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Bidirectional Reflectance of a Macroscopically Flat, High-Albedo Particulate Surface: An Efficient Radiative Transfer Solution and Applications to Regoliths

Many remote sensing applications rely on accurate knowledge of the bidirectional reflection function (BRF) of surfaces composed of discrete, randomly positioned scattering particles. Theoretical computations of BRFs for plane-parallel particulate layers are usually reduced to solving the radiative transfer equation (RTE) using one of existing exact or approximate techniques. Since semi-empirical approximate approaches are notorious for their low accuracy, violation of the energy conservation law, and ability to produce unphysical results, the use of numerically exact solutions of RTE has gained justified popularity. For example, the computation of BRFs for macroscopically flat particulate surfaces in many geophysical publications is based on the adding-doubling (AD) and discrete ordinate (DO) methods. A further saving of computer resources can be achieved by using a more efficient technique to solve the plane-parallel RTE than the AD and DO methods. Since many natural particulate surfaces can be well represented by the model of an optically semi-infinite, homogeneous scattering layer, one can find the BRF directly by solving the Ambartsumian's nonlinear integral equation using a simple iterative technique. In this way, the computation of the internal radiation field is avoided and the computer code becomes highly efficient and very accurate and compact. Furthermore, the BRF thus obtained fully obeys the fundamental physical laws of energy conservation and reciprocity. In this paper, we discuss numerical aspects and the computer implementation of this technique, examine the applicability of the Henyey-Greenstein phase function and the sigma-Eddington approximation in BRF and flux calculations, and describe sample applications demonstrating the potential effect of particle shape on the bidirectional reflectance of flat regolith surfaces. Although the effects of packing density and coherent backscattering are currently neglected, they can also be incorporated. The FORTRAN implementation of the technique is available on the World Wide Web, and can be applied to a wide range of remote sensing problems. BRF computations for undulated (macroscopically rough) surfaces are more complicated and often rely on time consuming Monte Carlo procedures. This approach is especially inefficient for optically thick, weakly absorbing media (e.g., snow and desert surfaces at visible wavelengths since a photon may undergo many internal scattering events before it exists the medium or is absorbed. However, undulated surfaces can often be represented as collections of locally flat tilted facets characterized by the BRF found from the traditional plane parallel RTE. In this way the MOnte Carlo procedure could be used only to evaluate the effects of surface shadowing and multiple surface reflections, thereby bypassing the time-consuming ray tracing inside the medium and providing a great savings of CPU time.

Mishchenko, Michael I.↗

The Effect Stacking Fault Segregation and Phase Transformations have on Creep Strength in Ni-based Superalloys

In this study, two commercially similar polycrystalline Ni-based disk superalloys (LSHR and ME3) were creep tested at 760C and 552MPa to approximately 0.3% plastic strain. LSHR consistently displayed superior creep properties at this stress/temperature regime even though the microstructural characteristics between the two alloys were comparable. High resolution structural and chemical analysis, however, revealed significant differences between the two alloys among active gamma prime shearing modes involving superlattice intrinsic and extrinsic stacking faults. In ME3, Co and Cr segregation and Ni and Al depletion were observed along the intrinsic faults - revealing a gamma prime -to- gamma phase transformation. Conversely in LSHR, an alloy with a higher W content, Co and W segregation was observed along the intrinsic faults. This observation combined with scanning transmission electron microscopy (STEM) simulations confirm a gamma prime-to-D019 phase transformation along the intrinsic faults in LSHR. Using experimental observations and density functional theory calculations, a novel local phase transformation strengthening mechanism is proposed that could be further utilized to improve the high temperature creep capabilities of Ni-base disk alloys.

Superalloy↗

The localized origin of equatorial F region irregularity patches

An intensive study of nighttime irregularities of electron density in the equatorial ionosphere was performed in October 1976 by making 50-MHz radar backscatter measurements at Jicamarca, Peru, and scintillation measurements of 249-MHz transmissions from Les 9 at two ground stations (Ancon and Huancayo, both in Peru) as well as by aircraft flying in the vicinity of the stations. The 137-MHz scintillations from the orbiting Wideband satellite were also recorded at Huancayo. The results of such measurements made on October 16-17, 1976, are discussed in this report. We find that on this particular night a large-scale irregularity patch evolved first in the west, as was detected by the radar at Jicamarca, and drifted eastward to cause successive onsets of scintillation activity on propagation paths from Ancon and Huancayo. The observations indicate the east-west dimension of the large-scale structure to be 400 km drifting eastward at a speed of approximately 100 m/s, having a lifetime of several hours, and containing a hierarchy of irregularity scale sizes in the range of kilometers to meters causing both scintillations at 249 MHz and radar backscatter at 50 MHz.

Aarons, J.↗

Diode Laser Sensors for Arc-Jet Characterization

The development and application of tunable diode laser (TDL) absorption sensors to monitor the health and operating conditions in the large-scale 60 MW arc-heated- plasma wind-tunnel at NASA Ames Research Center is reported. The interactive heating facility (THF) produces re-entry flow conditions by expanding the gas heated in a constricted plasma arc-heater to flow at high velocity over a model located in a test cabin. This facility provides the conditions needed to test thermal protective systems for spacecraft re-entering the earth s atmosphere. TDL sensors are developed to monitor gas flows in both the high-temperature constricted flow and the supersonic expansion flow into test cabin. These sensors utilize wavelength-tuned diode lasers to measure absorption transitions of atomic oxygen near 777.2 nm, atomic nitrogen near 856.8 nm, and atomic copper near 793.3 nm. The oxygen and nitrogen sensors measure the population density in exited electronic states of these atoms. The measurements combined with the assumption of local thermal and chemical equilibrium yield gas temperature (typically near 7,000K). The nitrogen and oxygen population temperatures are redundant, and their close agreement provides an important test of the local thermal equilibrium assumption. These temperature sensors provide time-resolved monitors of the operating conditions of the arc-heater and can be used to verify and control the test conditions. An additional TDL sensor was developed to monitor the copper concentration in the arc-heater flow yielding values as high as 13 ppm. Measurements of copper in the flow can identify flow conditions with unacceptably rapid electrode erosion, and hence this sensor provides valuable information needed to schedule maintenance to avoid costly arc-heater failure. TDL sensors were also developed for measurements in the test cabin, where absorption measurements of the populations of argon and molecular nitrogen in excited metastable electronic states established that the number density of these excited species is much lower than estimated using frozen-chemistry approximations. This key finding suggests that in the post-expansion region there is not a significant energy sequestration in electronically excited species. Finally, TDL measurements of atomic potassium seeded into the test cabin flow were used to directly measure the static temperature of the test gas. The results of this study illustrate the high potential of time-resolved TDL measurements for routine and economical sensing of arc-heater health (gas temperature and electrode erosion) as well as the time-resolved test-cabin-flow conditions in front of the model.

Hanson, Ronald K.↗

Cometary water-group ions in the region surrounding Comet Giacobini-Zinner - Distribution functions and bulk parameter estimates

The bulk parameters (number density and thermal energy density) of cometary water-group ions in the region surrounding Comet Giacobini-Zinner have been derived using data from the EPAS instrument on the ICE spacecraft. The derivation is based on the assumption that the pick-up ion distribution function is isotropic in the frame of the bulk flow, an approximation which has previously been shown to be reasonable within about 400,000 km of the comet nucleus along the spacecraft trajectory. The transition between the pick-up and mass-loaded regions occurs at the cometary shock, which was traversed at a cometocentric distance of about 100,000 km along the spacecraft track. Examination of the ion distribution functions in this region, transformed to the bulk flow frame, indicates the occurrence of a flattened distribution in the vicinity of the local pick-up speed, and a steeply falling tail at speeds above, which may be approximated as an exponential in ion speed.

Staines, K.↗

Non-LTE model atmosphere analysis of Nova Cygni 1992

We use spherically symmetric non-local thermodynamic equilibrium (non-LTE), line-blanketed, expanding model atmospheres to analyze the International Ultraviolet Explorer (IUE) and optical spectra of Nova Cygni 1992 during the early phases of its outburst. We find that the first IUE spectrum obtained just after discovery on 1992 February 20, is best reproduced by a model atmosphere with a steep density gradient and homologous expansion, whereas the IUE and optical spectra obtained on February 24 show an extended, optically thick, wind structure. Therefore, we distinguish two phases of the early evolution of the nova photosphere: the initial, rapid, 'fireball' phase and the subsequent, much longer, optically thick 'wind' phase. The importance of line-blanketing in nova spectra is demonstrated. Our preliminary abundance analysis implies that hydrogen is depeleted in the ejecta, corresponding to abundance enhancements of Fe by a factor of approximately 2 and of CNO by more than a factor of 10 when compared to solar abundances. The synthetic spectra reproduce both the observed pseudo-continua as well as most of the observed features from the UV to the optical spectral range and demonstrate the importance of obtaining nearly simultaneous UV and optical spectra for performing accurate analyses of expanding stellar atmospheres (for both novae and supernovae).

Hauschildt, P. H.↗

Energetic protons associated with interplanetary active regions 1-5 AU from the sun

Pioneer 11 has yielded data on approximately 100 energetic proton events at heliocentric distances between 1 and 2 AU. Measurements of absolute intensities, anisotropies, and crude energy spectra are studied in connection with interplanetary active regions (IAR's). It is found that in close vicinity to IAR's, the number of events observed per unit time interval is 10 times greater than in other areas of interplanetary space, and that the frequency of events has a maximum at plus or minus 5 hours of the time IAR edges are crossed. It is also noted that events in IAR vicinity have greater particle densities, softer energy spectra, and smaller time widths than other events. For many events associated with IAR's, particle anisotropies correspond to the net flow of particles along the interplanetary magnetic field toward the sun. This suggests that a mechanism in MHD shocks is responsible for local acceleration in the interplanetary medium.

Pesses, M. E.↗

Low-Velocity Impact Response of Sandwich Beams with Functionally Graded Core

The problem of low-speed impact of a one-dimensional sandwich panel by a rigid cylindrical projectile is considered. The core of the sandwich panel is functionally graded such that the density, and hence its stiffness, vary through the thickness. The problem is a combination of static contact problem and dynamic response of the sandwich panel obtained via a simple nonlinear spring-mass model (quasi-static approximation). The variation of core Young's modulus is represented by a polynomial in the thickness coordinate, but the Poisson's ratio is kept constant. The two-dimensional elasticity equations for the plane sandwich structure are solved using a combination of Fourier series and Galerkin method. The contact problem is solved using the assumed contact stress distribution method. For the impact problem we used a simple dynamic model based on quasi-static behavior of the panel - the sandwich beam was modeled as a combination of two springs, a linear spring to account for the global deflection and a nonlinear spring to represent the local indentation effects. Results indicate that the contact stiffness of thc beam with graded core Increases causing the contact stresses and other stress components in the vicinity of contact to increase. However, the values of maximum strains corresponding to the maximum impact load arc reduced considerably due to grading of thc core properties. For a better comparison, the thickness of the functionally graded cores was chosen such that the flexural stiffness was equal to that of a beam with homogeneous core. The results indicate that functionally graded cores can be used effectively to mitigate or completely prevent impact damage in sandwich composites.

Apetre, N. A.↗

Gravity investigation of the Manson impact structure, Iowa

The Manson crater, of probable Cretaceous/Tertiary age, is located in northwestern Iowa (center at 42 deg. 34.44 min N; 94 deg. 33.60 min W). A seismic reflection profile along an east west line across the crater and drill hole data indicate a crater about 35 km in diameter having the classic form for an impact crater, an uplifted central peak composed of uplifted Proterozoic crystalline bedrock, surrounded by a 'moat' filled with impact produced breccia and a ring graben zone composed of tilted fault blocks of the Proterozoic and Paleozoic country rocks. The structure has been significantly eroded. This geologic structure would be expected to produce a significant gravity signature and study of that signature would shed additional light on the details of the crater structure. A gravity study was undertaken to better resolve the crustal structure. The regional Bouguer gravity field is characterized by a southeastward decreasing field. To first order, the Bouguer gravity field can be understood in the context of the geology of the Precambrian basement. The high gravity at the southeast corner is associated with the mid-continent gravity high; the adjacent low to the northwest results from a basin containing low-density clastic sediments shed from the basement high. Modeling of a simple basin and adjacent high predicts much of the observed Bouguer gravity signature. A gravity signature due to structure associated with the Manson impact is not apparent in the Bouguer data. To resolve the gravity signature of the impact, a series of polynomial surfaces were fit to the Bouguer gravity field to isolate the small wavelength residual anomalies. The residual gravity obtained after subtracting a 5th- or 6th-order polynomial seems to remove most of the regional effects and isolate local anomalies. The pattern resolved in the residual gravity is one of a gravity high surrounded by gravity lows and in turn surrounded by isolated gravity highs. The central portion of the crater is characterized by two positive anomalies having amplitudes of about plus 4 mGal separated by a gentle saddle located approximately at the crater center.

Plescia, J. B.↗

Heat transfer on a flat plate in continuum to rarefied hypersonic flows at Mach numbers of 19.2 and 25.4

Local heat transfer rates were measured on a flat steel plate (10 in. wide and 16 in. long) with sharp and blunt leading edges (0.001 and 0.010 in.) in the transition from the strong interaction boundary layer regime with no slip at the surface to the free molecule regime. The tests were conducted in a combustion driven hypersonic shock tunnel, with the nominal free stream Mach numbers of 19.2 and 25.4, and a reflected stagnation temperature of approximately 2340 R. The twelve heat transfer gages were made of platinum sputtered on a Pyrex backing to a thickness of approximately 350 A, insulated by a silicone dioxide film. For both Mach numbers the heat transfer data agreed reasonably well with the strong interaction prediction of Li and Nagamatsu (1953, 1955) for unit Reynolds numbers greater than approximately 100,000 and leading edge Knudsen numbers less than approximately 4. At lower density conditions the rarefied flow effects began to dominate the flow phenomena near the leading edge region of the sharp flat plate. A systematic reduction in the heat transfer rate close to the leading edge was observed for both Mach number tests as the leading edge density was reduced and the mean free path was increased.

Nagamatsu, H. T.↗

Quiet-time intensifications along the poleward auroral boundary near midnight

Radar and optical measurements from Sondrestrom are combined with satellite and Goose Bay data in a study of the poleward edge of the nightside auroral oval during a quiet period. The B(sub y) and B(sub z) components of the interplanetary magnetic field were close to zero, and the B(sub x) component was approximately 8 nT for more than 24 hours. On a large scale, the convection and precipitation patterns remained almost constant during this period; on a small scale, however, the conditions were quite dynamic. At 10- to 20-min intervals the arc that marked the poleward auroral boundary intensified, and a new arc appeared poleward of it. About once per hour, stronger intensifications were observed. One such event is examined in detail. The auroral arcs first appeared to dim, and then they brightened, with a factor of 10 increase in E region electron density. At the time of the brightening a new arc formed poleward of all the arcs. The arcs then drifted southward at velocities of approximately 270 m/s. A plasma drift disturbance, characterized by a doubling of the southward velocity and a reversal in the east-west component, propagated westward at 900 m/s through the fields of view of the Sondrestrom and Goose Bay radars. A simultaneous satellite overpass close to the radars revealed the presence of an energetic ion event similar to the 'velocity dispersed ion structures' observed on the Aureol satellite and presumed to be the signature of fast ion beams within the plasma sheet boundary layer. The stronger arc intensification events observed by the Sondrestrom radar are associated with an increase in plasma flow across the boundary between open and closed magnetic field lines. We interpret this increased flow as the ionospheric signature of abrupt, localized increases in the reconnection rate in the midnight sector.

De La Beaujardiere, O.↗

Photoelectrons and solar wind/lunar limb interaction.

It is suggested that boundary conditions for solar wind/lunar limb interactions are active. The 'whole-moon' limb does not evoke a shock cone, because warm (approximately equal 13 eV/electron) solar wind electrons are replaced by cool (less than or equal to 2 eV/electron) photoelectrons that are ejected from the generally smooth areas of the lunar terminator illuminated at glazing angles by the sun. A localized volume of low thermal pressure is created in the solar wind by these cool photoelectrons. The solar wind expands into this turbulence-suppressive volume without shock production. Conversely, directly illuminated highland areas exchange hot photoelectrons (greater than 20 eV/electron) for warm solar wind electrons. The hot electrons generate a localized pressure increase in the adjacent solar wind flow which evokes a shock streamer in the solar wind. Shock streamers are identifiable by a coincident increase in the magnitude of the solar wind magnetic field immediately external to the lunar wake. Shock occurrence is controlled by lunar topography, solar activity in the hard ultraviolet (greater than 20 eV), solar wind electron density and thermal velocity, and the intensity of the solar wind magnetic field.

Criswell, D. R.↗

SAGE II stratospheric density and temperature retrieval experiment

This paper describes a stratospheric density and temperature retrieval experiment based on the solar occultation measurement of the Stratospheric Aerosol and Gas Experiment (SAGE II). The entire retrieval analysis involves two inversion steps: the vertical structure inversion, which derives the profile of local atmospheric extinction from SAGE II limb optical depth data, and the species inversion, which inverts the concentration of air molecules, aerosols, ozone, and nitrogen dioxide from the derived atmospheric extinction at five SAGE II short wavelengths (0.385, 0.448, 0.453, 0.525, and 0.600 microns). The derived density profile is then used to infer the temperature distribution, assuming that the atmosphere is in hydrostatic equilibrium and obeys the ideal gas law. The temperature profiles retrieved from the SAGE II observations are compared with near-coincident, in both time and space, French Rayleigh lidar and NASA Wallops Flight Facility rocket datasonde soundings as well as the National Meteorological Center (NMC) data analyses. The results indicate that the mean SAGE II temperature agrees with the mean lidar measurements to within 2 C at altitudes from 30.5 to 52.5 km. The SAGE II and datasonde observations agree to within about 4 C in approximately the same altitude region.

Wang, Pi-Huan↗

Observational and Numerical Diagnostics of Galaxy Cluster Outer Regions

Aims. We present the analysis of a local (z = 0.04 - 0.2) sample of 31 galaxy clusters with the aim of measuring the density of the X-ray emitting gas in cluster outskirts. We compare our results with numerical simulations to set constraints on the azimuthal symmetry and gas clumping in the outer regions of galaxy clusters. Methods. We exploit the large field-of-view and low instrumental background of ROSAT/PSPC to trace the density of the intracluster gas out to the virial radius. We perform a stacking of the density profiles to detect a signal beyond r(sub 200) and measure the typical density and scatter in cluster outskirts. We also compute the azimuthal scatter of the profiles with respect to the mean value to look for deviations from spherical symmetry. Finally, we compare our average density and scatter profiles with the results of numerical simulations. Results. As opposed to several recent results, we observe a steepening of the density profiles beyond approximately 0.3r(sub 500). Comparing our density profiles with simulations, we find that non-radiative runs predict too steep density profiles, whereas runs including additional physics and/or gas clumping are in better agreement with the observed gas distribution. We note a systematic difference between cool-core and non-cool core clusters beyond approximately 0.3r(sub 200), which we explain by a different distribution of the gas in the two classes. Beyond approximately r(sub 500), galaxy clusters deviate significantly from spherical symmetry, with only little differences between relaxed and disturbed systems. We find good agreement between the observed and predicted scatter profiles, but only when the 1% densest clumps are filtered out in the simulations. Conclusions. The general trend of steepening density around the virial radius indicates that the shallow density profiles found in several recent works were probably obtained along particular directions (e.g., filaments) and are not representative of the typical behavior of clusters. Comparing our results with numerical simulations, we find that non-radiative simulations fail to reproduce the gas distribution, even well outside cluster cores. Therefore, a detailed treatment of gas cooling, star formation, clumping, and AGN feedback is required to construct realistic models of cluster outer regions.

Eckert, D.↗

The formation phase of the solar nebula

Hydrodynamical calculations of the collapse of an axisymmetric, rotating protostellar cloud, with radiation transport and without magnetic fields, are presented. The collapse is assumed to start from a centrally condensed sphere of radius 5 x 10 to the 15th cm, a mean density of 4 x 10 to the -15th g/cu cm, a total mass of 1 solar mass, and a total angular momentum of 10 to the 53rd g sq cm per sec. The numerical grid is chosen to resolve the region of disk formation between 1 and 60 AU from the center. Frequency-dependent radiative transfer calculations show how the emergent spectrum of the structure depends upon viewing angle with respect to the rotation axis and how the observed isophotal contours should depend on wavelength and viewing angle. The central part of the protostar, interior to 1 AU, is not resolved numerically but is modeled approximately. At the end of the calculation, this region is found to have a mass of 0.6 solar mass and a ratio of rotational to gravitational energy of about 0.4, sufficiently large to be unstable to nonaxisymmetric perturbations. Although the disk is gravitationally stable according to the local Toomre criterion, the nonaxisymmetric structure in the center is likely to lead to angular momentum transport.

Bodenheimer, Peter↗

Self-consistent solar wind-LISM interaction including neutrals: A 4-fluid model

We investigate the interaction between the heliosphere and local interstellar medium, including charge-exchange self-consistently between the neutral and ionized components. The system is modelled as consisting of neutral hydrogen, protons and electrons. The electrons are assumed to contribute to the dynamics via a pressure that is everywhere equal to the proton pressure. The fluid equations for the protons and hydrogen are solved self-consistently, assuming the two species interact only via charge exchange. The protons are treated as a single isotropic fluid. The hydrogen is decomposed into three isotropic fluids: hydrogen of interstellar origin with characteristic temperature of approximately 104 K, hydrogen produced via charge exchange in the post-shock solar wind, with characteristic temperature of approximately 105 K, and hydrogen produced via charge exchange in the supersonic solar wind, relatively cool but expanding radially outward with the solar wind speed. Two-Dimensional time-dependent numerical simulations of this system are performed to calculate the number density, temperature and bulk velocity of neutral hydrogen and plasma everywhere in the heliosphere. These quantities are compared to relevant solar wind observations. Implications for the pick-up process in the supersonic solar wind, as well as heating and deceleration of the wind, are addressed.

Williams, L. L.↗

Conductive damping of coronal motions

Time-dependent motions in the solar corona are subject to modification through conductive transport of thermal energy away from compression regions and into rarefaction regions. The evolution depends on how the energy is introduced - by an essentially thermal perturbation or by a density perturbation. The net effect of thermal conduction is to convert local kinetic energy in the transient into widely distributed thermal energy, which then may in turn be partially converted into kinetic energy of the bulk flow. Local motion in the transient is thereby damped. These effects are modeled here using a numerical solution of the time-dependent solar wind equations for single-fluid thermally conductive flow. The simulations of transient phenomena are made for ordinary collisional thermal conduction and compared with similar simulations done using polytropic flow. Between one and five solar radii, the gross character in the two cases is very similar. However, there are effects with thermal conduction that cannot be modeled with polytropic flow, illustrated here by temperature and velocity forerunners and a delayed and large-amplitude trailing velocity low. If thermal conduction is artificially inhibited by simple reduction of the thermal conduction coefficient, relative velocity amplitudes remain larger in approximate proportion to the amount the coefficient is reduced.

Suess, S. T.↗

The Gas Distribution in the Outer Regions of Galaxy Clusters

Aims. We present our analysis of a local (z = 0.04 - 0.2) sample of 31 galaxy clusters with the aim of measuring the density of the X-ray emitting gas in cluster outskirts. We compare our results with numerical simulations to set constraints on the azimuthal symmetry and gas clumping in the outer regions of galaxy clusters. Methods. We have exploited the large field-of-view and low instrumental background of ROSAT/PSPC to trace the density of the intracluster gas out to the virial radius, We stacked the density profiles to detect a signal beyond T200 and measured the typical density and scatter in cluster outskirts. We also computed the azimuthal scatter of the profiles with respect to the mean value to look for deviations from spherical symmetry. Finally, we compared our average density and scatter profiles with the results of numerical simulations. Results. As opposed to some recent Suzaku results, and confirming previous evidence from ROSAT and Chandra, we observe a steepening of the density profiles beyond approximately r(sub 500). Comparing our density profiles with simulations, we find that non-radiative runs predict density profiles that are too steep, whereas runs including additional physics and/ or treating gas clumping agree better with the observed gas distribution. We report high-confidence detection of a systematic difference between cool-core and non cool-core clusters beyond approximately 0.3r(sub 200), which we explain by a different distribution of the gas in the two classes. Beyond approximately r(sub 500), galaxy clusters deviate significantly from spherical symmetry, with only small differences between relaxed and disturbed systems. We find good agreement between the observed and predicted scatter profiles, but only when the 1% densest clumps are filtered out in the ENZO simulations. Conclusions. Comparing our results with numerical simulations, we find that non-radiative simulations fail to reproduce the gas distribution, even well outside cluster cores. Although their general behavior agrees more closely with the observations, simulations including cooling and star formation convert a large amount of gas into stars, which results in a low gas fraction with respect to the observations. Consequently, a detailed treatment of gas cooling, star formation, AGN feedback, and consideration of gas clumping is required to construct realistic models of the outer regions of clusters.

Eckert, D.↗