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72-528: Asteroseismology is closely related to helioseismology , the study of stellar pulsation specifically in the Sun . Though both are based on the same underlying physics, more and qualitatively different information is available for the Sun because its surface can be resolved. By linearly perturbing the equations defining the mechanical equilibrium of a star (i.e. mass conservation and hydrostatic equilibrium ) and assuming that

144-517: A Cepheid by measuring its oscillation period, computing its luminosity, and comparing this to its observed brightness. Cepheid pulsations are excited by the kappa mechanism acting on the second ionization zone of helium. RR Lyraes are similar to Cepheid variables but of lower metallicity (i.e. Population II ) and much lower masses (about 0.6 to 0.8 time solar). They are core helium-burning giants that oscillate in one or both of their fundamental mode or first overtone. The oscillation are also driven by

216-416: A density stratified liquid can be observed in the 'Magic Cork' movie here . The concept derives from Newton's Second Law when applied to a fluid parcel in the presence of a background stratification (in which the density changes in the vertical - i.e. the density can be said to have multiple vertical layers). The parcel, perturbed vertically from its starting position, experiences a vertical acceleration. If

288-489: A few days, understood to be high-order gravity modes excited by the kappa mechanism. Beta Cephei variables are slightly hotter (and thus more massive), also have modes excited by the kappa mechanism and additionally oscillate in low-order gravity modes with periods of several hours. Both classes of oscillators contain only slowly rotating stars. Subdwarf B (sdB) stars are in essence the cores of core-helium burning giants who have somehow lost most of their hydrogen envelopes, to

360-415: A few g modes could substantially increase our knowledge of the deep interior of the Sun. However, no individual g mode has yet been unambiguously measured, although indirect detections have been both claimed and challenged. Additionally, there can be similar gravity modes confined to the convectively stable atmosphere. Surface gravity waves are analogous to waves in deep water, having the property that

432-536: A more general formulation used in meteorology is: Since θ = T ( P 0 / P ) R / c P {\displaystyle \theta =T(P_{0}/P)^{R/c_{P}}} , where P 0 {\displaystyle P_{0}} is a constant reference pressure, for a perfect gas this expression is equivalent to: where in the last form γ = c P / c V {\displaystyle \gamma =c_{P}/c_{V}} ,

504-668: A parcel of water or gas that has density ρ 0 {\displaystyle \rho _{0}} . This parcel is in an environment of other water or gas particles where the density of the environment is a function of height: ρ = ρ ( z ) {\displaystyle \rho =\rho (z)} . If the parcel is displaced by a small vertical increment z ′ {\displaystyle z'} , and it maintains its original density so that its volume does not change, it will be subject to an extra gravitational force against its surroundings of: where g {\displaystyle g}

576-414: 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 r {\displaystyle r} and a spherical harmonic Y l m ( θ , ϕ ) {\displaystyle Y_{l}^{m}(\theta ,\phi )} , and consequently can be characterized by

648-446: A star. By plotting the curves ω = N {\displaystyle \omega =N} and ω = S ℓ {\displaystyle \omega =S_{\ell }} (for given ℓ {\displaystyle \ell } ), we expect p-modes to resonate at frequencies below both curves or frequencies above both curves. Under fairly specific conditions, some stars have regions where heat

720-508: Is difficult to infer from them the structure of the solar core. 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. It has long been recognized that measurement of even just

792-488: Is important, or in fresh water lakes near freezing, where density is not a linear function of temperature: N ≡ − g ρ d ρ d z {\displaystyle N\equiv {\sqrt {-{g \over {\rho }}{d\rho \over {dz}}}}} where ρ {\displaystyle \rho } , the potential density , depends on both temperature and salinity. An example of Brunt–Väisälä oscillation in

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864-469: Is now known as the tachocline and is thought to be a key component for the solar dynamo . 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. Helioseismology benefits most from continuous monitoring of

936-677: Is only about these variables that information can be derived directly. The square of the adiabatic sound speed, c 2 = γ 1 p / ρ {\displaystyle c^{2}=\gamma _{1}p/\rho } , is such commonly adopted function, because that is the quantity upon which acoustic propagation principally depends. Properties of other, non-seismic, quantities, such as helium abundance, Y {\displaystyle Y} , or main-sequence age t ⊙ {\displaystyle t_{\odot }} , can be inferred only by supplementation with additional assumptions, which renders

1008-403: Is positive, then there is run away growth – i.e. the fluid is statically unstable. For a gas parcel, the density will only remain fixed as assumed in the previous derivation if the pressure, P {\displaystyle P} , is constant with height, which is not true in an atmosphere confined by gravity. Instead, the parcel will expand adiabatically as the pressure declines. Therefore

1080-720: Is the Lamb frequency. The last two are defined by N 2 = g ( 1 Γ 1 P d p d r − 1 ρ d ρ d r ) {\displaystyle N^{2}=g\left({\frac {1}{\Gamma _{1}P}}{\frac {dp}{dr}}-{\frac {1}{\rho }}{\frac {d\rho }{dr}}\right)} and S ℓ 2 = ℓ ( ℓ + 1 ) c s 2 r 2 {\displaystyle S_{\ell }^{2}={\frac {\ell (\ell +1)c_{s}^{2}}{r^{2}}}} respectively. By analogy with

1152-487: Is the gravitational acceleration, and is defined to be positive. We make a linear approximation to ρ ( z + z ′ ) − ρ ( z ) = ∂ ρ ( z ) ∂ z z ′ {\displaystyle \rho (z+z')-\rho (z)={\frac {\partial \rho (z)}{\partial z}}z'} , and move ρ 0 {\displaystyle \rho _{0}} to

1224-459: Is the radial co-ordinate in the star, ω {\displaystyle \omega } is the angular frequency of the oscillation mode, c s {\displaystyle c_{s}} is the sound speed inside the star, N {\displaystyle N} is the Brunt–Väisälä or buoyancy frequency and S ℓ {\displaystyle S_{\ell }}

1296-450: 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

1368-451: Is therefore the action of pressure forces p {\displaystyle p} (plus putative Maxwell stresses) against matter with inertia density ρ {\displaystyle \rho } , which itself depends upon the relation between them under adiabatic change, usually quantified via the (first) adiabatic exponent γ 1 {\displaystyle \gamma _{1}} . The equilibrium values of

1440-510: Is transported by radiation and the opacity is a sharply decreasing function of temperature. This opacity bump can drive oscillations through the κ {\displaystyle \kappa } -mechanism (or Eddington valve ). Suppose that, at the beginning of an oscillation cycle, the stellar envelope has contracted. By expanding and cooling slightly, the layer in the opacity bump becomes more opaque, absorbs more radiation, and heats up. This heating causes expansion, further cooling and

1512-412: The Brunt–Väisälä frequency , or buoyancy frequency , is a measure of the stability of a fluid to vertical displacements such as those caused by convection . More precisely it is the frequency at which a vertically displaced parcel will oscillate within a statically stable environment. It is named after David Brunt and Vilho Väisälä . It can be used as a measure of atmospheric stratification. Consider

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1584-606: The Kepler satellite revealed eccentric binary systems in which oscillations are excited during the closest approach. These systems are known as heartbeat stars because of the characteristic shape of the lightcurves. Because solar oscillations are driven by near-surface convection, any stellar oscillations caused similarly are known as solar-like oscillations and the stars themselves as solar-like oscillators . However, solar-like oscillations also occur in evolved stars (subgiants and red giants), which have convective envelopes, even though

1656-520: The Schwarzschild criterion for stability against convection (or the Ledoux criterion if there is compositional stratification) is equivalent to the statement that N 2 {\displaystyle N^{2}} should be positive. The Brunt–Väisälä frequency commonly appears in the thermodynamic equations for the atmosphere and in the structure of stars. In the ocean where salinity

1728-679: The adiabatic index . Using the ideal gas law , we can eliminate the temperature to express N 2 {\displaystyle N^{2}} in terms of pressure and density: This version is in fact more general than the first, as it applies when the chemical composition of the gas varies with height, and also for imperfect gases with variable adiabatic index, in which case γ ≡ γ 01 = ( ∂ ln ⁡ P / ∂ ln ⁡ ρ ) S {\displaystyle \gamma \equiv \gamma _{01}=(\partial \ln P/\partial \ln \rho )_{S}} , i.e.

1800-470: The p spectral subtype). Their dense mode spectra are understood in terms of the oblique pulsator model : the mode's frequencies are modulated by the magnetic field, which is not necessarily aligned with the star's rotation (as is the case in the Earth). The oscillation modes have frequencies around 1500 μHz and amplitudes of a few mmag. Slowly pulsating B (SPB) stars are B-type stars with oscillation periods of

1872-598: The Lagrangian pressure perturbation is essentially zero. They are of high degree ℓ {\displaystyle \ell } , penetrating a characteristic distance R / ℓ {\displaystyle R/\ell } , where R {\displaystyle R} is the solar radius. To good approximation, they obey the so-called deep-water-wave dispersion law: ω 2 = g k h {\displaystyle \omega ^{2}=gk_{\rm {h}}} , irrespective of

1944-597: The RHS: The above second-order differential equation has the following solution: where the Brunt–Väisälä frequency N {\displaystyle N} is: For negative ∂ ρ ( z ) ∂ z {\displaystyle {\frac {\partial \rho (z)}{\partial z}}} , the displacement z ′ {\displaystyle z'} has oscillating solutions (and N gives our angular frequency). If it

2016-434: 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 . The first inversions of

2088-476: The Sun's structure were made using Duvall's law and later using Duvall's law linearized about a reference solar model. These results were subsequently supplemented by analyses that linearize the full set of equations describing the stellar oscillations about a theoretical reference model and are now a standard way to invert frequency data. The inversions demonstrated differences in solar models that were greatly reduced by implementing gravitational settling :

2160-749: The Sun, which began first with uninterrupted observations from near the South Pole over the austral 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 several decades. Solar oscillation modes are interpreted as resonant vibrations of

2232-417: 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. 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

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2304-543: The Sun. Evidently, the oscillation frequencies ω {\displaystyle \omega } depend only on the seismic variables ρ ( p , Ω , B ) {\displaystyle \rho (p,\Omega ,{\rm {B)}}} , γ 1 {\displaystyle \gamma _{1}} , Ω {\displaystyle \Omega } and B {\displaystyle {\rm {B}}} , or any independent set of functions of them. Consequently it

2376-453: The acceleration is back towards the initial position, the stratification is said to be stable and the parcel oscillates vertically. In this case, N > 0 and the angular frequency of oscillation is given N . If the acceleration is away from the initial position ( N < 0 ), the stratification is unstable. In this case, overturning or convection generally ensues. The Brunt–Väisälä frequency relates to internal gravity waves : it

2448-425: The air parcel will not move any further. If the air parcel is pushed up and N 2 < 0 {\displaystyle N^{2}<0} , (i.e. the Brunt–Väisälä frequency is imaginary), then the air parcel will rise and rise unless N 2 {\displaystyle N^{2}} becomes positive or zero again further up in the atmosphere. In practice this leads to convection, and hence

2520-412: The base of a surface convection zone is sharp and the convective timescales slower than the pulsation timescales, the convective flows react too slowly to perturbations that can build up into large, coherent pulsations. This mechanism is known as convective blocking and is believed to drive pulsations in the γ {\displaystyle \gamma } Doradus variables. Observations from

2592-522: The behaviour of simple harmonic oscillators, this implies that oscillating solutions exist when the frequency is either greater or less than both S ℓ {\displaystyle S_{\ell }} and N {\displaystyle N} . We identify the former case as high-frequency pressure modes (p-modes) and the latter as low-frequency gravity modes (g-modes). This basic separation allows us to determine (to reasonable accuracy) where we expect what kind of mode to resonate in

2664-458: The component waves near the Sun's surface. 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 . The experimental discovery of neutrino oscillations

2736-419: 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. 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

2808-453: The derivative is taken at constant entropy , S {\displaystyle S} . If a gas parcel is pushed up and N 2 > 0 {\displaystyle N^{2}>0} , the air parcel will move up and down around the height where the density of the parcel matches the density of the surrounding air. If the air parcel is pushed up and N 2 = 0 {\displaystyle N^{2}=0} ,

2880-404: 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, and local helioseismology , which studies the propagation of

2952-402: The extent that there is no hydrogen-burning shell. They have multiple oscillation periods that range between about 1 and 10 minutes and amplitudes anywhere between 0.001 and 0.3 mag in visible light. The oscillations are low-order pressure modes, excited by the kappa mechanism acting on the iron opacity bump. White dwarfs are characterized by spectral type, much like ordinary stars, except that

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3024-566: The field to maturity, and the state of the field was summarized in a 1996 special issue of Science magazine . 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. The agreement between standard solar models and helioseismic inversions

3096-405: The gradual separation of heavier elements towards the solar centre (and lighter elements to the surface to replace them). 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

3168-421: The interior are p modes, with frequencies between about 1 and 5 millihertz and angular degrees ranging from zero (purely radial motion) to order 10 3 {\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

3240-588: The kappa mechanism acting through the second ionization of helium. Many RR Lyraes, including RR Lyrae itself, show long period amplitude modulations, known as the Blazhko effect . Delta Scuti variables are found roughly where the classical instability strip intersects the main sequence. They are typically A- to early F-type dwarfs and subgiants and the oscillation modes are low-order radial and non-radial pressure modes, with periods ranging from 0.25 to 8 hours and magnitude variations anywhere between. Like Cepheid variables,

3312-459: The kappa mechanism; V777 Herculis stars by convective blocking. A number of past, present and future spacecraft have asteroseismology studies as a significant part of their missions (order chronological). The Variable Star package (in R language) provides the main functions to analyzed patterns on the oscillation modes of variable stars. An UI for experimentation with synthetic data is also provided. Helioseismology Helioseismology

3384-402: The latter, but the results were not wholly convincing. It was not until Tom Duvall and Jack Harvey 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- Y {\displaystyle Y} model was established, thereby suggesting at that early stage that the resolution of

3456-473: The layer becomes even more opaque. This continues until the material opacity stops increasing so rapidly, at which point the radiation trapped in the layer can escape. The star contracts and the cycle prepares to commence again. In this sense, the opacity acts like a valve that traps heat in the star's envelope. Pulsations driven by the κ {\displaystyle \kappa } -mechanism are coherent and have relatively large amplitudes. It drives

3528-457: The low-order pressure modes. Gamma Doradus oscillations are generally thought to be high-order gravity modes, excited by convective blocking. Following results from Kepler , it appears that many Delta Scuti stars also show Gamma Doradus oscillations and are therefore hybrids. Rapidly oscillating Ap stars have similar parameters to Delta Scuti variables, mostly being A- and F-type, but they are also strongly magnetic and chemically peculiar (hence

3600-427: 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

3672-502: The oscillation mode frequencies vary with the Sun's magnetic activity cycle . 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, and the Global Oscillation Network Group ) from which the Sun would always be visible to at least one node. Long, uninterrupted observations brought

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3744-459: The oscillations are driven by the kappa mechanism acting on the second ionization of helium. SX Phoenicis variables are regarded as metal-poor relatives of Delta Scuti variables. Gamma Doradus variables occur in similar stars to the red end of the Delta Scuti variables, usually of early F-type. The stars show multiple oscillation frequencies between about 0.5 and 3 days, which is much slower than

3816-413: The oscillations might be global modes of the Sun and predicted that the modes would form clear ridges in two-dimensional power spectra. The ridges were subsequently confirmed in observations of high-degree modes in the mid 1970s, and mode multiplets of different radial orders were distinguished in whole-disc observations. At a similar time, Jørgen Christensen-Dalsgaard and Douglas Gough suggested

3888-527: The oscillations, granulation produces a stronger signal in intensity than line-of-sight velocity, so the latter is preferred for helioseismic observatories. Local helioseismology—a term coined by Charles Lindsey, Doug Braun and Stuart Jefferies in 1993 —employs several different analysis methods to make inferences from the observational data. 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

3960-471: The outcome more uncertain. 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

4032-1050: The perturbation to the gravitational potential is negligible (the Cowling approximation) and that the star's structure varies more slowly with radius than the oscillation mode, the equations can be reduced approximately to one second-order equation for the radial component of the displacement eigenfunction ξ r {\displaystyle \xi _{r}} , d 2 ξ r d r 2 = ω 2 c s 2 ( 1 − N 2 ω 2 ) ( S ℓ 2 ω 2 − 1 ) ξ r {\displaystyle {\frac {d^{2}\xi _{r}}{dr^{2}}}={\frac {\omega ^{2}}{c_{s}^{2}}}\left(1-{\frac {N^{2}}{\omega ^{2}}}\right)\left({\frac {S_{\ell }^{2}}{\omega ^{2}}}-1\right)\xi _{r}} where r {\displaystyle r}

4104-537: The perturbations are adiabatic, one can derive a system of four differential equations whose solutions give the frequency and structure of a star's modes of oscillation. The stellar structure is usually assumed to be spherically symmetric, so the horizontal (i.e. non-radial) component of the oscillations is described by spherical harmonics , indexed by an angular degree ℓ {\displaystyle \ell } and azimuthal order m {\displaystyle m} . In non-rotating stars, modes with

4176-422: The poles. 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: 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

4248-593: The potential of using individual mode frequencies to infer the interior structure of the Sun. They calibrated solar models against the low-degree data finding two similarly good fits, one with low Y {\displaystyle Y} and a corresponding low neutrino production rate L ν {\displaystyle L_{\nu }} , the other with higher Y {\displaystyle Y} and L ν {\displaystyle L_{\nu }} ; earlier envelope calibrations against high-degree frequencies preferred

4320-489: The pulsations in many of the longest-known variable stars, including the Cepheid and RR Lyrae variables . In stars with surface convection zones, turbulent fluids motions near the surface simultaneously excite and damp oscillations across a broad range of frequency. Because the modes are intrinsically stable, they have low amplitudes and are relatively short-lived. This is the driving mechanism in all solar-like oscillators. If

4392-759: The relationship between spectral type and effective temperature does not correspond in the same way. Thus, white dwarfs are known by types DO, DA and DB. Cooler types are physically possible but the Universe is too young for them to have cooled enough. White dwarfs of all three types are found to pulsate. The pulsators are known as GW Virginis stars (DO variables, sometimes also known as PG 1159 stars), V777 Herculis stars (DB variables) and ZZ Ceti stars (DA variables). All pulsate in low-degree, high-order g-modes. The oscillation periods broadly decrease with effective temperature, ranging from about 30 min down to about 1 minute. GW Virginis and ZZ Ceti stars are thought to be excited by

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4464-423: 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

4536-512: The same angular degree must all have the same frequency because there is no preferred axis. The angular degree indicates the number of nodal lines on the stellar surface, so for large values of ℓ {\displaystyle \ell } , the opposing sectors roughly cancel out, making it difficult to detect light variations. As a consequence, modes can only be detected up to an angular degree of about 3 in intensity and about 4 if observed in radial velocity. By additionally assuming that

4608-403: The stars are not Sun-like . Cepheid variables are one of the most important classes of pulsating star. They are core-helium burning stars with masses above about 5 solar masses. They principally oscillate at their fundamental modes, with typical periods ranging from days to months. Their pulsation periods are closely related to their luminosities, so it is possible to determine the distance to

4680-617: The stratification of the Sun, where ω {\displaystyle \omega } is the angular frequency, g {\displaystyle g} is the surface gravity and k h = ℓ / R {\displaystyle k_{\rm {h}}=\ell /R} is the horizontal wavenumber, and tend asymptotically to that relation as k h → ∞ {\displaystyle k_{\rm {h}}\rightarrow \infty } . The oscillations that have been successfully utilized for seismology are essentially adiabatic. Their dynamics

4752-449: The study of low degree modes (angular degree ℓ ≤ 3 {\displaystyle \ell \leq 3} ). This makes inversion much more difficult but upper limits can still be achieved by making more restrictive assumptions. Solar oscillations were first observed in the early 1960s as a quasi-periodic intensity and line-of-sight velocity variation with a period of about 5 minutes. Scientists gradually realized that

4824-429: 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

4896-417: The three quantum numbers which label: 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 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

4968-511: The variables p {\displaystyle p} and ρ {\displaystyle \rho } (together with the dynamically small angular velocity Ω {\displaystyle \Omega } and magnetic field B {\displaystyle {\rm {B}}} ) are related by the constraint of hydrostatic support, which depends upon the total mass M {\displaystyle M} and radius R {\displaystyle R} of

5040-481: Was disrupted by new measurements of the heavy element content of the solar photosphere based on detailed three-dimensional models. Though the results later shifted back towards the traditional values used in the 1990s, the new abundances significantly worsened the agreement between the models and helioseismic inversions. The cause of the discrepancy remains unsolved and is known as the solar abundance problem . Space-based observations by SoHO have continued and SoHO

5112-483: 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. Brunt%E2%80%93V%C3%A4is%C3%A4l%C3%A4 frequency In atmospheric dynamics , oceanography , asteroseismology and geophysics ,

5184-428: Was recognized by the 2015 Nobel Prize for Physics . 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

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