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At least 109 records · Page 6
All sourcesNASA NTRSDOE OSTI
DOE OSTI2014

MAGNETOHYDRODYNAMIC MODELING OF THE ACCRETION SHOCKS IN CLASSICAL T TAURI STARS: THE ROLE OF LOCAL ABSORPTION IN THE X-RAY EMISSION

We investigate the properties of X-ray emission from accretion shocks in classical T Tauri stars (CTTSs), generated where the infalling material impacts the stellar surface. Both observations and models of the accretion process reveal several aspects that are still unclear: the observed X-ray luminosity in accretion shocks is below the predicted value, and the density versus temperature structure of the shocked plasma, with increasing densities at higher temperature, deduced from the observations, is at odds with that proposed in the current picture of accretion shocks. To address these open issues, we investigate whether a correct treatment of the local absorption by the surrounding medium is crucial to explain the observations. To this end, we describe the impact of an accretion stream on a CTTS by considering a magnetohydrodynamic model. From the model results, we synthesize the X-ray emission from the accretion shock by producing maps and spectra. We perform density and temperature diagnostics on the synthetic spectra, and we directly compare the results with observations. Our model shows that the X-ray fluxes inferred from the emerging spectra are lower than expected because of the complex local absorption by the optically thick material of the chromosphere and of the unperturbed stream. Moreover, our model, including the effects of local absorption, explains in a natural way the apparently puzzling pattern of density versus temperature observed in the X-ray emission from accretion shocks.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2015

AS ABOVE, SO BELOW: EXPLOITING MASS SCALING IN BLACK HOLE ACCRETION TO BREAK DEGENERACIES IN SPECTRAL INTERPRETATION

Over the past decade, evidence has mounted that several aspects of black hole (BH) accretion physics proceed in a mass-invariant way. One of the best examples of this scaling is the empirical “fundamental plane of BH accretion” relation linking mass, radio, and X-ray luminosity over eight orders of magnitude in BH mass. The currently favored theoretical interpretation of this relation is that the physics governing power output in weakly accreting BHs depends more on relative accretion rate than on mass. In order to test this theory, we explore whether a mass-invariant approach can simultaneously explain the broadband spectral energy distributions from two BHs at opposite ends of the mass scale but that are at similar Eddington accretion fractions. We find that the same model, with the same value of several fitted physical parameters expressed in mass-scaling units to enforce self-similarity, can provide a good description of two data sets from V404 Cyg and M81*, a stellar and supermassive BH, respectively. Furthermore, only one of several potential emission scenarios for the X-ray band is successful, suggesting it is the dominant process driving the fundamental plane relation at this accretion rate. This approach thus holds promise for breaking current degeneracies in the interpretation of BH high-energy spectra and for constructing better prescriptions of BH accretion for use in various local and cosmological feedback applications.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2016

The reflares and outburst evolution in the accreting millisecond pulsar SAX J1808.4–3658: A disk truncated near co-rotation?

The accreting millisecond X-ray pulsar SAX J1808.4–3658 shows peculiar low luminosity states known as “reflares” after the end of the main outburst. During this phase the X-ray luminosity of the source varies by up to three orders of magnitude in less than 1–2 days. The lowest X-ray luminosity observed reaches a value of ∼10{sup 32} erg s{sup −1}, only a factor of a few brighter than its typical quiescent level. We investigate the 2008 and 2005 reflaring state of SAX J1808.4–3658 to determine whether there is any evidence for a change in the accretion flow with respect to the main outburst. We perform a multiwavelength photometric and spectral study of the 2005 and 2008 reflares with data collected during an observational campaign covering the near-infrared, optical, ultra-violet and X-ray band. We find that the NIR/optical/UV emission, expected to come from the outer accretion disk, shows variations in luminosity over an order of magnitude. The corresponding X-ray luminosity variations are instead much deeper, spanning about 2–3 orders of magnitude. The X-ray spectral state observed during the reflares does not change substantially with X-ray luminosity, indicating a rather stable configuration of the accretion flow. We investigate the most likely configuration of the innermost regions of the accretion flow and we infer an accretion disk truncated at or near the co-rotation radius. We interpret these findings as due to either a strong outflow (due to a propeller effect) or a trapped disk (with limited/no outflow) in the inner regions of the accretion flow.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2016

TESTING THE PROPAGATING FLUCTUATIONS MODEL WITH A LONG, GLOBAL ACCRETION DISK SIMULATION

The broadband variability of many accreting systems displays characteristic structures; log-normal flux distributions, root-mean square (rms)-flux relations, and long inter-band lags. These characteristics are usually interpreted as inward propagating fluctuations of the mass accretion rate in an accretion disk driven by stochasticity of the angular momentum transport mechanism. We present the first analysis of propagating fluctuations in a long-duration, high-resolution, global three-dimensional magnetohydrodynamic (MHD) simulation of a geometrically thin ( h / r ≈ 0.1) accretion disk around a black hole. While the dynamical-timescale turbulent fluctuations in the Maxwell stresses are too rapid to drive radially coherent fluctuations in the accretion rate, we find that the low-frequency quasi-periodic dynamo action introduces low-frequency fluctuations in the Maxwell stresses, which then drive the propagating fluctuations. Examining both the mass accretion rate and emission proxies, we recover log-normality, linear rms-flux relations, and radial coherence that would produce inter-band lags. Hence, we successfully relate and connect the phenomenology of propagating fluctuations to modern MHD accretion disk theory.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2017

The accretion rate dependence of burst oscillation amplitude

Neutron stars in low-mass X-ray binaries exhibit oscillations during thermonuclear bursts, attributed to asymmetric brightness patterns on the burning surfaces. All models that have been proposed to explain the origin of these asymmetries (spreading hotspots, surface waves, and cooling wakes) depend on the accretion rate. By analysis of archival RXTE data of six oscillation sources, we investigate the accretion rate dependence of the amplitude of burst oscillations. This more than doubles the size of the sample analyzed previously by Muno et al., who found indications for a relationship between accretion rate and oscillation amplitudes. We find that burst oscillation signals can be detected at all observed accretion rates. Moreover, oscillations at low accretion rates are found to have relatively small amplitudes (A{sub rms}⩽0.10) while oscillations detected in bursts observed at high accretion rates cover a broad spread in amplitudes (0.05⩽A{sub rms}⩽0.20). In this paper we present the results of our analysis and discuss these in the light of current burst oscillation models. Additionally, we investigate the bursts of two sources without previously detected oscillations. Despite the fact that these sources have been observed at accretion rates where burst oscillations might be expected, we find their behavior not to be anomalous compared to oscillation sources.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2017

Radiation-driven Turbulent Accretion onto Massive Black Holes

Accretion of gas and interaction of matter and radiation are at the heart of many questions pertaining to black hole (BH) growth and coevolution of massive BHs and their host galaxies. To answer them, it is critical to quantify how the ionizing radiation that emanates from the innermost regions of the BH accretion flow couples to the surrounding medium and how it regulates the BH fueling. In this work, we use high-resolution three-dimensional (3D) radiation-hydrodynamic simulations with the code Enzo , equipped with adaptive ray-tracing module Moray , to investigate radiation-regulated BH accretion of cold gas. Our simulations reproduce findings from an earlier generation of 1D/2D simulations: the accretion-powered UV and X-ray radiation forms a highly ionized bubble, which leads to suppression of BH accretion rate characterized by quasi-periodic outbursts. A new feature revealed by the 3D simulations is the highly turbulent nature of the gas flow in vicinity of the ionization front. During quiescent periods between accretion outbursts, the ionized bubble shrinks in size and the gas density that precedes the ionization front increases. Consequently, the 3D simulations show oscillations in the accretion rate of only ∼2–3 orders of magnitude, significantly smaller than 1D/2D models. We calculate the energy budget of the gas flow and find that turbulence is the main contributor to the kinetic energy of the gas but corresponds to less than 10% of its thermal energy and thus does not contribute significantly to the pressure support of the gas.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2022

Flux Eruption Events Drive Angular Momentum Transport in Magnetically Arrested Accretion Flows

We evolve two high-resolution general relativistic magnetohydrodynamic simulations of advection-dominated accretion flows around nonspinning black holes (BHs), each over a duration ~3 × 10 5 GM BH /c 3 . One model captures the evolution of a weakly magnetized (SANE) disk and the other that of a magnetically arrested disk (MAD). Magnetic flux eruptions in the MAD model push out gas from the disk and launch strong winds with outflow efficiencies at times reaching 10% of the incoming accretion power. Despite the substantial power in these winds, average mass outflow rates remain low out to a radius ~100GM BH /c 2 , only reaching ~60%–80% of the horizon accretion rate. The average outward angular momentum transport is primarily radial in both modes of accretion, but with a clear distinction: magnetic flux eruption–driven disk winds cause a strong vertical flow of angular momentum in the MAD model, while for the SANE model, the magnetorotational instability (MRI) moves angular momentum mostly equatorially through the disk. Further, we find that the MAD state is highly transitory and nonaxisymmetric, with the accretion mode often changing to a SANE-like state following an eruption before reattaining magnetic flux saturation with time. The Reynolds stress changes directions during such transitions, with the MAD (SANE) state showing an inward (outward) stress, possibly pointing to intermittent MRI-driven accretion in MADs. Pinning down the nature of flux eruptions using next-generation telescopes will be crucial in understanding the flow of mass, magnetic flux, and angular momentum in sub-Eddington accreting BHs like M87* and Sagittarius A*.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2024

Planet Formation by Gas-assisted Accretion of Small Solids

We compute the accretion efficiency of small solids, with radii 1 cm ≤ R s ≤ 10 m, on planets embedded in gaseous disks. Planets have masses 3 ≤ M p ≤ 20 Earth masses (M ⊕ ) and orbit within 10 au of a solar mass star. Disk thermodynamics is modeled via 3D radiation-hydrodynamics calculations that typically resolve the planetary envelopes. Both icy and rocky solids are considered, explicitly modeling their thermodynamic evolution. The maximum efficiencies of 1 ≤ R s ≤ 100 cm particles are generally ≲10%, whereas 10 m solids tend to accrete efficiently or be segregated beyond the planet’s orbit. A simplified approach is applied to compute the accretion efficiency of small cores, with masses M p ≤ 1 M ⊕ and without envelopes, for which efficiencies are approximately proportional to $M^{2/3}_{p}$. The mass flux of solids, estimated from unperturbed drag-induced drift velocities, provides typical accretion rates dM p /dt ≲ 10 -5 M ⊕ yr -1 . In representative disk models with an initial gas-to-dust mass ratio of 70–100 and total mass of 0.05–0.06 M ⊙ , the solids’ accretion falls below 10 -6 M ⊕ yr -1 after 1–1.5 Myr. The derived accretion rates, as functions of time and planet mass, are applied to formation calculations that compute dust opacity self-consistently with the delivery of solids to the envelope. Assuming dust-to-solid coagulation times of ≈0.3 Myr and disk lifetimes of ≈3.5 Myr, heavy-element inventories in the range 3–7 M ⊕ require that ≈90–150 M ⊕ of solids cross the planet’s orbit. The formation calculations encompass a variety of outcomes, from planets a few times M ⊕ , predominantly composed of heavy elements, to giant planets. The peak luminosities during the epoch of the solids’ accretion range from ≈10 -7 to ≈10 -6 L ⊙ .

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2017

On the accretion rates and radiative efficiencies of the highest-redshift quasars

We estimate the accretion rates onto the supermassive black holes that power 20 of the highest-redshift quasars, at z≳5.8, including the quasar with the highest redshift known to date—ULAS J1120 at z = 7.09. The analysis is based on the observed (rest-frame) optical luminosities and reliable “virial” estimates of the BH masses of the quasars, and utilizes scaling relations derived from thin accretion disk theory. The mass accretion rates through the postulated disks cover a wide range, M-dot {sub disk}≃4--190 M{sub ⊙} yr{sup −1}, with most of the objects (80%) having M-dot {sub disk}≃10--65 M{sub ⊙} yr{sup −1}, confirming the Eddington-limited nature of the accretion flows. By combining our estimates of M-dot {sub disk} with conservative, lower limits on the bolometric luminosities of the quasars, we investigate which alternative values of η best account for all the available data. We find that the vast majority of quasars (∼85%) can be explained with radiative efficiencies in the range η≃0.03--0.3, with a median value close to the commonly assumed η = 0.1. Within this range, we obtain conservative estimates of η≳0.14 for ULAS J1120 and SDSS J0100 (at z = 6.3), and of ≳0.19 for SDSS J1148 (at z=6.41; assuming their BH masses are accurate). The implied accretion timescales are generally in the range t{sub acc}≡M{sub BH}/ M-dot {sub BH}≃0.1--1 Gyr, suggesting that most quasars could have had ∼1--10 mass e-foldings since BH seed formation. Our analysis therefore demonstrates that the available luminosities and masses for the highest-redshift quasars can be explained self-consistently within the thin, radiatively efficient accretion disk paradigm. Episodes of radiatively inefficient, “super-critical” accretion may have occurred at significantly earlier epochs (i.e., z≳10).

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2023

Stable accretion in young stars: the cases of EX Lupi and TW Hya

ABSTRACT We examine the long-term spectroscopic and photometric variability of EX Lupi and TW Hya, studying the presence of stable accretion and the role it plays in the observed variability. Analysing the velocity modulations of the emission lines with star-melt, we obtain information on the structure of the accretion columns and the disc–star connection. The emission-line radial velocities reveal that TW Hya, like EX Lupi, has a remarkably stable or slow-varying accretion column footprint, locked to the star for several years. The line-emitting regions are non-polar for both EX Lupi and TW Hya, and species with different energies differ in position. In contrast, the continuum emission as observed in the photometry is very variable and can be modelled by hotspot(s) that change over time in phase, shape, temperature, size, and location with respect to the emission-line region. The continuum emission region may not be limited to the stellar surface, especially during episodes of high accretion. The broad-line emission observed in EX Lupi during episodes of increased accretion reveals a further structure, which can be fitted by non-axisymmetric disc in Keplerian rotation inwards of the corotation radius. Since the radial velocity modulation due to accretion footprints is so stable, we used it to search for further velocity modulations. While no residual modulation (other than caused by stellar rotation) is found in these objects, a similar analysis could help to identify young planets/companions. Therefore, determining whether stable accretion footprints are common among young stars is a key to detect young planets.

Sicilia-Aguilar, A.↗
DOE OSTI2017

The Formation of Uranus and Neptune: Fine-tuning in Core Accretion

Uranus and Neptune are ice giants with ∼15% atmospheres by mass, which places them in a category intermediate between rocky planets and gas giants. These atmospheres are too massive to have been primarily outgassed, but they never underwent runaway gas accretion. The ice giants never reached critical core mass (M{sub crit}) in a full gas disk, but their cores are ≳M{sub crit}, suggesting that their envelopes were mainly accreted at the end of the disk lifetime. Pebble accretion calls into question traditional slow atmospheric growth during this phase. We show that the full-sized ice giants predominantly accreted gas from a disk depleted by at least a factor of ∼100. Such a disk dissipates in ≲10{sup 5} years. Why would both cores stay subcritical for the entire ∼Myr disk lifetime, only to reach M{sub crit} in the final 10{sup 5} years? This is fine tuned. Ice giants in the outer disk have atmospheric mass fractions comparable to the disk gas-to-solid ratio during the bulk of their gas accretion. This point in disk evolution coincides with a dynamical upheaval: the gas loses its ability to efficiently damp the core random velocities, allowing them to be gravitationally excited by Jupiter and Saturn. We suggest that the ice giants’ cores began growing on closer-in orbits (staying subcritical), and migrated out during this dynamical instability. There, their orbits circularized after accreting much of their mass in solids. Finally, they accreted their envelopes from a depleted nebula, where the sparseness of feeding-zone gas prevented runaway.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2017

HST/COS observations of ionized gas accretion at the disk-halo interface of M33

We report the detection of accreting ionized gas at the disk–halo interface of the nearby galaxy M33. We analyze Hubble Space Telescope/Cosmic Origins Spectrograph absorption-line spectra of seven ultraviolet-bright stars evenly distributed across the disk of M33. We find Si iv absorption components consistently redshifted relative to the bulk M33's ISM absorption along all the sightlines. The Si iv detection indicates an enriched, disk-wide, ionized gas inflow toward the disk. This inflow is most likely multi-phase as the redshifted components can also be observed in ions with lower ionization states (e.g., S ii, P ii, Fe ii, Si ii). Kinematic modeling of the inflow is consistent with an accreting layer at the disk–halo interface of M33, which has an accretion velocity of 110{sub −20}{sup +15} km s{sup −1} at a distance of 1.5{sub −1.0}{sup +1.0} kpc above the disk. The modeling indicates a total mass of ∼3.9 × 10{sup 7} M {sub ⊙} for the accreting material at the disk–halo interface on the near side of the M33 disk, with an accretion rate of ∼2.9 M {sub ⊙} yr{sup −1}. The high accretion rate and the level of metal enrichment suggest the inflow is likely to be the fallback of M33 gas from a galactic fountain and/or the gas pulled loosed during a close interaction between M31 and M33. Our study of M33 is the first to unambiguously reveal the existence of a disk-wide, ionized gas inflow beyond the Milky Way, providing a better understanding of gas accretion in the vicinity of a galaxy disk.

79 ASTRONOMY AND ASTROPHYSICS↗
NASA NTRS1974

A model for accretion of the terrestrial planets

One possible origin of the terrestrial planets involves their formation by gravitational accretion of particles originally in Keplerian orbits about the sun. Some implications of this theory are considered. A formal expression for the rate of mass accretion by a planet is developed. The formal singularity of the gravitational collision cross section for low relative velocities is shown to be without physical significance when the accreting bodies are in heliocentric orbits. The distribution of particle velocities relative to an accreting planet is considered; the mean velocity increases with time. The internal temperature of an accreting planet is shown to depend simply on the accretion rate. A simple and physically reasonable approximate expression for a planetary accretion rate is proposed.

Weidenschilling, S. J.↗
NASA NTRS1990

Stability analysis of the accretion line

The stability of the flow along the accretion line in the Bondi-Hoyle-Lyttleton-type accretion flow is studied. The WKB approximation is used to analyze short wavelength linear perturbations of the axisymmetric steady state flow. The viscosity terms in the gasdynamical equations are included in the analysis. The radial modes are studied both in two- and three-dimensional accretion flow. Since a displaced accretion line cannot be defined in three-dimensional flow, the tangential modes are only analyzed in the two-dimensional flow. The analysis of the perturbed gasdynamical equations are made simpler as a result of the radial and tangential modes being decoupled in the linear regime. As was shown before, the three-dimensional accretion flow is unstable against radial modes. The two-dimensional accretion flow is found to be unstable against tangential modes, as well as against radial modes. Using typical numerical parameters which are in common use, the wavelength which gives the maximum growth rate of a small perturbation is found to be about 0.5 r(a), where r(a) is the accretion radius.

Soker, Noam↗
NASA NTRS1991

Steady spherical hypercritical accretion onto neutron stars

The present study of hypercritical accretion flows onto neutron stars considers steady-state, spherically symmetric flows whose accretion rate range is characterized by the carrying away of gravitational-accretion energy by neutrinos. The models used encompass pair production, radiation diffusion, and general relativistic effects. While pair pressure dominates throughout the accretion envelope when accretion rates above about 100 solar masses/yr, radiation diffusion becomes important when the accretion rate falls below about 0.001 solar masses/yr. At the lower accretion rates, free fall toward the neutron-star surface stops and an extended, quasi-static, radiation pressure-supported envelope emerges which is probably dynamically unstable.

Houck, John C.↗
NASA NTRS1993

Angular momentum regulation in low-mass young stars surrounded by accretion disks

From study of a sample of 34 T Tauri stars with photometrically derived rotation periods and spectral types later than KS, we find that the observed periods appear to be related to the presence or absence of an accretion disk. Those stars which we infer to be surrounded by accretion disks have rotation periods P(rot) over 4 days with a most probable P(rot) of about 8.5 days, while those stars which lack accretion disk signatures cover a wide range of P(rot) from 1.5 to 16 days, including a significant number of objects with P(rot) less than 4 days. This suggests the possibility that the 'initial' angular momentum of a star is not established until it dissipates its circumstellar accretion disk. During the disk accretion phase, the stellar angular velocity appears to be regulated at a low value, countering the tendency of the star to spin up both from contraction toward the main sequence and from the accretion of inner disk material of high specific angular momentum. When the accretion disk is dissipated, this regulation mechanism will cease to function. At this point, the star is no longer maintained at a low angular velocity, but is 'free' to conserve its angular momentum, and thus to increase its angular velocity in response to contraction and changes in moment of inertia. This hypothesis, combined with a spread in disk dispersal time scales, provides a context for explaining the observed distribution of stellar rotational velocities for stars on the ZAMS in young clusters.

Edwards, Suzan↗
NASA NTRS1993

The evidence for clumpy accretion in the Herbig Ae star HR 5999

Analysis of IUE high- and low-dispersion spectra of the young Herbig Ae star HR 5999 (HD 144668) covering 1978-1992 revealed dramatic changes in the Mg II h and k (2795.5, 2802.7 A) emission profiles, changes in the column density and distribution in radial velocity of accreting gas, and flux in the Ly(alpha), O I, and C IV emission lines, which are correlated with the UV excess luminosity. Variability in the spectral type inferred from the UV spectral energy distribution, ranging from A5 IV-III in high state to A7 III in the low state, was also observed. The trend of earlier inferred spectral type with decreasing wavelength and with increasing UV continuum flux has previously been noted as a signature of accretion disks in lower mass pre-main sequence stars (PMS) and in systems undergoing FU Orionis-type outbursts. Our data represent the first detection of similar phenomena in an intermediate mass (M greater than or equal to 2 solar mass) PMS star. Recent IUE spectra show gas accreting toward the star with velocities as high as plus 300 km/s, much as is seen toward beta Pic, and suggest that we also view this system through the debris disk. The absence of UV lines with the rotational broadening expected given the optical data (A7 IV, V sini=180 plus or minus 20 km/s for this system) also suggests that most of the UV light originates in the disk, even in the low continuum state. The dramatic variability in the column density of accreting gas, is consistent with clumpy accretion, such as has been observed toward beta Pic, is a hallmark of accretion onto young stars, and is not restricted to the clearing phase, since detectable amounts of accretion are present for stars with 0.5 Myr less than t(sub age) less than 2.8 Myr. The implications for models of beta Pic and similar systems are briefly discussed.

Perez, M. R.↗
NASA NTRS1994

Evolution of dynamo-generated magnetic fields in accretion disks around compact and young stars

Geometrically thin, optically thick, turbulent accretion disks are believed to surround many stars. Some of them are the compact components of close binaries, while the others are throught to be T Tauri stars. These accretion disks must be magnetized objects because the accreted matter, whether it comes from the companion star (binaries) or from a collapsing molecular cloud core (single young stars), carries an embedded magnetic field. In addition, most accretion disks are hot and turbulent, thus meeting the condition for the MHD turbulent dynamo to maintain and amplify any seed field magnetic field. In fact, for a disk's magnetic field to persist long enough in comparison with the disk viscous time it must be contemporaneously regenerated because the characteristic diffusion time of a magnetic field is typically much shorter than a disk's viscous time. This is true for most thin accretion disks. Consequently, studying magentic fields in thin disks is usually synonymous with studying magnetic dynamos, a fact that is not commonly recognized in the literature. Progress in studying the structure of many accretion disks was achieved mainly because most disks can be regarded as two-dimensional flows in which vertical and radial structures are largely decoupled. By analogy, in a thin disk, one may expect that vertical and radial structures of the magnetic field are decoupled because the magnetic field diffuses more rapidly to the vertical boundary of the disk than along the radius. Thus, an asymptotic method, called an adiabatic approximation, can be applied to accretion disk dynamo. We can represent the solution to the dynamo equation in the form B = Q(r)b(r,z), where Q(r) describes the field distribution along the radius, while the field distribution across the disk is included in the vector function b, which parametrically depends on r and is normalized by the condition max (b(z)) = 1. The field distribution across the disk is established rapidly, while the radial distribution Q(r) evolves on a considerably longer timescale. It is this evolution that is the subject of this paper.

Stepinski, Tomasz F.↗