximatively by

$$\theta_1(\omega,l;r) \approx \omega \tau \qquad \tau(r)=\int_0^r \frac{{\rm d}r}{c}$$ (54)

where $\tau$, is the so-called acoustic radius. However, the phase stays practically constant on the evanescent regions.

Alternatively, the gravity modes are given by Eq. (37), corresponding to the smaller frequencies ( $\omega \rightarrow 0$), the discriminants $f_{\rm p}$ and $f_{\rm g}$ can be approximated by

$$f_{\rm p}\approx -\frac{L^2}{\omega}\frac{c}{r^2}\;\; {\rm and}\;\; f_{\rm g} \approx -\frac{1}{\omega} \frac{N^2}{c}\cdot$$ (55)

In that case, we have a competition between these two terms, which give the waviness pattern, characteristic of the phase function of gravity waves. It is possible to demonstrate that the asymptotic leading term, in this case leads to $\sqrt{f_{\rm p} f_{\rm g}}$ $\approx L/\omega \; N/r$.

In the example presented above, mixing of modes does not occur (no overlap of the propagative region), but if this happens, the phase function presents a similar behaviour to the acoustic and gravity modes for each one of the respective propagation regions (see Fig. 2b).