uncertainty in extracting individual frequencies in a single-site power spectrum complicated by side lobes. In particular, the indications in the echelle diagram of effects of g-mode trapping is clearly extremely tentative. On the other hand, the close agreement between the observed and computed value of the l= 0−2 frequency separation is suggestive. The difference of 10µHz in the reference frequencyν0required to obtain agreement between the location of the modes in Figure 5.20 is clearly a concern; however, it should be noted that such differences may be induced by errors that are likely present as a result of the simplified treatment of the near-surface layers in the modelling of the star and its oscillations (cf.
Section 5.1.2). In fact, is of a similar magnitude to the differences observed in comparisons of solar observed and computed frequencies and attributed to errors in the treatment of the superficial layers (see Figure 5.25). Similar effects might be expected forη Bootis (see also Christensen-Dalsgaardet al. 1995b).
5.4 Oscillations in stellar atmospheres
In the preceding section I neglected terms of the order of the inverse scale heights in equi-librium quantities. As pointed out, this is invalid near the stellar surface, where the scale heights become small compared with the wavelength of the oscillations. In the present sec-tion I discuss these effects in the particularly simple case of an isothermal atmosphere. This has the significant advantage of allowing an analytical solution of the oscillation equations.
Furthermore, it is a reasonable approximation to a realistic stellar atmosphere where the temperature variation is substantially slower than the variations in pressure and density.
In particular, in the solar case the temperature has a minimum at an optical depth of about 10−4, corresponding to an altitude of about 500 km above the visible surface. An early treatment of this problem was given by Biermann (1947); for a somewhat more recent review, see Schatzman & Souffrin (1967). The extension of the results obtained here to a more general model is discussed in Chapter 7.
I neglect effects of ionization and treat the gas in the atmosphere as ideal, so that the equation of state is given by equation (3.19), where the mean molecular weightµ is taken to be constant. Then equation (3.33) of hydrostatic support gives
dp
dr =−gρ=− p Hp
, (5.36)
where the pressure and density scale heights Hp and H (which are evidently the same in this case) are given by
Hp=H= kBT
gµmu . (5.37)
As the extent of the atmosphere of at least main-sequence stars is much smaller than the stellar radius, g can be taken to be constant. Then H is constant, and the solution to equation (5.36) is
104 CHAPTER 5. PROPERTIES OF SOLAR AND STELLAR OSCILLATIONS.
Here I have introduced the altitude h=r−R, where R is the photospheric radius (corre-sponding to the visible surface of the star,e.g. defined as the point where the temperature equals the effective temperature), and ps and ρs are the values of pand ρ ath= 0.
We now consider the oscillations. As argued in Section 3.1.4, the motion becomes strongly nonadiabatic in the stellar atmosphere. Nonetheless, for simplicity, I shall here use the adiabatic approximation in the atmosphere. This preserves the most important features of the atmospheric behaviour of the oscillations, at least qualitatively. The study of atmospheric waves and oscillations, with full consideration of effects of radiative transfer, is a very complex and still incompletely developed area (e.g. Christensen-Dalsgaard &
Frandsen 1983b). It might be noticed that the waves are in fact approximately adiabatic in the upper part of the atmosphere. Here the diffusion approximation (upon which the argument in Section 3.1.4 was based) is totally inadequate, as the gas is optically thin;
indeed the density is so low that the gas radiates, and hence loses energy, very inefficiently, and the motion is nearly adiabatic.
I use the Cowling approximation, equations (5.12) and (5.13). Due to the small extent of the atmosphere I neglect the term in 2/r(this is consistent with assuminggto be constant, and corresponds to regarding the atmosphere as plane-parallel). Then the equations may be written as are constant. In accordance with the plane-parallel approximation I have introduced the horizontal wave numberkh instead of the degreel, using equation (4.51). These equations may be combined into a single, second-order equation forξr:
d2ξr
is a characteristic frequency for the atmosphere. In the solar atmosphereHis approximately equal to 120 km, and ωa is about 0.03 s−1, corresponding to a cyclic frequency of about 5 mHz, or a period of about 3 min.
Exercise 5.3:
Carry through the derivation of equations (5.44) and (5.45). Furthermore, consider the dependence on stellar parameters of ωa, measured in units of the characteristic dynamical frequency (GM/R3)1/2 (cf. equation 5.10).
5.4. OSCILLATIONS IN STELLAR ATMOSPHERES 105 Equation (5.44) has constant coefficients, and so the solution can be written down immediately as
These equations have been the subject of extensive studies in connection with early attempts to interpret observations of solar 5-minute oscillations of high degree (see e.g. Stein &
Leibacher 1974). From the expression for λ± one may qualitatively expect two regimes:
one where the frequency is relatively large, kh is relatively small and the first two terms in {· · ·} dominate; the second where the frequency is small, kh is large and the last term in {· · ·} dominates. These correspond to atmospheric acoustic waves and gravity waves, respectively. From the point of view of global stellar oscillations interest centres on waves with a wavelength much larger than the scale height of the atmosphere. Thus I neglect the last term, reducing equation (5.47) to
λ±= 1
This is clearly the relation for purely vertical waves.
Equation (5.48) shows the physical meaning of ωa. When ω < ωa, λ± are real, and the motion behaves exponentially in the atmosphere. Whenω > ωa, λ± are complex, and the motion corresponds to a wave propagating through the atmosphere. Thus ωa is the minimum frequency of a propagating wave, and is consequently known as the acoustical cut-off frequency (Lamb 1909). The exponential behaviour in the former case provides the upper reflection of p modes, which was absent in the simplified asymptotic analysis of Section 5.3. To study this in more detail, I consider the boundary conditions for an atmosphere of infinite extent. Here the energy density in the motion must be bounded as h tends to infinity. The energy density is proportional to ρξr2, which for the two solutions behaves as
Therefore only the λ− solution is acceptable, and here the energy density decreases expo-nentially. This gives rise to the atmospheric reflection. Thus only modes with frequencies below ωa are trapped in the stellar interior. At higher frequencies the waves propagate through the atmosphere; oscillations at such frequencies would therefore rapidly lose en-ergy (such waves, generated in the convection zone, may contribute to the heating of the solar chromosphere). In fact, the observed spectrum of oscillations stops at frequencies of about 5 mHz. It should be noticed thatλ−>0, so that the displacementincreaseswith al-titude. This increase can in fact be observed by comparing oscillation amplitudes obtained in spectral lines formed at different levels in the atmosphere.
From this solution we may obtain a more realistic boundary condition, to replace the condition (4.68) discussed earlier. The condition of adiabaticity (4.58) and the continuity equation (3.41) give
106 CHAPTER 5. PROPERTIES OF SOLAR AND STELLAR OSCILLATIONS.
or
p0 =δp−ξrdp dh = 1
Hp(1−Γ1λ−)ξr , (5.51) where, for simplicity, I neglected the horizontal part of the divergence ofδδδr(this could, quite simply, be included). This provides a boundary condition that may be used in numerical computations, in place of equation (4.68). Typically it is applied at a suitable point, such as the temperature minimum, in the atmosphere of the stellar model.
When ω is small compared withωa, we can approximateλ− by
Then Γ1λ− can be neglected in equation (5.51), compared with 1, and we recover the boundary condition in equation (4.68). In this limit the displacement is almost constant throughout the atmosphere. Also, it follows from equation (5.50) that the Lagrangian perturbations to pressure and density, and consequently also to temperature, are small.
Physically, this means that the atmosphere is just lifted passively up and down by the oscillation, without changing its structure. Only when the frequency is quite close to the acoustical cut-off frequency does the oscillation have a dynamical effect on the atmosphere.
In reality the Sun, and likely many other stars, is surrounded by a high-temperature corona; in the solar case the temperature exceeds 106K. Here the scale heights are large, and the acoustical cut-off frequency correspondingly small. A typical temperature of 1.6×106K, corresponds to ωa '0.001 s−1. This is below the frequencies of the modes observed in the solar 5-minute region, and these are therefore propagating in the corona. This atmospheric structure may be approximated by representing it by two isothermal layers, corresponding to the inner atmosphere and the corona, respectively. In the corona the two solutions obtained from equation (5.48) are waves propagating outwards into, and inwards from, the corona. Of these only the former solution is physically realistic (unless one assumes that there is a source of waves in the outer corona!). It may be written, as a function of altitude and time, as
δr(h, t) =Aρ−1/2exp[i(krh−ωt)], (5.53) where the radial wave number is
kr =H−1 ω2 ωa2 −1
!1/2
. (5.54)
This solution may also be used to obtain a boundary condition on numerical solutions of the oscillation equations in the inner atmosphere, by requiring that the displacement and the pressure perturbation be continuous at the interface between the atmosphere and the corona. I shall not write down this condition in detail. Note, however, that as the solution (5.53) is complex, the relation obtained between ξr and p0 has complex coefficients. Thus the solution and the eigenfrequency are no longer purely real. Physically, the propagating wave in the corona carries away energy from the oscillation, which is therefore damped, even in the adiabatic approximation. This damping gives rise to an imaginary part in the eigenfrequency, which corresponds to an exponential decay of the solution with time.
To estimate the importance of these effects we return to the solution (5.46), in the inner atmosphere. Because of the effect of the corona both terms must now be included. By
5.5. THE FUNCTIONAL ANALYSIS OF ADIABATIC OSCILLATIONS 107