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At least 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2016

STOCHASTIC TRANSIENTS AS A SOURCE OF QUASI-PERIODIC PROCESSES IN THE SOLAR ATMOSPHERE

Solar dynamics and turbulence occur at all heights of the solar atmosphere and could be described as stochastic processes. We propose that finite-lifetime transients recurring at a certain place could trigger quasi-periodic processes in the associated structures. In this study, we developed a mathematical model for finite-lifetime and randomly occurring transients, and found that quasi-periodic processes with periods longer than the timescale of the transients, are detectable intrinsically in the form of trains. We simulate their propagation in an empirical solar atmospheric model with chromosphere, transition region, and corona. We found that, due to the filtering effect of the chromospheric cavity, only the resonance period of the acoustic resonator is able to propagate to the upper atmosphere; such a scenario is applicable to slow magnetoacoustic waves in sunspots and active regions. If the thermal structure of the atmosphere is less wild and acoustic resonance does not take place, the long-period oscillations could propagate to the upper atmosphere. Such a case would be more likely to occur in polar plumes.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2013

ON THE COMBINATION OF IMAGING-POLARIMETRY WITH SPECTROPOLARIMETRY OF UPPER SOLAR ATMOSPHERES DURING SOLAR ECLIPSES

We present results from imaging polarimetry (IP) of upper solar atmospheres during a total solar eclipse on 2012 November 13 and spectropolarimetry of an annular solar eclipse on 2010 January 15. This combination of techniques provides both the synoptic spatial distribution of polarization above the solar limb and spectral information on the physical mechanism producing the polarization. Using these techniques together we demonstrate that even in the transition region, the linear polarization increases with height and can exceed 20%. IP shows a relatively smooth background distribution in terms of the amplitude and direction modified by solar structures above the limb. A map of a new quantity that reflects direction departure from the background polarization supplies an effective technique to improve the contrast of this fine structure. Spectral polarimetry shows that the relative contribution to the integrated polarization over the observed passband from the spectral lines decreases with height while the contribution from the continuum increases as a general trend. We conclude that both imaging and spectral polarimetry obtained simultaneously over matched spatial and spectral domains will be fruitful for future eclipse observations.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2013

VELOCITY-SHEAR-INDUCED MODE COUPLING IN THE SOLAR ATMOSPHERE AND SOLAR WIND: IMPLICATIONS FOR PLASMA HEATING AND MHD TURBULENCE

We analytically consider how velocity shear in the corona and solar wind can cause an initial Alfven wave to drive up other propagating signals. The process is similar to the familiar coupling into other modes induced by non-WKB refraction in an inhomogeneous plasma, except here the refraction is a consequence of velocity shear. We limit our discussion to a low-beta plasma, and ignore couplings into signals resembling the slow mode. If the initial Alfven wave is propagating nearly parallel to the background magnetic field, then the induced signals are mainly a forward-going (i.e., propagating in the same sense as the original Alfven wave) fast mode, and a driven signal propagating like a forward-going Alfven wave but polarized like the fast mode; both signals are compressive and subject to damping by the Landau resonance. For an initial Alfven wave propagating obliquely with respect to the magnetic field, the induced signals are mainly forward- and backward-going fast modes, and a driven signal propagating like a forward-going Alfven wave but polarized like the fast mode; these signals are all compressive and subject to damping by the Landau resonance. A backward-going Alfven wave, thought to be important in the development of MHD turbulence, is also produced, but it is very weak. However, we suggest that for oblique propagation of the initial Alfven wave the induced fast-polarized signal propagating like a forward-going Alfven wave may interact coherently with the initial Alfven wave and distort it at a strong-turbulence-like rate.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2017

Solar Energetic Particle Acceleration by a Shock Wave Accompanying a Coronal Mass Ejection in the Solar Atmosphere

Solar energetic particle acceleration by a shock wave accompanying a coronal mass ejection (CME) is studied. The description of the accelerated particle spectrum evolution is based on the numerical calculation of the diffusive transport equation with a set of realistic parameters. The relation between the CME and shock speeds, which depend on the initial CME radius, is determined. Depending on the initial CME radius, its speed, and the magnetic energy of the scattering Alfvén waves, the accelerated particle spectrum is established 10–60 minutes from the beginning of CME motion. The maximum energies of particles reach 0.1–10 GeV. The CME radii of 3–5 R{sub ⊙} and the shock radii of 5–10 R{sub ⊙} agree with observations. The calculated particle spectra agree with the observed ones in events registered by ground-based detectors if the turbulence spectrum in the solar corona significantly differs from the Kolmogorov one.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2011

Chromospheric anemone jets and magnetic reconnection in partially ionized solar atmosphere

The solar optical telescope onboard Hinode with temporal resolution of less than 5 s and spatial resolution of 150 km has observed the lower solar atmosphere with an unprecedented detail. This has led to many important findings, one of them is the discovery of chromospheric anemone jets in the solar chromosphere. The chromospheric anemone jets are ubiquitous in solar chromosphere and statistical studies show that the typical length, life time and energy of the chromospheric anemone jets are much smaller than the coronal events (e.g., jets/flares/CMEs). Among various observational parameters, the apparent length and maximum velocity shows good correlation. The velocity of chromospheric anemone jets is comparable to the local Alfven speed in the lower solar chromosphere. Since the discovery of chromospheric anemone jets by Hinode, several evidences of magnetic reconnection in chromospheric anemone jets have been found and these observations are summarized in this paper. These observations clearly suggest that reconnection occurs quite rapidly as well as intermittently in the solar chromosphere. In the solar corona ({lambda}{sub i} > {delta}{sub SP}), anomalous resistivity arises due to various collisionless processes. Previous MHD simulations show that reconnection becomes fast as well as strongly time-dependent due to anomalous resistivity. Such processes would not arise in the solar chromosphere which is fully collisional and partially-ionized. So, it is unclear how the rapid and strongly time-dependent reconnection would occur in the solar chromosphere. It is quite likely that the Hall and ambipolar diffusion are present in the solar chromosphere and they could play an important role in driving such rapid, strongly time-dependent reconnection in the solar chromosphere.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2017

Solar atmospheric neutrinos and the sensitivity floor for solar dark matter annihilation searches

Cosmic rays interacting in the solar atmosphere produce showers that result in a flux of high-energy neutrinos from the Sun. These form an irreducible background to indirect solar WIMP self-annihilation searches, which look for heavy dark matter particles annihilating into final states containing neutrinos in the Solar core. This background will eventually create a sensitivity floor for indirect WIMP self-annihilation searches analogous to that imposed by low-energy solar neutrino interactions for direct dark matter detection experiments. We present a new calculation of the flux of solar atmospheric neutrinos with a detailed treatment of systematic uncertainties inherent in solar atmospheric shower evolution, and we use this to derive the sensitivity floor for indirect solar WIMP annihilation analyses. We find that the floor lies less than one order of magnitude beyond the present experimental limits on spin-dependent WIMP-proton cross sections for some mass points, and that the high-energy solar atmospheric neutrino flux may be observable with running and future neutrino telescopes.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2011

RAPID FLUCTUATIONS IN THE LOWER SOLAR ATMOSPHERE

The Rapid Oscillations in the Solar Atmosphere instrument reveals solar atmospheric fluctuations at high frequencies. Spectra of variations of the G-band intensity (I{sub G} ) and Ca II K-line intensity (I{sub K} ) show correlated fluctuations above white noise to frequencies beyond 300 mHz and 50 mHz, respectively. The noise-corrected G-band spectrum for f = 28-326 mHz shows a power law with exponent -1.21 {+-} 0.02, consistent with the presence of turbulent motions. G-band spectral power in the 25-100 mHz ('UHF') range is concentrated at the locations of magnetic bright points in the intergranular lanes and is highly intermittent in time. The intermittence of the UHF G-band fluctuations, shown by a positive kurtosis {kappa}, also suggests turbulence. Combining values of I{sub G} , I{sub K} , UHF power, and {kappa} reveals two distinct states of the solar atmosphere. State 1, including almost all the data, is characterized by low I{sub G} , I{sub K} , and UHF power and {kappa} Almost-Equal-To 6. State 2, including only a very small fraction of the data, is characterized by high I{sub G} , I{sub K} , and UHF power and {kappa} Almost-Equal-To 3. Superposed epoch analysis shows that the UHF power peaks simultaneously with spatio-temporal I{sub G} maxima in either state. For State 1, I{sub K} shows 3.5 minute chromospheric oscillations with maxima occurring 21 s after I{sub G} maxima implying a 150-210 km effective height difference. However, for State 2 the I{sub K} and I{sub G} maxima are simultaneous; in this highly magnetized environment sites of G-band and K-line emission may be spatially close together.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2011

WHAT DO SPECTRAL LINE PROFILE ASYMMETRIES TELL US ABOUT THE SOLAR ATMOSPHERE?

Recently, analysis of solar spectra obtained with the EUV Imaging Spectrograph (EIS) onboard the Hinode satellite has revealed the ubiquitous presence of asymmetries in transition region (TR) and coronal spectral line profiles. These asymmetries have been observed especially at the footpoints of coronal loops and have been associated with strong upflows that may play a significant role in providing the corona with hot plasma. Here, we perform a detailed study of the various processes that can lead to spectral line asymmetries, using both simple forward models and state-of-the-art three-dimensional radiative MHD simulations of the solar atmosphere using the Bifrost code. We describe a novel technique to determine the presence and properties of faint secondary components in the wings of spectral line profiles. This method is based on least-squares fitting of observed so-called R(ed)B(lue) asymmetry profiles with pre-calculated RB asymmetry profiles for a wide variety of secondary component properties. We illustrate how this method could be used to perform reliable double Gaussian fits that are not over- or under-constrained. We also find that spectral line asymmetries appear in TR and coronal lines that are synthesized from our three-dimensional MHD simulations. Our models show that the spectral asymmetries are a sensitive measure of the velocity gradient with height in the TR of coronal loops. The modeled TR shows a large gradient of velocity that increases with height: this occurs as a consequence of ubiquitous, episodic heating at low heights in the model atmosphere. We show that the contribution function of spectral lines as a function of temperature is critical for sensitivity to velocity gradients and thus line asymmetries: lines that are formed over a temperature range that includes most of the TR are the most sensitive. As a result, lines from lithium-like ions (e.g., O VI) are found to be the most sensitive to line asymmetries. We compare the simulated line profiles directly with line profiles observed in the quiet Sun with SOHO/SUMER and Hinode/EIS and find that the shape of the profiles is very similar. In addition, the simulated profiles with the strongest blueward asymmetry occur in footpoint regions of coronal loops, which is similar to what we observe with SUMER and EIS. There is however a significant discrepancy between the simulations and observations: the simulated RB asymmetries are an order of magnitude smaller than the observations. We discuss the possible reasons for this discrepancy. In summary, our analysis shows that observations of spectral line asymmetries can provide a powerful new diagnostic to help constrain coronal heating models.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2012

PROPAGATING WAVE PHENOMENA DETECTED IN OBSERVATIONS AND SIMULATIONS OF THE LOWER SOLAR ATMOSPHERE

We present high-cadence observations and simulations of the solar photosphere, obtained using the Rapid Oscillations in the Solar Atmosphere imaging system and the MuRAM magnetohydrodynamic (MHD) code, respectively. Each data set demonstrates a wealth of magnetoacoustic oscillatory behavior, visible as periodic intensity fluctuations with periods in the range 110-600 s. Almost no propagating waves with periods less than 140 s and 110 s are detected in the observational and simulated data sets, respectively. High concentrations of power are found in highly magnetized regions, such as magnetic bright points and intergranular lanes. Radiative diagnostics of the photospheric simulations replicate our observational results, confirming that the current breed of MHD simulations are able to accurately represent the lower solar atmosphere. All observed oscillations are generated as a result of naturally occurring magnetoconvective processes, with no specific input driver present. Using contribution functions extracted from our numerical simulations, we estimate minimum G-band and 4170 A continuum formation heights of 100 km and 25 km, respectively. Detected magnetoacoustic oscillations exhibit a dominant phase delay of -8 Degree-Sign between the G-band and 4170 A continuum observations, suggesting the presence of upwardly propagating waves. More than 73% of MBPs (73% from observations and 96% from simulations) display upwardly propagating wave phenomena, suggesting the abundant nature of oscillatory behavior detected higher in the solar atmosphere may be traced back to magnetoconvective processes occurring in the upper layers of the Sun's convection zone.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2012

HOW MUCH DOES A MAGNETIC FLUX TUBE EMERGE INTO THE SOLAR ATMOSPHERE?

The emergence process of the magnetic field into the solar atmosphere plays an essential role in determining the configuration of the magnetic field and its activity on the Sun. This paper focuses on how much the magnetic flux contained by a flux tube emerges into the solar atmosphere, which is the key to understanding the physical mechanism of solar eruptions. By comparing a kinematic model of an emerging flux tube to a series of magnetohydrodynamic simulations, we derive the characteristics of the emergence process, showing how the process depends on the pre-emerged state of the magnetic field such as the radius of a flux tube, field strength, field-line twist, and wavelength of undulation assumed by the flux tube. We also discuss the relationship between magnetic configurations and their stability on the Sun.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2014

Statistical equilibrium of copper in the solar atmosphere

Non-local thermodynamic equilibrium (NLTE) line formation for neutral copper in the one-dimensional solar atmospheres is presented for the atomic model, including 96 terms of Cu I and the ground state of Cu II. The accurate oscillator strengths for all the line transitions in model atom and photoionization cross sections were calculated using the R-matrix method in the Russell-Saunders coupling scheme. The main NLTE mechanism for Cu I is the ultraviolet overionization. We find that NLTE leads to systematically depleted total absorption in the Cu I lines and, accordingly, positive abundance corrections. Inelastic collisions with neutral hydrogen atoms produce minor effects on the statistical equilibrium of Cu I in the solar atmosphere. For the solar Cu I lines, the departures from LTE are found to be small, the mean NLTE abundance correction of ∼0.01 dex. It was found that the six low-excitation lines, with excitation energy of the lower level E {sub exc} ≤ 1.64 eV, give a 0.14 dex lower mean solar abundance compared to that from the six E {sub exc} > 3.7 eV lines, when applying experimental gf-values of Kock and Richter. Without the two strong resonance transitions, the solar mean NLTE abundance from 10 lines of Cu I is log ε{sub ☉}(Cu) = 4.19 ± 0.10, which is consistent within the error bars with the meteoritic value 4.25 ± 0.05 of Lodders et al. The discrepancy between E {sub exc} = 1.39-1.64 eV and E {sub exc} > 3.7 eV lines can be removed when the calculated gf-values are adopted and a mean solar abundance of log ε{sub ☉}(Cu) = 4.24 ± 0.08 is derived.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2016

HEATING MECHANISMS IN THE LOW SOLAR ATMOSPHERE THROUGH MAGNETIC RECONNECTION IN CURRENT SHEETS

We simulate several magnetic reconnection processes in the low solar chromosphere/photosphere; the radiation cooling, heat conduction and ambipolar diffusion are all included. Our numerical results indicate that both the high temperature (≳8 × 10{sup 4} K) and low temperature (∼10{sup 4} K) magnetic reconnection events can happen in the low solar atmosphere (100–600 km above the solar surface). The plasma β controlled by plasma density and magnetic fields is one important factor to decide how much the plasma can be heated up. The low temperature event is formed in a high β magnetic reconnection process, Joule heating is the main mechanism to heat plasma and the maximum temperature increase is only several thousand Kelvin. The high temperature explosions can be generated in a low β magnetic reconnection process, slow and fast-mode shocks attached at the edges of the well developed plasmoids are the main physical mechanisms to heat the plasma from several thousand Kelvin to over 8 × 10{sup 4} K. Gravity in the low chromosphere can strongly hinder the plasmoid instability and the formation of slow-mode shocks in a vertical current sheet. Only small secondary islands are formed; these islands, however, are not as well developed as those in the horizontal current sheets. This work can be applied to understand the heating mechanism in the low solar atmosphere and could possibly be extended to explain the formation of common low temperature Ellerman bombs (∼10{sup 4} K) and the high temperature Interface Region Imaging Spectrograph (IRIS) bombs (≳8 × 10{sup 4}) in the future.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2015

STABILITY OF ROTATING MAGNETIZED JETS IN THE SOLAR ATMOSPHERE. I. KELVIN–HELMHOLTZ INSTABILITY

Observations show various jets in the solar atmosphere with significant rotational motions, which may undergo instabilities leading to heat ambient plasma. We study the Kelvin–Helmholtz instability (KHI) of twisted and rotating jets caused by the velocity jumps near the jet surface. We derive a dispersion equation with appropriate boundary conditions for total pressure (including centrifugal force of tube rotation), which governs the dynamics of incompressible jets. Then, we obtain analytical instability criteria of KHI in various cases, which were verified by numerical solutions to the dispersion equation. We find that twisted and rotating jets are unstable to KHI when the kinetic energy of rotation is more than the magnetic energy of the twist. Our analysis shows that the azimuthal magnetic field of 1–5 G can stabilize observed rotations in spicule/macrospicules and X-ray/extreme-ultraviolet (EUV) jets. On the other hand, nontwisted jets are always unstable to KHI. In this case, the instability growth time is several seconds for spicule/macrospicules and a few minutes (or less) for EUV/X-ray jets. We also find that standing kink and torsional Alfvén waves are always unstable near the antinodes, owing to the jump of azimuthal velocity at the surface, while the propagating waves are generally stable. Kelvin–Helmholtz (KH) vortices may lead to enhanced turbulence development and heating of surrounding plasma; therefore, rotating jets may provide energy for chromospheric and coronal heating.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2011

THERMAL FINE STRUCTURE AND MAGNETIC FIELDS IN THE SOLAR ATMOSPHERE: SPICULES AND FIBRILS

The relationship between observed structures in the solar atmosphere and the magnetic fields threading them is known only for the solar photosphere, even then imprecisely. We suggest that some of the fine structures in the more tenuous chromosphere and corona-specifically some populations of spicules and fibrils-correspond to warps in two-dimensional sheet-like structures, as an alternative to conventional interpretations in terms of tube-like structures. The sheets are perhaps related to magnetic tangential discontinuities, which Parker has argued arise naturally in low-{beta} conditions. Some consequences of this suggestion, if it can be confirmed, are discussed.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2014

The generation and damping of propagating MHD kink waves in the solar atmosphere

The source of the non-thermal energy required for the heating of the upper solar atmosphere to temperatures in excess of a million degrees and the acceleration of the solar wind to hundreds of kilometers per second is still unclear. One such mechanism for providing the required energy flux is incompressible torsional Alfvén and kink magnetohydrodynamic (MHD) waves, which are magnetically dominated waves supported by the Sun's pervasive and complex magnetic field. In particular, propagating MHD kink waves have recently been observed to be ubiquitous throughout the solar atmosphere, but, until now, critical details of the transport of the kink wave energy throughout the Sun's atmosphere were lacking. Here, the ubiquity of the waves is exploited for statistical studies in the highly dynamic solar chromosphere. This large-scale investigation allows for the determination of the chromospheric kink wave velocity power spectra, a missing link necessary for determining the energy transport between the photosphere and corona. Crucially, the power spectra contain evidence for horizontal photospheric motions being an important mechanism for kink wave generation in the quiescent Sun. In addition, a comparison with measured coronal power spectra is provided for the first time, revealing frequency-dependent transmission profiles, suggesting that there is enhanced damping of kink waves in the lower corona.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2014

The transverse and rotational motions of magnetohydrodynamic kink waves in the solar atmosphere

Magnetohydrodynamic (MHD) kink waves have now been observed to be ubiquitous throughout the solar atmosphere. With modern instruments, they have now been detected in the chromosphere, interface region, and corona. The key purpose of this paper is to show that kink waves do not only involve purely transverse motions of solar magnetic flux tubes, but the velocity field is a spatially and temporally varying sum of both transverse and rotational motion. Taking this fact into account is particularly important for the accurate interpretation of varying Doppler velocity profiles across oscillating structures such as spicules. It has now been shown that, as well as bulk transverse motions, spicules have omnipresent rotational motions. Here we emphasize that caution should be used before interpreting the particular MHD wave mode/s responsible for these rotational motions. The rotational motions are not necessarily signatures of the classic axisymmetric torsional Alfvén wave alone, because kink motion itself can also contribute substantially to varying Doppler velocity profiles observed across these structures. In this paper, the displacement field of the kink wave is demonstrated to be a sum of its transverse and rotational components, both for a flux tube with a discontinuous density profile at its boundary, and one with a more realistic density continuum between the internal and external plasma. Furthermore, the Doppler velocity profile of the kink wave is forward modeled to demonstrate that, depending on the line of sight, it can either be quite distinct or very similar to that expected from a torsional Alfvén wave.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2013

THE FORMATION OF IRIS DIAGNOSTICS. II. THE FORMATION OF THE Mg II h and k LINES IN THE SOLAR ATMOSPHERE

NASA's Interface Region Imaging Spectrograph (IRIS) small explorer mission will study how the solar atmosphere is energized. IRIS contains an imaging spectrograph that covers the Mg II h and k lines as well as a slit-jaw imager centered at Mg II k. Understanding the observations requires forward modeling of Mg II h and k line formation from three-dimensional (3D) radiation-magnetohydrodynamic (RMHD) models. This paper is the second in a series where we undertake this modeling. We compute the vertically emergent h and k intensity from a snapshot of a dynamic 3D RMHD model of the solar atmosphere, and investigate which diagnostic information about the atmosphere is contained in the synthetic line profiles. We find that the Doppler shift of the central line depression correlates strongly with the vertical velocity at optical depth unity, which is typically located less than 200 km below the transition region (TR). By combining the Doppler shifts of the h and k lines we can retrieve the sign of the velocity gradient just below the TR. The intensity in the central line depression is anti-correlated with the formation height, especially in subfields of a few square Mm. This intensity could thus be used to measure the spatial variation of the height of the TR. The intensity in the line-core emission peaks correlates with the temperature at its formation height, especially for strong emission peaks. The peaks can thus be exploited as a temperature diagnostic. The wavelength difference between the blue and red peaks provides a diagnostic of the velocity gradients in the upper chromosphere. The intensity ratio of the blue and red peaks correlates strongly with the average velocity in the upper chromosphere. We conclude that the Mg II h and k lines are excellent probes of the very upper chromosphere just below the TR, a height regime that is impossible to probe with other spectral lines. They also provide decent temperature and velocity diagnostics of the middle chromosphere.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2011

PROBING THE SOLAR ATMOSPHERE USING OSCILLATIONS OF INFRARED CO SPECTRAL LINES

Oscillations were observed across the whole solar disk using the Doppler shift and line center intensity of spectral lines from the CO molecule near 4666 nm with the National Solar Observatory's McMath/Pierce solar telescope. Power, coherence, and phase spectra were examined, and diagnostic diagrams reveal power ridges at the solar global mode frequencies to show that these oscillations are solar p-modes. The phase was used to determine the height of formation of the CO lines by comparison with the IR continuum intensity phase shifts as measured in Kopp et al.; we find that the CO line formation height varies from 425 km < z < 560 km as we move from disk center toward the solar limb 1.0 > {mu} > 0.5. The velocity power spectra show that while the sum of the background and p-mode power increases with height in the solar atmosphere as seen in previous work, the power in the p-modes only (background subtracted) decreases with height. The CO line center intensity weakens in regions of stronger magnetic fields, as does the p-mode oscillation power. Across most of the solar surface the phase shift is larger than the expected value of 90{sup 0} for an adiabatic atmosphere. We fit the phase spectra at different disk positions with a simple atmospheric model to determine that the acoustic cutoff frequency is about 4.5 mHz with only small variations, but that the thermal relaxation frequency drops significantly from 2.7 to 0 mHz at these heights in the solar atmosphere.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗