Helioseismology is the process of inferring the internal structure and kinematics of the Sun from the propagation of seismic waves, particularly acoustic waves (p waves) and surface gravity waves (f waves). It was developed by analogy to geoseismology (originally called simply seismology), and subsequently there emerged asteroseismology, in which seismic waves are analysed to constrain the internal structures of other stars. Because the Sun is fluid, to a first approximation it cannot support shear waves (s-waves), unlike the seismic waves on Earth. An exception is the magneto-acoustic waves which appear to be important only in the atmosphere. The helioseismic waves are generated by the turbulence in the convection zone immediately beneath the Sun's surface. Certain frequencies are amplified by constructive interference, leading to resonance. In other words, the turbulence "rings" the sun like a bell. The resonant waves are reflected near the photosphere, the visible surface of the sun, where they can be observed. The oscillations are detectable in almost any time series of solar images, but are best observed by measuring the Doppler shift of atmospheric absorption lines. Details of the propagation of the seismic waves through the Sun, inferred from the resonant frequencies, reveal the Sun's inner structure, allowing astrophysicists to develop an extremely detailed representation of the hydrostatic stratification and the internal angular velocity. That has permitted the evaluation of the quadrupole moment,
J
2
=
1.8
×
10
−
7
, and higher-order moments of the Sun's external gravitational potential. It is a more accurate and more robust procedure than trying to infer it from the oblateness of the visible disc. Together with measurements of the orbits of Mercury and of spacecraft, the result is consistent with General Relativity.
Helioseismology has been able to rule out the possibility that the solar neutrino problem was a result of incorrect static models of the interior of the Sun. Features revealed by helioseismology include that the outer convective zone and the inner radiative zone rotate differently, which is thought by some to generate the main magnetic field at least in the outer layers of the Sun by a dynamo effect. Broadly speaking, the angular velocity of the convection zone decreases from equator to the poles, varying only weakly with depth; the radiative envelope beneath rotates almost uniformly. These two regions are separated by a shear layer called the tachocline, which is too thin to be resolved directly by seismological analysis alone. The convective zone has "jet streams" of plasma (called torsional oscillations) thousands of kilometers below the surface. The jet streams form broad fronts at the equator, breaking into smaller cyclonic storms at high latitudes. Torsional oscillations are the time variation in solar differential rotation. They are alternating bands of faster and slower rotation. So far there is no generally accepted theoretical explanation for them, even though a close relation to the solar cycle is evident, as they have a period of eleven years, as was known since they were first observed in 1980.
Helioseismology can also be used to image the far side of the Sun from the Earth, including sunspots. In simple terms, sunspots both absorb and deflect helioseismic waves, causing a seismic deficit where next they encounter the photosphere. To facilitate spaceweather forecasting, seismic images of the central portion of the solar far side have been produced nearly continuously since late 2000 by analysing data from the SOHO spacecraft, and since 2001 the entire far side has been imaged with these data.
Individual oscillations in the Sun are damped; in the absence of continual excitation they would die out in a few days. Resonating interference between propagating waves produces global standing waves, known also as normal modes. Analysis of these modes constitutes the discipline of global helioseismology.
Solar oscillation modes are divided up into three basic categories, according to their dominant restoring force: pressure dominates in p modes, and buoyancy in gravity modes, both internal (g modes) and surface (f modes):
p-mode dynamics is determined by the variation of the speed of sound inside the sun. Oscillations with amplitudes great enough to be detectable have frequencies between about 1 and 5 mHz, and are particularly strong in the 2-4 mHz range, where they are often referred to as "5-minute oscillations". (Note: 5 minutes per cycle is 1/300 cycles per second = 3.33 mHz.) At the solar surface, individual p modes have velocity amplitudes of order 10 cm/s, implying displacement amplitudes of a few meters and causing intensity fluctuations of a few parts per million, and are readily detectable with Doppler imaging or sensitive spectral line intensity imaging. Thousands of p modes of high and intermediate degree l (see below for the wavenumber degree l) have been detected by both GONG and the Michelson Doppler Imager (MDI) instrument aboard the SOHO spacecraft, with those of degree l below 200 clearly separated and higher degree modes ridged together. Modes of very low degree have been observed most successfully in light integrated over the entire image of the Sun, by both ground-based networks of observatories such as BiSON distributed about the world to obtain continuous temporal coverage, and by the GOLF instrument aboard the SOHO spacecraft.
g modes are standing internal gravity waves whose principal restoring force is negative buoyancy of vertically displaced material, hence the name "g mode". They are of relatively low frequency (0-0.4 mHz). They are confined either to the interior of the sun below the convection zone (the inner 70 per cent by radius), or to the atmosphere. Because they cannot propagate through convectively unstable regions (in which the temperature gradient exceeds the adiabatic lapse rate, buoyancy is positive and the force on a displaced element of fluid is not restoring), the former are practically unobservable at the surface. The g modes are said to be evanescent in the convection zone, and are thought to have residual velocity amplitudes of only millimeters per second at the photosphere, though perhaps they are more prominent as temperature perturbations. Since the 1980s, there have been several claims of g-mode detection, including one claimed in 2007 using the GOLF data. At the GONG2008 / SOHO XXI conference held in Boulder, the Phoebus group reported that it could not confirm these findings, putting an upper limit of 3 mm/s on g-mode amplitudes, right at the detection limit of the GOLF instrument. The Phoebus group has recently published a review of the current state of g-mode knowledge.
f modes are surface gravity waves, and, aside from the modes of lowest degree l, are confined to the near-surface layers of the Sun, penetrating to a depth of about R/l beneath the photosphere, where R is the radius of the Sun. The frequencies of high-degree f modes are determined essentially by just the surface gravity and the horizontal wavelength, and depend only very weakly on the structure of the Sun. Their terrestrial counterparts are deep water waves, and in the limit of high degree l they share the same dispersion relation. Deviations from that limiting relation provide information principally about the density stratification of the surface layers of the Sun Doppler shifts brought about by advection have been measured by MDI (see below), to set constraints on the horizontal subsurface flow
Helioseismic waves are of very low amplitude, and can be described as a superposition of solutions of the linearized wave equation. Because the Sun is very nearly spherical, the spatial structure of those solutions can be represented, with respect to spherical polar coordinates
(
r
,
θ
,
ϕ
)
, as products of orthonormal surface harmonics, of
θ
and
ϕ
, and an amplitude function of
r
. It is usual to adopt as a basis for the surface harmonics a product of exp(i
m
ϕ
) and the associated Legendre function of cos
θ
of degree
l
and (azimuthal) order
m
. Globally, the background structure hardly changes over an oscillation period, so the temporal variation is simply a multiplicative sinusoidal function of
t
, whose frequencies
ω
are a sequence of eigenvalues of the wave equation, and are labeled by the order
n
. The degree
l
is the total number of nodal circles on a surface of constant
r
, and the azimuthal order
m
is the number of complete nodal circles crossing the equator; by convention the order
n
is zero for f modes, and counts upwards/downwards for p/g modes according roughly to the number of radial nodes in the eigenfunction; frequency
ω
is a strictly increasing function of
n
at constant
l
and
m
. An example of such a mode is illustrated at the top right of this article.
The data from time-series of solar spectra contain all the oscillations overlapping. Thousands of modes have been detected (with the true number being in the millions). The mathematical technique of Fourier analysis is used to recover information about individual modes from this mass of data. The idea is that any bounded function
f
in a bounded domain can be written as a weighted sum of orthogonal harmonic functions (the basis functions), which in one dimension are the simplest periodic functions, namely sines and cosines (with different frequencies). To determine how much (the amplitude) of each basis function contributes to
f
, one applies the Fourier transform: essentially the projection (functional scalar product) of
f
onto the basis functions over the domain, although in practice the technique of accomplishing that task is more sophisticated, and faster, than carrying out the projections explicitly.
Note that if the Sun were spherically symmetric, the eigenfrequencies
ω
n
l
m
would be degenerate with respect to the azimuthal order
m
, because all chosen spherical polar coordinate systems would be indistinguishable. The Sun's rotation creates an equatorial bulge, which, along with other aspherical perturbations such as sunspots, break that symmetry, and lift the degeneracy. Therefore, in general the frequencies
ω
n
l
m
of stellar oscillations depend on all three quantum numbers
n
,
l
and
m
. It is convenient to separate the frequency into the multiplet frequency
ω
n
l
, the uniformly weighted average over
m
, corresponding to the spherically symmetric structure of the star, and the frequency splitting
δ
ω
n
l
m
=
ω
n
l
m
−
ω
n
l
, which is determined by the asphericity.
Analyses of oscillation data are aimed at separating these different frequency components. In the case of the Sun the oscillations can be observed directly as functions of position on the solar disc as well as time. Projection onto the spatial eigenfunctions goes some way towards isolating
l
and
m
, although the outcome contains contributions from many other harmonics, partly because in practice only about one-third of the full surface area of the Sun can usefully be measured. The average over the stellar surface implicit in observations of stellar oscillations can be thought of as an example of such spatial filtering, just as are the full-disc solar observations by BiSON and GOLF. Each projection is followed by a Fourier transform in time, from which, with adequate resolution, the frequencies of the modes can be determined.
Note that the oscillation data, rather than being continuous functions, are actually discrete samples in space and time, and are subject to observational error. When computing transforms, interpolation is implied, a process which inevitably introduces further errors.
This discussion is adapted from the Jørgen Christensen-Dalsgaard lecture notes on stellar oscillations.
Information about helioseismic waves (such as mode frequencies and frequency-splitting) collected by transforming the oscillation data can be used to infer numerical details of internal features of the Sun such as the internal sound speed and the internal differential rotation. Equations and analytic relations such as integrals can be manipulated to relate the desired internal properties to the transformed data. The numerical methods used are adapted to the particular internal features examined so as to extract the maximum amount of information, with the least error, from the oscillations about the internal features. This process is termed helioseismic inversion.
As an example in slightly more detail, the oscillation frequency splitting can be related, via an integral, to the angular velocity within the sun.
Helioseismic observations reveal the inner uniformly rotating zone and the differentially rotating envelope of the Sun, roughly corresponding to the radiation and convection zones, respectively. See the diagram on the right. The transition layer is called the tachocline.
The age of the sun can be inferred with helioseismic studies. This is because the propagation of acoustic waves deep within the sun depends on the composition of the sun, in particular the relative abundance of helium and hydrogen in the core. Since the sun has been fusing hydrogen into helium throughout its lifetime, the present day abundance of helium in the core can be used to infer the age of the sun, using numerical models of stellar evolution applied to the Sun (standard solar model). This method provides verification of the age of the solar system gathered from the radiometric dating of meteorites.
The goal of local helioseismology, a term first used in 1993, is to interpret the full wave field observed at the surface, not just the mode (more precisely, eigenmode) frequencies. Another way to look at it, is that global helioseismology studies standing waves of the entire Sun and local helioseismology studies propagating waves in parts of the Sun. A variety of solar phenomena are being studied, including sunspots, plage, supergranulation, giant cell convection, magnetically active region evolution, meridional circulation, and solar rotation. Local helioseismology provides a three-dimensional view of the solar interior, which is important to understand large-scale flows, magnetic structures, and their interactions in the solar interior.
There are many techniques used in this new and expanding field, which include:
Fourier–Hankel spectral method, first introduced by Braun and Duvall, was originally used to search for wave absorption by sunspots.
Ring-diagram analysis, first introduced by F. Hill, is used to infer the speed and direction of horizontal flows below the solar surface by observing the Doppler shifts of ambient acoustic waves from power spectra of solar oscillations computed over patches of the solar surface (typically 15° × 15°). Thus ring analysis is a generalization of global helioseismology applied to local areas on the Sun (as opposed to half of the Sun). For example, sound speed and adiabatic index can be compared within magnetically active and inactive (quiet Sun) regions.
Time-distance helioseismology, introduced by Duvall et al., aims to measure and interpret the travel times of solar waves between any two locations on the solar surface. A travel time anomaly contains the seismic signature of buried inhomogeneities within the proximity of the ray path that connects two surface locations. An inverse problem must then be solved to infer the local structure and dynamics of the solar interior.
Helioseismic holography, introduced in detail by Lindsey and Braun for the purpose of far-side (magnetic) imaging, a special case of phase-sensitive holography. The idea is to use the wavefield on the visible disk to learn about active regions on the far side of the Sun. The basic idea in helioseismic holography is that the wavefield, e.g., the line-of-sight Doppler velocity observed at the solar surface, can be used to make an estimate of the wavefield at any location in the solar interior at any instant in time. In this sense, holography is much like seismic migration, a technique in geophysics that has been in use since the 1940s. As another example, this technique has been used to give a seismic image of a solar flare. Acoustic holography, applied to MDI data, is ideal for the detection of sources and sinks of acoustic waves on the Sun. Braun and Fan discovered a region of lower acoustic emission in the 3 – 4 mHz frequency band which extends far beyond the sunspots (the ‘acoustic moat’). Acoustic moats extend beyond magnetic regions into the quiet Sun. In addition, Braun and Lindsey discovered high-frequency emission (‘acoustic glories’) surrounding active regions.
Direct modelling, after Woodard. Here the idea is to estimate subsurface flows from direct inversion of the frequency-wavenumber correlations seen in the wavefield in the Fourier domain. Woodard gave a practical demonstration of the ability of the technique to recover near-surface flows from the f-mode part of the spectrum.
This section is adapted from Laurent Gizon and Aaron C. Birch, "Local Helioseismology", Living Rev. Solar Phys. 2, (2005), 6. online article (cited on November 22, 2009).
An internal jet stream moving behind schedule may explain the delayed start to the solar cycle in 2009.
