Helioseismology

Helioseismology, a term coined by Douglas Gough, 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 (also coined by Gough), which are respectively the studies of the Earth or stars through their oscillations. While the Sun's oscillations were first detected in the early 1960s, it was only in the mid-1970s that it was realized 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 directly, ^[1] and local helioseismology, which studies the propagation of the component waves near the Sun's surface. ^[2]

Helioseismology has contributed to a number of scientific breakthroughs. The most notable 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] [5]} The experimental discovery of neutrino oscillations was recognized by the 2015 Nobel Prize for Physics. ^[6] Helioseismology also allowed accurate measurements of the quadrupole (and higher-order) moments of the Sun's gravitational potential, ^{[7] [8] [9]} 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 known as the tachocline ^[10] and is thought to be a key component for the solar dynamo. ^[11] Although it roughly coincides with the base of the solar convection zone — also inferred through helioseismology — it is conceptually distinct, being a boundary layer in which there is a meridional flow connected with the convection zone and driven by the interplay between baroclinicity and Maxwell stresses. ^[12]

Helioseismology benefits most from continuous monitoring of the Sun, which began first with uninterrupted observations from near the South Pole over the austral summer. ^{[13] [14]} 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 several decades.

Types of solar oscillation

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. 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.

Solar oscillation modes are interpreted as resonant vibrations of a roughly spherically symmetric self-gravitating fluid in hydrostatic equilibrium. Each mode can then be represented approximately as the product of a function of radius ${\displaystyle r}$ and a spherical harmonic ${\displaystyle Y_{l}^{m}(\theta ,\phi )}$, and consequently can be characterized by the three quantum numbers which label:

• the number of nodal shells in radius, known as the radial order${\displaystyle n}$;
• the total number of nodal circles on each spherical shell, known as the angular degree${\displaystyle \ell }$; and
• the number of those nodal circles that are longitudinal, known as the azimuthal order${\displaystyle m}$.

It can be shown that the oscillations are separated into two categories: interior oscillations and a special category of surface oscillations. More specifically, there are:

Pressure modes (p modes)

Pressure modes are in essence standing sound waves. The dominant restoring force is the pressure (rather than buoyancy), hence the name. All the solar oscillations that are used for inferences about the interior are p modes, with frequencies between about 1 and 5 millihertz and angular degrees ranging from zero (purely radial motion) to order ${\displaystyle 10^{3}}$. Broadly speaking, their energy densities vary with radius inversely proportional to the sound speed, so their resonant frequencies are determined predominantly by the outer regions of the Sun. Consequently it is difficult to infer from them the structure of the solar core.

A propagation diagram for a standard solar model 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.

Gravity modes (g modes)

Gravity modes are confined to convectively stable regions, either the radiative interior or the atmosphere. The restoring force is predominantly buoyancy, and thus indirectly gravity, from which they take their name. They are evanescent in the convection zone, and therefore interior modes have tiny amplitudes at the surface and are extremely difficult to detect and identify. ^[17] It has long been recognized that measurement of even just a few g modes could substantially increase our knowledge of the deep interior of the Sun. ^[18] However, no individual g mode has yet been unambiguously measured, although indirect detections have been both claimed ^{[19] [20]} and challenged. ^{[21] [22]} Additionally, there can be similar gravity modes confined to the convectively stable atmosphere.

For more, see Phoebus group.

Surface gravity modes (f modes)

Surface gravity waves are analogous to waves in deep water, having the property that the Lagrangian pressure perturbation is essentially zero. They are of high degree ${\displaystyle \ell }$, penetrating a characteristic distance ${\displaystyle R/\ell }$, where ${\displaystyle R}$ is the solar radius. To good approximation, they obey the so-called deep-water-wave dispersion law: ${\displaystyle \omega ^{2}=gk_{\rm {h}}}$, irrespective of the stratification of the Sun, where ${\displaystyle \omega }$ is the angular frequency, ${\displaystyle g}$ is the surface gravity and ${\displaystyle k_{\rm {h}}=\ell /R}$ is the horizontal wavenumber, ^[23] and tend asymptotically to that relation as ${\displaystyle k_{\rm {h}}\rightarrow \infty }$.

What seismology can reveal

The oscillations that have been successfully utilized for seismology are essentially adiabatic. Their dynamics is therefore the action of pressure forces ${\displaystyle p}$ (plus putative Maxwell stresses) against matter with inertia density ${\displaystyle \rho }$, which itself depends upon the relation between them under adiabatic change, usually quantified via the (first) adiabatic exponent ${\displaystyle \gamma _{1}}$. The equilibrium values of the variables ${\displaystyle p}$ and ${\displaystyle \rho }$ (together with the dynamically small angular velocity ${\displaystyle \Omega }$ and magnetic field ${\displaystyle {\rm {B}}}$) are related by the constraint of hydrostatic support, which depends upon the total mass ${\displaystyle M}$ and radius ${\displaystyle R}$ of the Sun. Evidently, the oscillation frequencies ${\displaystyle \omega }$ depend only on the seismic variables ${\displaystyle \rho (p,\Omega ,{\rm {B)}}}$, ${\displaystyle \gamma _{1}}$, ${\displaystyle \Omega }$ and ${\displaystyle {\rm {B}}}$, or any independent set of functions of them. Consequently it is only about these variables that information can be derived directly. The square of the adiabatic sound speed, ${\displaystyle c^{2}=\gamma _{1}p/\rho }$, is such commonly adopted function, because that is the quantity upon which acoustic propagation principally depends. ^[24] Properties of other, non-seismic, quantities, such as helium abundance, ^[25] ${\displaystyle Y}$, or main-sequence age ^[26] ${\displaystyle t_{\odot }}$, can be inferred only by supplementation with additional assumptions, which renders the outcome more uncertain.

Data analysis

Global helioseismology

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 (${\displaystyle 0\leq \ell <300}$) solar oscillations, computed for 144 days of data from the MDI instrument aboard SOHO. 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 the raw seismic data 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. Spatially resolved data are usually projected onto desired spherical harmonics to obtain time series which are then Fourier transformed. Helioseismologists typically combine the resulting one-dimensional power spectra into a two-dimensional 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, so the latter is preferred for helioseismic observatories.

Local helioseismology

Local helioseismology—a term coined by Charles Lindsey, Doug Braun and Stuart Jefferies in 1993 ^[28] —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. ^[29]
• Ring-diagram analysis, first introduced by Frank Hill, ^[30] 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. ^[31]
• Time-distance helioseismology ^[32] aims to measure and interpret the travel times of solar waves between any two locations on the solar surface. Inhomogeneities 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. ^[33]
• Helioseismic holography, introduced in detail by Charles Lindsey and Doug Braun for the purpose of far-side (magnetic) imaging, ^[34] 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. ^[35]
• 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 ^[36] demonstrated the ability of the technique to recover near-surface flows the f modes.

Inversion

Introduction

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 between its mode frequencies and those of the Sun, if small, are weighted averages of the differences between the Sun's structure and that of the reference model. The frequency differences can then be used to infer those structural differences. The weighting functions of these averages are known as kernels.

Structure

The first inversions of the Sun's structure were made using Duvall's law ^[37] and later using Duvall's law linearized about a reference solar model. ^[38] These results were subsequently supplemented by analyses that linearize the full set of equations describing the stellar oscillations about a theoretical reference model ^{[18] [39] [40]} and are now a standard way to invert frequency data. ^{[41] [42]} 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). ^{[43] [44]}

Rotation

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 angular velocity through 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. ^[45] For example, if the Sun were rotating uniformly throughout, all the p modes would be split by approximately the same amount. Actually, the angular velocity is not uniform, as can be seen at the surface, where the equator rotates faster than the poles. ^[46] The Sun rotates slowly enough that a spherical, non-rotating model is close enough to reality for deriving the rotational kernels.

Helioseismology has shown that the Sun has a rotation profile with several features: ^[47]

• 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

Geoseismology

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

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 ${\displaystyle \ell \leq 3}$). This makes inversion much more difficult but upper limits can still be achieved by making more restrictive assumptions.

History

Solar oscillations were first observed in the early 1960s ^{[48] [49]} as a quasi-periodic intensity and line-of-sight velocity variation with a period of about 5 minutes. Scientists gradually realized that the oscillations might be global modes of the Sun and predicted that the modes would form clear ridges in two-dimensional power spectra. ^{[50] [51]} The ridges were subsequently confirmed in observations of high-degree modes in the mid 1970s, ^{[52] [53]} and mode multiplets of different radial orders were distinguished in whole-disc observations. ^{[13] [54]} 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. ^[55] They calibrated solar models against the low-degree data ^[56] finding two similarly good fits, one with low ${\displaystyle Y}$ and a corresponding low neutrino production rate ${\displaystyle L_{\nu }}$, the other with higher ${\displaystyle Y}$ and ${\displaystyle L_{\nu }}$; earlier envelope calibrations against high-degree frequencies ^{[57] [58]} preferred the latter, but the results were not wholly convincing. It was not until Tom Duvall and Jack Harvey ^[14] connected the two extreme data sets by measuring modes of intermediate degree to establish the quantum numbers associated with the earlier observations that the higher-${\displaystyle Y}$ model was established, thereby suggesting at that early stage that the resolution of the neutrino problem must lie in nuclear or particle physics.

New methods of inversion developed in the 1980s, allowing researchers to infer the profiles sound speed and, less accurately, density throughout most of the Sun, corroborating the conclusion that residual errors in the inference of the solar structure is not the cause of the neutrino problem. Towards the end of the decade, observations also began to show that the oscillation mode frequencies vary with the Sun's magnetic activity cycle. ^[59]

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, or BiSON, ^{[60] [61]} and the Global Oscillation Network Group ^[62] ) 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 . ^[63] 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 seismic observations began to allow analysis of multiple solar activity cycles. ^[64] The agreement between standard solar models and helioseismic inversions ^[65] was disrupted by new measurements of the heavy element content of the solar photosphere based on detailed three-dimensional models. ^[66] Though the results later shifted back towards the traditional values used in the 1990s, ^[67] the new abundances significantly worsened the agreement between the models and helioseismic inversions. ^[68] The cause of the discrepancy remains unsolved ^[24] 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

• 160-minute solar cycle
• Coronal seismology
• Differential rotation
• Diskoseismology
• Frequency separation
• Magnetogravity wave
• Moreton wave
• Solar neutrino problem
• Solar tower (astronomy)
• Stellar rotation

References

1. Gough, D.O.; Kosovichev, A.G.; Toomre, J.; et al. (1996), "The seismic structure of the Sun", Science, 272 (5266): 1296–1300, Bibcode:1996Sci...272.1296G, doi:10.1126/science.272.5266.1296, PMID   8662458, S2CID   15996636
2. Gizon, L.; Birch, A. C. (2005), "Local Helioseismology", Living Reviews in Solar Physics, 2 (1): 6, Bibcode:2005LRSP....2....6G, doi: 10.12942/lrsp-2005-6
3. Fukuda, Y.; Super-Kamiokande Collaboration (1998), "Evidence for oscillation of atmospheric neutrinos", Phys. Rev. Lett., 81 (8): 1562–1567, arXiv: hep-ex/9807003 , Bibcode:1998PhRvL..81.1562F, doi: 10.1103/PhysRevLett.81.1562
4. Bahcall, J. N.; Concha, Gonzalez-Garcia M.; Pe, na-Garay C. (2001), "Global analysis of solar neutrino oscillations including SNO CC measurement", Journal of High Energy Physics, 2001 (8): 014, arXiv: hep-ph/0106258 , Bibcode:2001JHEP...08..014B, doi:10.1088/1126-6708/2001/08/014, S2CID   6595480
5. Bahcall, J. N. (2001), "High-energy physics: Neutrinos reveal split personalities", Nature, 412 (6842): 29–31, Bibcode:2001Natur.412...29B, doi:10.1038/35083665, PMID   11452285, S2CID   205018839
6. Webb, Jonathan (6 October 2015). "Neutrino 'flip' wins physics Nobel Prize". BBC News.
7. Duvall, T.L. Jr; Dziembowski, W.A.; Goode, P.R.; Gough, D.O.; Harvey, J.W.; Leibacher, J.W. (1984), "The internal rotation of the Sun", Nature, 310 (5972): 22–25, Bibcode:1984Natur.310...22D, doi:10.1038/310022a0, S2CID   4310140
8. Pijpers, F.P. (1998), "Helioseismic determination of the solar gravitational quadrupole moment", Mon. Not. R. Astron. Soc., 297 (3): L76–L80, arXiv: astro-ph/9804258 , Bibcode:1998MNRAS.297L..76P, doi:10.1046/j.1365-8711.1998.01801.x, S2CID   14179539
9. Antia, H.M.; Chitre, S.M.; Gough, D.O. (2008), "Temporal variations in the Sun's rotational kinetic energy", Astron. Astrophys., 477 (2): 657–663, arXiv: 0711.0799 , Bibcode:2008A&A...477..657A, doi: 10.1051/0004-6361:20078209
10. Spiegel, E. A.; Zahn, J.-P. (1992), "The solar tachocline", Astronomy and Astrophysics, 265: 106, Bibcode:1992A&A...265..106S
11. Fan, Y. (2009), "Magnetic Fields in the Solar Convection Zone", Living Reviews in Solar Physics, 6 (1): 4, Bibcode:2009LRSP....6....4F, doi: 10.12942/lrsp-2009-4
12. Gough, D.O.; McIntyre, M.E. (1998), "Inevitability of a magnetic field in the Sun's interior", Nature, 394 (6695): 755, Bibcode:1998Natur.394..755G, doi:10.1038/29472, S2CID   1365619
13. Grec, G.; Fossat, E.; Pomerantz, M. (1980), "Solar oscillations: full disk observations from the geographic South Pole", Nature, 288 (5791): 541–544, Bibcode:1980Natur.288..541G, doi:10.1038/288541a0, S2CID   4345313
14. Duvall, Jr. T. L.; Harvey, J. W. (1983), "Observations of solar oscillations of low and intermediate degree", Nature, 302 (5903): 24, Bibcode:1983Natur.302...24D, doi:10.1038/302024a0, S2CID   4274994
15. Christensen-Dalsgaard, J.; Dappen, W.; Ajukov, S. V.; Anderson, E. R.; Antia, H. M.; Basu, S.; Baturin, V. A.; Berthomieu, G.; Chaboyer, B.; Chitre, S. M.; Cox, A. N.; Demarque, P.; Donatowicz, J.; Dziembowski, W. A.; Gabriel, M.; Gough, D. O.; Guenther, D. B.; Guzik, J. A.; Harvey, J. W.; Hill, F.; Houdek, G.; Iglesias, C. A.; Kosovichev, A. G.; Leibacher, J. W.; Morel, P.; Proffitt, C. R.; Provost, J.; Reiter, J.; Rhodes, Jr. E. J.; Rogers, F. J.; Roxburgh, I. W.; Thompson, M. J.; Ulrich, R. K. (1996), "The Current State of Solar Modeling", Science, 272 (5266): 1286–92, Bibcode:1996Sci...272.1286C, doi:10.1126/science.272.5266.1286, PMID   8662456, S2CID   35469049
16. Christensen-Dalsgaard, J.; Dappen, W.; Ajukov, S. V. and (1996), "The Current State of Solar Modeling", Science, 272 (5266): 1286–1292, Bibcode:1996Sci...272.1286C, doi:10.1126/science.272.5266.1286, PMID   8662456, S2CID   35469049
17. Appourchaux, T.; Belkacem, K.; Broomhall, A.-M.; Chaplin, W. J.; Gough, D. O.; Houdek, G.; Provost, J.; Baudin, F.; Boumier, P.; Elsworth, Y.; Garc\'\ia, R. A.; Andersen, B. N.; Finsterle, W.; Fr\ohlich, C.; Gabriel, A.; Grec, G.; Jiménez, A.; Kosovichev, A.; Sekii, T.; Toutain, T.; Turck-Chi\`eze, S. (2010), "The quest for the solar g modes", Astronomy and Astrophysics Review, 18 (1–2): 197, arXiv: 0910.0848 , Bibcode:2010A&ARv..18..197A, doi:10.1007/s00159-009-0027-z, S2CID   119272874
18. Gough, D.O. (1984), "Solar inverse theory", Solar Seismology from Space (Ed. R.K. Ulrich, JPL Publ., Pasadena), 84–84: 49–78, Bibcode:1984sses.nasa...49G
19. Garc\'\ia, R. A.; Turck-Chi\`eze, S.; Jiménez-Reyes, S. J.; Ballot, J.; Pallé, P. L.; Eff-Darwich, A.; Mathur, S.; Provost, J. (2007), "Tracking Solar Gravity Modes: The Dynamics of the Solar Core", Science, 316 (5831): 1591–3, Bibcode:2007Sci...316.1591G, doi:10.1126/science.1140598, PMID   17478682, S2CID   35285705
20. Fossat, E.; Boumier, P.; Corbard, T.; Provost, J.; Salabert, D.; Schmider, F. X.; Gabriel, A. H.; Grec, G.; Renaud, C.; Robillot, J. M.; Roca-Cortés, T.; Turck-Chi\`eze, S.; Ulrich, R. K.; Lazrek, M. (2017), "Asymptotic g modes: Evidence for a rapid rotation of the solar core", Astronomy and Astrophysics, 604: A40, arXiv: 1708.00259 , Bibcode:2017A&A...604A..40F, doi:10.1051/0004-6361/201730460, S2CID   53498421
21. Schunker, H.; Schou, J.; Gaulme, P.; Gizon, L. (2018), "Fragile Detection of Solar g-Modes by Fossat et al.", Solar Physics, 293 (6): 95, arXiv: 1804.04407 , Bibcode:2018SoPh..293...95S, doi: 10.1007/s11207-018-1313-6
22. Scherrer, P. H.; Gough, D. O. (2019), "A critical evaluation of recent claims concerning solar rotation", Astrophysical Journal, 877 (1): 42–53, arXiv: 1904.02820 , Bibcode:2019ApJ...877...42S, doi: 10.3847/1538-4357/ab13ad , S2CID   102351083
23. Gough, D.O. (1982), "A review of the theory of solar oscillations and its implications concerning the internal structure of the Sun", In Pulsations in Classical and Cataclysmic Variable Stars (Ed. J.P. Cox & C.J. Hansen, JILA, Boulder): 117–137, Bibcode:1982pccv.conf..117G
24. Gough, D.O. (2003), "What have we learned from helioseismology, what have we really learned, and what do we aspire to learn?", Solar Physics, 287 (1–2): 9–41, arXiv: 1210.0820 , doi:10.1007/s11207-012-0099-1, S2CID   119291920
25. Kosovichev, A.G.; Christensen-Dalsgaard, J.; Däeppen, W.; Dziembowski, W.A.; Gough, D.O.; Thompson, M.J. (1992), "Sources of uncertainty in direct seismological measurements of the solar helium abundance" (PDF), Mon. Not. R. Astron. Soc., 259 (3): 536–558, Bibcode:1992MNRAS.259..536K, doi: 10.1093/mnras/259.3.536
26. Houdek, G.; Gough, D.O. (2011), "On the seismic age and heavy-element abundance of the Sun", Mon. Not. R. Astron. Soc., 418 (2): 1217–1230, arXiv: 1108.0802 , Bibcode:2011MNRAS.418.1217H, doi: 10.1111/j.1365-2966.2011.19572.x
27. Rhodes, Jr. E. J.; Kosovichev, A. G.; Schou, J.; et al. (1997), "Measurements of Frequencies of Solar Oscillations from the MDI Medium-l Program", Solar Physics, 175 (2): 287, Bibcode:1997SoPh..175..287R, doi:10.1023/A:1004963425123, S2CID   51790986
28. Lindsey, C.; Braun, D.C.; Jefferies, S.M. (January 1993). T.M. Brown (ed.). "Local Helioseismology of Subsurface Structure" in "GONG 1992. Seismic Investigation of the Sun and Stars". Astronomical Society of the Pacific Conference Series. Vol. 42. pp. 81–84. Bibcode:1993ASPC...42...81L. ISBN   978-0-937707-61-6.{{cite book}}: |journal= ignored (help)
29. Braun, D.C.; Duvall Jr., T.L.; Labonte, B.J. (August 1987). "Acoustic absorption by sunspots". The Astrophysical Journal. 319: L27–L31. Bibcode:1987ApJ...319L..27B. doi:10.1086/184949.
30. Hill, F. (October 1988). "Rings and trumpets - Three-dimensional power spectra of solar oscillations". Astrophysical Journal. 333: 996–1013. Bibcode:1988ApJ...333..996H. doi:10.1086/166807.
31. Basu, S.; Antia, H.M.; Bogart, R.S. (August 2004). "Ring-Diagram Analysis of the Structure of Solar Active Regions". The Astrophysical Journal. 610 (2): 1157–1168. Bibcode:2004ApJ...610.1157B. doi: 10.1086/421843 .
32. Duvall Jr., T.L.; Jefferies, S.M.; Harvey, J.W.; Pomerantz, M.A. (April 1993). "Time-distance helioseismology". Nature. 362 (6419): 430–432. Bibcode:1993Natur.362..430D. doi:10.1038/362430a0. hdl: 2060/20110005678 . S2CID   4244835.
33. Jensen, J. M. (2003), "Time-distance: what does it tell us?", Gong+ 2002. Local and Global Helioseismology: The Present and Future, 517: 61, Bibcode:2003ESASP.517...61J
34. Braun, D. C.; Lindsey, C. (2001), "Seismic Imaging of the Far Hemisphere of the Sun", Astrophysical Journal Letters, 560 (2): L189, Bibcode:2001ApJ...560L.189B, doi: 10.1086/324323
35. Donea, A.-C.; Braun, D.C.; Lindsey, C. (March 1999). "Seismic Images of a Solar Flare". The Astrophysical Journal. 513 (2): L143–L146. Bibcode:1999ApJ...513L.143D. doi: 10.1086/311915 .
36. Woodard, M. F. (January 2002). "Solar Subsurface Flow Inferred Directly from Frequency-Wavenumber Correlations in the Seismic Velocity Field". The Astrophysical Journal. 565 (1): 634–639. Bibcode:2002ApJ...565..634W. CiteSeerX   10.1.1.513.1704 . doi:10.1086/324546. S2CID   122970114.
37. Christensen-Dalsgaard, J.; Duvall, Jr. T. L.; Gough, D. O.; Harvey, J. W.; Rhodes, Jr. E. J. (1985), "Speed of sound in the solar interior", Nature, 315 (6018): 378, Bibcode:1985Natur.315..378C, doi:10.1038/315378a0, S2CID   4338576
38. Christensen-Dalsgaard, J.; Thompson, M. J.; Gough, D. O. (1989), "Differential asymptotic sound-speed inversions", MNRAS, 238 (2): 481–502, Bibcode:1989MNRAS.238..481C, doi: 10.1093/mnras/238.2.481
39. Dziembowski, W.A.; Pamyatnykh, A.A.; Sienkiewicz, R. (1990), "Solar model from helioseismology and the neutrino flux problem", Mon. Not. R. Astron. Soc., 244: 542–550, Bibcode:1990MNRAS.244..542D
40. Antia, H. M.; Basu, S. (1994), "Nonasymptotic helioseismic inversion for solar structure.", Astronomy & Astrophysics Supplement Series, 107: 421, Bibcode:1994A&AS..107..421A
41. Gough, D.O.; Thompson, M.J. (1991), "The inversion problem", in A. N. Cox; W. C. Livingston; M. S. Matthews (eds.), Solar interior and atmosphere, Tucson: University of Arizona Press, pp. 519–561, Bibcode:1991sia..book..519G
42. Basu, S. (2016), "Global seismology of the Sun", Living Reviews in Solar Physics, 13 (1): 2, arXiv: 1606.07071 , Bibcode:2016LRSP...13....2B, doi:10.1007/s41116-016-0003-4, S2CID   118486913
43. Cox, A. N.; Guzik, J. A.; Kidman, R. B. (1989), "Oscillations of solar models with internal element diffusion", Astrophysical Journal, 342: 1187, Bibcode:1989ApJ...342.1187C, doi:10.1086/167675, S2CID   122535514
44. Christensen-Dalsgaard, J.; Proffitt, C. R.; Thompson, M. J. (1993), "Effects of diffusion on solar models and their oscillation frequencies" (PDF), Astrophysical Journal Letters, 403: L75, Bibcode:1993ApJ...403L..75C, doi:10.1086/186725
45. Thompson, M. J.; Christensen-Dalsgaard, J.; Miesch, M. S.; Toomre, J. (2003), "The Internal Rotation of the Sun", Annual Review of Astronomy & Astrophysics, 41: 599–643, Bibcode:2003ARA&A..41..599T, doi:10.1146/annurev.astro.41.011802.094848, S2CID   123622875
46. Beck, J. G. (2000), "A comparison of differential rotation measurements - (Invited Review)", Solar Physics, 191 (1): 47–70, Bibcode:2000SoPh..191...47B, doi:10.1023/A:1005226402796, S2CID   118030329
47. Howe, R. (2009), "Solar Interior Rotation and its Variation", Living Reviews in Solar Physics, 6 (1): 1, arXiv: 0902.2406 , Bibcode:2009LRSP....6....1H, doi:10.12942/lrsp-2009-1, S2CID   10532243
48. Leighton, R. B.; Noyes, R. W.; Simon, G. W. (1962), "Velocity Fields in the Solar Atmosphere. I. Preliminary Report.", Astrophysical Journal, 135: 474, Bibcode:1962ApJ...135..474L, doi:10.1086/147285
49. Evans, J. W.; Michard, R. (1962), "Observational Study of Macroscopic Inhomogeneities in the Solar Atmosphere. III. Vertical Oscillatory Motions in the Solar Photosphere.", Astrophysical Journal, 136: 493, Bibcode:1962ApJ...136..493E, doi: 10.1086/147403
50. Leibacher, J. W.; Stein, R. F. (1971), "A New Description of the Solar Five-Minute Oscillation", Astrophysical Letters, 7: 191, Bibcode:1971ApL.....7..191L
51. Ulrich, R. K. (1970), "The Five-Minute Oscillations on the Solar Surface", Astrophysical Journal, 162: 993, Bibcode:1970ApJ...162..993U, doi:10.1086/150731, S2CID   17225920
52. Deubner, F.-L. (1975), "Observations of low wavenumber nonradial eigenmodes of the sun", Astronomy and Astrophysics, 44 (2): 371, Bibcode:1975A&A....44..371D
53. Rhodes, Jr. E. J.; Ulrich, R. K.; Simon, G. W. (1977), "Observations of nonradial p-mode oscillations on the sun", Astrophysical Journal, 218: 901, Bibcode:1977ApJ...218..901R, doi:10.1086/155745, S2CID   115143527
54. Claverie, A.; Isaak, G. R.; McLeod, C. P.; van, der Raay H. B.; Cortes, T. R. (1979), "Solar structure from global studies of the 5-minute oscillation", Nature, 282 (5739): 591–594, Bibcode:1979Natur.282..591C, doi:10.1038/282591a0, S2CID   4342247
55. Christensen-Dalsgaard, J.; Gough, D. O. (1976), "Towards a heliological inverse problem", Nature, 259 (5539): 89, Bibcode:1976Natur.259...89C, doi:10.1038/259089a0, S2CID   10540902
56. Christensen-Dalsgaard, J.; Gough, D. O. (1981), "Comparison of the observed solar whole-disk oscillation frequencies with the predictions of a sequence of solar models", Astron. Astrophys., 104 (2): 173–176, Bibcode:1981A&A...104..173C
57. Gough, D.O. (1977), "Random remarks on solar hydrodynamics", Proc. IAU Colloq. 36: 3–36, Bibcode:1977ebhs.coll....3G
58. Rhodes, Jr. E. J.; Ulrich, R. K. (1977), "The sensitivity of nonradial p mode eigenfrequencies to solar envelope structure", Astrophysical Journal, 218: 521–529, Bibcode:1977ApJ...218..521U, doi: 10.1086/155705
59. Libbrecht, K. G.; Woodard, M. F. (1990), "Solar-cycle effects on solar oscillation frequencies", Nature, 345 (6278): 779, Bibcode:1990Natur.345..779L, doi:10.1038/345779a0, S2CID   4305062
60. Aindow, A.; Elsworth, Y. P.; Isaak, G. R.; McLeod, C. P.; New, R.; Vanderraay, H. B. (1988), "The current status of the Birmingham solar seismology network", Seismology of the Sun and Sun-Like Stars, 286: 157, Bibcode:1988ESASP.286..157A
61. Chaplin, W. J.; Elsworth, Y.; Howe, R.; Isaak, G. R.; McLeod, C. P.; Miller, B. A.; van, der Raay H. B.; Wheeler, S. J.; New, R. (1996), "BiSON Performance", Solar Physics, 168 (1): 1, Bibcode:1996SoPh..168....1C, doi:10.1007/BF00145821, S2CID   189828557
62. Harvey, J. W.; Hill, F.; Kennedy, J. R.; Leibacher, J. W.; Livingston, W. C. (1988), "The Global Oscillation Network Group (GONG)", Advances in Space Research, 8 (11): 117, Bibcode:1988AdSpR...8k.117H, doi:10.1016/0273-1177(88)90304-3 )
63. "Special Issue: GONG Helioseismology", Science, 272 (5266), 1996
64. Chaplin, W. J.; Elsworth, Y.; Miller, B. A.; Verner, G. A.; New, R. (2007), "Solar p-Mode Frequencies over Three Solar Cycles", Astrophysical Journal, 659 (2): 1749, Bibcode:2007ApJ...659.1749C, doi: 10.1086/512543
65. Bahcall, J. N.; Pinsonneault, M. H.; Basu, S. (2001), "Solar Models: Current Epoch and Time Dependences Neutrinos and Helioseismological Properties", Astrophysical Journal, 555 (2): 990–1012, arXiv: astro-ph/0010346 , Bibcode:2001ApJ...555..990B, doi:10.1086/321493, S2CID   13798091
66. Asplund, M.; Grevesse, N.; Sauval, A. J. (2005), "The Solar Chemical Composition", Cosmic Abundances as Records of Stellar Evolution and Nucleosynthesis, 336: 25, Bibcode:2005ASPC..336...25A
67. Asplund, M.; Grevesse, N.; Sauval, A. J.; Scott, P. (2009), "The Chemical Composition of the Sun", Annual Review of Astronomy & Astrophysics, 47 (1): 481–522, arXiv: 0909.0948 , Bibcode:2009ARA&A..47..481A, doi:10.1146/annurev.astro.46.060407.145222, S2CID   17921922
68. Bahcall, J. N.; Basu, S.; Pinsonneault, M.; Serenelli, A. M. (2005), "Helioseismological Implications of Recent Solar Abundance Determinations", Astrophysical Journal, 618 (2): 1049–1056, arXiv: astro-ph/0407060 , Bibcode:2005ApJ...618.1049B, doi:10.1086/426070, S2CID   2412268

External links

• Non-technical description of helio- and asteroseismology retrieved November 2009
• Gough, D.O. (2003). "Solar Neutrino Production". Annales Henri Poincaré. 4 (S1): 303–317. Bibcode:2003AnHP....4..303G. doi: 10.1007/s00023-003-0924-z . S2CID   195335212.
• Gizon, Laurent; Birch, Aaron C. (2005). "Local Helioseismology". Living Rev. Sol. Phys. 2 (1): 6. Bibcode:2005LRSP....2....6G. doi: 10.12942/lrsp-2005-6 .
• Scientists Issue Unprecedented Forecast of Next Sunspot Cycle National Science Foundation press release, March 6, 2006
• Miesch, Mark S. (2005). "Large-Scale Dynamics of the Convection Zone and Tachocline". Living Rev. Sol. Phys. 2 (1): 1. Bibcode:2005LRSP....2....1M. doi: 10.12942/lrsp-2005-1 .
• European Helio- and Asteroseismology Network (HELAS)
• Farside and Earthside images of the Sun
• Living Reviews in Solar Physics Archived 2010-09-29 at the Wayback Machine
• Helioseismology and Asteroseismology at MPS

Satellite instruments

• VIRGO
• SOI/MDI
• SDO/HMI
• TRACE

Ground-based instruments

• BiSON
• Mark-1
• GONG
• HiDHN

Further reading

• Christensen-Dalsgaard, Jørgen. "Lecture notes on stellar oscillations" . Retrieved 5 June 2015.
• Pijpers, Frank P. (2006). Methods in Helio- and Asteroseismology. London: Imperial College Press. ISBN   978-1-8609-4755-1.