arXiv subjects

Liubin Pan

Publications and source records attributed to Liubin Pan.

At least 19 recordsLinked to original sources

arXiv2025

Widespread Hot Molecular Gas Heated by Shear-induced Turbulence in the Galactic Center

We observed NH3 metastable inversion lines from (3, 3) to (18, 18) toward G0.66-0.13 in the Galactic center with the Shanghai Tianma 65m radio telescope and Yebes 40 m telescope. Highly-excited lines of NH3 (17, 17), (18, 18) were detected in emission for the first time in the interstellar medium, with upper energy levels up to 3100 K. Mapping observations reveal widespread hot molecular gas traced by NH3 (13, 13) toward G0.66-0.13. The rotation temperatures of hot gas traced by NH3 exceed 400 K, which amounts to five percent of the total NH3 in the Galactic Center. Hot gas (>400 K) and warm gas (100-140 K) are found in distinct clumps, with the hot gas located at the interfacing regions between different warm clouds. The theory of intermittency in turbulence reproduces the complex temperature structure in the central molecular zone, especially the hot gas observed here. The results presented here demonstrate that turbulence heating dominates the heating of the molecular gas in the Central Molecular Zone, while the turbulence is induced by the shear-motion of molecular clouds under the gravitational potential of the nuclear star clusters and the supermassive black hole. Our results suggest that shear-induced turbulence heating could be a widespread factor influencing galactic evolution.

astro-ph.GA↗

arXiv2025

An Improved Fit to the Density Distribution in Supersonic Isothermal Turbulence

The density distribution of supersonic isothermal turbulence plays a critical role in many astrophysical systems. It is commonly approximated by a lognormal distribution with a variance of $\sigma_{s,{\rm V}}^2 \approx \ln(1 + b^2 M_{\rm V}^2),$ where $s \equiv \ln \rho/\rho_0,$ $M_{\rm V}$ is the rms volume-weighted Mach number, and $b$ is a parameter that depends on the driving mechanism, which can be solenoidal (divergence-free), compressive (curl-free), or a mix of both. However, this fit neglects the driving correlation time, $\tau_{\rm a}$, which plays a key role when compressive driving is significant. Here we conduct turbulence simulations spanning a wide range of Mach numbers, driving mechanisms, and $\tau_{\rm a}$ values. In the compressive case, $\sigma_{s,{\rm V}}^2$ is not well fit by the standard expression. Instead, it scales approximately linearly with $M_{\rm V},$ and its dependence on $\tau_{\rm a}$ is $\sigma_{s,{\rm V}}^2 \approx M_{\rm V} [1 + \frac{2}{3}(1 + \lambda_{\rm a})\Theta(1 + \lambda_{\rm a})]$, where $\lambda_{\rm a} \equiv \ln(\tau_{\rm a}/\tau_{\rm e})$, $\tau_{\rm e}$ is the eddy turnover time, and $\Theta$ is the Heaviside step function. Mixed-driven turbulence shows a weak dependence on $\tau_{\rm a},$ and for solenoidally-driven turbulence, $\sigma_{s,{\rm V}}^2 \approx \frac{1}{3}M_{\rm V}$, which is consistent with the standard expression when $M_{\rm V} \lesssim 8.$ The volume-weighted mean and skewness also show systematic trends with $M_{\rm V}$ and $\tau_{\rm a}$, deviating from lognormal expectations. The mass-weighted density distribution displays significant broadening and skewness in compressively-driven cases, especially at large $\tau_{\rm a}/\tau_{\rm e}$. These results provide a refined framework for modeling astrophysical turbulence.

astro-ph.GA↗

arXiv2025

The Density Distribution of Compressively-Forced Supersonic Turbulence Depends on the Driving Correlation Time

Supersonic turbulence plays a critical role in shaping astrophysical systems, from molecular clouds to the circumgalactic medium. Key properties of this turbulence include the Mach number, driving scale, and nature of the driving mechanism, which can be solenoidal (divergence-free), compressive (curl-free), or a mix of the two. A less studied property is the correlation time of the driving accelerations, $\tau_{\rm a}.$ While this timescale has a minimal impact on solenoidally-driven turbulence, we show that it has a strong impact on compressively-driven turbulence. Using high-resolution simulations with tracer particles, we analyze the evolution of density fluctuations, focusing on the PDF of the logarithmic density, $s$, and its rate of change, $\frac{ds}{dt},$ and the conditional statistics of $\frac{ds}{dt}$ and $\frac{d^2s}{dt^2}$. When the driving correlation time is comparable to the eddy turnover time, $\tau_{\rm a} \approx \tau_{\rm e},$ compressive driving leads to the formation of large, low-density voids in which the variance of $\frac{ds}{dt}$ is large. These are directly linked to sustained accelerated expansions, which results in a strong correlation between density and the divergence of the driving acceleration field. In contrast, when $\tau_{\rm a} \approx 0.1 \, \tau_{\rm e}$, compressive driving does not produce such voids, resulting in a narrower, less skewed distribution. We show using analytical estimates that $\tau_{\rm a}$ is may be significantly less than $\tau_{\rm e}$ in supernova-driven turbulence, highlighting the need to better understand the role of the driving correlation time in shaping the density structure of turbulent astrophysical systems.

astro-ph.GA↗

arXiv2024

Understanding Density Fluctuations in Supersonic, Isothermal Turbulence

Supersonic turbulence occurs in many environments, particularly in astrophysics. In the crucial case of isothermal turbulence, the probability density function (PDF) of the logarithmic density, $s$, is well measured, but a theoretical understanding of the processes leading to this distribution remains elusive. We investigate these processes using Lagrangian tracer particles to track $s$ and $\frac{ds}{dt}$ in direct numerical simulations, and we show that their evolution can be modeled as a stochastic differential process with time-correlated noise. The temporal correlation functions of $s$ and $\frac{ds}{dt}$ decay exponentially, as predicted by the model, and the decay timescale is $\approx$ 1/6 the eddy turnover time. The behavior of the conditional averages of $\frac{ds}{dt}$ and $\frac{d^2s}{dt^2}$ is also well explained by the model, which shows that the density PDF arises from a balance between stochastic compressions/expansions, which tend to broaden the PDF, and the acceleration/deceleration of shocks by density gradients, which tends to narrow it.

astro-ph.GA↗

arXiv2024

The Formation of Protoplanetary Disks through Pre-Main Sequence Bondi-Hoyle Accretion

Protoplanetary disks are traditionally described as finite mass reservoirs left over by the gravitational collapse of the protostellar core, a view that strongly constrains both disk evolution and planet formation models. We propose a different scenario where protoplanetary disks of pre-main sequence stars are primarily assembled by Bondi-Hoyle accretion from the parent gas cloud. We demonstrate that Bondi-Hoyle accretion can supply not only the mass, but also the angular momentum necessary to explain the observed size of protoplanetary disks. Additionally, we predict how the specific angular momentum of protoplanetary disks scales with stellar mass. Our conclusions are based on a new analytical derivation of the scaling of the angular momentum in turbulent flows, which we confirm with a numerical simulation of supersonic turbulence. A key outcome of our analysis is the recognition that density fluctuations in supersonic turbulence--previously overlooked in studies of cloud and core rotation--lead to a significant increase in angular momentum at disk-forming scales. This revised understanding of disk formation and evolution alleviates several longstanding observational discrepancies and compels substantial revisions to current models of disk and planet formation.

astro-ph.GA↗

arXiv2020

Collapsing Index: A New Method to Identify Star-forming Cores Based on ALMA Images

Stars form through the gravitational collapse of molecular cloud cores. Before collapsing, the cores are supported by thermal pressure and turbulent motions. A question of critical importance for the understanding of star formation is how to observationally discern whether a core has already initiated gravitational collapse or is still in hydrostatic balance. The canonical method to identify gravitational collapse is based on the observed density radial profile, which would change from a Bonnor-Ebert type toward power laws as the core collapses. In practice, due to the projection effect, the resolution limit, and other caveats, it has been difficult to directly reveal the dynamical status of cores, particularly in massive star-forming regions. We here propose a novel, straight-forward diagnostic, namely, the collapsing index (CI), which can be modeled and calculated based on the radial profile of the line width of dense gas. A meaningful measurement of CI requires spatially and spectrally resolved images of optically thin and chemically stable dense gas tracers. ALMA observations are making such data sets increasingly available for massive star-forming regions. Applying our method to one of the deepest dense-gas spectral images ever taken toward such a region, namely, the Orion molecular cloud, we detect the dynamical status of selected cores therein. We observationally distinguished a collapsing core in a massive star-forming region from a hydrostatical one. Our approach would help significantly improve our understanding of the interaction between gravity and turbulence within molecular cloud cores in the process of star formation.

astro-ph.GA↗

arXiv2020

The Origin of Massive Stars: The Inertial-Inflow Model

We address the problem of the origin of massive stars, namely the origin, path and timescale of the mass flows that create them. Based on extensive numerical simulations, we propose a scenario where massive stars are assembled by large-scale, converging, inertial flows that naturally occur in supersonic turbulence. We refer to this scenario of massive-star formation as the "Inertial-Inflow Model". This model stems directly from the idea that the mass distribution of stars is primarily the result of turbulent fragmentation. Under this hypothesis, the statistical properties of the turbulence determine the formation timescale and mass of prestellar cores, posing definite constraints on the formation mechanism of massive stars. We quantify such constraints by the analysis of a simulation of supernova-driven turbulence in a 250-pc region of the interstellar medium, describing the formation of hundreds of massive stars over a time of approximately 30 Myr. Due to the large size of our statistical sample, we can say with full confidence that massive stars in general do not form from the collapse of massive cores, nor from competitive accretion, as both models are incompatible with the numerical results. We also compute synthetic continuum observables in Herschel and ALMA bands. We find that, depending on the distance of the observed regions, estimates of core mass based on commonly-used methods may exceed the actual core masses by up to two orders of magnitude, and that there is essentially no correlation between estimated and real core masses.

astro-ph.GA↗

arXiv2020

The Effect of Supernovae on the Turbulence and Dispersal of Molecular Clouds

While the importance of supernova feedback in galaxies is well established, its role on the scale of molecular clouds is still debated. In this work, we focus on the impact of supernovae on individual clouds, using a high-resolution magneto-hydrodynamic simulation of a region of 250 pc where we resolve the formation of individual massive stars. The supernova feedback is implemented with real supernovae that are the natural evolution of the resolved massive stars, so their position and timing are self-consistent. We select a large sample of molecular clouds from the simulation to investigate the supernova energy injection and the resulting properties of molecular clouds. We find that molecular clouds have a lifetime of a few dynamical times, less then half of them contract to the point of becoming gravitationally bound, and the dispersal time of bound clouds, of order one dynamical time, is a factor of two shorter than that of unbound clouds. We stress the importance of internal supernovae, that is massive stars that explode inside their parent cloud, in setting the cloud dispersal time, and their huge overdensity compared to models where the supernovae are randomly distributed. We also quantify the energy injection efficiency of supernovae as a function of supernova distance to the clouds. We conclude that intermittent driving by supernovae can maintain molecular-cloud turbulence and may be the main process of cloud dispersal. The role of supernovae in the evolution of molecular clouds cannot be fully accounted for without a self-consistent implementation of their feedback.

astro-ph.GA↗

arXiv2019

The Probability Distribution of Density Fluctuations in Supersonic Turbulence

We study density fluctuations in supersonic turbulence using both theoretical methods and numerical simulations. A theoretical formulation is developed for the probability distribution function (PDF) of the density at steady state, connecting it to the conditional statistics of the velocity divergence. Two sets of numerical simulations are carried out, using either a Riemann solver to evolve the Euler equations or a finite-difference method to evolve the Navier-Stokes (N-S) equations. After confirming the validity of our theoretical formulation with the N-S simulations, we examine the effects of dynamical processes on the PDF, showing that the nonlinear term in the divergence equation amplifies the right tail of the PDF and reduces the left one, the pressure term reduces both the right and left tails, and the viscosity term, counter-intuitively, broadens the right tail of the PDF. Despite the inaccuracy of the velocity divergence from the Riemann runs, as found in our previous work, we show that the density PDF from the Riemann runs is consistent with that from the N-S runs. Taking advantage of their much higher effective resolution, we then use the Riemann runs to study the dependence of the PDF on the Mach number, $\mathcal{M}$, up to $\mathcal{M}\sim30$. The PDF width, $σ_{s}$, follows the relation $σ_{s}^2 = \ln (1+b^2 {\mathcal M}^2)$, with $b\approx0.38$. However, the PDF exhibits a negative skewness that increases with increasing $\mathcal{M}$, so much of the growth of $σ_{s}$ is accounted for by the growth of the left PDF tail, while the growth of the right tail tends to saturate. Thus, the usual prescription that combines a lognormal shape with the standard variance-Mach number relation greatly overestimates the right PDF tail at large $\mathcal{M}$, which may have a significant impact on theoretical models of star formation.

astro-ph.GA↗

arXiv2019

Inaccuracy of Spatial Derivatives in Simulations of Supersonic Turbulence

We examine the accuracy of spatial derivatives computed from numerical simulations of supersonic turbulence. Two sets of simulations, carried out using a finite-volume code that evolves the hydrodynamic equations with an approximate Riemann solver and a finite-difference code that solves the Navier-Stokes equations, are tested against a number of criteria based on the continuity equation, including exact results at statistically steady state. We find that the spatial derivatives in the Navier-Stokes runs are accurate and satisfy all the criteria. In particular, they satisfy our exact results that the conditional mean velocity divergence, $\langle \nabla \cdot {\bs u}|s\rangle$, where $s$ is the logarithm of density, and the conditional mean of the advection of $s$, $\langle {\bs u} \cdot \nabla s|s\rangle$, vanish at steady state for all density values, $s$. On the other hand, the Riemann solver simulations fail all the tests that require accurate evaluation of spatial derivatives, resulting in apparent violation of the continuity equation, even if the solver enforces mass conservation. In particular, analysis of the Riemann simulations may lead to the incorrect conclusion that the $\pdv$ work tends to preferentially convert kinetic energy into thermal energy, inconsistent with the exact result that the energy exchange by $\pdv$ work is symmetric in barotropic supersonic turbulence at steady state. The inaccuracy of spatial derivatives is a general problem in the post-processing of simulations of supersonic turbulence with Riemann solvers. Solutions from such simulations must be used with caution in post-processing studies concerning the spatial gradients.

physics.flu-dyn↗

arXiv2018

Detailed Balance and Exact Results for Density Fluctuations in Supersonic Turbulence

The probabilistic approach to turbulence is applied to investigate density fluctuations in supersonic turbulence. We derive kinetic equations for the probability distribution function (PDF) of the logarithm of the density field, $s$, in compressible turbulence in two forms: a first-order partial differential equation involving the average divergence conditioned on the flow density, $\langle \nabla \cdot {\bs u} | s\rangle$, and a Fokker-Planck equation with the drift and diffusion coefficients equal to $-\langle {\bs u} \cdot \nabla s | s\rangle$ and $\langle {\bs u} \cdot \nabla s | s\rangle$, respectively. Assuming statistical homogeneity only, the detailed balance at steady state leads to two exact results, $\langle \nabla \cdot {\bs u} | s \rangle =0$, and $\langle {\bs u} \cdot \nabla s | s\rangle=0$. The former indicates a balance of the flow divergence over all expanding and contracting regions at each given density. The exact results provide an objective criterion to judge the accuracy of numerical codes with respect to the density statistics in supersonic turbulence. We also present a method to estimate the effective numerical diffusion as a function of the flow density and discuss its effects on the shape of the density PDF.

astro-ph.GA↗

arXiv2016

Following The Cosmic Evolution Of Pristine Gas I: Implications For Milky Way Halo Stars

We make use of new subgrid model of turbulent mixing to accurately follow the cosmological evolution of the first stars, the mixing of their supernova ejecta, and the impact on the chemical composition of the Galactic Halo. Using the cosmological adaptive mesh refinement code RAMSES, we implement a model for the pollution of pristine gas as described in Pan et al. Tracking the metallicity of Pop III stars with metallicities below a critical value allows us to account for the fraction of Z < Zcrit stars formed even in regions in which the gas' average metallicity is well above Zcrit. We demonstrate that such partially-mixed regions account for 0.5 to 0.7 of all Pop III stars formed up to z = 5. Additionally, we track the creation and transport of "primordial metals" (PM) generated by Pop III supernovae (SNe). These neutron-capture deficient metals are taken up by second-generation stars and likely lead to unique abundance signatures characteristic of carbon-enhanced, metal-poor (CEMP-no) stars. As an illustrative example, we associate primordial metals with abundance ratios used by Keller et al. to explain the source of metals in the star SMSS J031300.36-670839.3, finding good agreement with the observed [Fe/H], [C/H], [O/H], and [Mg/Ca] ratios in CEMP-no Milky Way halo stars. Similar future simulations will aid in further constraining the properties of Pop III stars using CEMP observations, as well as improve predictions of the spatial distribution of Pop III stars, as will be explored by the next generation of ground- and space-based telescopes.

astro-ph.GA↗

arXiv2016

Supernova Driving. II. Compressive Ratio in Molecular-Cloud Turbulence

The compressibility of molecular cloud (MC) turbulence plays a crucial role in star formation models, because it controls the amplitude and distribution of density fluctuations. The relation between the compressive ratio (the ratio of powers in compressive and solenoidal motions) and the statistics of turbulence has been previously studied systematically only in idealized simulations with random external forces. In this work, we analyze a simulation of large-scale turbulence (250 pc) driven by supernova (SN) explosions that has been shown to yield realistic MC properties. We demonstrate that SN driving results in MC turbulence with a broad lognormal distribution of the compressive ratio, with a mean value $\approx 0.3$, lower than the equilibrium value of $\approx 0.5$ found in the inertial range of isothermal simulations with random solenoidal driving. We also find that the compressibility of the turbulence is not noticeably affected by gravity, nor are the mean cloud radial (expansion or contraction) and solid-body rotation velocities. Furthermore, the clouds follow a general relation between the rms density and the rms Mach number similar to that of supersonic isothermal turbulence, though with a large scatter, and their average gas density PDF is described well by a lognormal distribution, with the addition of a high-density power-law tail when self-gravity is included.

astro-ph.GA↗

arXiv2016

Supernova Driving. III. Synthetic Molecular Cloud Observations

We present a comparison of molecular clouds (MCs) from a simulation of supernova-driven interstellar medium (ISM) turbulence with real MCs from the Outer Galaxy Survey. The radiative transfer calculations to compute synthetic CO spectra are carried out assuming the CO relative abundance depends only on gas density, according to four different models. Synthetic MCs are selected above a threshold brightness temperature value, $T_{\rm B,min}=1.4$ K, of the $J=1-0$ $^{12}$CO line, generating 16 synthetic catalogs (four different spatial resolutions and four CO abundance models), each containing up to several thousands MCs. The comparison with the observations focuses on the mass and size distributions and on the velocity-size and mass-size Larson relations. The mass and size distributions are found to be consistent with the observations, with no significant variations with spatial resolution or chemical model, except in the case of the unrealistic model with constant CO abundance. The velocity-size relation is slightly too steep for some of the models, while the mass-size relation is a bit too shallow for all models only at a spatial resolution $dx\approx 1$ pc. The normalizations of the Larson relations show a clear dependence on spatial resolution, for both the synthetic and the real MCs. The comparison of the velocity-size normalization suggests that the SN rate in the Perseus arm is approximately 70\% or less of the rate adopted in the simulation. Overall, the realistic properties of the synthetic clouds confirm that supernova-driven turbulence can explain the origin and dynamics of MCs.

astro-ph.GA↗

arXiv2016

Supernova Driving. I. The Origin of Molecular Cloud Turbulence

Turbulence is ubiquitous in molecular clouds (MCs), but its origin is still unclear because MCs are usually assumed to live longer than the turbulence dissipation time. Interstellar medium (ISM) turbulence is likely driven by SN explosions, but it has never been demonstrated that SN explosions can establish and maintain a turbulent cascade inside MCs consistent with the observations. In this work, we carry out a simulation of SN-driven turbulence in a volume of (250 pc)$^3$, specifically designed to test if SN driving alone can be responsible for the observed turbulence inside MCs. We find that SN driving establishes a velocity scaling consistent with the usual scaling laws of supersonic turbulence, suggesting that previous idealized simulations of MC turbulence, driven with a random, large-scale volume force, were correctly adopted as appropriate models for MC turbulence, despite the artificial driving. We also find that the same scaling laws extend to the interior of MCs, and that the velocity-size relation of the MCs selected from our simulation is consistent with that of MCs from the Outer-Galaxy Survey, the largest MC sample available. The mass-size relation and the mass and size probability distributions also compare successfully with those of the Outer Galaxy Survey. Finally, we show that MC turbulence is super-Alfvénic with respect to both the mean and rms magnetic-field strength. We conclude that MC structure and dynamics are the natural result of SN-driven turbulence.

astro-ph.GA↗

arXiv2014

Turbulence-Induced Relative Velocity of Dust particles IV: the Collision Kernel

Motivated by its importance for modeling dust particle growth in protoplanetary disks, we study turbulence-induced collision statistics of inertial particles as a function of the particle friction time, tau_p. We show that turbulent clustering significantly enhances the collision rate for particles of similar sizes with tau_p corresponding to the inertial range of the flow. If the friction time, tau_p,h, of the larger particle is in the inertial range, the collision kernel per unit cross section increases with increasing friction time, tau_p,l, of the smaller particle, and reaches the maximum at tau_p,l = tau_p,h, where the clustering effect peaks. This feature is not captured by the commonly-used kernel formula, which neglects the effect of clustering. We argue that turbulent clustering helps alleviate the bouncing barrier problem for planetesimal formation. We also investigate the collision velocity statistics using a collision-rate weighting factor to account for higher collision frequency for particle pairs with larger relative velocity. For tau_p,h in the inertial range, the rms relative velocity with collision-rate weighting is found to be invariant with tau_p,l and scales with tau_p,h roughly as ~ tau_p,h^(1/2). The weighting factor favors collisions with larger relative velocity, and including it leads to more destructive and less sticking collisions. We compare two collision kernel formulations based on spherical and cylindrical geometries. The two formulations give consistent results for the collision rate and the collision-rate weighted statistics, except that the spherical formulation predicts more head-on collisions than the cylindrical formulation.

astro-ph.EP↗

arXiv2014

Turbulence-Induced Relative Velocity of Dust Particles III: The Probability Distribution

Motivated by its important role in the collisional growth of dust particles in protoplanetary disks, we investigate the probability distribution function (PDF) of the relative velocity of inertial particles suspended in turbulent flows. Using the simulation from our previous work, we compute the relative velocity PDF as a function of the friction timescales, tau_p1 and tau_p2, of two particles of arbitrary sizes. The friction time of particles included in the simulation ranges from 0.1 tau_eta to 54T_L, with tau_eta and T_L the Kolmogorov time and the Lagrangian correlation time of the flow, respectively. The relative velocity PDF is generically non-Gaussian, exhibiting fat tails. For a fixed value of tau_p1, the PDF is the fattest for equal-size particles (tau_p2~tau_p1), and becomes thinner at both tau_p2 tau_p1. Defining f as the friction time ratio of the smaller particle to the larger one, we find that, at a given f in 1/2 >T_L). These features are successfully explained by the Pan & Padoan model. Using our simulation data and some simplifying assumptions, we estimated the fractions of collisions resulting in sticking, bouncing, and fragmentation as a function of the dust size in protoplanerary disks, and argued that accounting for non-Gaussianity of the collision velocity may help further alleviate the bouncing barrier problem.

astro-ph.EP↗

arXiv2014

Turbulence-Induced Relative Velocity of Dust Particles II: The Bidisperse Case

We extend our earlier work on turbulence-induced relative velocity between equal-size particles (Pan and Padoan, Paper I) to particles of arbitrarily different sizes. The Pan and Padoan (PP10) model shows that the relative velocity between different particles has two contributions, named the generalized shear and acceleration terms, respectively. The generalized shear term represents the particles' memory of the spatial flow velocity difference across the particle distance in the past, while the acceleration term is associated with the temporal flow velocity difference on individual particle trajectories. Using the simulation of Paper I, we compute the root-mean-square relative velocity, ^1/2, as a function of the friction times, tau_p1 and tau_p2, of the two particles, and show that the PP10 prediction is in satisfactory agreement with the data, confirming its physical picture. For a given tau_p1 below the Lagrangian correlation time of the flow, T_L, ^1/2 as a function of tau_p2 shows a dip at tau_p2~tau_p1, indicating tighter velocity correlation between similar particles. Defining a ratio f=tau_pl/tau_ph, with tau_pl and tau_ph the friction times of the smaller and larger particles, we find that ^1/2 increases with decreasing f due to the generalized acceleration contribution, which dominates at f<1/4. At a fixed f, our model predicts that ^1/2 scales as tau_ph^1/2 for tau_p,h in the inertial range of the flow, stays roughly constant for T_L >T_L/f. The acceleration term is independent of the particle distance, r, and thus reduces the r-dependence of ^1/2 in the bidisperse case.

astro-ph.EP↗