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At least 145 records · Page 8

Cryogenic Tank Modeling for the Saturn AS-203 Experiment

A computational fluid dynamics (CFD) model is developed for the Saturn S-IVB liquid hydrogen (LH2) tank to simulate the 1966 AS-203 flight experiment. This significant experiment is the only known, adequately-instrumented, low-gravity, cryogenic self pressurization test that is well suited for CFD model validation. A 4000-cell, axisymmetric model predicts motion of the LH2 surface including boil-off and thermal stratification in the liquid and gas phases. The model is based on a modified version of the commercially available FLOW3D software. During the experiment, heat enters the LH2 tank through the tank forward dome, side wall, aft dome, and common bulkhead. In both model and test the liquid and gases thermally stratify in the low-gravity natural convection environment. LH2 boils at the free surface which in turn increases the pressure within the tank during the 5360 second experiment. The Saturn S-IVB tank model is shown to accurately simulate the self pressurization and thermal stratification in the 1966 AS-203 test. The average predicted pressurization rate is within 4% of the pressure rise rate suggested by test data. Ullage temperature results are also in good agreement with the test where the model predicts an ullage temperature rise rate within 6% of the measured data. The model is based on first principles only and includes no adjustments to bring the predictions closer to the test data. Although quantitative model validation is achieved or one specific case, a significant step is taken towards demonstrating general use of CFD for low-gravity cryogenic fluid modeling.

Grayson, Gary D.↗

Identifying Meteorologic and Oceanic Conditions Contributing to a Unique Phytoplankton Bloom Occurrence in Micronesia during October 2013

On the first several days of October 2013, daily chlorophyll a (chl a) data acquired by the Moderate Resolution Imaging Spectroradiometer (MODIS) onboard the National Aeronautics and Space Administration (NASA) Aqua satellite detected a marked increase in chlorophyll a concentrations (chl a) in the vicinity of Chuuk Lagoon and the Federated States of Micronesia. Such an increase, likely indicative of a phytoplankton bloom, has not been observed in this location at any other time during the MODIS-Aqua mission, which commenced in 2002 and continues to present. Examination of sea surface wind data from the Modern Era Retrospective analysis for Research and Applications Version 2 (MERRA-2) reanalysis dataset indicated that the region experienced a sequence of alternating elevated and very low wind speed events prior to the observation of the bloom. The influence of the winds can be seen in MERRA-2 sea surface skin temperature data. Elevated windspeeds for several days likely induced a mixing of deeper waters with higher nutrient levels to the surface, which was followed by stratification and phytoplankton growth during low wind intervals and finally transport induced by a brief high windspeed event. Analysis of hourly MERRA-2 maximum windspeed data over a 40-year period indicated that this sequence was climatologically rare.

Remote Sensing↗

Prediction of the ullage gas thermal stratification in a NASP vehicle propellant tank experimental simulation using FLOW-3D

As part of the National Aero-Space Plane (NASP) project, the multi-dimensional effects of gravitational force, initial tank pressure, initial ullage temperature, and heat transfer rate on the 2-D temperature profiles were studied. FLOW-3D, a commercial finite difference fluid flow model, was used for the evaluation. These effects were examined on the basis of previous liquid hydrogen experimental data with gaseous hydrogen pressurant. FLOW-3D results were compared against an existing 1-D model. In addition, the effects of mesh size and convergence criteria on the analytical results were investigated. Suggestions for future modifications and uses of FLOW-3D for modeling of a NASP tank are also presented.

Hardy, Terry L.↗

Zeno: Critical Fluid Light Scattering Experiment

The Zeno (Critical Fluid Light Scattering) experiment is the culmination of a long history of critical fluid light scattering in liquid-vapor systems. The major limitation to making accurate measurements closer to the critical point was the density stratification which occurs in these extremely compressible fluids. Zeno was to determine the critical density fluctuation decay rates at a pair of supplementary angles in the temperature range 100 mK to 100 (mu)K from T(sub c) in a sample of xenon accurately loaded to the critical density. This paper gives some highlights from operating the instrument on two flights March, 1994 on STS-62 and February, 1996 on STS-75. More detail of the experiment Science Requirements, the personnel, apparatus, and results are displayed on the Web homepage at http://www.zeno.umd.edu.

Gammon, Robert W.↗

An Examination of Issues Related to a Europa Subsurface Component for the JIMO Mission

The Galileo Europa data set served to revolutionize our view of Europa. In particular the strong evidence of a large, cold, salty Ocean beneath 5-30 km of ice has profoundly altered the significance of Europa in our thinking, especially of context of habitability in the solar system. While much remains to be learned from spacecraft observations of several sorts, there are significant questions answerable only by in-situ techniques; these relate to the formation of Europa, the nature of its ocean, and the prospects for life in its ocean, sediments, and ice. We feel that wide-ranging discussion of an in-situ subsurface mission to Europa, as part of JIMO, should proceed. The science objective of the mission is to characterize the icy shell of Europa to resolve its provenance, estimate the composition of brine of the Europa ocean, and search for evidence of Earth-like life. Probably anyone would agree that an in-situ mission to Europa would be of great value, but he or she would also immediately take the position that such a mission is utterly impractical. We take the position here of defining the least complex mission that can nonetheless justify its cost and to argue that such a mission is realistic enough that it should be seriously considered. Our mission thinking has been: 1) Soft landing. A soft lander is required on a site sufficiently flat to offer a stable platform; no further site selectivity is required. 2) Subsurface exploration. The Europa subsurface must be examined. Surficial processes on Europa arguably have exposed the upper 200 m of shell to chemical effects from the Jovian radiation belts as well as cometary infall, etc; to examine native ice we must descend below that point to, for discussion, 300 m. At that depth we argue that the ice is characteristic of ice at depth and possibly is effectively sea ice. 3) Science data. A few simple measurements at various depths and at 300 m constitute a scientifically successful mission. Measurements would include analysis of meltwater for a few inorganic ions and amino acids and an optical examination of the borehole wall. 4) Communication. Transmission of data to an orbiter is essential, but we will constrain the landed mission to a daily communication over a few days. 5) Subsurface access. Drilling to 300 m is a significant challenge; it can be addressed by several means: Thermal Probe (Cryobot) which permits water to refreeze above the vehicle. This is our tentative choice with plutonium as the fuel to generate thermal energy for drilling and electrical power for operations. Open Hole Drill, a thermal system in which the meltwater is removed for greater thermal efficiency. Meltwater removal on Europa is both a complexity and a risk, but analysis is improved. Mechanical Drilling in which cutting or grinding generates ice chips which are removed. This is too complex at Europa temperatures. The measurement objectives for the mission will be: Obj. 1: Determine the concentration of simple inorganic salts in the Europa Ice Shell and, by extrapolation, of the ocean. These data will also validate spaceborne sensors. Obj. 2: Determine the nature and abundance of amino acids in the ice such that cometary infall material in the upper ice can be compared to material at depth. Obj. 3: Optically examine the ice to resolve inclusion structure, particulate content, and stratification. Access to 300 m depth is a significant if not audacious plan; we are aware that this has not been done on any planetary body. Our approach is the use of a plutonium heat source; to overcome Europa's surface temperature and to melt ice a significant amount of plutonium is needed, and significant shielding and other protective steps will be required. The quantity of plutonium is a key concern. The mission will require subsurface collection and processing of samples for in situ analysis, calling for a miniature, high pressure micro-sampling system designed to meet needs of instruments that require low presses for operation. The inlet system itself collects a micro-sample in the external high pressure environment, then transfers it into a protected low pressure environment for analysis.

Carsey, F. D.↗

Apollo cryogenic integrated systems program

The integrated systems program is capable of simulating both nominal and anomalous operation of the Apollo cryogenics storage system (CSS). Two versions of the program exist; one for the Apollo 14 configuration and the other for J Type Mission configurations. The program consists of two mathematical models which are dynamically coupled. A model of the CSS components and lines determines the oxygen and hydrogen flowrate from each storage tank given the tank pressures and temperatures, and the electrical power subsystem and environmental control subsystem flow demands. Temperatures and pressures throughout the components and lines are also determined. A model of the CSS tankage determines the pressure and temperatures in the tanks given the flowrate from each tank and the thermal environment. The model accounts for tank stretch and includes simplified oxygen tank heater and stratification routines. The program is currently operational on the Univac 1108 computer.

Seto, R. K. M.↗

Secondary Sonic Boom Ray Path Prediction using sBOOM

Sonic boom ray path equations are solved numerically to show the extent of secondary booms compared to primary sonic boom carpet. Secondary booms are shown to reach altitudes of ~125 km before turning toward the ground and impinging at lateral distances of ~50 km and beyond. Exact closed-form solutions of the ray paths and their partial derivatives with respect to certain relevant parameters are derived for the simplified case of windless atmospheres and compared against numerical solutions. Ray tube areas are computed using the Jacobian of a single ray as opposed to the traditional technique of using 4 rays that are temporally and azimuthally perturbed. The ray tube areas corresponding to secondary booms can get 1000 times larger than those from primary booms. The numerically computed ray tube areas are verified using exact expressions for the special case of windless atmospheres. Finally, pressure scaling due to ray tube area changes and atmospheric stratification is presented and the shortcomings associated with the current atmospheric interpolation techniques are discussed. It is determined that piece-wise linear interpolation of the atmospheric temperature may be inadequate and could lead to numerical noise in the prediction of secondary booms.

Sriram K Rallabhandi↗

IUE observations of a hot DAO white dwarf: Implications for diffusion theory and photospheric stratification

Observations of the DAO white dwarf PG1210+533, including the first high dispersion spectrum of a hybrid H-He object of this nature were obtained by IUE. In contrast with hot DAs in the 50,000 K temperature range, PG1210+533 shows no narrow interstellar-like metal lines, in spite of an optically observed He/H abundance of 0.1. This lack of metal makes accretion from the ISM an unlikely source for the He in the PG1210+533 photosphere. A significant discovery in the high dispersion spectrum is the existence of a sharp, non-LTE like, core seen in the He II 1640 line. Such features are detected in DO white dwarfs. A small aperture SWP low dispersion observation reveals the Lyman alpha profile of PG1210+533 to be surprisingly weak and narrow. Fits of this profile using pure H models yielded a T(eff) = 56,000 K. Fits of the Balmer H gamma profile however, yield T(eff) = 42,300 K and log g = 8.5 + or - 0.5 for the same models. It is unlikely that homogeneously mixed H-He atmospheres can resolve the inconsistency between the Lyman alpha and H gamma features in this star. Stratified models involving thin H photospheres may be necessary to explain these results.

Holberg, J. B.↗

Tri-Gas Pressurization System Testing and Modeling for Cryogenic Applications

The use of Tri‐gas in rocket propulsion systems is somewhat of a new technology. This paper defines Tri‐gas as a mixture of gases composed largely of helium with a small percentage of a stoichiometric mixture of hydrogen and oxygen. When exposed to a catalyst the hydrogen and oxygen in the mixture combusts, significantly raising the temperature of the mixture. The increase in enthalpy resulting from the combustion process significantly decreases the required quantity of gas needed to pressurize the ullage of the vehicle propellant tanks. The objective of this effort was to better understand the operating characteristics of Tri‐gas in a pressurization system with low temperature applications. In conjunction with ongoing programs at NASA Marshall Space Flight Center, an effort has been undertaken to evaluate the operating characteristics of Tri‐gas through modeling and bench testing. Through improved understanding of the operating characteristics, the risk of using this new technology in a launch vehicle propulsion system was reduced. Bench testing of Tri‐gas was a multistep process that targeted gas characteristics and performance aspects that pose a risk to application in a pressurization system. Pressurization systems are vital to propulsion system performance. Keeping a target ullage pressure in propulsions tanks is necessary to supply propellant at the conditions and flow rates required to maintain desired engine functionality. The first component of testing consisted of sampling Tri‐gas sources that had been stagnant for various lengths of time in order to determine the rate at which stratification takes place. Second, a bench test was set up in which Tri‐gas was sent through a catalyst bed. This test was designed to evaluate the performance characteristics of Tri‐gas, under low temperature inlet temperatures, in a flight‐like catalyst bed reactor. The third, most complex, test examined the performance characteristics of Tri‐gas at low temperature temperatures in a test configuration built to more closely resemble a vehicle pressurization system. The results of these bench tests address various risks that all relate to underperformance of Tri‐gas pressurization systems.

Taylor, B.↗

Optical Spectroscopy and X-Ray Observations of the D-Type Symbiotic Star EF Aql

We performed high-resolution optical spectroscopy and X-ray observations of the recently identified Mira-type symbiotic star EF Aql. Based on high-resolution optical spectroscopy obtained with SALT, we determine the temperature (~55 000 K) and the luminosity (~5.3 L⨀) of the hot component in the system. The heliocentric radial velocities of the emission lines in the spectra reveal possible stratification of the chemical elements. We also estimate the mass-loss rate of the Mira donor star. Our Swift observation did not detect EF Aql in X-rays. The upper limit of the X-ray observations is 10(exp -12) erg cm(exp -2) s(exp -1), which means that EF Aql is consistent with the faintest X-ray systems detected so far. Otherwise, we detected it with the UVOT instrument with an average UVM2 magnitude of 14.05. During the exposure, EF Aql became approximately 0.2 UVM2 magnitudes fainter. The periodogram analysis of the V-band data reveals an improved period of 320.4±0.3 d caused by the pulsations of the Mira-type donor star. The spectra are available upon request from the authors.

Symbiotic binary stars↗

Surface heating and patchiness in the coastal ocean off central California during a wind relaxation event

The difference between the temperature of the ocean at 4-cm and 2-m depth was continuously monitored during a cruise to the coastal transition zone off Point Arena, California, during June 1987. The two temperatures were coincident most of the time but diverged during one nearshore leg of the cruise where large temperature differences of up to 4.7 C were observed between the 4-cm and 2-m sensors, in areas which were separated by regions where the two temperatures were coincident as usual. The spatial scale of this 'patchy' thermal structure was about 5-10 km. A mixed layer model (Garwood, 1977) was used to simulate the near surface stratification when forced by the observed wind stress, surface heating, and optical clarity of the water. The model produced a thin strongly stratified surface layer at stations where exceptionally high turbidity was observed but did not produce such features otherwise. This simple model could not explain the horizontal patchiness in the thermal structure, which was likely due to patchiness in the near-surface chlorophyll distributions or to submesoscale variability of the surface wind stress.

Ramp, Steven R.↗

Inviscid Interactions Between Wake Vortices and Shear Layers

Aircraft trailing vortices can be influenced significantly by atmospheric conditions such as crosswind, turbulence, and stratification. According to the NASA 1994 and 1995 field measurement program in Memphis, Tennessee, the descending aircraft wake vortices could stall or be deflected at the top of low-level temperature inversions that usually produce pronounced shear zones. Numerical simulations of vortex/shear interactions with ground effects have been performed by several groups. Burnham used a series of evenly spaced line vortices at a particular altitude to model the ground shear layer of the cross- wind. He found that the wind shear was swept up around the downwind vortex and caused the downwind vortex to move upward, and claimed that the effect was actually produced by the vertical gradient in the wind shear rather than by the wind shear directly, because uniformly distributed wind-shear vortices would have no effect on the trailing vortex vertical motion. Recently, Proctor et al. numerically tested the effects of narrow shear zones on the behavior of the vortex pair, motivated by the observation of the Memphis field data. The shear-layer sensitivity tests indicated that the downwind vortex was more sensitive and deflected to a higher altitude than its upwind counterpart. The downstream vortex contained vorticity of opposite sign to that of the shear. There was no detectable preference for the downwind vortex (or upwind vortex) to weaken (or strengthen) at a greater rate.

Zheng, Z. C.↗

Cryogenic Flow Boiling in Microgravity: Effects of Reduced Gravity on Two-Phase Fluid Physics and Heat Transfer

With the growing interest in space exploration, cryogenic technologies involving two-phase flow and heat transfer are in high demand to successfully procure advanced space applications such as fuel depots and nuclear thermal propulsion (NTP) systems for deep space missions. However, the unique and extreme thermal properties of cryogenic fluids introduce distinct flow boiling fluid physics and energy transport phenomena, which differ significantly from those observed with conventional fluids. Understanding the unique two-phase physics in cryogenic flow boiling remains an ongoing challenge. Furthermore, the lack of readily available microgravity cryogenic steady-state heat transfer data hinders the assessment of gravitational effects on cryogenic flow boiling. This study aims to elucidate the gravitational effects on two-phase fluid physics and heat transfer by conducting the first-ever experimental measurement of cryogenic flow boiling performance using a steady-state heated method in a reduced gravity environment. Parabolic flight experiments were performed to acquire both heat transfer measurements and high-speed video of interfacial behaviors, under varying gravity levels (microgravity, hypergravity, Lunar gravity, and Martian gravity). The experiments involved flow boiling of liquid nitrogen (LN 2 ) with a near-saturated inlet along a circular heated tube of dimensions 8.5-mm inner diameter and 680-mm heated length. The operating parameters varied are mass velocity of 398.3 - 1342.8 kg/m2s, inlet quality of -0.08 to -0.01, and inlet pressure of 413.68 - 689.48 kPa. Captured microgravity flow patterns range from bubbly to annular, all having vapor structures that are larger than those under higher gravity levels. Under microgravity, absence of buoyancy yields symmetrical vapor structures without flow stratification, laying a physical foundation for the distinct two-phase heat transfer trends during LN 2 flow boiling in microgravity. Transient data collected during the flight parabolas exhibited decreasing heated wall temperature as the aircraft transitioned from hypergravity to microgravity phases. The temperature variation indicated an enhancement in flow boiling heat transfer with decreasing gravity levels and a reduction with increasing gravity levels. The effect of reduced gravity on cryogenic flow boiling heat transfer coefficient (HTC) is discussed based on steady state heat transfer analysis. Seminal HTC correlations are evaluated against the measured microgravity HTC data, of which one is identified for superior accuracy in predicting microgravity data. Finally, a new HTC correlation is proposed to improve accuracy of microgravity predictions, yet there still exists room for further improvement with future terrestrial flow boiling experiments at different flow orientations relative to Earth gravity.

Microgravity↗

Magnetohydrodynamic thermal instabilities in cool inhomogeneous atmospheres

The stability of magnetic loops to current-driven filamentation instabilities is investigated. The unperturbed atmosphere is assumed to be composed of an (upper) isothermal optically thin low-density portion and a (lower) higher-density portion which is in radiative equilibrium; in both cases, the atmosphere is in hydrostatic equilibrium, so that gravitational stratification is taken into account. In order to provide specific equilibrium conditions for evaluation of the dispersion relation, conditions appropriate for the surface of a solar-type star are adopted; i.e., a fairly low temperature (T = 5000 K) appropriate for a 'precoronal' state associated, for example, with magnetic flux emerging from photospheric levels under the action of magnetic buoyancy. A linear stability analysis is performed, and numerical results show that physically plausible current densities, which would be generated by typical loop-footpoint motions, are effective in driving MHD instabilities in such a plasma. The instability growth rates are strongly dependent on the assumed current density distribution and on the density scale height.

Bodo, G.↗

Planetary-scale circulations in the presence of climatological and wave-induced heating

Interaction between the large-scale circulation and the convective pattern is investigated in a coupled system governed by the linearized primitive equations. Convection is represented in terms of two components of heating: A 'climatological component' is prescribed stochastically to represent convection that is maintained by fixed distributions of land and sea and sea surface temperature (SST). An 'induced component' is defined in terms of the column-integrated moisture flux convergence to represent convection that is produced through feedback with the circulation. Each component describes the envelope organizing mesoscale convective activity. As SST on the equator is increased, induced heating amplifies in the gravest zonal wavenumbers at eastward frequencies, where positive feedback offsets dissipation. Under barotropic stratification, a critical SST of 29.5 C results in positive feedback exactly cancelling dissipation in wavenumber 1 for an eastward phase speed of 6 m/s. Sympathetic interaction between the circulation and the induced heating is the basis for 'frictional wave-Conditional Instability of the Second Kind (CISK)', which is distinguished from classical wave-CISK by rendering the gravest zonal dimensions most unstable. Under baroclinic stratification, the coupled system exhibits similar behavior. The critical SST is only 26.5 C for conditions representative of equinox, but in excess of 30 C for conditions representative of solstice. Having the form of an unsteady Walker circulation, the disturbance produced by frictional wave-CISK compares favorably with the observed life cycle of the Madden-Julian oscillation (MJO). SST above the critical value produces an amplifying disturbance in which enhanced convection coincides with upper-tropospheric westerlies and is positively correlated with temperature and surface convergence. Conversely, SST below the critical value produces a decaying disturbance in which enhanced convection coincides with upper-tropospheric easterlies and is nearly in quadrature with temperature and surface convergence. While sharing essential features with the MJO in the Eastern Hemisphere, frictional wave-CISK does not explain observed behavior in the Western Hemisphere, where the convective signal is largely absent. Comprised of Kelvin structure with the same frequency, observed behavior in the Western Hemisphere can be understood as a propagating response that is excited in and radiates away from the fluctuation of convection in the Eastern Hemisphere.

Salby, Murry L↗

A Dynamical Model for The Jupiter Thermocline

A new conceptual model for the observed banded wind structure of the giant outer planets is proposed, assuming a smoothly distributed potential vorticity (PV) atop a layer of deeply seated, latitudinally variable stratification. With a vanishing horizontal entropy contrast presumably enforced at the bottom of the flow layer by the underlying convection, the thermal-wind balance of observed cloud-top motions implies a mapping of constant potential temperature surfaces, nearly vertical at upper tropospheric levels, down to a deeper, flatter, but variably sloped 'thermocline.' For a generally colder-poleward distribution of isentropes consistent with the strong equatorial jets of both Jupiter and Saturn, this kind of mapping would also imply a poleward decreasing static stability. The proposed temperature-stability distribution is just the reverse of the Earth's troposphere, where the strongest latitudinal potential temperature gradients are at the bottom instead of the top, with static stability generally increasing toward the pole. The warmer-stabler correlation for Jupiter would be consistent, however, with a nearly monotonic distribution of PV from low to high latitudes, over planetary scales for which the planetary vorticity dominates the relative vorticity of the jets. In this way efficient PV mixing for the isentropically bounded thermocline can account for the dynamical maintenance of the cyclonic flanks of the equatorial jet. Over latitudinal intervals comparable to the internal deformation scale, however, the gradient of the absolute vorticity is dominated by the flow curvature, with the local latitudinal gradient of the static stability proportional to the local potential temperature gradient and therefore the local variation in geostrophic velocity. The resulting correlation of velocities and vorticity gradients implies an alternation of the flow over the internal deformation radius set by the vertical entropy contrast.

Allison, Michael↗

Microwave Remote Sensing of Snow: Advances over Ice Sheet, Land, and Sea Ice

Satellite microwave radiometers have enabled us to observe the cryosphere and its changes. A variety of algorithms have been developed since the late 1970s. These convert observed microwave radiation into geophysical, glaciological properties relevant to study ice sheets’ snow accumulation and melt, terrestrial snow water equivalent and freeze/thaw state, and various sea ice cover characteristics like extent, concentration, and thickness. In spite of the continuous availability of satellite microwave radiometers for the past 40 years, potential for advancing our understanding of the relationship between microwave radiation and snow/ice properties still exists. Original empirical algorithms have matured and are becoming more physically based. This presentation offers insights into some advances made during the past 10 years in monitoring ice sheet, terrestrial snow, and sea ice. These advances made it possible to provide new, more reliable climate-related variables to the community for the satellite era using the typical 18-37 GHz frequency range (e.g., retrievals of grain size profiles in Antarctica, snow cover stratification, and therefore accumulation with applications to climate studies). Recent NASA instruments have recorded low microwave frequencies (at 1.4 GHz). These observations have a large penetration depth, they emanate from deep into the ice, where properties including temperature are very stable. Nonetheless, it has been found at both Dome C, Antarctica and Summit, Greenland that changes in surface snow properties significantly influence these observations. Compared to ice sheets, terrestrial snow and sea ice present higher spatial heterogeneities. Over land, presence of canopy and lakes, though with known locations, add ambiguities in the retrievals of snow properties. Over sea ice, ridges, leads, changes in salinity, and sea ice drift augment further the level of difficulty in obtaining robust geophysical properties. Assessing the quality of satellite retrievals, often requiring field activities, is a necessary step in designing the next generation of microwave algorithms to monitor changes in the cryosphere.

Brucker, Ludovic↗

Internal Waves in CVX

Near the liquid-vapor critical point, density stratification supports internal gravity waves which affect 1-g viscosity measurements in the CVX (Critical Viscosity of Xenon) experiment. Two internal-wave modes were seen in the horizontal viscometer. The frequencies of the two modes had different temperature dependences: with decreasing temperature, the higher frequency increased monotonically from 0.7 to 2.8 Hz, but the lower frequency varied non-monotonically, with a maximum of 1.0 Hz at 20 mK above the critical temperature. The measured frequencies agree with independently calculated frequencies to within 15%.

Berg, Robert F.↗