3 Phase analysis representation of the equation of motion

The phase method proposed here is based in a generalization of a technique normally used to study the solutions of a particular type of self-adjoint second-order linear differential equation with homogeneous boundary conditions, the so-called Sturm-Liouville problem. This method consists in representing the eigenmodes in a convenient Poincaré phase plane (in terms of polar coordinates), normally used to study an autonomous system of differential equations.

In a first step, we present two complementary self-adjoint forms of the standard second-order differential equation that describes the adiabatic nonradial oscillations of a spherically symmetric star (Eq. (4)). Second, we present the phase analysis method to determine the eigenstates of each of these self-adjoint differential equations, where we introduce a classification scheme that is build using the boundary conditions.

Figure 2: Propagation diagram for Polytropic models of index $n_{\rm e}=3$, $n_{\rm e}=4.5$ and adiabatic index $\gamma =5/3$, for modes of degree $\ell =1$. The representation adopted is the same as in previous Fig. 1.

   
3.1 Complementary self-adjoint differential equations

The standard second-order differential equation of linear adiabatic stellar oscillations (Eq. (4)), can be written into a pair of self-adjoint differential equations. As we will discuss later, these equations should then be regarded as being complementary to each other, for the study of the local properties, related with the two driving mechanisms, pressure and gravity. The transformation of the second-order differential equation into a self-adjoint form will be done, taking into account the previous decomposition related with the two propagation cavities for a given wave. This is accomplished by the transformation,

 

$$\Psi={\cal M}_i \; {\cal Y}_i,$$ (28)

where i=1,2 defines the two possible transformations, related with two propagative regions. We point out that, in the most general case, under these transformations ${\cal Y}_i$ does not conserve the location and the number of zeros of the wave function $\Psi$. We consider also that ${\cal M}_i$ is a real function. However, if the function ${\cal M}_i$, is conveniently chosen (with no zeros in the whole interval considered) it is possible to determine the location of the zeros for the wave function $\Psi$, by the location of zeros of ${\cal Y}_i$. Resulting in the following self-adjoint equation of motion,

 

$$\frac{\rm d}{{\rm d}r}\left[{\cal P}_i (r) \frac{{\rm d}{\cal Y}_i}{{\rm d}r}\right] + {\cal Q}_i(r) {\cal Y}_i=0,$$ (29)

where ${\cal P}_i$, is given by

$${\cal P}_i (r)={\rm sgn}({\cal M}_i)\;\vert{\cal M}_i\vert^2$$ (30)

and ${\cal Q}_i$, is given by

$${\cal Q}_i (r)={\cal P}_i\left(k_{\rm r}^2+\frac{1}{{\cal M}_i}\frac{{\rm d}^2{\cal M}_i}{{\rm d}r^2}\right)\cdot$$ (31)

The function ${\cal M}_i$, can take two possible choices, which will be in the mainframe of our analysis:

 

$${\cal M}_1={\rm sgn}(f_{\rm g})\vert f_{\rm g}\vert^{-1/2}$$ (32)

and

 

$${\cal M}_2={\rm sgn}(f_{\rm p})\vert f_{\rm p}\vert^{-1/2}.$$ (33)

These two differential equations are equivalent to Eq. (4), where $k_{\rm r}^2$ is an invariant of that family of equivalent differential equations. The determination of ${\cal M}_i$ presented previously, has been done under the consideration that $f_{\rm g}$ and $f_{\rm p}$ are real functions. In the case they are complex conjugates then ${\cal M}_i$ is defined as the real part of $f_{\rm g}$ and $f_{\rm p}$, respectively.

These complementary equations will be very useful in the analysis of the zeros of the wave function, $\Psi$. The location of the turning points will determine what is the most convenient formulation to study the properties of a given wave of frequency, $\omega$ and degree, l, i.e., the formulation is chosen such that ${\cal M}_i$ is strictly positive in the interval considered. In fact, the localization of the singular points depends on $\omega$ and l, and they correspond to the zeros of $f_{\rm p}=0$ or $f_{\rm g}=0$. Choosing the convenient self-adjoint formulation for a given wave of parameters $\omega$ and l, the function ${\cal P}^{-1}_i(r)$ is regular ( ${\cal P}^{-1}_i(r)\ne \infty$) in the interval considered. In this sense, in waves for which $\omega^2 > \omega_{\pm}^2$ ( $f_{\rm p}>0$) in all the interval (acoustic type), the location of zeros of $\Psi$, corresponds to the location of zeros of ${\cal Y}_1$, provided that ${\cal M}_1$ does not have zeros on the interval considered. Equation (29) with i=1, is the convenient formulation to study the oscillatory motions. Conversely, in waves for which $\omega^2 < \omega_{\pm}^2$ ( $f_{\rm g} <0$) in the interval considered (gravity type), the location of zeros of $\Psi$corresponds to the location of zeros of ${\cal Y}_2$. Equation (29) with i=2, is then the convenient formulation to study the oscillatory motions. As we will discuss in Sect. 3.2, for some l and $\omega$a mixing mode can appear and both descriptions will need to be used. Under a correct choice of the self-adjoint form, the differential Eq. (4) for each of the propagative regions, the number and the location of the zeros of $\Psi$, can all be determined from a convenient self-adjoint form, even in such case.

In the following sections, when we are referring to a particular formulation, we will use the underscript i on the variables ${\cal P}_i$, ${\cal Q}_i$ and ${\cal Y}_i$, with i=1 for the acoustic description and i=2 for the gravity description. This is a first step to the classification scheme that will be defined like it was originally proposed by Cowling (1941), taking into account the two driving mechanisms, the gravity and the pressure.

   
3.2 Complementary phase equations

The phase analysis proposed here, can be considered as a generalization of the method proposed by Eckart (1960), Scuflaire (1974), Osaki (1975), Gabriel & Scuflaire (1979) and Gough (1993) to discuss the classification scheme of stellar oscillations. The method proposed by these authors is based on the algebraic counting of the number of nodes according to the behaviour of p' relatively to $\delta p$ (or the other two wave functions). In our case the classification scheme will be determined for a generic wave function that can be related with $\delta p$ (for example), and the order of the mode will be determined by the boundary conditions which can be, or not, reliable with the number of nodes of this wave function.

The motion equation for linear adiabatic nonradial oscillations (Eq. (4)) can be transformed into two self-adjoint differential equations (Eq. (29)) to determine the properties of the oscillations. Using the phase transformation (with i=1, 2) for each of the self-adjoint forms (see details about this phase analysis in Appendix A), given by

 

$${\cal Y}_i=R_i \sin{\theta_i}$$ (34)

and

 

$${\cal P}_i{\cal Y}'_i =R_i \cos{\theta}_i$$ (35)

where $R_i(\omega,l;r)$ and $\theta_i(\omega,l;r)$are the amplitude and phase of the wave function ${\cal Y}_i$. After some straightforward manipulation, we obtain the following two first-order differential equations for the phase (phase equations):

 

$$\frac{{\rm d}\theta_1}{{\rm d}r}= \left(f_{\rm p}+{\cal T}(f_{\rm g})\right) \sin^2{\theta_1} +f_{\rm g} \cos^2{\theta_1}$$ (36)

and

 

$$\frac{{\rm d}\theta_2}{{\rm d}r}= \left(f_{\rm g}+{\cal T}(f_{\rm p})\right) \sin^2{\theta_2} +f_{\rm p} \cos^2{\theta_2},$$ (37)

where ${\cal T}(x)$, is the real function given by

$${\cal T}(x)={\rm sgn}(x)\;\vert x\vert^{-1/2}\frac{{\rm d}^2\vert x\vert^{-1/2}}{{\rm d}r^2}\cdot$$ (38)

The amplitudes in each case, are given by

$$R_1 =R_{1,{\rm o}}\exp{\left[ \frac{1}{2} \int_{r_{\rm o}}^r \lef... ...}-{\cal T}(f_{\rm g}) \right) \sin{\left(2\theta_1\right)} \; {\rm d}r \right]}$$ (39)

and

$$R_2 =R_{2,{\rm o}}\exp{\left[ \frac{1}{2} \int_{r_{\rm o}}^r \lef... ...}-{\cal T}(f_{\rm p}) \right) \sin{\left(2\theta_2\right)} \; {\rm d}r \right]}$$ (40)

where $R_{i,{\rm o}}$ is the initial amplitude.

At this level, it is important to point out that Eqs. (36) and (37) are two different phase representations of the same physical system described by Eq. (4), written in a way that it avoids singular points for a given wave of degree l and frequency $\omega$, in a given region. It is possible to determine a relation between the two phases, which is obtained by imposing a matching of the logarithmic derivatives of $\Psi$ (Eq. (28)) between both descriptions. This is given by

$$\frac{\rm d}{{\rm d}r}\ln{\vert{\cal M}_1{\cal Y}_1\vert}=\frac{\rm d}{{\rm d}r}\ln{\vert{\cal M}_2{\cal Y}_2\vert}.$$ (41)

Using Eqs. (32) and (33) with the phase relation A.5, we obtain the following relation between phase $\theta_1$ and $\theta_2$:

 

$$\cot{\theta_1} =\frac{f_{\rm p}}{f_{\rm g}} \cot{\theta_2} -\frac... ...{1}{f_{\rm g}}\frac{\rm d}{{\rm d}r}\ln{\vert\frac{f_{\rm p}}{f_{\rm g}}\vert}.$$ (42)

This relation can be used to determine the phase $\theta_1$ (of the acoustic description) knowing the phase $\theta_2$ (of the gravity description). We observe that when we want to convert $\theta_2$ into $\theta_1$ (go from the gravity description to the acoustic description), the point where $f_{\rm g}=0$ is singular, so that $\cot{\theta_1}=-\infty$, and consequently $\theta_1=0 + n\pi$. Finally, we note that Eq. (42) is undefined in the radius where $f_{\rm p}=0$. However this situation never happens within this transformation.

In short, the propagation of a wave of degree l and frequency $\omega$ can be explicitly determined for any of the three types of waves (which can be defined from the propagation diagrams) using the appropriate phase equation. For the pure acoustic type waves, the phase can be determined from Eq. (36) (acoustic description) because no singular points occur inside.

For the pure gravity type waves, the phase can be determined by Eq. (37) (gravity description). Furthermore, for mixed type waves which correspond to a mode that propagates in two different regions, Eq. (36) can be used to determine the phase propagation on the gravity region and Eq. (37) to determine the phase propagation on the acoustic region.

The matching condition is made in the region where both descriptions, are accepted (without singular points). A possible choise is a point $r^\star$, where $f_{\rm g}(r^\star)=f_{\rm p}(r^\star)$ and $f_{\rm g}^\prime(r^\star)=f_{\rm p}^\prime(r^\star)$, in that case from Eq. (42) we obtain $\theta_1(r^\star)=\theta_2(r^\star)$. To make the matching we integrate each of the equations from each side of the interval and match them in the point $r=r^\star$, where both solutions are valid.

   
3.3 Boundary conditions

The stellar oscillation eigenmodes can be determined by using one of the phase equations (Eqs. (36) or (37)), with a convenient set of boundary conditions at the endpoints, the centre and the surface. The phase equations are equivalent one another and the choice of which discription to use is matter of the tast of the author as well the type of singularities that can be enconter. Indeed, a certain phase equation with the respective boundary conditions at the center and the surface, constitutes an eigen-value problem. In the following, we will indicate how it is possible to determine the boundary conditions for each representation.

The phase at the endpoints is determined by using one of two procedures. One consists in transforming the boundary conditions of the eigenfunction $\Psi$ or $\delta p$(see Sect. 2.2) into an equivalent endpoint condition for the phase function $\theta_i$ of the phase associated system, related with the differential Eq. (29). Independent of the approximation made on the determination of the local wavenumber, defined on the equation of motion (see Sect. 2), it is possible to determine the phase at the endpoints from the Lagrangian perturbation of pressure $\delta p$and the Eq. (A.5). In the Appendix B, we present the determination of the boundary condition in the case of the planar approximation (Deubner & Gough 1984).

An alternatively method consists in determining the phase at the endpoints by imposing the regularity of the phase function $\theta_i$. A detailed analysis of the phase Eqs. (36) and (37), indicates that these ones are singular at the endpoints, r=0 (for all stars) and r=R (only for polytropes). This implies that the boundary conditions must be applied such that the singular solution at each of the endpoints is eliminated. This is obtained by expanding the solution, $\theta_i$, as a power series of r, relatively to the critical endpoint. It is necessary also to expand the equilibrium quantities around the endpoints, r=0 and r=R. It is worth to notice that for each description it is possible to study the properties of the gravity waves, as well as of the acoustic waves, even if it is more convenient to use a gravity description to study the eigenstates of gravity perturbations and similarly the acoustic description to study the eigenstates of acoustic perturbations. In that case for each description, we will refered to the perturbation as characterized by the main contributer for the restoring force that is present in a certain region inside the star. In this context, we will refered to an acoustic behaviour or the gravity behaviour, as it is the gravity or the pressure the main contributer for the restoring force. The mixed modes can be treated as a combination of these two "asymptotic'' cases (as discussed in the previous section), as an acoustic type mode or gravity type mode for each of the region where one of these character prevails. In particular, the inner and outer boundary conditions for mixed modes are determined based in which is the dominant wave character near the center or near the surface of the star.

The inner boundary condition near the center is obtained by imposing that the solution to the phase equation is regular, i.e. $\theta_i$ has a finite value. Now, we will illustrate that by considering the case corresponding to the acoustic description, i.e., we are interested in determine the solution, $\theta_1$ as given by Eq. (36). Furthermore, we will starting by considering a wave that has an acoustic behaviour near the stellar center, it follows that $\theta_1$ must be a regular function at the center, as the phase has a finite value. We notice that the first-member of Eq. (36) is singular, consequently, the finite value of ${\rm d}\theta_1/{\rm d}r$ is obtained by choosing all the multi-value solutions, such as $\theta_1=0 + n\pi$where n is a positive or a negative integer (or in an equivalent form $\theta_1=0\;\;({\rm mod}\;\; \pi)$; see also Appendix B). Similary in the case of the gravity description (Eq. (36)), a wave with a gravity behaviour at the center has a regular solution if $\theta_2=0\;\;({\rm mod}\;\; \pi)$. It follows that the same boundary conditions can be presented in both descriptions, given that $\theta_1=\theta_2+\pi/2$. This relation between phases is illustrated in the next section. In resume, the boundary condition for the center require that

$$\left( \frac{{\rm d}\theta_i}{{\rm d}r}\right)_{r=0}= \left\{ \be... ...i) \\ & \\ \displaystyle +\infty \;\; ({\rm mod}\;\; \pi) \end{array}\right..$$ (43)

Alternativelly, from the phase Eqs. (36) and (37) it follows in the case of the acoustic description,

 

$$\theta_1(\omega,l;0) = \left\{ \begin{array}{ll} \displaystyle 0 ... ...}{2} \;\; ({\rm mod}\;\; \pi) & {\rm gravity \;\; behaviour} \end{array}\right.$$ (44)

for modes with acoustic and gravity behaviour near the centre of the star, and in the case of the gravity description,

 

$$\theta_2(\omega,l;0) = \left\{ \begin{array}{ll} \displaystyle \f... ...le 0 \;\; ({\rm mod}\;\; \pi) & {\rm gravity \;\; behaviour} \end{array}\right.$$ (45)

for modes with acoustic or gravity behaviour.

The outer boundary condition, can be obtained also by a series expansion around the endpoint r=R. In the case of polytropic equilibrium structures, because the sound speed is zero at r=R, the phase Eqs. (36) and (37) can be singular. It follows that

$$\left( \frac{{\rm d}\theta_i}{{\rm d}r}\right)_{r=R}= \left\{ \be... ...i) \\ & \\ \displaystyle +\infty \;\; ({\rm mod}\;\; \pi) \end{array}\right.,$$ (46)

then we obtain the following type of conditions for the surface,

 

$$\tan^2{\theta_1(\omega,l;R)}= -\frac{f_{\rm g}}{f_{\rm p}+{\cal T} (f_{\rm g}) } \;\; ({\rm mod} \;\;\pi)$$ (47)

and

 

$$\tan^2{\theta_2(\omega,l;R)}= -\frac{f_{\rm p}}{f_{\rm g}+{\cal T} (f_{\rm p}) } \;\; ({\rm mod} \;\;\pi).$$ (48)

An approximative solution can be obtain by using the results of Appendix B. We start bu notice, that the sound speed vanish at the surface, consequently, we have $\tan^2{\theta_1}\sim -(1-N^2\omega^{-2})$ and $\tan^2{\theta_2}\sim -1/(1-N^2\omega^{-2})$. In case of acoustic description if we consider the asymptotic limit, $\omega \rightarrow \infty$, then $\tan^2{\theta_1}=1$, or $\theta_1=0$ (mod $\pi$) and in the case of gravity description if we consider the asymptotic limit $\omega \rightarrow 0$ then $\tan^2{\theta_2}=1$ or $\theta_2=0$ (mod $\pi$). Consequently, following a similar procedure to study the solution in the center of the star, we can obtain the expression for the two type of waves in both descriptions. It follows,

$$\theta_1(\omega,l;R)\rightarrow \left\{ \begin{array}{ll} \displa... ...}{2} \;\; ({\rm mod}\;\; \pi) & {\rm gravity \;\; behaviour} \end{array}\right.$$ (49)

and

$$\theta_2(\omega,l;R)\rightarrow \left\{ \begin{array}{ll} \displa... ... 0 \;\; ({\rm mod}\;\; \pi) & {\rm gravity \;\; behaviour} \end{array}\right. .$$ (50)

In real stars this outer boundary condition can be slightly different, but the main conclusion remains the same.

We observe that with these boundary conditions for the inner endpoint, in both descriptions both solutions are regular because the amplitude and the phase have finite values for r=0.

   
3.4 Simplified phase equations

Ledoux & Walraven (1958) have derived a particular solution, which presents two asymptotic limits for very high and very small eigenfrequencies. In this case the system of stellar oscillations under the Cowling approximation tends toward a Sturm-Liouville type solution. Here, an analogous situation takes place, when the term with the form ${\cal T}(x)$, in Eqs. (36) and (37) can be neglected. This is a good approximation in most of the stellar interior, because this term remains small compared with the other terms almost everywhere, except near the turning points (where $f_{\rm p}$ or $f_{\rm g}$ becomes 0) and the stellar surface. In that case, the nodes of ${\cal Y}_1$ (or ${\cal Y}_2$) correspond to extremes of ${\cal Y}_2$ (or ${\cal Y}_1$). The phase equations take the simple form:

 

$$\frac{{\rm d}\theta_1}{{\rm d}r}=f_{\rm p}\sin^2{\theta_1}+f_{\rm g} \cos^2{\theta_1}$$ (51)

and

 

$$\frac{{\rm d}\theta_2}{{\rm d}r}= f_{\rm g} \sin^2{\theta_2}+f_{\rm p} \cos^2{\theta_2}.$$ (52)

In that case the phase between the two prescriptions has the very simple relation, $\theta_1=\theta_2\pm\pi/2$. It is also important to remark that, in a given representation when a solution at the initial endpoint is singular, it is regular at the other phase representation. This lead us to use a convenient phase repesentation to avoid this singular behaviour, which is the origin of the acoustic and gravity description presented in the last section.

It is important to stress that this is a very good representation for the phase equations and from that we can say that the difference between the acoustic and gravity waves presents a shift of $\pi/2$. In this case, it is very clear the fact that both phase equations are strictly equivalent.

For a given l, if $\omega ^2$ is large this suggests the existence of a spectrum of indefinitely increasing eigenvalues, corresponding to the eigensolution denoted by acoustic modes or p-modes. Conversely, for $\omega ^2$ very small, this tends to define a Sturm-Liouville problem with the parameter $1/\omega^2$, which corresponds to the eigensolutions denoted by gravity modes or g-modes.