Helioseismology

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 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 term was coined by Douglas Gough in the 90s. 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 that the anomaly in 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.

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.

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.

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 discrepancy, known as the solar abundance problem, was restored with a 2022 reanalysis which corrected the amount of heavy elements back to the old value.^[69]

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.

• 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 PhysicsArchived 2010-09-29 at the Wayback Machine
• Helioseismology and Asteroseismology at MPS

• Christensen-Dalsgaard, Jørgen. "Lecture notes on stellar oscillations". Archived from the original on 1 July 2018. Retrieved 5 June 2015.
• Pijpers, Frank P. (2006). Methods in Helio- and Asteroseismology. London: Imperial College Press. ISBN 978-1-8609-4755-1.