Appendix A: Phase analysis

Here, we introduce a technique which allows to investigate the properties of the eigen-solutions of a formally self-adjoint differential equation, which was been developed initially, to analyze the properties of the solutions of the Sturm-Liouville problem (Prüfer 1926). This method is convenient to find out how often the solution of Eq. (29) (corresponding to both descriptions), oscillates in the interval under consideration, which corresponds to the number of zeros of that solution.

A.1 Basic transformation

The linear adiabatic wave equation of stellar oscillations (Eq. (29)), is written as a system of two differential equations of first-order. This is accomplished by the transformation

$${\cal Z}_i(r) = \frac{1}{{\cal P}_i (r)}\frac{{\rm d}{\cal Y}_i}{{\rm d}r}$$ (A.1)

resulting in the system

 

$$\left\{ \begin{array}{ll} \frac{ {\rm d} {\cal Y}_i } { {\rm d}r ... ...m d} {\cal Z}_i } { {\rm d}r } =- {\cal Q}_i(r) {\cal Y}_i \end{array}\right. .$$ (A.2)

It is possible to reduce this system to a new one, where the two equations are coupled only in the amplitude equation, leaving a single phase equation to determine the eigenfrequencies. This is accomplished by using polar coordinates (phase transformation),

$${\cal Y}_i=R_i \sin{\theta_i} \qquad {\cal Z}_i={\cal P}_i{\cal Y}'_i=R_i \cos{\theta}_i$$ (A.3)

where ${\cal R }_i=R_i(r)$, is the amplitude of the wave function ${\cal Y}_i$, given by

 

$$R_i^2={\cal Y}_i^2+\left({\cal P}_i{\cal Y}'_i\right)^2$$ (A.4)

and $\theta_i=\theta_i(r)$, the phase of the wave function ${\cal Y}_i$, given by

 

$$\theta_i=\arctan{\left(\frac{{\cal Y}_i}{{\cal P}_i{\cal Y}'_i}\right)}\cdot$$ (A.5)

Solving the system (A.2) for R_i and $\theta_i$, yields

 

$$\frac{ {\rm d} \theta_i } { {\rm d} r} = {\cal Q}_i(r) \sin^2{\theta_i} + \frac{1}{{\cal P}_i(r)} \cos^2{\theta_i}$$ (A.6)

and

 

$$R_i =R_{i,{\rm o}}\exp{\left[ \frac{1}{2} \int_{r_{\rm o}}^r \lef... ...r) } - {\cal Q}_i(r) \right) \sin{\left(2\theta_i\right)} \; {\rm d}r \right]}.$$ (A.7)

Equations (A.6) and (A.7) are called the phase associated system. Each solution of the original self-adjoint differential Eq. (29) is determined up by two constants: the initial amplitude $R_{i,{\rm o}}=R_i(r_{\rm o})$ and the initial phase $\theta_{i,{\rm o}}=\theta_i(r_{\rm o})$, where $r_{\rm o}$ corresponds to the endpoint where the boundary condition is applied. The change of the initial phase $R_{i,{\rm o}}$, corresponds to multiply the solution ${\cal Y}_i$, by a constant factor; thus, the zeros of the wave function ${\cal Y}_i$can be located by studying the differential Eq. (A.6). In this sense, this transformation preserves the zeros of the original wave function, and their respective location. The eigenmodes can be determined by the solution of Eq. (A.6) with the convenient boundary conditions. We remember that this analysis is applied to the linear adiabatic non-radial oscillations and consequently the amplitude of waves can be determined up to an arbitrary constant.

We point out that the transformation of the standard differential equation into a self-adjoint form does not conserve necessarily the number of zeros or the location of the original differential equation. The same remains true for all the transformations between equivalent differential equations. However, under the two transformations presented previously to the differential Eq. (4), the self-adjoint transformation and the phase transformation, the number and the location of zeros of the wave function $\Psi$are preserved. In the case of the self-adjoint form it is necessary to take the convenient formulation, for which the function ${\cal M}_i$, does not presents zeros in the interval, where the self-adjoint form is applied.

Furthermore, we point out that the number of zeros of a given eigenfunction is not a unique way to classify modes (see Sect. 5). In fact, a classification scheme can be determined independent of that and the number of zeros can be used to classify the modes only in some particular cases.

   
A.2 Local phase properties

The properties of the eigenfunction and of the correspondent eigenfrequencies can be determined from Eq. (A.6), with the correspondent boundary conditions. It is possible to build a graph for each eigenmode of a given eigenfrequency $\omega$ and degree l, where the variation of the phase with the stellar radius is represented. We call this the linear phase diagram (see Fig. 4), and we will return later to this point.

A solution ${\cal Y}_i(r)$ of the Eq. (29) has a zero at a point r=r₁ if and only if $\theta_i(r_1)= n\pi$, where n is always an integer. At each of these points $\cos{\theta_i}=1$ and $\frac{{\rm d}\theta_i}{{\rm d}r}={\cal P}_i^{-1}(r)$. On the propagation regions where $k_{\rm r}^2>0$, this means geometrically that each curve $({\cal P}_i{\cal Y}'_i,{\cal Y}_i )$in the plane $({\cal P}_i{\cal Y}'_i,{\cal Y}_i )$, corresponding to a solution ${\cal Y}_i$ of the differential Eq. (29) (for a given eigenvalue $\omega$), can cross the ${\cal P}_i{\cal Y}'_i$ axis ( $\theta= n\pi$) counterclockwise when ${\cal P}_i > 0$, and clockwise when ${\cal P}_i < 0$. The solution ${\cal Y}_i(r)$ of the Eq. (29) (or equivalently of the Eq. (A.6)) for a given interval can lead to one of two possible cases: If sgn $\left({\cal P}_i(r)\right)= {\rm sgn}\left({\cal Q}_i(r)\right)$then it has a discrete number of zeros, if any at all. Alternatively, if sgn $\left({\cal P}_i(r)\right)\ne {\rm sgn}\left({\cal Q}_i(r)\right)$then it has either one zero or no zero at all. The waves are propagative only, where sgn $\left({\cal P}_i(r)\right)= {\rm sgn}\left({\cal Q}_i(r)\right)$and they are evanescent elsewhere. In the case of the acoustic propagative region ( $f_{\rm p}>0$ and $f_{\rm g}>0$) the phase is increasing with the radius ( $\frac{{\rm d}\theta_i}{{\rm d}r} > 0$) and in the case of gravity propagative region ( $f_{\rm p}<0$ and $f_{\rm g} <0$) the phase is decreasing with the radius ( $\frac{{\rm d}\theta_i}{{\rm d}r}< 0$). In case of pure acoustic modes, the phase $\theta_i$ is always increasing ( $\frac{{\rm d}\theta_i}{{\rm d}r} > 0$) in all the interval where the wave is propagative, and decreasing elsewhere. This works conversely for the pure gravity modes.

In resume, we can identify a particular phase signature of modes related with the propagative behavior in a given region. Then locally a gravity type wave and a acoustic type wave, propagates depending upon whether the phase point is traveling clockwise (gravity or g-modes, where $\frac{{\rm d}\theta_i}{{\rm d}r} > 0$), or counterclockwise (acoustic or p-modes, where $\frac{{\rm d}\theta_i}{{\rm d}r}< 0$), at points of $\theta_i(r_1)= n\pi$, as r increases. The global behavior of the eigenmodes is discussed in the Sect. 3.2.