## Ac 2Hp CR1 lT g

where R is the gas constant. Near the stellar surface, as T decreases rapidly, the cutoff frequency rises fast. This fast growth constitutes the limit of acoustic waves near the surface. In the atmosphere of the Sun, ®ac « 0.03 s-1, which corresponds to a period P = 1/v = 2n/a « 200 s.

Let us now examine the behavior of the other key frequencies which determine the nature of the waves.

- The Brunt-Vaisala Nad (Sect. 5.1). Figure 16.1 shows its behavior in a solar model. At the very center, the gravity tends toward zero, which makes Nad ^ 0. N2d is then positive, as expected in a radiative zone. Slightly away from the center, Nad reaches its highest value due to the ^ gradient near the center. Then, Nad decreases because the ^ gradient vanishes and also because the gravity decreases. Indeed for the law of perfect gas, one can write N2d = (g2Q /P)(Vad - V + VM). As r increases, g2 decreases while (q/P) does not change very much; thus N2d slightly decreases outward. It sharply becomes zero at the edge of radiative zone to become negative in the outer solar convective envelope. _

- The Lamb frequency Sf. It is defined by (16.56) and scales like \JT/(^r2) and thus it continuously decreases outward as shown in Fig. 16.1. The acoustic wave with a given spherical harmonic degree f must lie above the corresponding curve Sf in Fig. 16.1. For f = 1, we see that the Sf curve in Fig. 16.1 is partly below that of Nad; this means that the wave of the corresponding frequencies have their inner turning point at the surface Nad = 0.

The corresponding domains are illustrated in Fig. 16.1. According to condition (1) in Sect. 16.2.3, the acoustic modes occur mainly in the convective envelope of the Sun and over a part of the inner radiative region, depending on the f value (with

Fig. 16.1 Variations of the Brunt-Vaisala Nad and of the Lamb Sf(r) frequencies for various values of f as a function of the radius in the solar model. The g-mode trapping region is indicated by a gray area. The outer convection extends from this zone to the surface. The trapping regions are also limited by the acoustic cutoff frequency (cf. Fig. 16.8). Adapted from J. Christensen-Dalsgaard [131]

Fig. 16.2 Left: the relative amplitude i,r/R as a function of the radius in a solar model, n = 20 and £ = 2. The amplitude decreases fast with depth. Right: oscillation amplitude for the case n = 20 and £ = 2, but with a normalization by the square root of the inner density. Courtesy of P. Eggenberger [169]

Fig. 16.2 Left: the relative amplitude i,r/R as a function of the radius in a solar model, n = 20 and £ = 2. The amplitude decreases fast with depth. Right: oscillation amplitude for the case n = 20 and £ = 2, but with a normalization by the square root of the inner density. Courtesy of P. Eggenberger [169]

variable amplitudes, cf. Fig. 16.2). Near the surface, the rise of (c constitutes the limit, as shown in Fig. 16.8.

Figure 16.1 shows where according to condition (2) the g modes can propagate. The corresponding zone lies where A^d is positive, i.e., in the radiative interior of the Sun. The domain of g modes is also limited by the Lamb frequency, depending on the £ number. However, except for £ = 1, 2 this makes no change, as it can be seen in Fig. 16.1.

16.2.5 The Degree £ and Radial Order n

Here we examine in more detail the significance of the various quantum numbers appearing in the above developments. The spherical harmonic degree £ appears in the system of equations (16.52) and (16.57) or in the second-order equation (16.70), and thus it directly influences the solutions for B,r and the other variables. From the properties of the Legendre polynomials and the study of their zeros, one learns that the degree £ is the number of nodes on the surface, i.e., the locations where the amplitudes are zero. This can also be seen in a simple way by the local identification (16.41) of the nonradial component with a plane wave at the stellar surface R:

For large £, this gives

which shows that there are about £ wavelengths Xh of frequency o over the stellar circumference.

The equations (16.52) and (16.57) or (16.70) subject to the boundary conditions (16.59) and (16.61) admit solutions only for some specific frequencies (n of "eigen-modes" corresponding to stationary waves. In a resonant cavity, interferences can be constructive or destructive. Stationary waves survive only for constructive interferences, which occur for some frequencies such that there is a specific number of half-wavelengths X/2 in the cavity. In principle, for a liquid in a basin, one should have a node at the two extremities, i.e. there should be an integer number of half-wavelengths in the cavity. In a gas, the situation is different, because there is no steady wall at the reflexion point. Beyond this point, the wave becomes evanescent with an amplitude decreasing exponentially. Thus at a reflexion point, the wave must match an exponential decline over a distance more or less of (1/4) X as shown in Fig. 16.3. This means that only oscillations with a number (n/2 + 1/4)X in the cavity can be resonant. Therefore the integration of the wave number radially for a stationary wave must satisfy the condition

This will be further studied in the asymptotic theory (Sect. 16.4.1). Number n is the number of zero along the stellar radius without counting the center and the surface. The value n = 0 corresponds to the fundamental radial mode, for which the period P0 is the travel time of the sound wave from the surface to the center and return,

Fig. 16.3 Schematic representation of the boundary of a propagation zone of p modes. There is an integer number of X/2, plus a fraction of ~ (1/4)X in the cavity. At the boundary, the stationary waves match an exponential decrease

This period is of the order of the dynamical timescale of the star (1.28).

The amplitude of a stationary wave for a nonradial p-mode oscillation is illustrated as a function of the radius in the Sun by Fig. 16.2, left. As the density increases with depth the oscillation amplitude declines. Figure 16.2, right, shows the same with a weighting by the square root of the inner density at the considered point, in agreement with the behavior of the energy of an oscillation mode.

The number m, which appears in the spherical harmonics (16.38), is absent in non-rotating star, since there is no particular axis of symmetry. It only appears in rotating stars (cf. Sect. 16.6); it is called the azimuthal number and corresponds to the number of nodal lines which cross the equator. In summary, the n, i, m numbers represent

- n: the radial order, which is the number of nodal lines (v = 0) along the stellar radius, without counting the center and the surface.

- i: the degree i is the number of nodal lines on the surface of the star.

- m: the azimuthal order, i.e., the number of nodal lines crossing the equator.

A few didactic examples of oscillations in the acoustic modes for a solar-type star are given in Figs. 16.4 and 16.5 for different values of i and m.

Fig. 16.4 Schematic illustration of some acoustic modes of different spherical harmonic degrees i and azimuthal numbers m. Number i gives the number of nodal lines (places where v = 0) and m the number of nodal lines crossing the equator. The darker areas moves upward while the clearer ones moves downward. The black dots show the point where the rotation axis crosses the surface

Fig. 16.4 Schematic illustration of some acoustic modes of different spherical harmonic degrees i and azimuthal numbers m. Number i gives the number of nodal lines (places where v = 0) and m the number of nodal lines crossing the equator. The darker areas moves upward while the clearer ones moves downward. The black dots show the point where the rotation axis crosses the surface

16.3 Properties of Acoustic or p Modes

The p modes are present between the surface and a inner point called the turning point rt where the wave frequency is equal to the Lamb frequency. Thus p modes occur in the outer convective envelope of the Sun and in a part of the radiative interior, as illustrated in Fig. 16.1.

16.3.1 Inner Turning Points ofp Modes

Expressions (16.70) together with (16.56) define the inner turning point:

Taking into account that for such modes rn > N, K(r) simplifies to

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