interior, and has constrained input physics (see, e.g., Gough et al. 1996).

From the discussion above, however, it seems that the study of solar dy-

namics bene¬ts from helioseismology much more than do stellar evolution

studies.

Seismology of solar rotation is a two-step process. First, we need rota-

tional splitting measurement. Second, we have to carry out inversion to

obtain a spatially resolved view of solar rotation. On accomplishing these

two steps, one can then proceed to investigate the physics of solar rotation.

Below I shall summarize the background and current status of, and discuss

problems in and around, seismology of solar rotation.

18.2 Helioseismic measurement of solar internal rotation

Helioseismology, as it is practiced today, is mainly about using eigenmodes of

acoustic oscillations as a probe into the solar interior. Wave propagation in

the sun is a¬ected by the solar rotation, and then the quantization condition

for forming standing waves changes, thereby shifting the eigenfrequencies.

This is done principally by advection in the case of high-frequency p modes.

Since the sun is close enough to a sphere, we can identify the normal

modes of the solar oscillation by radial order n, spherical-harmonic degree

l and azimuthal order m (= 0, ±1, ±2, . . . , ±l). Without rotation or any

other symmetry-breaking agent, the eigenfrequency depends only on n and

l. We may write this (hypothetical) frequency as ωnl . In the presence

of rotation, however, this degeneracy with respect to m is lifted and the

Seismology of solar rotation 265

frequency depends on m as well. Let us denote this frequency by ωnlm . If

we forget magnetic ¬eld and other structural asphericity for the moment,

we can say that the di¬erence between ωnlm and ωnl is caused by rotation.

Let us write down, in spherical polar coordinates (r, θ, φ), the displacement

vector associated with mode (nlm) as ξ nlm (r, θ, φ) exp(iωnlm t), where t is

time. Then the pulsation equation can be written in the form

Lξ nlm = ρωnlm ξ nlm ,

2

where ρ is the density and L is a linear operator. For the expression of L

see, e.g., Unno et al. (1989).

The frequency shift due to rotation „¦rot (r, θ) = „¦(r, θ)ez (ez is a unit

vector in the z-direction) is calculated by taking account, by linear per-

turbation theory, of the perturbation ∆L(ξ) = 2iωnl ρ(v rot · ∇)ξ nlm to the

linear operator in the pulsation equation caused by the rotational velocity

vrot . This results in

m i

ξ — · („¦rot — ξnlm )ρdV ,

∆ωnlm ≡ ωnlm ’ ωnl = |ξ nlm |2 „¦ρdV ’ nlm

Inl Inl

where dV = r 2 sin θdrdθdφ and

|ξ nlm |2 ρdV ,

Inl =

which does not depend on m (integrations are over the whole volume of the

sun). This is often rewritten in the following form:

∆ωnlm = m Knlm (r, θ)„¦(r, θ)drdθ . (18.1)

The rotational shift of the frequency ∆ωnlm is essentially a weighted average

of „¦(r, θ), and the weighting function Knlm (r, θ) is the splitting kernel, which

can be derived from an equilibrium model and its eigenfunctions after solving

the eigenvalue problem under appropriate boundary conditions. In the solar

case ∆ωnlm = m „¦ ∼ m — 400 nHz, which already gives a measure of

the solar rotation rate without any further analysis (here · · · denotes an

average). However, to disentangle contributions from di¬erent parts of the

sun, so that we have any spatial resolution, we need an inversion procedure.

18.3 Inversion for internal rotation

Since in equation (18.1) ∆ω is expressed as a linear functional of „¦, the

inverse problem for „¦ is linear. There are various inversion methods that

have been proposed or used for the rotation inversion, such as i) asymptotic

Sekii

266

methods, which rely on asymptotic approximations to yield semi-analytically

invertible expressions, ii) methods based on optimally localized averaging

(OLA), and iii) methods based on least-squares ¬tting, such as regularized

least-squares ¬tting (RLSF). Normally, model ¬tting is distinguished from

inversion but the boundary can be fuzzy, as can be seen from the inclusion of

the RLSF in the list above. There are certain important properties of linear

inverse problems, some very general, some depending on which linear inverse

problem one attempts to solve, and some depending on which method one

uses. For any linear inverse problem,

• there exist functions orthogonal to all the kernels (the space spanned by

such functions is called the annihilator),

• there is a trade-o¬ between resolution and error magni¬cation,

• the region where no kernel has amplitude is inaccessible, and

• data redundancy can be caused by kernels having similar structure, even

locally

which all limits the amount of information we can extract. Properties spe-

ci¬c to the inversion of rotational splitting are derived from the properties

of the p-mode splitting kernels Knlm , which are

• Knlm has no amplitude close to the centre or to the pole,

• Knlm has large amplitude near the surface,

• Knlm contains only those terms that are even in m ,

• Knlm is sensitive only to the north-south symmetric component.

It is interesting to compare the ¬rst two of the list immediate above, and

the last two of the previous list. From the third item in the above, we see

that any even component in the frequency shift is due to magnetic ¬eld or

asphericity, including the second-order e¬ect of rotation (cf. Dziembowski &

Goode 1992). Observationally, this even component is small but detectable.

In order to remove any e¬ect of the even component from rotation inversion,

we can use ∆ωnlm ≡ (ωnlm ’ ωnl,’m)/2 as the new de¬nition of ∆ωnlm.

For the convenience of a more formal discussion, let us rewrite the set

of linear constraints on „¦(r, θ) in a more symbolic fashion. Introducing

discretization at this point is not quite necessary, but it will save the need of

switching between discussion of functions and discussion of vectors of ¬nite

dimension, in what follows. Suppose we discretize equation (18.1) using N

grid points and obtain

N

Kij „¦j = di ,

j=1

Seismology of solar rotation 267

where the mode index i, which corresponds to a certain set of (nlm), runs

from 1 to M (the number of the modes). Here the right-hand side represents

∆ωnlm/m for the mode i. Then we can introduce an M — N matrix K, an

N -vector „¦ and an M -vector d to write

K„¦ = d . (18.2)

Since in reality any observation is associated with error, we must replace d

by

ˆ

d=d+e ,

where the term e represents observational error.

Solving a linear inverse problem by a linear process is equivalent to ¬nding,

ˆ

somehow, another matrix R to obtain an estimate of „¦ as

ˆ

ˆ

„¦ = Rd . (18.3)

ˆ

Obviously, the component of „¦ that is due to the term e is Re. We would

like δ„¦ ≡ |Re| as small as possible. However, this alone does not guarantee

ˆ

ˆ

that „¦ resembles „¦. For this there are two other important aspects we

ˆ ˆ

need to check. One is the degree of mis¬t: |d ’ K „¦| = |(IM ’ KR)d|,

ˆ

where IM is the identity matrix of size M . The other is resolution. Since

ˆ ˆ

„¦ = RK„¦ + δ„¦, we can examine RK to see how well features in „¦ are

ˆ

resolved in „¦.

Simplistically put, methods based on optimally localized averaging (OLA)

aims at rendering RK as diagonal as possible (RK ’ IN ), while keeping

the magnitude of |Re| at a reasonable level.

On the other hand, methods based on least-squares ¬tting aims to mini-

mize the mis¬t (KR ’ IM ), also without increasing the magnitude of |Re|