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Illustration of a solar pressure mode (p-mode) with radial order n=14, angular degree l=20 and azimuthal order m=16. The surface shows the corresponding spherical harmonic. The interior shows the radial displacement computed using a standard solar model.[1] Note that the increase in the speed of sound as waves approach the center of the sun causes a corresponding increase in the acoustic wavelength.

Helioseismology is the study of the structure and dynamics of the Sun through its oscillations. These are principally caused by sound waves that are continuously driven and damped by convection near the Sun's surface. It is similar to geoseismology or asteroseismology, which are respectively the studies of the Earth or stars through their respective oscillations. While the Sun's oscillations were first detected in the early 1960s, it was only in the mid-1970s that it was realised that the oscillations propagated throughout the Sun and could allow scientists to study the Sun's deep interior. The modern field is separated into global helioseismology, which studies the Sun's resonant modes, and local helioseismology, which studies all the waves propagating at the Sun's surface.[2]

Helioseismology has contributed to a number of scientific breakthroughs. The most notable breakthrough was to show the predicted neutrino flux from the Sun could not be caused by flaws in stellar models and must instead be a problem of particle physics. The so-called solar neutrino problem was ultimately resolved by neutrino oscillations.[3][4] The experimental discovery of neutrino oscillations was recognized by the 2015 Nobel Prize for Physics.[5] Helioseismology also allowed accurate measurements of the quadrupole (and higher-order) moments of the Sun's gravitational potential, which are consistent with general relativity. The first helioseismic calculations of the Sun's internal rotation profile showed a rough separation into a rigidly-rotating core and differentially-rotating envelope. The boundary layer is now know as the tachocline[6] and is thought to be a key component for the solar dynamo[7]. Although it roughly coincides with the base of the solar convection zone—also inferred through helioseismology—it is conceptually a distinct entity.

Helioseismology benefits most from continuous monitoring of the Sun, which began first with uninterrupted observations from near the South Pole over the southern summer. In addition, observations over multiple solar cycles have allowed helioseismologists to study changes in the Sun's structure over decades. These studies are made possible by global telescope networks like the Global Oscillations Network Group (GONG) and the Birmingham Solar Oscillations Network (BiSON), which have been operating for over 20 years.

Types of solar oscillation

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A propagation diagram for a standard solar model[8] showing where oscillations have a g-mode character (blue) or where dipole modes have a p-mode character (orange). The dashed line shows the acoustic cut-off frequency, computed from more precise modelling, and above which modes are not trapped in the star, and roughly-speaking do not resonate.

Solar oscillation modes are interpreted (to first order) as vibrations of a spherically symmetric self-gravitating fluid in hydrostatic equilibrium. Each mode is then characterised by three numbers:

  • the number of nodal shells in radius, known as the radial order ;
  • the total number of nodal circles on each spherical shell, known as the angular degree ; and
  • the number of those nodal circles that are longitudinal, known as the azimuthal order .

The latter two correspond to the quantum numbers of the spherical harmonics. Under these assumptions, it can be shown that the oscillations are separated into two categories of interior oscillations and a third special category of surface modes.

Pressure modes (p-modes)

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Pressure modes are in essence standing sound waves. The dominant restoring force is the pressure (rather than buoyancy), hence the name. Most solar oscillations, and all that are used for inferences about the interior, are p-modes, with frequencies between about 1 and 5 millihertz. They span angular degrees from zero (purely radial motion) to several hundred.

Gravity modes (g-modes)

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Gravity modes are lower-frequency modes that are confined to the non-convective interior. The restoring force is buoyancy and thus indirectly gravity, from which they take their name. They are evanescent in the convection zone and therefore have tiny amplitudes at the surface.[9] No individual g-modes have been detected but indirect detections have been claimed.[10][11]

Surface gravity modes (f-modes)

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Surface gravity waves are analogous to waves on deep water. To good approximation, they follow the same dispersion law: , where is the angular frequency, is the surface gravity and is the horizontal wavenumber. Surface gravity modes observed on the Sun are all of very high degree ().

Data analysis

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Global helioseismology

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Power spectrum of the Sun using data from instruments aboard the Solar and Heliospheric Observatory on double-logarithmic axes. The three passbands of the VIRGO/SPM instrument show nearly the same power spectrum. The line-of-sight velocity observations from GOLF are less sensitive to the red noise produced by granulation. All the datasets clearly show the oscillation modes around 3mHz.
Power spectrum of the Sun around where the modes have maximum power, using data from the GOLF and VIRGO/SPM instruments aboard the Solar and Heliospheric Observatory. The low-degree modes (l<4) show a clear comb-like pattern with a regular spacing.
Power spectrum of medium angular degree () solar oscillations, computed for 144 days of data from the MDI instrument aboard SOHO.[12] The colour scale is logarithmic and saturated at one hundredth the maximum power in the signal, to make the modes more visible. The low-frequency region is dominated by the signal of granulation. As the angular degree increases, the individual mode frequencies converge onto clear ridges, each corresponding to a sequence of low-order modes.

The chief tool for analysing data in global helioseismology is the Fourier transform. To good approximation, each mode is a damped harmonic oscillator, for which the power as a function of frequency is a Lorentz function. Resolved data is usually integrated over the desired spherical harmonic to obtain a single timeseries that can be Fourier transformed. In this way, helioseismologists combined a power spectrum for each spherical harmonic into a two-dimensional power spectrum.

The lower frequency range of the oscillations is dominated by the variations caused by granulation. This must first be filtered out before (or at the same time that) the modes are analysed. Granular flows at the solar surface are mostly horizontal, from the centres of the rising granules to the narrow downdrafts between them. Relative to the oscillations, granulation produces a stronger signal in intensity than line-of-sight velocity, hence the latter is preferred for helioseismic observatories.

Local helioseismology

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Local helioseismology—a term coined by Charles Lindsey, Doug Braun and Stuart Jefferies in 1993[13]—employs several different analysis methods to make inferences from the observational data.[2]

  • The Fourier–Hankel spectral method was originally used to search for wave absorption by sunspots.[14]
  • Ring-diagram analysis, first introduced by Frank Hill,[15] 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-diagram analysis is a generalization of global helioseismology applied to local areas on the Sun (as opposed to half of the Sun). For example, the sound speed and adiabatic index can be compared within magnetically active and inactive (quiet Sun) regions.[16]
  • Time-distance helioseismology[17] aims to measure and interpret the travel times of solar waves between any two locations on the solar surface. Inhomogenities near the ray path connecting the two locations perturb the travel time between those two points. An inverse problem must then be solved to infer the local structure and dynamics of the solar interior.[18]
  • Helioseismic holography, introduced in detail by Charles Lindsey and Doug Braun for the purpose of far-side (magnetic) imaging,[19] is 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.[20]
  • In direct modelling, the idea is to estimate subsurface flows from direct inversion of the frequency-wavenumber correlations seen in the wavefield in the Fourier domain. Woodard[21] demonstrated the ability of the technique to recover near-surface flows the f-modes.

Inversion

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Introduction

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The Sun's oscillation modes represent a discrete set of observations that are sensitive to its continuous structure. This allows scientists to formulate inverse problems for the Sun's interior structure and dynamics. Given a reference model of the Sun, the differences in the mode frequencies are weighted averages of the differences between the structure of the reference model and the real Sun. The difference in the mode frequencies can then be used to infer the differences in the structures. The weighting of these averages are known as kernels.

Structure

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The first inversions of the Sun's structure were made using Duvall's law[22] and later using Duvall's law linearised about a reference solar model.[23] These results were superceded by analyses that use the full set of equations describing the stellar oscillations[24] and are now the standard way to compute inversions.[25] The inversions demonstrated differences in solar models that were greatly reduced by implementing gravitational settling: the gradual separation of heavier elements towards the solar centre (and lighter elements to the surface to replace them).[26][27]

Rotation

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The internal rotation profile of the Sun inferred using data from the Helioseismic and Magnetic Imager aboard the Solar Dynamics Observatory. The inner radius has been truncated where the measurements are less certain than 1%, which happens around 3/4 of the way to the core. The dashed line indicates the base of the solar convection zone, which happens to coincide with the boundary at which the rotation profile changes, known as the tachocline.

If the Sun were perfectly spherical, the modes with different azimuthal orders m would have the same frequencies. Rotation, however, breaks this degeneracy, and the modes frequencies differ by rotational splittings that are weighted-averages of the rotation rate throughout the Sun. Different modes are sensitive to different parts of the Sun and, given enough data, these differences can be used to infer the rotation rate throughout the Sun.[28] For example, if the Sun rotated with the same rotational frequency throughout, then all the modes would be split by the same amount. In reality, different layers of the Sun rotate at different speeds, as can be seen at the surface, where the equator rotates faster than the poles.[29] The Sun rotates slowly enough that a spherical, non-rotating model serves as close enough to reality to derive the rotational kernels.

Helioseismology has shown that the Sun has a rotation profile with several features:[30]

  • a rigidly-rotating radiative (i.e. non-convective) zone, though the rotation rate of the inner core is not well known;
  • a thin shear layer, known as the tachocline, which separates the rigidly-rotating interior and the differentially-rotating convective envelope;
  • a convective envelope in which the rotation rate varies both with depth and latitude; and
  • a final shear layer just beneath the surface, in which the rotation rate slows down towards the surface.

Relationship to other fields

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Geoseismology

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Main article: seismology

Helioseismology was born from analogy with geoseismology but several important differences remain. First, the Sun lacks a solid surface and therefore cannot support shear waves. From the data analysis perspective, global helioseismology differs from geoseismology by studying only normal modes. Local helioseismology is thus somewhat closer in spirit to geoseismology in the sense that it studies the complete wavefield.

Asteroseismology

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Main article: asteroseismology

Because the Sun is a star, helioseismology is closely related to the study of oscillations in other stars, known as asteroseismology. Helioseismology is most closely related to the study of stars whose oscillations are also driven and damped by their outer convection zones, known as solar-like oscillators, but the underlying theory is broadly the same for other classes of variable star.

The principal difference is that oscillations in distant stars cannot be resolved. Because the brighter and darker sectors of the spherical harmonic cancel out, this restricts asteroseismology almost entirely to the study of low degree modes (angular degree . This makes inversion much more difficult but upper limits can still be achieved by making more restrictive assumptions.

History

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Solar oscillations were first observed in the early 1960s[31][32] as a quasi-periodic intensity and line-of-sight velocity variation with a period of about 5 minutes. Scientists gradually realised that the oscillations might be global modes of the Sun and predicted that the modes would form clear ridges in two-dimensional power spectra.[33][34] The ridges were subsequently confirmed in observations in the mid 1970s[35][36] although individual mode frequencies were not measured until several years later.[37] At a similar time, Jørgen Christensen-Dalsgaard and Douglas Gough suggested the potential of using individual mode frequencies to infer the interior structure of the Sun.[38]

Helioseismology developed rapidly in the 1980s. A major breakthrough was made when observations from Antarctica linked the frequencies for modes of low and intermediate angular degree, from which the radial orders of the modes could be identified.[39] In addition, new methods of inversion developed, allowing researchers to infer the sound speed profile of the Sun. Towards the end of the decade, observations also began to show that the oscillation mode frequencies vary with the Sun's magnetic activity cycle.[40]

To overcome the problem of not being able to observe the Sun at night, several groups had begun to assemble networks of telescopes (e.g. the Birmingham Solar Oscillations Network[41][42] and the Global Oscillation Network Group[43]) from which the Sun would always be visible to at least one node. Long, uninterrupted observations brought the field to maturity, and the state of the field was summarized in a 1996 special issue of Science magazine.[44] This coincided with the start of normal operations of the Solar and Heliospheric Observatory (SoHO), which began producing high-quality data for helioseismology.

The subsequent years saw the resolution of the solar neutrino problem and the long observations began to allow analysis of multiple solar activity cycles.[45] The agreement between standard solar models and helioseismic inversions[46] was disrupted by new measurements of the heavy element content of the solar photosphere based on detailed three-dimensional models.[47] Though the results later shifted back towards the traditional values used in the 1990s[48], the new abundances significantly worsened the agreement between the models and helioseismic inversions.[49] The cause of the discrepancy remains unsolved and is known as the solar abundance problem.

Space-based observations by SoHO have continued and SoHO was joined in 2010 by the Solar Dynamics Observatory (SDO), which has also been monitoring the Sun continuously since its operations began. In addition, ground-based networks (notably BiSON and GONG) continue to operate, providing nearly continuous data from the ground too.

See also

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References

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  49. ^ Bahcall, J. N.; Basu, S.; Pinsonneault, M.; Serenelli, A. M. (2005), "Helioseismological Implications of Recent Solar Abundance Determinations", The Astrophysical Journal, 618 (2): 1049, arXiv:astro-ph/0407060, Bibcode:2005ApJ...618.1049B, doi:10.1086/426070, S2CID 2412268
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