Assumption Of Dynamic Equilibrium Research Articles

Measuring Galactic dark matter through unsupervised machine learning

ABSTRACTMeasuring the density profile of dark matter in the Solar neighbourhood has important implications for both dark matter theory and experiment. In this work, we apply autoregressive flows to stars from a realistic simulation of a Milky Way-type galaxy to learn – in an unsupervised way – the stellar phase space density and its derivatives. With these as inputs, and under the assumption of dynamic equilibrium, the gravitational acceleration field and mass density can be calculated directly from the Boltzmann equation without the need to assume either cylindrical symmetry or specific functional forms for the galaxy’s mass density. We demonstrate our approach can accurately reconstruct the mass density and acceleration profiles of the simulated galaxy, even in the presence of Gaia-like errors in the kinematic measurements.

Magellan/IMACS Spectroscopy of Grus I: A Low Metallicity Ultra-faint Dwarf Galaxy* *This paper includes data gathered with the 6.5 m Magellan Telescopes located at Las Campanas Observatory, Chile.

We present a chemodynamical study of the Grus I ultra-faint dwarf galaxy (UFD) from medium-resolution (R ∼ 11,000) Magellan/IMACS spectra of its individual member stars. We identify eight confirmed members of Grus I, based on their low metallicities and coherent radial velocities, and four candidate members for which only velocities are derived. In contrast to previous work, we find that Grus I has a very low mean metallicity of 〈[Fe/H]〉 = −2.62 ± 0.11 dex, making it one of the most metal-poor UFDs. Grus I has a systemic radial velocity of −143.5 ± 1.2 km s−1 and a velocity dispersion of km s−1, which results in a dynamical mass of M ⊙ and a mass-to-light ratio of M/L V = M ⊙/L ⊙. Under the assumption of dynamical equilibrium, our analysis confirms that Grus I is a dark-matter-dominated UFD (M/L > 80 M ⊙/L ⊙). However, we do not resolve a metallicity dispersion (σ [Fe/H] < 0.44 dex). Our results indicate that Grus I is a fairly typical UFD with parameters that agree with mass–metallicity and metallicity-luminosity trends for faint galaxies. This agreement suggests that Grus I has not lost an especially significant amount of mass from tidal encounters with the Milky Way, in line with its orbital parameters. Intriguingly, Grus I has among the lowest central densities ( M ⊙ kpc−3) of the UFDs that are not known to be tidally disrupting. Models of the formation and evolution of UFDs will need to explain the diversity of these central densities, in addition to any diversity in the outer regions of these relic galaxies.

Three-dimensional structure of the Sagittarius dwarf spheroidal core from RR Lyrae

ABSTRACT We obtain distances to a sample of RR Lyrae in the central core of the Sagittarius dwarf spheroidal galaxy from OGLE data. We use these distances, along with RR Lyrae from Gaia DR2, to measure the shape of the stellar distribution within the central ∼2 kpc. The best-fitting stellar distribution is triaxial, with axis ratios 1 : 0.76 : 0.43. A prolate-spheroid model is ruled out at high statistical significance relative to the triaxial model. The major axis is aligned nearly parallel to the sky plane as seen by an Earth-based observer and is nearly perpendicular to the direction of the Galactic Centre. This result may be compared to cosmological simulations which generally predict that the major axis of the dark matter distribution of subhalos is aligned with the Galactic Centre. The triaxial structure that we obtain can provide important constraints on the Sagittarius progenitor, as well as the central dark matter distribution under the assumption of dynamical equilibrium.

WASP-128b: a transiting brown dwarf in the dynamical-tide regime

Massive companions in close orbits around G dwarfs are thought to undergo rapid orbital decay due to runaway tidal dissipation. We report here the discovery of WASP-128b, a brown dwarf discovered by the WASP survey transiting a G0V host on a $2.2\,\mathrm{d}$ orbit, where the measured stellar rotation rate places the companion in a regime where tidal interaction is dominated by dynamical tides. Under the assumption of dynamical equilibrium, we derive a value of the stellar tidal quality factor $\log{Q_\star'} = 6.96 \pm 0.19$. A combined analysis of ground-based photometry and high-resolution spectroscopy reveals a mass and radius of the host, $M_\star = 1.16 \pm 0.04\,M_\odot$, $R_\star = 1.16 \pm 0.02\,R_\odot$, and for the companion, $M_\mathrm{b} =37.5 \pm 0.8\,M_\mathrm{Jup}$, $R_\mathrm{b} = 0.94 \pm 0.02\,R_\mathrm{Jup}$, placing WASP-128b in the driest parts of the brown dwarf desert, and suggesting a mild inflation for its age. We estimate a remaining lifetime for WASP-128b similar to that of some ultra-short period massive hot Jupiters, and note it may be a propitious candidate for measuring orbital decay and testing tidal theories.

Charge dynamics of 57Fe probe atoms in La2Li0.5Cu0.5O4

The objective of this study is to characterize the electronic state and local surrounding of 57Fe Mössbauer probe atoms within iron-doped layered perovskite La2Li0.5Cu0.5O4 containing transition metal in unusual formal oxidation states “+3”. An approach based on the qualitative energy diagrams analysis and the calculations within the cluster configuration interaction method have been developed. It was shown that a large amount of charge is transferred via Cu-O bonds from the O: 2p bands to the Cu: 3d orbitals and the ground state is dominated by the d9L configuration (“Cu2+-O-” state). The dominant d9L ground state for the (CuO6) sublattice induces in the environment of the 57Fe probe cations a charge transfer Fe3+ + O−(L) → Fe4+ + O2−, which transforms “Fe3+” into “Fe4+” state. The experimental spectra in the entire temperature range 77–300 K were described with the use of the stochastic two-level model based on the assumption of dynamic equilibrium between two Fe3+↔Fe4+ valence states related to the iron atom in the [Fe(1)O4]4- center. The relaxation frequencies and activation energies of the corresponding charge fluctuations were estimated based on Mössbauer data. The results are discussed assuming a temperature-induced change in the electronic state of the [CuO4]5- clusters in the layered perovskite.

Impacts of inadequate historical disturbance data in the early twentieth century on modeling recent carbon dynamics (1951–2010) in conterminous U.S. forests

AbstractStand age and disturbance data have become more available in recent years and can facilitate modeling studies that integrate and quantify effects of disturbance and nondisturbance factors on carbon dynamics. Since high‐quality disturbance and forest age data to support forest dynamic modeling are lacking before 1950, we assumed dynamic equilibrium (carbon neutrality) for the starting conditions of forests with unknown historical disturbance and forest age information. The impacts of this assumption on forest carbon cycle estimation for recent decades have not been systematically examined. In this study, we tested an assumption of disequilibrium conditions for forests with unknown disturbance and age data by randomly assigning ages to them in the model initial year (1900) and analyzed uncertainties for 1951–2010 carbon dynamic simulations compared with the equilibrium assumption. Results show that with the dynamic equilibrium assumption, the total net biome productivity (NBP) of conterminous U.S. forests was 188 ± 60 Tg C yr−1 with 185 ± 56 Tg C yr−1 in living biomass and 3 ± 23 Tg C yr−1 in soil. The C release due to disturbance on average was about 68 ± 55 Tg C yr−1. The disequilibrium assumption causes annual NBP from 1951 to 2010 in conterminous U.S. forests to vary by an average of 13% with the largest impact on the soil carbon component. Uncertainties related to nondisturbance factors have relatively small impacts on NBP estimation (within 10%), while uncertainties related to disturbances cause biases in NBP of 4 to 28%. We conclude that the dynamic equilibrium assumption for the model initialization in 1900 is acceptable for simulating 1951–2010 forest carbon dynamics as long as disturbance and age data are reliable for this later period, although caution should be taken regarding the prior‐1950 simulation results because of their greater uncertainties.

Dimensional Analysis and Dissipation Rate Estimation in the Near Wake of a Circular Cylinder

A particle image velocimetry (PIV) experiment is performed for dissipation rate estimation in the near wake behind a circular cylinder with diameter D of 12 mm and corresponding Reynolds number of 7100. Considering the limitation of PIV resolution, a large eddy PIV method based on idea of large eddy simulation (LES), is used for more accurate estimation of dissipation rate. Based on the dynamic equilibrium assumption in the inertial subrange, the dissipation rate of the subgrid scales is approximated by the subgrid scale (SGS) flux, computed from PIV velocity fields and Smagorinsky model for SGS stress. A dimensional analysis about the integral length scale and the Kolmogorov length scale is discussed firstly to verify whether the dynamic equilibrium assumption holds or not.

MEASURING THE MASS DISTRIBUTION IN GALAXY CLUSTERS

Cluster mass profiles are tests of models of structure formation. Only two current observational methods of determining the mass profile, gravitational lensing and the caustic technique, are independent of the assumption of dynamical equilibrium. Both techniques enable determination of the extended mass profile at radii beyond the virial radius. For 19 clusters, we compare the mass profile based on the caustic technique with weak lensing measurements taken from the literature. This comparison offers a test of systematic issues in both techniques. Around the virial radius, the two methods of mass estimation agree to within about 30%, consistent with the expected errors in the individual techniques. At small radii, the caustic technique overestimates the mass as expected from numerical simulations. The ratio between the lensing profile and the caustic mass profile at these radii suggests that the weak lensing profiles are a good representation of the true mass profile. At radii larger than the virial radius, the lensing mass profile exceeds the caustic mass profile possibly as a result of contamination of the lensing profile by large-scale structures within the lensing kernel. We highlight the case of the closely neighboring clusters MS0906+11 and A750 to illustrate the potential seriousness of contamination of the the weak lensing signal by unrelated structures.

Hydraulic conductivity of the High Plains Aquifer re‐evaluated using surface drainage patterns

The High Plains Aquifer (HPA), underlying parts of 8 states from South Dakota to Texas, is one of the largest fresh water aquifers in the world and accounts for 30% of the groundwater used for irrigation in the US. Determining the distribution of HPA's hydraulic conductivity (K) is critical for water management and addressing water quality issues. K is traditionally estimated from well pumping data coupled with computer modeling and is known to be highly variable, spanning several orders of magnitude for the same type of rock. Here we show that applying our innovative method of determining effective horizontal K to HPA (based on surface drainage patterns and a dynamic equilibrium assumption) produced results generally consistent with those from traditional methods but reveals much more detailed spatial variation. With the exception of a few places such as the Sand Hills area, our results also show for the first time (to the best of our knowledge) a distinct relationship between surface stream drainage density and subsurface aquifer K in a major aquifer system on a regional scale. Because aquifer particle size strongly controls K, our results can be used to study patterns of past sediment movement and deposition.

Estimation of Viscous Dissipative Stresses Induced by a Mechanical Heart Valve Using PIV Data

Among the clinical complications of mechanical heart valves (MHVs), hemolysis was previously thought to result from Reynolds stresses in turbulent flows. A more recent hypothesis suggests viscous dissipative stresses at spatial scales similar in size to red blood cells may be related to hemolysis in MHVs, but the resolution of current instrumentation is insufficient to measure the smallest eddy sizes. We studied the St. Jude Medical (SJM) 27 mm valve in the aortic position of a pulsatile circulatory mock loop under physiologic conditions with particle image velocimetry (PIV). Assuming a dynamic equilibrium assumption between the resolved and sub-grid-scale (SGS) energy flux, the SGS energy flux was calculated from the strain rate tensor computed from the resolved velocity fields and the SGS stress was determined by the Smagorinsky model, from which the turbulence dissipation rate and then the viscous dissipative stresses were estimated. Our results showed Reynolds stresses up to 80 N/m2 throughout the cardiac cycle, and viscous dissipative stresses below 12 N/m2. The viscous dissipative stresses remain far below the threshold of red blood cell hemolysis, but could potentially damage platelets, implying the need for further study in the phenomenon of MHV hemolytic complications.

ASSESSING RECOVERY FOLLOWING ENVIRONMENTAL ACCIDENTS: ENVIRONMENTAL VARIATION, ECOLOGICAL ASSUMPTIONS, AND STRATEGIES

Gauging whether or when a population, species, or community recovers from an environmental accident or disturbance, such as an oil spill or forest fire, is complicated by environmental variation in time and space, and therefore depends on the assumptions one makes about equilibrium. These ecological assumptions about equilibrium affect how one designs and interprets studies to assess recovery from environmental accidents or disturbances. We use examples from studies conducted following the Exxon Valdezoil spill to illustrate several approaches to assessing recovery and their sensitivity to the form of equilibrium one assumes. Baseline study designs, which compare levels of a resource after the disturbance to pre-disturbance levels for impact data only, are generally inadequate because they rest on the unrealistic assumption of steady-state equilibrium. Since data for the impact area only are used, recovery and temporal variation are confounded. Unlike baseline designs, before-after control-impact (BACI) designs use impact and reference data, and relax this sensitivity by incorporating both temporal and spatial variation. Studies that compare impacted with reference areas in a single year following the disturbance assume spatial equilibrium and therefore may confound recovery with systematic spatial differences between the areas. Sampling and analytical strategies such as stratified random sampling or the use of environmental measures as covariates may lessen the sensitivity to this as- sumption. Multiyear studies that include comparisons between impacted and reference areas or that sample areas along a gradient of disturbance rest on the more realistic assumption of dynamic equilibrium. Understanding the underlying assumptions and how they relate to the approach one uses must be part of assessing the recovery of biological resources from an environmental accident. Because the dynamics of different populations, species, and communities and the environments they occupy vary and exhibit different dependencies on the scale of distur- bance (and the scale of analysis), there is no single ''best'' approach to assessing recovery. Discussions about recovery should include an explicit and honest consideration of the underlying ecological assumptions, the likelihood that they hold in the system being studied, and the consequences if the assumptions are violated.

New Insights on the Draco Dwarf Spheroidal Galaxy from the Sloan Digital Sky Survey: A Larger Radius and No Tidal Tails

We have investigated the spatial extent and structure of the Draco dwarf spheroidal galaxy by using deep wide-field multicolor CCD photometry from the Sloan Digital Sky Survey (SDSS). Our study covers an area of 27 deg2 around the center of the Draco dwarf and reaches 2 mag below the level of Draco's horizontal branch. The SDSS photometry allows very effective filtering in color-magnitude space. With such filtering the density of the foreground of Galactic field stars is decreased by more than an order of magnitude, and the stellar population of the Draco dwarf galaxy thus stands out with much higher contrast than in former investigations. We show that the spatial distribution of Draco's red giants, red horizontal-branch stars, and subgiants down to i* = 21.7 mag does not provide evidence for the existence of tidally induced tails or a halo of unbound stars. The projected surface density of the dwarf galaxy is flattened with a nearly constant ellipticity of 0.29 ± 0.02 at position angle 88° ± 3°. The radial profile can be fitted by King models, as well as by a generalized exponential. Using the empirical King profile, the core radius and the limiting (or tidal) radius along the major axis are rc = 77 and rt = 401, respectively; the latter means that the size of the Draco dwarf galaxy is 40% larger than previously measured. Fitting the profile of King's theoretical models yields a still larger limiting radius of rt = 495. There is no clear indication of a taillike extension of the Draco population beyond this radius. A break in the radial surface density profile, which might indicate a halo of extratidal stars, is also not found in our Draco data. We conclude that down to the above magnitude limit tidal effects can exist only at a level of 10-3 of the central surface density of Draco or below. The regular structure of Draco found from the new data argues against its being a portion of an unbound tidal stream and lends support to the assumption of dynamical equilibrium, which is the basis for mass estimation. The changes in the values for the core radius and limiting radius imply that the total mass of Draco is higher by more than a factor of 2. Using a King spherical model of equivalent size as a reference and adopting a line-of-sight velocity dispersion of either 10.7 or 8.5 km s-1, we derive estimates of the total mass within radius rt of (3.5 ± 0.7) × 107 M⊙ and (2.2 ± 0.5) × 107 M⊙, respectively. From the combined i-band flux of all possible Draco members that lie within major-axis radius rt we determine the total luminosity of the Draco dwarf galaxy as (L/L⊙)i = (2.4 ± 0.5) × 105. This includes corrections for the flux of the foreground stars and the unseen fainter part of the Draco population. We thus obtain overall mass-to-light ratios M/Li of 146 ± 42, or 92 ± 28 in solar units. In summary, our results strengthen the case for a strongly dark matter–dominated, bound stellar system.

Saline water intrusion adjacent to the Fraser River, Richmond, British Columbia

During periods of average to low flow in the Fraser River, seawater from Georgia Strait migrates with a rising tide up the river channel, reaching distances up to 16 km from the river's outlet to the ocean. This relatively dense water infiltrates through the river bed and circulates as a stable wedge of saline water within permeable deltaic deposits adjacent to the river. At the Kidd2 site in Richmond, the wedge extends approximately 500 m inland from the river. The top of the transition zone between fresh and saline water is 10 m below ground surface. The bottom of the saline wedge is located at the contact with underlying low-permeability delta slope deposits, resulting in a wedge with a vertical extent of approximately 10 m. Maximum salinity values observed in the wedge (16–17 parts per thousand) reflect the time-averaged response to density-driven flow under a complex sequence of daily and seasonal variations in salinity within the Fraser River and mixing with fresh river water that may enter the aquifer during the summer freshet. Under the assumption of dynamic equilibrium, simulation of the saline wedge at the Kidd2 site suggests (1) the effective, long-term average salinity at the base of the river channel is approximately 19 parts per thousand; (2) the velocity of groundwater, directed inland in the central portion of the wedge, is in the range from 0.5 to 6 m/year; and (3) fresh water in the confined deltaic sands moves from inland areas toward the Fraser River under a hydraulic gradient of approximately 3 × 10–4.Key words: salt water intrusion, deltaic sediments, modeling.

Mass estimation in the outer regions of galaxy clusters

We present a technique for estimating the mass in the outskirts of galaxy clusters where the usual assumption of dynamical equilibrium is not valid. The method assumes that clusters form through hierarchical clustering and requires only galaxy redshifts and positions on the sky. We apply the method to dissipationless cosmological N-body simulations where galaxies form and evolve according to semi-analytic modelling. The method recovers the actual cluster mass profile within a factor of 2 to several megaparsecs from the cluster centre. This error originates from projection effects, sparse sampling, and contamination by foreground and background galaxies. In the absence of velocity biases, this method can provide an estimate of the mass-to-light ratio on scales ∼1–10 h−1 Mpc where this quantity is still poorly known.

Cenozoic dynamics of the African plate with emphasis on the Africa‐Eurasia collision

We investigate the forces acting on the African plate from about 56 Ma to the present‐day. We aim (1) to establish a basis for quantification of the overall dynamics of the Mediterranean/Middle East region and (2) to understand how Africa reacts to temporal variation in the forces associated with the Africa‐Eurasia collision. We use the assumption of dynamical equilibrium which implies that at any given time, the net torque of forces on Africa's northern margin must have been balanced by the other torques acting on the plate. Moreover, temporal variation in the forces on Africa's northern margin must have been balanced by contemporaneous changes in one or more of the other torques. By quantifying those “other” torques we may derive a constraint on the forces at the northern margin. We expect the Africa‐Eurasia collision to have been associated with changes in the associated forces and examine how these changes are balanced. Using alternative parameterizations, we calculate the net torque of ridge push, transform fault resistance, and basal drag. We also quantify changes in the torque of forces associated with the continent‐ocean transition. The results constitute a framework for detailed dynamic models of the Africa‐Eurasia convergence zone. Temporal changes in the ridge push torque prove to be small. The torque of transform resistance does change, but these changes do not relate to forces at the northern margin. Evidence for a relation with collision is found only for the case of basal drag. The results point to a mechanism where the plate maintains its dynamical equilibrium mainly through collision‐induced changes in absolute motion.

A Century of Turnover: Community Dynamics at Three Timescales

We calculated the observed species turnover of the bird communities on 13 small islands off the coast of the British Isles and the Republic of Ireland for different census intervals. For seven of these islands, the maximum intervals exceeded 80 years. We developed a non-linear, asymptotic model to describe how observed turnover should change with census interval. Our assumptions were traditional ones, based upon the assumption of dynamic equilibrium in familiar island biogeography theory, even though we knew that such equilibrium was rare in these islands. Furthermore, the model considered the average dynamics of the species present and not the dynamics of the individual species. This model showed that neither the observed turnover calculated over different intervals, nor the turnover rate obtained by dividing this observed turnover by the interval, are statistics that permit comparison between islands. Observed turnover increased over time, thus 1-year turnover underestimated the turnovers over a decade or a century. The increase was less than linear, however, so dividing observed turnover by the number of years in its calculation produced a statistic that declined progressively with that number. The model provided a significant overall fit to the data, but underestimated turnover at both the shortest and longest census intervals. We modified the model to reduce the amount of underestimation, by incorporating the long-term changes in the number of species on each island and hence removing the assumption of equilibrium. This non-equilibrium model provided a much improved fit to the data, but it still failed to describe turnover at the very shortest intervals. These, however, are known from other studies to be inflated by individuals-floaters-that nest only once or twice on the islands. The improved, non-equilibrium model made good predictions of the observed turnover over a 4-year interval. These predictions may be used to compare islands, including those for which empirical data on 4-year turnovers are sparse. We divided the non-equilibrium model into an intrinsic and an extrinsic component, representing the influence of within-community and external factors, respectively. Even the intrinsic components of turnover are large, involving differences in species composition of 6-36% between widely separated censuses. How these intrinsic components of turnover vary from island to island is not clear because previous studies have been unable to compare turnover at different time scales. The number of islands in this study was too few for this purpose and we leave a more broadly based comparison for a future paper.

Quantification of soil production and downslope creep rates from cosmogenic 10Be accumulations on a hillslope profile

Research Article| April 01, 1993 Quantification of soil production and downslope creep rates from cosmogenic 10Be accumulations on a hillslope profile James A. McKean; James A. McKean 1Department of Geology and Geophysics, University of California, Berkeley, California 94720 Search for other works by this author on: GSW Google Scholar William E. Dietrich; William E. Dietrich 1Department of Geology and Geophysics, University of California, Berkeley, California 94720 Search for other works by this author on: GSW Google Scholar Robert C. Finkel; Robert C. Finkel 2Center for Accelerator Mass Spectrometry, Lawrence Livermore National Laboratory, Livermore, California 94550 Search for other works by this author on: GSW Google Scholar John R. Southon; John R. Southon 2Center for Accelerator Mass Spectrometry, Lawrence Livermore National Laboratory, Livermore, California 94550 Search for other works by this author on: GSW Google Scholar Marc W. Caffee Marc W. Caffee 2Center for Accelerator Mass Spectrometry, Lawrence Livermore National Laboratory, Livermore, California 94550 Search for other works by this author on: GSW Google Scholar Geology (1993) 21 (4): 343–346. https://doi.org/10.1130/0091-7613(1993)021<0343:QOSPAD>2.3.CO;2 Article history first online: 02 Jun 2017 Cite View This Citation Add to Citation Manager Share Icon Share Facebook Twitter LinkedIn MailTo Tools Icon Tools Get Permissions Search Site Citation James A. McKean, William E. Dietrich, Robert C. Finkel, John R. Southon, Marc W. Caffee; Quantification of soil production and downslope creep rates from cosmogenic 10Be accumulations on a hillslope profile. Geology 1993;; 21 (4): 343–346. doi: https://doi.org/10.1130/0091-7613(1993)021<0343:QOSPAD>2.3.CO;2 Download citation file: Ris (Zotero) Refmanager EasyBib Bookends Mendeley Papers EndNote RefWorks BibTex toolbar search Search Dropdown Menu toolbar search search input Search input auto suggest filter your search All ContentBy SocietyGeology Search Advanced Search Abstract Average soil transport rates over a period of ∼3500 yr on a convex soil-mantled hillslope have been quantified using a mass-balance model that incorporates the soil concentration of the cosmogenic isotope 10Be. The 10Be model results support the assumption used in most geomorphic models that the soil creep rate is proportional to surface gradient. The predicted diffusion coefficient is 360 ±55 cm3 ⋅ yr-1 ⋅ cm-1 contour length and the average rate of soil production is 0.026 ±0.007 cm/yr. Within the uncertainty of this technique, the data do not reject G. K. Gilbert's hypothesis that some hillslopes may exist in a condition of dynamic equilibrium with a uniform soil production rate. However, the model does not require an assumption of dynamic equilibrium and may be an approach that uniquely allows the quantification of a local soil-production rate law. This content is PDF only. Please click on the PDF icon to access. First Page Preview Close Modal You do not have access to this content, please speak to your institutional administrator if you feel you should have access.

Structure and Evolution of Globular Clusters.

view Abstract Citations References Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Structure and Evolution of Globular Clusters. Michie, Richard W. Abstract A three-parameter distribution function in terms of energy and angular momentum and with a cutoff at the energy of escape is used to describe the structure of a spherical stellar system. The corresponding density distribution for a specific set of parameter values is obtained numerically and the result agrees very well with observations of 47 Tuc from one to 50 parsecs by Gascoigne and Burr. In obtaining the evolution of a spherical cluster, the basic assumption is that the continuous evolutionary process can be approximated by a series of successive equilibrium states. Therefore, the distribution function is assumed invariant in form and the problem is the calculation of the changes in the three parameters after a time interval. This is done by taking three moments of the Boltzmann equation with the assumption of dynamical equilibrium during the interval of time. These moments are mass, the sum over all stars of the kinetic energy, and the sum over all stars of the angular momentum squared. An evolutionary time step of three quarters of a billion years is obtained numerically with an IBM 650. The results show a flow of stars inward within one parsec from the center and an expansion beyond twenty parsecs. The core of the cluster (within 1.6 parsecs from the center) contracts. In the inner regions the mean-square velocity decreases, and beyond twenty parsecs the mean-square velocity increases. The isotropy of the distribution in velocity space increases in the inner regions and decreases for the outer sections. This result can be explained by a few stars suffering encounters and being thrown into orbits of high energy and low angular momentum. Publication: The Astronomical Journal Pub Date: 1961 DOI: 10.1086/108494 Bibcode: 1961AJ.....66Q..49M full text sources ADS |