Computational efficiency demands that the distribution of mesh points be chosen appro-priately. It is immediately obvious from the eigenfunctions (cf. Figures 5.8 and 5.10 that a mesh uniform in r is far from optimal; also the distribution of points should clearly be different for p and for g modes.
Procedures exist that determine the optimal mesh as part of the numerical solution of a set of differential equations (Gough, Spiegel & Toomre 1975). In the present case, however, the requirements on the mesh are essentially driven by the behaviour of the modes of high
124 CHAPTER 6. NUMERICAL TECHNIQUES radial order, whose eigenfunctions are given, with considerable precision, by the asymptotic expression (5.22). Thus to have a roughly constant number of meshpoints between the nodes in the eigenfunction, the mesh should be approximately uniformly spaced in terms of the integral in this equation.
To define a flexible method for setting up the mesh I have adopted a simplified version of the procedure developed by Gough et al. (1975). Thus I introduce a variablez, with a range from 0 to 1, such that the mesh is uniform inz, and determined by
dz
dr =λH(r) ; (6.8)
here λ is a normalization constant, relating the ranges of z and r, and the function H determines the properties of the mesh. GivenH,z is obtained as
z(r) =λ Z r
0
H(r0)dr0, (6.9)
with
λ= Z R
0 H(r)dr
!−1
. (6.10)
The mesh {r(n), n= 1, . . . , Nme} is finally determined by solving the equations z(r(n)) = n−1
Nme−1 , (6.11)
by interpolating in the computed z(r).
The choice of the functionH must be guided by the asymptotic behaviour of the modes, as described by the functionK in equation (5.22). Specifically, I have used
H(r)2=R−2+c1ωg2
c2 +c2|N|2 ωg2r2 +c3
d lnp dr
2
, (6.12)
where ωg2 ≡ GM/R3 = t−dyn2 is a characteristic squared frequency. Here the term in c1 results from noting that in the limit of an extreme p mode K ∝c−2 [cf. equation (5.29)], whereas the term inc2similarly corresponds to the extreme g-mode case, whereK ∝N2/r2 [cf. equation (5.33)]. The term in c3 provides extra meshpoints near the surface, where the reflection of the p modes takes place. Finally the constant term ensures a reasonable resolution of regions where the other terms are small.
Table 6.1
c1 c2 c3 p-mode mesh 10. 0.01 0.015 g-mode mesh 0.025 0.1 0.0001
Table 6.1: Parameters in equation (6.12) for determination of meshes suitable for computing p and g modes in a model of the present Sun.
6.7. THE DETERMINATION OF THE MESH 125 The parameters in this expression can be determined by testing the numerical accuracy of the computed frequencies (e.g. Christensen-Dalsgaard & Berthomieu 1991). For nor-mal solar models reasonable choices, based on a fairly extensive (but far from exhaustive) set of calculations, are given in Table 6.1. In the p-mode case the mesh is predominantly determined by the variation of sound speed, with the term inN giving a significant contri-bution near the centre and the term in d lnp/dr contributing very near the surface. The g-mode mesh is dominated by the term in N in most of the radiative interior, whereas in the convection zone, where |N| is generally small, the constant term dominates; the term in d lnp/dr is again important in the surface layers.
126 CHAPTER 6. NUMERICAL TECHNIQUES
Chapter 7
Asymptotic theory of stellar oscillations
In Chapter 5 I discussed in a qualitative way how different modes of oscillation are trapped in different regions of the Sun. However, the simplified analysis presented there can, with a little additional effort, be made more precise and does in fact provide quite accurate quantitative information about the oscillations.
The second-order differential equation (5.17) derived in the previous chapter cannot be used to discuss the eigenfunctions. Thus in Section 7.1 I derive a more accurate second-order differential equation for ξr. In Section 7.2 the asymptotic solution of such equations by means of the JWKB method is briefly discussed, with little emphasis on mathematical rigour; the results are used to obtain asymptotic expressions for the eigenfrequencies and eigenfunctions. They are used in Sections 7.3 and 7.4 to discuss p and g modes. This approximation, however, is invalid near the surface, and furthermore suffers from critical points in the stellar interior where it formally breaks down. In Section 7.5 I discuss an asymptotic formulation derived by D. O. Gough (cf. Deubner & Gough 1984) that does not suffer from these problems; in particular, it incorporates the atmospheric behaviour of the oscillations analyzed in Section 5.4. On the other hand, it uses a dependent variable with a less obvious physical meaning. This method gives a unified asymptotic treatment of the oscillations throughout the Sun, although still under certain simplifying assumptions.
A similar, but even more complete, treatment was developed by Gough (1993), although this appears so far not to have been substantially applied to numerical calculations.
One of the most important results of the asymptotic analysis is the so-called Duvall relation, which was first discovered by Duvall (1982) from analysis of observed frequencies of solar oscillation. A rough justification for the relation is given in Section 7.3, and a more rigorous derivation is presented in Section 7.5. It is shown that frequencies of p modes approximately satisfy
Z R rt
1− L2c2 ω2r2
!1/2
dr
c = [n+α(ω)]π
ω . (7.1)
This is evidently a very special dependence of the frequencies on nand l. As discussed in Section 7.7, this relation gives considerable insight into the dependence of the frequencies on the sound speed, and it provides the basis for approximate, but quite accurate, methods for inferring the solar internal sound speed on the basis of observed frequencies.
127
128 CHAPTER 7. ASYMPTOTIC THEORY OF STELLAR OSCILLATIONS
7.1 A second-order differential equation for ξ
rTo obtain this equation I go back to the two equations (5.12) and (5.13) in the Cowling approximation. By differentiating equation (5.12) and eliminating dp0/dr using equation (5.13) we obtain
Herep0 may be expressed in terms ofξr and its derivative by means of equation (5.12). The result is
where K is still given by equation (5.21). All other terms inξr are lumped together in ˜h;
these contain derivatives of equilibrium quantities, and so may be assumed to be negligible compared with K (except, as usual, near the surface). Equation (7.3) may also be written
as d2ξr
It should be noticed that the principal difference between equation (7.4) and equation (5.20) derived previously is the presence of a term in dξr/dr. This occurs because I have now not neglected the term inξron the right-hand side of equation (5.12), and the corresponding term in p0 in equation (5.13). These terms cannot be neglected ifξr is rapidly varying, as assumed.
It is convenient to work with an equation without a first derivative, on the form of equation (5.20). I introduce ˆξr by
ξr(r) =f(r)1/2ξˆr(r) ; (7.6)
7.2. THE JWKB ANALYSIS 129