2 Basic equation of oscillatory motion

Formally, the nonradial adiabatic oscillations of stars can be described as the solution of the simple linear homogeneous adiabatic wave equation:

$${\put(0,0){\framebox (6,6){$\;\;$ }}}\hspace{0.3cm}\Psi=\left[\frac{\partial^2}{\partial t^2}+{\cal L} \right]\Psi =0$$ (1)

where ${\put(0,0){\framebox (6,6){$\;\;$ }}}\hspace{0.3cm}$ is the adiabatic wave operator and $\Psi$ is some wave function depending on the position vector $\vec{r}$ and the time t, that characterizes the oscillations (Gough 1993, 1996). The oscillatory motion under study is supposed to be of such low amplitude that linearization of the full equations of fluid dynamics is valid, allowing to determine the wave equation that describes the oscillatory movements.

Therefore, we are ignoring the interactions between waves, non-adiabatic effects which are important only in beneath the atmosphere, and effects of the turbulence in the convective zone. We also consider a non-magnetic and non-rotating star. The adiabatic wave operator depends on the structure of the solar model, which we will refer to as the background state. We have assumed that a frame of reference exists in which the background state is independent of time. In this case, there are genuinely separable solutions $\Psi(\vec{r},t)={\cal R}\left[ \Psi(\vec{r}) {\rm e}^{-i\omega t} \right]$of the simple homogeneous adiabatic equation. Considering that the wave has a pure dependence on t with frequency $\omega$, the spatial part of the wave function, $\Psi(\vec{r})$ satisfies:

 

$${\cal L}_\omega \Psi =0$$ (2)

where the three-dimensional spatial differential operator ${\cal L}_\omega$is obtained from the full wave operator ${\put(0,0){\framebox (6,6){$\;\;$ }}}\hspace{0.3cm}$ by replacing $\frac{\partial}{\partial t}$by $-i\omega$. We will use $\Psi_r$ to represent the factor depending on r in the separated form $\Psi (\vec{r})=\Psi_r (r) Y_{l}^{m}(\theta,\phi)$ with respect to the spherical polar coordinates $(r,\theta,\phi)$ when the background state is spherically symmetric, and where $Y_{l}^{m}(\theta,\phi)=P_{l}^{m}(\cos{\theta})\; {\rm e}^{i m\phi}$is a spherical harmonic of degree l and azimuthal order m, P^m_l being the associated Legendre function of first kind. In this case, the separation of variables into radial and angular parts is possible for all background variables, with the angular dependence of $Y_{l}^{m}(\theta,\phi)$ satisfying the eigenvalue equation,

$$r^2\nabla_{\perp}^2 Y_{l}^{m}(\theta,\phi)=-L^2 Y_{l}^{m}(\theta,\phi)$$ (3)

where $\nabla_{\perp}$ is the horizontal Laplace operator, m=-l,...,+l and L²=l(l+1). In that case ${\cal L}_\omega$ will represent the radial part of the corresponding three-dimensional operator with the same name. In the spherically symmetric case, the solutions of Eq. (2) together with appropriate boundary conditions, admit discrete eigenfrequencies $\omega_{nl}$, where n is the radial wave number. The spherical symmetry of the background state is responsable by the degeneracy of $\omega_{nl}$ with respect to m.

In the case of a spherically symmetrical background state, the homogeneous differential equation (Eq. (2)) representing adiabatic oscillations is of fourth-order and has to be solved subject to two regularity conditions at the coordinate singularity r=0, and to two boundary conditions at the surface r=R (Unno et al. 1989; Gough 1993).

Taking into account the mathematical structure of the operator ${\cal L}_\omega$, it is possible to reduce this one to the standard second-order differential equation. This equation of motion can be obtained directly from a linearized Eulerian momentum equation and from the Poisson equation that describes the oscillatory motion by making a convenient transformation (Gough 1993). In the following, we make a very brief presentation of approximations to the second-order motion equations.

   
2.1 The standard form

The linear adiabatic nonradial stellar oscillations of a spherically symmetrical background state can be written in a standard form. Generically we can write the motion equation of adiabatic nonradial oscillations as

 

$$\frac{{\rm d}^2\Psi}{{\rm d}r^2} +k_{\rm r}^2 \Psi = 0$$ (4)

where the radial component of the local wave number, $k_{\rm r}$, is given by

 

$$k_{\rm r}^2 = \frac{\omega^2-\omega_{\rm c}^{2}}{c^2} -\frac{L^2}{r^2}\left(1-\frac{{\cal N}^{2}}{\omega^2} \right)$$ (5)

where c, is the radial distribution of sound speed, $\omega_{\rm c}$, defines a generalized critical frequency and ${\cal N}$, a generalized Brunt-Väisälä frequency, that takes into account the geometrical terms and gravity contribution for the local wave number, which will be defined for each of the approximative motion equations. $\Psi$ is the wave function, which is proportional to the Lagrangian perturbation of pressure and will take a particular dependence, in agreement with the approximation used.

In the following analysis we will always consider this general case given by Eq. (4), unless we say explicitly the contrary.

   
2.1.1 The Planar approximation

In 1984, Deubner & Gough determined a linear second-order differential equation in the standard form of the Eq. (4), to describe the dynamics of adiabatic non-radial oscillations. This equation of motion has been obtained by a procedure analogous to the Lamb (1932) method. This approximation can be made for waves with wavelength much smaller than the solar radius and where the local effects of spherical geometry on the oscillatory motion can be ignored. Additionally, the perturbations of the gravitational potential have been ignored. In this case, the wave function is given by,

 

$$\delta p=\rho^{1/2} \Psi$$ (6)

where $\rho$, is the density and $\delta p$, is the Lagrangian perturbation of pressure. In this case, the local wave number, $k_{\rm r}$, is given by Eq. (5), where the critical frequency, $\omega_{\rm c}$, is given by

$$\omega^2_{\rm c}=\frac{c^2}{4}\frac{1}{H^2}\left(1-2\frac{{\rm d}H}{{\rm d}r}\right),$$ (7)

where H, is the density scale height. The generalized Brunt-Väisälä frequency, ${\cal N}$, is reduced to the classical Brunt-Väisälä frequency, N, given by

$$N^2=g\left(H^{-1}-\frac{g}{c^2}\right),$$ (8)

where g, is the gravitational acceleration. This model is particularly interesting because of the simplicity in the interpretation of the characteristic frequencies of the equilibrium structure that determine the cavities of propagative waves. All the main properties of the local behavior of waves, can be well defined in this approximation.

2.1.2 The Cowling approximation

The equation of motion of stellar oscillations can be obtained by reducing the fourth-order system of stellar oscillations to a second-order differential equation, where the Eulerian perturbation of the gravitational potential is neglected (Gough 1993). Cowling (1941) showed the interest of this approximation, by pointing out that it has a relatively minor effect on the modes, with exception to the modes of low degree and low order. Under the approximation proposed, i.e., neglecting the contribution of the Eulerian pertubation of gravitational potential, $\Phi^\prime$, the initial fourth-order system of adiabatic stellar oscillations is reduced to a second-order motion equation, by the following transformation

$$\delta p = {\rm sgn}(f) \left\vert\frac{g\rho f }{r^3} \right\vert^{1/2} \Psi,$$ (9)

where sgn(f), is the sign of the discriminant f. This discriminant, is given by

 

$$f=4+\left(\omega^2-\omega_J^2\right)\frac{r}{g}-\frac{L^2}{\omega^2}\frac{g}{r}\cdot$$ (10)

$\omega^2_{\rm J}$, is the square of the Jeans frequency, that is equal to $4\pi G\rho$, where G is the gravitational constant. Using this transformation, the motion equation of adiabatic nonradial stellar oscillations, can be written in the form of Eq. (4). In that case, the generalized Brunt-Väisälä frequency, ${\cal N}$, is given by

 

$${\cal N}^{2}=g \left({\cal H}^{-1}-\frac{g}{c^2}-2 h^{-1} \right)$$ (11)

and the critical frequency, $\omega_{\rm c}$, is given by

 

$${\omega}_{\rm c}^{2}= \frac{c^2}{4} \left({\cal H}^{-2}+2 \frac{{\rm d}{\cal H}^{-1}}{{\rm d}r} \right) -\frac{g}{h},$$ (12)

where h is the scale height of g/r² and ${\cal H}$ is a generalized scale height. The scale height h, is given by

$$h^{-1} =4r^{-1}-\frac{\omega^2_{\rm J}}{g}$$ (13)

and ${\cal H}$, is given by

$${\cal H}^{-1}= 3r^{-1}+H^{-1}+H_{\rm g}^{-1}+H_f^{-1},$$ (14)

where $H_{\rm g}$, is the gravity scale height and H_f, is the scale height of discriminant f. A detailed discussion about these variables is presented in Gough (1993).

The difference between the generalization of the Brunt-Väisälä frequency, ${\cal N}$ and the critical frequency, $\omega_{\rm c}$ relatively to these quantities in the Deubner & Gough (1984) approach, is due to the contribution of geometrical terms, that can be important for the most penetrative modes in very dense stars.

2.1.3 The first post-Cowling approximation

A more general solution to the initial system of stellar oscillations can be obtained, which takes into account a major contribution related with $\Phi '$. This is called the first Post-Cowling approximation (Gough 1993; Dziembowski & Gough, private comunication). Under this approximation the Brunt-Väisälä frequency, ${\cal N}$, is given by

$${\cal N}^{2}= g \left(\frac{1}{{\cal H}}-\frac{g}{c^2}-\frac{2}{h} \right)$$ (15)

and the critical frequency, $\omega_{\rm c}$, is given by

$$\omega_{\rm c}^{2}=\frac{c^2}{4{\cal H}^{2}} \left(1-2 \frac{{\rm d}{\cal H}}{{\rm d}r} \right) -\frac{g}{h}-\omega^2_{\rm J}.$$

The total scale height ${\cal H}$, is given by

$${\cal H}^{-1}=3r^{-1}+H^{-1}+H_{\rm g}^{-1}+H_f^{-1}+H_{\varphi}^{-1},$$ (16)

where the scale height $H_{\varphi}$ takes into account the contribution related with the perturbation of the gravitational potential (Gough 1993; Dziembowski & Gough, private comunication; Lopes 2001). The scale height $H_{\varphi}$, is given by

$$H_{\varphi}^{-1}=\frac{\omega_{\rm J}^2 r}{g f H}\cdot$$ (17)

The wave function, $\delta p$, can be determined as

$$\delta p = {\rm sgn}(f) \left\vert\frac{g\rho f \varphi}{r^3} \right\vert^{1/2} \Psi$$ (18)

where $\varphi$, is a structure term related with the background state, given by

$$\varphi(r) = \int_0^r H^{-1}_\varphi\; {\rm d}r.$$ (19)

These new expressions for the characteristic frequencies come from a contribution related with the perturbation of the gravitational potential which can be obtained from the linearized Poisson equation. The magnitude of the effect relatively to the classical Cowling approximation strongly depends on the density distribution in the central region of the star.

   
2.2 Boundary conditions

The second-order motion equation together with two boundary conditions, one at the centre and the other at the surface, forms an eigen-value problem. We will present these boundary conditions here, as they will be needed later.

The center is a regular singular point. The regularity condition for the perturbation of the Lagrangian variation of pressure, $\delta p$, is given by

 

$$\frac{{\rm d}\delta p}{{\rm d}r}-\frac{l}{r}\delta p\rightarrow 0$$ (20)

as $r\rightarrow 0$. The outer boundary condition, is fixed in the atmosphere and we consider that the inertia of the corona is so high that $\delta p$, is given approximately by

 

$$\delta p= 0$$ (21)

at r=R. These simple conditions are sufficient to the discussion proposed in this article. These are the so-called "zero-boundary conditions'' (Unno et al. 1989). The consideration of more realistic boundary conditions does not modify significantly the method proposed to classify modes in stars.

   
2.3 Propagation diagram

Figure 1: Propagation diagram for a polytropic model with index $n_{\rm e}=3.0$, adiabatic index $\gamma =5/3$. The curves correspond to a dimensionless circular frequency relatively to the dimensionless radius. These curves delimit the propagation region for modes of degree $\ell =2$: a) The solid curves represent $\omega _{\pm }^2$ computed in the Cowling approximation, the dashed curves represent planar values of $\omega _{\pm }^2$, and the dotted curves represent the Brünt-Väisäla frequency, N, (increasing from the center to the surface) and the Lamb frequency, $S_\ell$, (decreasing from the center to the surface). b) The solid curves represent $\omega _{\pm }^2$ computed in the first Post-Cowling approximation, and the dashed curve represents $\omega _{\pm }^2$ in the Cowling approximation.

The direction and magnitude of the acoustic-gravity wave is determined by the competition between the radial and tangential components of the local wave number $\vec{k}$, given by

$$\vec{k} =k_{\rm r}\vec{e}_{\rm r}+k_{\rm h}\vec{e}_{\rm h},$$ (22)

where $\vec{e}_{\rm r}$, $\vec{e}_{\rm h}$ are unitary vectors in the radial and horizontal directions and $k_{\rm h}$ ( $\equiv L^2/r^2$), is the tangential component of the local wave number (Unno et al. 1989). The radial component of the local wave number, is determined by the local properties of the equilibrium structure. In this sense, it is possible to build a diagram for the radial component of the local wave number, which defines the regions of propagative or evanescent behavior of a given acoustic-gravity wave of frequency $\omega$ and degree l. This is the so-called propagation diagram. Waves can propagate where the square of the vertical component of the local wave $k_{\rm r}$ is greater than zero, and are evanescent elsewhere. This leads to two asymptotic limits, for the very high and for the very small frequencies. In the case of very high frequencies, $k_{\rm r} \gg k_{\rm h}$, the propagation is predominantly in the radial direction. In the other case, $k_{\rm r} \ll k_{\rm h}$, then the propagation is predominantly in the horizontal direction. This corresponds also to the spectral regions where the restoring forces are dominated by gravity (through the buoyancy) or pressure. Because of that, it is convenient to write the local wave number as a function of the critical frequencies $\omega _{\pm }^2$, in which case $k_{\rm r}$, is given by

 

$$k_{\rm r}^2 =f_{\rm g} \;\; f_{\rm p}$$ (23)

where $f_{\rm g}$ is a discriminant, given by

$$f_{\rm g} =\frac{\omega}{c} \left(1-\frac{\omega_{-}^{2}}{\omega^2}\right)$$ (24)

and $f_{\rm p}$ is a discriminant, given by

$$f_{\rm p}=\frac{\omega}{c} \left(1-\frac{\omega_{+}^{2}}{\omega^2}\right)\cdot$$ (25)

The underscript g or p is fixed by the propagative regions that are separated by $\omega_{-}$ or $\omega_{+}$, respectively.

It follows from Eq. (23) that $k_{\rm r}^2>0$, $\omega _{\pm }^2$ are real and $f_{\rm p}>0$ and $f_{\rm g}>0$(acoustic or p-region, locally behaves like an acoustic type wave) or $f_{\rm p}<0$ and $f_{\rm g} <0$ (gravity or g-region, locally behaves like a gravity type wave). Here we will be mainly concerned with the case where $\omega_{+}^2$and $\omega_{-}^2$ are both real. In the case where $\omega_{-}^2$ and $\omega_{+}^2$are complex conjugates it is also possible to use the same analysis. It will slightly complicate the algebra, but the physical conclusions remain the same.

The critical frequencies $\omega _{\pm }^2$, define the turning points of the standard second-order differential equation, previously presented, i.e., $k_{\rm r}^2=0$. In the more general case, $\omega _{\pm }^2$, can be determined numerically. However the critical frequencies $\omega_{\pm}^{2}(\omega,l;r)$can be obtained, by the relation,

 

$$\omega_{\pm}^{2}(\omega,l;r)= \frac{1}{2}\left(S_{\rm l}^2+\omega... ...c{1}{4}\left(S_{\rm l}^2+\omega_{\rm c}^{2}\right)^2 -{\cal N}^{2}S_{\rm l}^2 }$$ (26)

where, the Lamb frequency $S_{\rm l}$, is given by

$$S_{\rm l}^2=l \left(l+1\right) \; \frac{c^2}{r^2}\cdot$$ (27)

In the case of Deubner & Gough (1984), these expressions determine an explicit relation of the critical frequencies as function of the background structure and independent of the frequency (see Sect. 2.1.1). In Figs. 1 and 2 we present the computed critical frequencies, $\omega _{\pm }^2$, for the different approximations presented in the previous section, in the case of polytropes of index $n_{\rm e}=3$ and $n_{\rm e}=4$ and adiabatic index $\gamma =5/3$. A discussion about the propagation of waves in polytropic equilibrium structure with a different adiabatic index can be found in Cowling (1941) and Scuflaire (1974). The contribution related with the geometric terms occuring in the variables ${\cal N}$ and $\omega_{\rm c}$ when computed in the Cowling approximation is more important for the more penetrative waves, i.e., waves with smaller degree, as it is illustrated in Figs. 1a and 2a. Similarly, the first Post-Cowling approximation, seems worth considering in the case of stars with high density, where the presence of the gravitational potential modifies significantly the values of $\omega _{\pm }^2$ in the central region of the star (cf. Figs. 1b and 2b).

In the next section, using the differential Eq. (4) with the local wave number given by Eq. (23), we will write the standard equation of motion in a convenient self-adjoint form, very appropriate for the phase analysis that we will present below.