Trapping of the modes

The equations in the Cowling approximation can be written as dξ_r

hence H_p is the pressure scale height, i.e., the distance, roughly, over which the pressure changes by a factor e. For oscillations of high radial order, the eigenfunctions vary much more rapidly than the equilibrium quantities; thus, e.g., the left hand side of equation (5.12) is much larger than the first term on the right hand side which contains derivatives of equilibrium quantities. As a first, very rough approximation, I simply neglect these terms, reducing the equations to

76 CHAPTER 5. PROPERTIES OF SOLAR AND STELLAR OSCILLATIONS.

These two equations can be combined into a single second-order differential equation for ξ_r; neglecting again derivatives of equilibrium quantities, the result is

d²ξ_r dr² = ω²

c² 1−N² ω²

! S_l² ω² −1

!

ξ_r. (5.17)

Figure 5.2: Buoyancy frequency N [cf. equation (4.63); continuous line] and characteristic acoustic frequencyS_l [cf. equation (4.60); dashed lines, labelled by the values of l], shown in terms of the corresponding cyclic frequencies, against fractional radius r/R for a model of the present Sun. The heavy horizontal lines indicate the trapping regions for a g mode with frequency ν= 100µHz, and for a p mode with degree 20 and ν = 2000µHz.

This equation represents the crudest possible approximation to the equations of non-radial oscillations. In fact the assumptions going into the derivations are questionable. In particular, the pressure scale height becomes small near the stellar surface (notice that H_p = p/(ρg) is proportional to temperature), and so derivatives of pressure and density cannot be neglected. I return to this question in Chapter 7. Similarly, the term in 2/r neglected in equation (5.12) is large very near the centre. Nevertheless, the equation is adequate to describe the overall properties of the modes of oscillation, and in fact gives a reasonably accurate determination of their frequencies.

From equation (5.17) it is evident that the characteristic frequencies S_l and N, defined in equations (4.60) and (4.63), play a very important role in determining the behaviour of the oscillations. They are illustrated in Figure 5.2 for a “standard” solar model. S_l tends to infinity as r tends to zero and decreases monotonically towards the surface, due to the decrease in c and the increase in r. As discussed in Section 3.3, N² is negative in convection zones (although generally of small absolute value), and positive in convectively stable regions. All normal solar models have a convection zone in the outer about 30 per

5.2. THE PHYSICAL NATURE OF THE MODES OF OSCILLATION 77

Figure 5.3: (a) Hydrogen content X against mass fraction m/M for three models in a 2.2M evolution sequence. The solid line is for age 0, the dotted line for age 0.47 Gyr and the dashed line for age 0.71 Gyr. Only the inner 40 per cent of the models is shown. (b) Scaled buoyancy frequency, expressed in terms of cyclic frequency, against m/M for the same three models. In the scaling factor, R and R₀ are the radii of the actual and the zero-age main sequence model, respectively. For the model of age 0.71 Gyr, the maximum value of (R/R₀)^3/2N/2πis 2400µHz. (c) Scaled buoyancy frequencyN (heavy lines) and characteristic acoustic frequency S_l for l = 2 (thin lines), for the same three models, plotted against fractional radiusr/R.

cent of the radius, whereas the entire interior is stable. The sharp maximum in N very near the centre is associated with the increase towards the centre in the helium abundance in the region where nuclear burning has taken place. Here, effectively, lighter material is

78 CHAPTER 5. PROPERTIES OF SOLAR AND STELLAR OSCILLATIONS.

on top of heavier material, which adds to the convective stability and hence increases N. This is most easily seen by using the ideal gas law for a fully ionized gas, equation (3.19) which is approximately valid in the interior of the stars, to rewrite N² as

N²' g²ρ In the region of nuclear burning, µ increases with increasing depth and hence increasing pressure, and therefore the term in∇µ makes a positive contribution toN².

The behaviour of N is rather more extreme in stars with convective cores; this is il-lustrated in Figure 5.3 for the case of a 2.2M evolution sequence. The convective core is fully mixed and here, therefore, the composition is uniform, with ∇µ = 0. However, in stars of this and higher masses the convective core generally shrinks during the evolution, leaving behind a steep gradient in the hydrogen abundance X, as shown in Figure 5.3a.

This causes a sharp peak in ∇µ and hence in N. When plotted as a function of mass fraction m/M, as in panel (b) of Figure 5.3, the location of this peak is essentially fixed although its width increases with the shrinking of the core¹. However, as illustrated in Figure 5.3c, the location shifts towards smaller radius: this is a consequence of the increase with evolution of the central density and hence the decrease in the radial extent of a region of given mass. This also causes an increase gravity gin this region and hence in N, visible in the figure. In accordance with equation (5.7), the characteristic frequencies have been scaled byR^3/2 in Figure 5.3: it is evident thatS_l, and N in the outer parts of the model, are then largely independent of evolution. Thus the stellar envelope essentially changes homologously, while this is far from the case for the core; it follows that stellar oscillations sensitive to the structure of the core might be expected to show considerable variation with evolution of the dimensionless frequencyσ introduced in equation (5.10). This is confirmed by the numerical results shown in Section 5.3.2.

To analyze the behaviour of the oscillations I write equation (5.17) as d²ξ_r

The local behaviour of ξ_r depends on the sign of K. Where K is positive, ξ_r is locally an oscillating function ofr, and whereK(r) is negative the solution is locally an exponentially increasing or decreasing function ofr. Indeed, as will be shown in more detail in Chapter 7, in the former case the solution may be written approximately as

ξ_r∼cos Z

K^1/2dr+φ

, K >0, (5.22)

1The erratic variation in N in the chemically inhomogeneous region is caused by small fluctuations, introduced by numerical errors, inX(m).

5.2. THE PHYSICAL NATURE OF THE MODES OF OSCILLATION 79 (φ being a phase determined by the boundary conditions) while in the latter case

ξ_r∼exp

± Z

|K|^1/2dr

, K <0. (5.23)

Thus according to this description the solution oscillates when

o1) |ω|>|N| and |ω|> S_l, (5.24) or

o2) |ω|<|N| and |ω|< S_l, (5.25) and it is exponential when

e1) |N|<|ω|< S_l , (5.26) or

e2) S_l<|ω|<|N|. (5.27) These possibilities are discussed graphically in Unno et al. (1989), Section 16.

For a given mode of oscillation there may be several regions where the solution oscillates, according to criterion o1) or o2), with intervening regions where it is exponential. However, in general one of these oscillating regions is dominant, with the solution decaying exponen-tially away from it. The solution is then said to be trappedin this region; its frequency is predominantly determined by the structure of the model in the region of trapping. The boundaries of the trapping region are generally at points where K(r) = 0; such points are known as turning points. From equation (5.22) it follows that within the trapping region the mode oscillates the more rapidly as a function of r, the higher the value of K. Thus, if the order of the mode is roughly characterized by the number of zeros in ξr2 the order generally increases with increasing K.

From the behaviour ofS_landN shown in Figure 5.2, and the conditions for an oscillating solution, we may expect two classes of modes:

i) Modes with high frequencies satisfying o1), labelled p modes.

ii) Modes with low frequencies satisfying o2), labelled g modes.

These are discussed separately below.

Typical trapping regions, for a p and a g mode in a model of the present Sun, are shown in Figure 5.2. I note also that in evolved stars with convective cores the large values of N at the edge of the core may lead to the condition o2) being satisfied even at quite high frequency. Thus here one might expect that the distinction in frequency between p and g modes becomes less clear. Some consequences of that are illustrated in Section 5.3.2.