Showing papers in "The Astrophysical Journal in 1977"
3,700 citations
TL;DR: In this paper, a self-consistent model of the interstellar medium was developed to account for the observed pressure of interstellar clouds, the galactic soft X-ray background, the O VI absorption line observations, the ionization and heating of much of the galaxy, and the motions of the clouds.
Abstract: Supernova explosions in a cloudy interstellar medium produce a three-component medium in which a large fraction of the volume is filled with hot, tenuous gas. In the disk of the galaxy the evolution of supernova remnants is altered by evaporation of cool clouds embedded in the hot medium. Radiative losses are enhanced by the resulting increase in density and by radiation from the conductive interfaces between clouds and hot gas. Mass balance (cloud evaporation rate=dense shell formation rate) and energy balance (supernova shock input=radiation loss) determine the density and temperature of the hot medium with (n, T) = (10/sup -2.5/, 10/sup 5.7/) being representative values. Very small clouds will be rapidly evaporated or swept up. The outer edges of ''standard'' clouds ionized by the diffuse UV and soft X-ray backgrounds provide the warm (approx.10/sup 4/ K) ionized and neutral components. A self-consistent model of the interstellar medium developed herein accounts for the observed pressure of interstellar clouds, the galactic soft X-ray background, the O VI absorption line observations, the ionization and heating of much of the interstellar medium, and the motions of the clouds. In the halo of the galaxy, where the clouds are relatively unimportant, we estimate (n, T)more » = (10/sup -3.3/, 10/sup 6.0/) below one pressure scale height. Energy input from halo supernovae is probably adequate to drive a galactic wind.« less
1,980 citations
TL;DR: In this paper, the problem of the gravitational collapse of isothermal spheres by applying the similarity method to the gas-dynamic flow is considered, and two types of similarity solutions are obtained: one is the prototype for starting states which correspond to unstable hydrostatic equilibrium; the other, for states where the mass of the cloud slightly exceeds the maximum limit allowable for hydrostatic equilibria.
Abstract: We consider the problem of the gravitational collapse of isothermal spheres by applying the similarity method to the gas-dynamic flow. We argue that a previous solution obtained by Larson and Penston to describe the stages prior to core formation is physically artificial; however, we find that the flow following core formation does exhibit self-similar properties.The latter similarity solution shows that the inflow in the dense central regions proceeds virtually at free-fall before the material is arrested by a strong radiating shock upon impact with the surface of the core. Two types of similarity solutions are obtained: one is the prototype for starting states which correspond to unstable hydrostatic equilibrium; the other, for states where the mass of the cloud slightly exceeds the maximum limit allowable for hydrostatic equilibrium. In both cases, an r/sup -2/ law holds for the density distribution in the static or nearly static outer envelope, and an r/sup -3///sup 2/ law holds for the freely falling inner envelope. Rapid infall is initiated at the head of the expansion wave associated with the dropping of the central regions from beneath the envelope. A numerical example is presented which is shown to be in good agreement with the envelopemore » dynamics obtained in previous studies of star formation using hydrodynamic codes.« less
1,747 citations
TL;DR: In this article, the detailed structure of the interaction of a strong stellar wind with the interstellar medium is presented, including the effects of thermal conduction between the hot interior and the cold shell of swept-up interstellar matter.
Abstract: The detailed structure of the interaction of a strong stellar wind with the interstellar medium is presented. First, an adiabatic similarity solution is given which is applicable at early times. Second, a similarity solution is derived which includes the effects of thermal conduction between the hot (about 1 million K) interior and the cold shell of swept-up interstellar matter. This solution is then modified to include the effects of radiative energy losses. The evolution of an interstellar bubble is calculated, including the radiative losses. The quantitative results for the outer-shell radius and velocity and the column density of highly ionized species such as O VI are within a factor 2 of the approximate results of Castor, McCray, and Weaver (1975). The effect of stellar motion on the structure of a bubble, the hydrodynamic stability of the outer shell, and the observable properties of the hot region and the outer shell are discussed.
1,652 citations
TL;DR: In this article, the structure and formation of a star-subgroup association were examined in view of radio and infrared observations of the adjacent molecular clouds, and it was shown that a layer of dense neutral material accumulates between the I and S fronts and eventually becomes gravitationally unstable.
Abstract: We reconsider the structure and formation of OB association in view of recent radio and infrared observations of the adjacent molecular clouds. As a result of this reexamination, we propose that OB subgroups are formed in a step-by-step process which involves the propagation of ionization (I) and shock (S) fronts through a molecular cloud complex. OB stars formed at the edge of a molecular cloud drive these I-S fronts into the cloud. A layer of dense neutral material accumulates between the I and S fronts and eventually becomes gravitationally unstable. This process is analyzed in detail. Several arguments concerning the temperature and mass of this layer suggest that a new OB subgroup will form. After approximately one-half million years, these stars will emerge from and disrupt the star-forming layer. A new shock will be driven into the remaining molecular cloud and will initiate another cycle of star formation.Several observed properties of OB associations are shown to follow from a sequential star-forming mechanism. These include the spatial separation and systematic differences in age of OB subgroups in a given association, the regularity of subgroup masses, the alignment of subgroups along the galactic plane, and their physical expansion. Detailed observations of ionizationmore » fronts, masers, IR sources, and molecular clouds are also in agreement with this model. Finally, this mechanism provides a means of dissipating a molecular cloud and exposing less massive stars (e.g., T Tauri stars) which may have formed ahead of the shock as part of the original cloud collapsed and fragmented.« less
1,044 citations
TL;DR: In this paper, the authors developed a theory of Newtonian accretion disks based on an explicit formulation of viscous stresses arising from turbulent magnetic fields which are generated by Keplerian differential rotation of plasmas.
Abstract: We develop a theory of Newtonian accretion disks based on an explicit formulation of viscous stresses arising from turbulent magnetic fields which are generated by Keplerian differential rotation of plasmas. It is particularly pointed out that two physically distinct states exist in the middle part of the accretion disk surrounding a black hole; a transition from one state to the other takes place when the physical conditions of the accreting gas near the disk outer boundary cross over the critical conditions separating the two states. Thermal stability of those states is examined. It is also found that within a certain inner radius the disk develops into a single, high-temperature state, irrespective of the critical conditions; copious emission of hard X-rays arises from this inner domain. We then apply those results to construct a definite numerical model of Cyg X-1 accounting for observed features of its transitions, such as magnitudes and spectral distributions of the X-ray luminosity, transition times, and variabilities. Possibility of transient X-ray sources associated with accretion into black holes is predicted.
767 citations
686 citations
TL;DR: In this paper, column densities of early-type stars were derived by fitting damping profiles to the observed spectra, most of which exhibit strong damping lines in the lower rotational levels surveyed.
Abstract: Data from the Copernicus satellite's ultraviolet telescope were used to survey column densities of atomic and molecular hydrogen from a large sample of early-type stars; these data have bearing on an eventual understanding of diffuse and dense interstellar clouds. Column densities are derived by fitting damping profiles to the observed spectra, most of which exhibit strong damping lines in the lower rotational levels surveyed. Plots of dust column density, fractional abundance of molecular hydrogen, and the logarithm of fractional abundance versus total gas column density are given for many of the stars; stars with abnormally large or small hydrogen column densities, as well as some distant stars at high galactic latitudes, are considered. Equilibrium and nonequilibrium theories accounting for the abundance of interstellar hydrogen are compared, and support is found in the data for an account which balances hydrogen formation on interstellar grains with destruction through photodissociation. Overall averages for atomic and molecular hydrogen levels in the galactic plane are also calculated.
664 citations
647 citations
TL;DR: In this paper, an outline of the physical processes involved in the emerging flux model, which appears to explain naturally many solar flare observations, is presented, and it is suggested that a (small) simple loop flare occurs if the new flux appears in a region where no great amount of magnetic energy in excess of potential is stored.
Abstract: An outline is presented of the physical processes involved in the emerging flux model, which appears to explain naturally many solar flare observations. The separate physical phases of the basic model include a preflare heating phase as the new flux emerges, an impulsive phase as high-energy particles are accelerated, a flash (or explosive) phase when the H-alpha intensity increases, and a main phase while it decreases. The extent and morphology of the main phase emission depend on the structure of the magnetic field region in which the new flux finds itself imbedded. It is suggested that a (small) simple loop flare occurs if the new flux appears in a region where no great amount of magnetic energy in excess of potential is stored. A two-ribbon flare occurs if the flux emerges near the polarity inversion line of an active region that has begun to develop filaments.
629 citations
TL;DR: In this paper, the two-and three-point angular correlation functions for the high-resolution (10') Shane-Wirtanen catalog of galaxies were estimated and corrections for plate-to-plate variations of limiting magnitude, counting error, and the effect of the 10' x 10' counting cell were given.
Abstract: We present estimates of the two- and three-point angular correlation functions for the high-resolution (10') Shane-Wirtanen catalog of galaxies. Special attention is given to statistical corrections for plate-to-plate variations of limiting magnitude, counting error, and the effect of the 10' x 10' counting cell. The two-point function is well approximated by a power law for theta9 or approx. =2/sup 0/.5, corresponding to a projected distance of approx.9h/sup -1/ Mpc (H=100h km s/sup -1/ Mpc/sup -1/), but breaks sharply below the power law at larger angles. Several arguments indicate that the break is an intrinsic feature of the galaxy distribution, not an artifact of the analysis. New scaling relations taking account of redshift and curvature are derived and used to compare the correlation functions estimated for the Zwicky, Shane-Wirtanen, and Jagellonian samples. Corrections for redshift effects are about 20% for scaling between the Zwicky and Shane-Wirtanen catalogs, and including them improves the agreement among the estimates. The two-point spatial function is estimated to be xi (r) = (r/sub 0//r)/sup 1.77/, with r/sub 0/=4.7h/sup -1/ Mpc, 0.05 Mpc9 or approx. =hr9 or approx. =9 Mpc. The three-point function at theta9 or approx. =3/sup 0/ is well represented by the model zeta (1,2,3)more » =Q(xi (1) xi (2)+xi (2) xi (3)+xi (3) xi (1)) for the spatial function, with Q=1.29 +- 0.21. The three-point function shows little or no evidence of a preference for linear features in the galaxy distribution.« less
TL;DR: In this article, the formation of galaxies is studied in the context of the fragmentation of massive diffuse ionized gas clouds, and the relationship between binding energy and mass is investigated for spheroidal galaxies.
Abstract: The formation of galaxies is studied in the context of the fragmentation of massive diffuse ionized gas clouds. A semiquantitative discussion of the role of cooling indicates that there is a characteristic mass of a galaxy (of the order of 500 billion solar masses) which can fragment out of an ionized medium of density not exceeding 10 to the -25th power g/cu cm, whereas more massive galaxies attain a characteristic radius of the order of 60 kpc. The relationship between binding energy and mass is investigated for spheroidal galaxies. The role of H2 and Ly-alpha cooling in the fragmentation of primordial gas clouds is evaluated, and relatively qualitative arguments indicate that the first stars have masses of at least 0.3 solar mass with a characteristic mass of approximately 20 solar masses. When the average heavy-element abundance by mass exceeds about 0.00001, heavy-element cooling prevails over Ly-alpha cooling, and subsequent fragmentation forms second-generation stars of low characteristic mass.
TL;DR: In this article, it is shown that for particles with rigidities greater than about 0.3 GV in the solar wind, they are larger than the solar-wind velocity over much of the heliosphere.
Abstract: Although gradient and curvature drifts are explicitly contained in the general equations of cosmic-ray transport, they have been almost universally neglected in applications of these equations. The drifts are evaluated explicitly for the Parker (1965) spiral magnetic field, and it is shown that for particles with rigidities greater than about 0.3 GV in the solar wind, they are larger than the solar-wind velocity over much of the heliosphere. Hence most current models of solar modulation and solar-flare particle events neglect terms which in many cases are as important as those retained. Calculations are presented which demonstrate the importance of the effects for simple modulation models. It is concluded that comparisons of presently available model calculations with observations do not provide a fair test of transport theory since they neglect drifts, which may substantially reduce the net modulation.
TL;DR: In this paper, the authors assume that the simple periodic orbits of a test particle in the restricted three-body problem are a very good approximation to the streamlines in the accretion disk with a very small pressure and viscosity.
Abstract: The simple periodic orbits of a test particle in the restricted three-body problem are a very good approximation to the streamlines in the accretion disk with a very small pressure and viscosity. The maximum size of such accretion disk is found as a function of mass ratio of a binary system. I assume that this is identical with the largest simple periodic orbit which does not intersect other orbits. The maximum size of a disk is much larger than that found by Kruszewski, by Flannery, or by Lubow and Shu, but it is always less than the Roche lobe. Optical accretion disks observed in 17 binaries have sizes intermediate between the two theoretical limits. Masses of cataclysmic variables were systematically overestimated in the past because of improper analysis of the observed radial velocity changes.
TL;DR: The isotopic composition of Mg was measured in different phases of a Ca-Al-rich inclusion in the Allende meteorite and large excesses of mg-26 of up to 10% were found.
Abstract: The isotopic composition of Mg was measured in different phases of a Ca-Al-rich inclusion in the Allende meteorite. Large excesses of Mg-26 of up to 10% were found. These excesses correlate strictly with the Al-27/Mg-24 ratio for four coexisting phases with distinctive chemical compositions. Models of in situ decay of Al-26 within the solar system and of mixing of interstellar dust grains containing fossil Al-26 with normal solar system material are presented. The observed correlation provides definitive evidence for the presence of Al-26 in the early solar system. This requires either injection of freshly synthesized nucleosynthetic material into the solar system immediately before condensation and planet formation, or local production within the solar system by intense activity of the early sun. Planets promptly produced from material with the inferred Al-26/Al-27 would melt within about 300,000 years.
TL;DR: In this paper, the authors considered the steady state distribution and consumption rate of stars orbiting a massive object at the center of a spherical, N-body stellar system and employed an approximate, analytic analysis of the two-dimensional Fokker-Planck equation describing diffusion in energy E and angular momentum J. The distribution of stars is determined by the consumption of low angular momentum stars which pass within a small distance r/sub t/ of the central mass M and by the relaxation processes associated with gravitational stellar encounters.
Abstract: We consider the steady-state distribution and consumption rate of stars orbiting a massive object at the center of a spherical, N-body stellar system. The distribution of stars is determined by the consumption of low angular momentum stars which pass within a small distance r/sub t/ of the central mass M and by the relaxation processes associated with gravitational stellar encounters. Our method employs an approximate, analytic analysis of the two-dimensional Fokker--Planck equation describing diffusion in energy E and angular momentum J. The basic results are the following: (1) Consumption of low angular momentum stars which have entered the ''loss-cone'' J > greater than or equal to r/sub t/, at which the root mean square angular momentum transferred to a star via stellar encounters in one orbital period equals J/sub min/. (2) The total consumption rate of stars by M is roughly the number of stars inside r/sub crit/ divided by the relaxation time at r/sub crit/. (3) A self-consistent solution can be found in which the distribution of stars is almost isotropic for high-J stars and varies only more » logarithmically with J for low-J stars. (4) The density of core stars has the following form: n(r) approx. =n (r/sub a/)(1+(r/sub a//r)/sup l/), wher the accretion radius r/sub a/ approx. GM/ >> r/sub t/, is the mean-squared velocity dispersion in the core outside r/sub a/, and l decreases slowly from approx. 1.75 at >> r/sub crit/ to approx. 1.60 at r approx. 10r/sub t/. Neglect of loss-cone effects gives a consumption rate too small by roughly the ratio r/sub t//r/sub crit/ and a constant exponent l = 1.75 for >> r/sub t/. These results are applied to massive black holes at the centers of globular clusters and galactic nuclei. (AIP) « less
TL;DR: In this article, the authors used a white-light coronagraph aboard Skylab to determine the boundaries of a coronal hole in the northern polar region and the three-dimensional density structure within the hole between heights of 2 and 5 solar radii.
Abstract: Observations with a white-light coronagraph aboard Skylab are used to determine the boundaries of a coronal hole in the northern polar region and the three-dimensional density structure within the hole between heights of 2 and 5 solar radii. The boundary of the hole is found to be essentially axisymmetric about the polar axis, nearly radial from 3 to 6 solar radii, and located near 25 deg latitude at these heights. The radiances arising from the hole are interpreted as resulting from an axisymmetric density distribution whose logarithmic radial gradient is independent of position within the hole and whose magnitude increases with angular distance away from the hole's axis. The velocity distribution within the hole is obtained from the continuity equation by assuming that the particle flux flowing outward in the hole is similar to that measured for high-speed solar-wind streams at 1 AU, and it is shown that the transition from subsonic to supersonic flow occurs between 2.2 and 3 solar radii.
TL;DR: In this article, the authors test the hypothesis that the solar p-modes are stabilized by damping due to turbulent viscosity in the convective zone and find that the interaction between a p-mode and the turbulent convection is such that the modal energy tends toward equipartition with the kinetic energy of turbulent eddies whose lifetimes are comparable to the modality period.
Abstract: We test the hypothesis that the solar p-modes are stabilized by damping due to turbulent viscosity in the convective zone. Starting from the assumption that the modes are stable, we calculate expectation values for the modal energies. We find that the interaction between a p-mode and the turbulent convection is such that the modal energy tends toward equipartition with the kinetic energy of turbulent eddies whose lifetimes are comparable to the modal period. From the calculated values of the modal energies, we compute rms surface velocity amplitudes. Our predicted rms surface velocities range from 0.01 cm/sec for the fundamental radial mode to 0.6 cm/sec for the radial mode whose period is approximately 5 minutes. The predicted surface velocities for the low order p-modes are much smaller than the velocities inferred from recent observations.
TL;DR: In this paper, the two-body tidal capture mechanism of Fabian, Pringle, and Rees has been studied for stars with an n=3 polytropic structure.
Abstract: We calculate in detail the two-body tidal capture mechanism of Fabian, Pringle, and Rees: when two unbound stars have a close encounter, they may become bound by the energy that each deposits into nonradial oscillations of the other. After dimensional scalings are removed, the process depends only on a single dimensionless parameter, and on the dimensionless envelope structure of the stars. General formulae are derived; for definiteness, we apply them to the specific case of stars with an n=3 polytropic structure. Capture cross sections as a function of velocity and capture rates for an isothermal distribution are given for the case of equal-mass stars; other cases can easily be computed from the formulae given.
TL;DR: In this paper, the steady state distribution of stars around a massive black hole in a globular cluster is determined by solving numerically the coupled time-dependent Boltzmann equations for a system containing stars of two different masses.
Abstract: The steady-state distribution of stars around a massive black hole in a globular cluster is determined by solving numerically the coupled time-dependent Boltzmann equations for a system containing stars of two different masses. Similar results are found for an arbitrary spectrum of masses with the aid of approximate analytic solutions of the time-independent equations. The effects of mass segregation are summarized by scaling laws that are derived both by analytic approximations and by numerical solutions. The detectability of a black hole in a globular cluster is discussed in terms of possible observations of the central star distributions.
TL;DR: In this article, the morphology of limb flares is investigated and the observed spatial structure is related with the other physical parameters of the region (temperature, density, characteristic times, and energy density).
Abstract: The data used in the investigation were obtained by the American Science and Engineering S-054 experiment on board Skylab. The instrument employed was a grazing incidence X-ray telescope which records images of the sun on film. The morphology of limb flares is investigated and the observed spatial structure is related with the other physical parameters of the region (temperature, density, characteristic times, and energy density). It is concluded that two physically distinct classes of flares exist that differ both in their morphology and in the physical properties which characterize the emitting regions.
TL;DR: In this article, the wave energy density as a function of heliocentric distance is derived for radially propagating acoustic, Alfven, and fast mode waves, and the effect of waves as a source of momentum is explored by examining critical points of the momentum equation for isothermal spherically symmetric flow.
Abstract: The fluid equations for the solar wind are presented in a form which includes the momentum and energy flux of waves in a general and consistent way. The concept of conservation of wave action is introduced and is used to derive expressions for the wave energy density as a function of heliocentric distance. The explicit form of the terms due to waves in both the momentum and energy equations are given for radially propagating acoustic, Alfven, and fast mode waves. The effect of waves as a source of momentum is explored by examining the critical points of the momentum equation for isothermal spherically symmetric flow. We find that the principal effect of waves on the solutions is to bring the critical point closer to the sun's surface and to increase the Mach number at the critical point. When a simple model of dissipation is included for acoustic waves, in some cases there are multiple critical points.
TL;DR: In this article, the authors derived general-relativistic equations of structure under the assumptions of hydrostatic and thermal equilibrium, spherical symmetry, no rotation and no magnetic field, and derived analytic expressions for the stellar structure in various interior regions.
Abstract: Stars with massive envelopes (M/sub env/approximately-greater-than1 M/sub sun/) and degenerate neutron cores (M/sub core/approx.1 M/sub sun/, R/sub core/approx.10 km) are analyzed theoretically: General-relativistic equations of structure are derived under the assumptions of hydrostatic and thermal equilibrium, spherical symmetry, no rotation and no magnetic field. Numerical models are constructed, and analytic expressions are derived for the stellar structure in various interior regions. It is argued that all nonrotating, equilibrium models probably resemble qualitatively those constructed in this paper. Brief discussions are given of the stability and evolution of the models, and of prospects for identifying such stars observationally.Viewed externally, our models are extreme M supergiants (Lapprox.3 x 10/sup 4/ to 1.3 x 10/sup 5/ L/sub sun/, T/sub photosphere/approx.2600 to 3100 K, R/sub photosphere/approx. =1000 R/sub sun/). The large, diffuse envelope of each model is separated from its compact core by a thin (approx.40 meter) energy-generation layer called the ''halo.'' The envelope convects from the outer edge of the halo all the way out to the photosphere. Matter contracts from the envelope through the halo and into the core at a rate of approx.1 x 10/sup -8/ M/sub sun/ yr/sup -1/. The contracting matter releases its gravitational energy and burns its hydrogenmore » and helium while passing through the halo. When the envelope mass exceeds approx.10 M/sub sun/, the hydrogen-burning shell occurs at the halo-envelope interface, and the products of hot (Tapprox. =1 x 10/sup 9/ K) nonequilibrium hydrogen burning are convected directly from the burning shell out to the photosphere, where they should be observable.« less
TL;DR: In this article, it was shown that there is a scale radius approx30 kpc associated with the collapse of the most massive galaxies and that it may be possible to explain the original inability of the material of galactic disks to fragment in terms of this scale radius.
Abstract: A number of arguments lead one to consider the possibility that the first massive galaxies were formed after redshift 10 The formation of galaxies at small redshifts can only have occurred if the formation process was dissipative This paper is concerned with one theory of dissipative galaxy formation: the theory which arises from the supposition that a protogalactic cloud would not have fragmented very extensively during the first phase of its infall and thus that the collapse of the cloud would be halted by the formation of shock fronts around a caustic surface Detailed calculations of the formation and evolution of these shock fronts yield values for the physical conditions of the protogalactic gas and the rate of radiation of galactic binding energy by such shocks The adiabaticity or isothermality of these shock fronts is found to be dependent on the characteristic infall velocity and the surface density associated with the shock front but not on the shape of the initial density profile perpendicular to the incipient shock surfaces Protogalaxies which recollapsed at high z are more likely to have generated isothermal shocks than ones which collapsed at low z It is found that this binding energy release could makemore » collapsing protogalactic clouds very high surface brightness objects It appears that there is a scale radius approx30 kpc associated with the collapse of the most massive galaxies and that it may be possible to explain the original inability of the material of galactic disks to fragment in terms of this scale radius A similar theory of the origin of Magellanic irregulars, low surface brightness spheroidals, and dwarf ellipticals is suggested« less
TL;DR: In this paper, the authors consider self-contained gas flows, in which mass ejected by galaxies in the outer regions of a cluster flows toward the core of the cluster and ultimately accretes onto the central galaxies.
Abstract: Radiation from gas within rich clusters of galaxies may regulate the central gas density to that value at which the cooling time equals the lifetime of the system, since at higher densities mass deposition by cooling in the central regions would exceed mass injection from the member galaxies and mass inflow from the outer regions of the cluster. We argue that in X-ray clusters a steady inflow of material takes place, in which the gravitational energy of the accreting material powers the X-ray luminosity of the source and accounts for the observed spatial extent of the gas distribution.In the present paper we consider self-contained gas flows, in which mass ejected by galaxies in the outer regions of the cluster flows toward the core of the cluster and ultimately accretes onto the central galaxies. This model, which depends only on a single parameter measuring the ratio of the mass-injection rate to the cooling in the central regions, accounts for the observed X-ray luminosities of the clusters, the surface-brightness profiles, and energy spectra in terms of thermal bremsstrahlung emission from the gas. We briefly argue that thermal conduction does not affect the observed properties of the source.We discuss the expected equivalent widthsmore » of observable emission lines, the morphology of the X-ray emission from the Perseus cluster, the formation of cD and active galaxies, and extinction and scattering by dust within the cluster.« less