Friction Timescale Research Articles

Dynamical Friction Models for Black Hole Binary Formation in Active Galactic Nucleus Disks

Stellar-mass black holes (sBHs) embedded in gaseous disks of active galactic nuclei (AGN) can be important sources of detectable gravitational radiation for LIGO/Virgo when they form binaries and coalesce due to orbital decay. In this paper, we study the effect of dynamical friction (DF) on the formation of BH binaries in AGN disks using N-body simulations. We employ two simplified models of DF, with the force on the BH depending on Δ v , the velocity of the sBH relative to the background Keplerian gas. We integrate the motion of two sBHs initially on circular orbits around the central supermassive black hole (SMBH) and evaluate the probability of binary formation under various conditions. We find that both models of DF (with different dependence of the frictional coefficient on ∣Δ v ∣) can foster the formation of binaries when the effective friction timescale τ satisfies ΩK τ ≲ 20–30 (where ΩK is the Keplerian frequency around the SMBH): prograde binaries are formed when the DF is stronger (smaller τ), while retrograde binaries dominate when the DF is weaker (larger τ). We determine the distribution of both prograde and retrograde binaries as a function of initial orbital separation and the DF strength. Using our models of DF, we show that for a given sBH number density in the AGN disk, the formation rate of sBH binaries increases with decreasing τ and can reach a moderate value with a sufficiently strong DF.

Dynamical friction due to fuzzy dark matter on satellites described by axisymmetric logarithmic potentials

A plausible dark matter candidate is an ultralight bosonic particle referred to as fuzzy dark matter. The equivalent mass-energy of the fuzzy dark matter boson is ∼ 10-22 eV and has a corresponding de Broglie wavelength of kiloparsec scale, thus exhibiting wave behaviour in scales comparable to a galactic core, which could not appear in conventional cold dark matter models. The presence of fuzzy dark matter in galactic clusters will impact the motion of their members through dynamical friction. In this work, we present simulations of the dynamical friction on satellites traversing an initially uniform fuzzy dark matter halo. We focus on the satellites whose shapes are beyond spherical symmetry described by ellipsoidal and logarithmic potentials. We find that the wakes created on the fuzzy dark matter halo due to the passage of such satellites are qualitatively different from those generated by spherically symmetric ones. Furthermore, we quantify the dynamical friction coefficient for such systems, finding that the same satellite may experience a drag differing by a factor of 5 depending on its ellipticity and the direction of motion. Finally, we find that the dynamical friction time-scale is close to Hubble time, assuming a satellite of 1011 M ⊙ traversing at 103 km/s a FDM halo whose mean density is ∼ 106 M ⊙ kpc-3.

Exploratory scenarios for the differential nuclear and tidal evolution of TZ Fornacis

Aims. TZ Fornacis is a double-lined eclipsing binary system with similar masses (2.057 ± 0.001 and 1.958 ± 0.001 M⊙) but characterized by very different radii (8.28 ± 0.22 and 3.94 ± 0.17 R⊙). This similarity in terms of mass makes it possible to study the system’s differential stellar evolution as well as some aspects of its tidal evolution. With regard to its orbital elements, it was recently confirmed that its orbit is circular with an orbital period of 75.7 days. The less massive component rotates about 17 times faster than the primary one, which is synchronized with the mean orbital angular velocity. Our main objective in this work is to study both the nuclear and the tidal evolution of the system. Methods. To model the TZ For system, we used the MESA package, computing the grids using the exact observed masses, radii, and effective temperatures as input, and then varying the metallicity, the core overshooting amount, and the mixing-length parameter. A χ2 statistic was used to infer the optimal values of the core overshooting and the mixing-length parameters. The same procedure was used to generate rotating models with the GRANADA code. The respective errors in the average age of TZ For were less than 5%. On the other hand, the differential equations that govern the tidal evolution were integrated using the fifth-order Runge–Kutta method, ith a tolerance of 1 × 10−7. Results. We explored two scenarios regarding the initial eccentricities: a high one (0.30) and a case of an initial circular orbit. A good agreement has been found between the observational values of the eccentricity, synchronism levels, and orbital period with the values predicted by the integration of the tidal evolution equations. The influence of the friction timescale on the evolution of the orbital elements of TZ For is also studied here. The orbital elements most affected by the uncertainties in the friction timescale are the synchronism levels of the two components. On the other hand, we used the properties of the rotating models generated by the GRANADA code as the initial angular velocities instead of using trial values. In this case, comparisons between the theoretical values of the orbital elements and their observed counterparts also lead to a good interagreement.

Effects of stellar-mass primordial black holes on first star formation

ABSTRACT We use cosmological hydrodynamic zoom-in simulations and semi-analytical models to study the effects of primordial black holes (PBHs) on first star formation. Our models self-consistently combine two competing effects: initial (isocurvature) perturbations induced by PBHs and BH accretion feedback. Focusing on PBHs with masses $\sim 30\, \rm M_{\odot }$, we find that the standard picture of first star formation in molecular-cooling minihaloes is not changed by PBHs, as the simulated star-forming gas clouds in the central parsec are very similar to those in the $\rm \Lambda CDM$ case when PBHs make up fPBH ∼ 10−4−0.1 of dark matter. With a dynamical friction time-scale of $\sim 2{\!-\!}10\ \rm Myr$ when the central gas density reaches $10^{5}\ \rm cm^{-3}$, it is also unlikely that PBHs can sink into star-forming discs and affect the evolution of protostars, although they may interact with the stars during the main-sequence stage. At larger scales, PBHs tend to shift star formation to more massive haloes and accelerate structure formation. The latter effect is stronger in regions with higher initial overdensities. For fPBH ∼ 10−4−0.01 (allowed by observational constraints), the collapsed mass fraction of haloes hosting Population III stars is similar (within a factor of ∼2 at z ≲ 30) to that in $\rm \Lambda CDM$, implying that the impact of stellar-mass PBHs on the cosmic star formation history at z ≳ 10 is small. We also find that the Lyman–Werner photons from PBH accretion in atomic-cooling haloes may facilitate the formation of direct-collapse BHs.

Dynamical friction from ultralight dark matter

We simulate the gravitational dynamics of a massive object interacting with ultralight/fuzzy dark matter (ULDM/FDM), nonrelativistic quantum matter described by the Schr\"odinger-Poisson equation. We first consider a point mass moving in a uniform background, and then a supermassive black hole (SMBH) moving within a ULDM soliton. After replicating simple dynamical friction scenarios to verify our numerical strategies, we demonstrate that the wake induced by a moving mass in a uniform medium may undergo gravitational collapse that dramatically increases the drag force, albeit in a scenario unlikely to be encountered astrophysically. We broadly confirm simple estimates of dynamical friction timescales for a black hole at the center of a halo but see that a large moving point mass excites coherent ``breathing modes'' in a ULDM soliton. These can lead to ``stone skipping'' trajectories for point masses which do not sink uniformly toward the center of the soliton, as well as stochastic motion near the center itself. These effects will add complexity to SMBH-ULDM interactions and to SMBH mergers in a ULDM universe.

Introducing a new multi-particle collision method for the evolution of dense stellar systems

Context.In a previous paper we introduced a new method for simulating collisional gravitationalN-body systems with linear time scaling onN, based on the multi-particle collision (MPC) approach. This allows us to easily simulate globular clusters with a realistic number of stellar particles (105 − 106) in a matter of hours on a typical workstation.Aims.We evolve star clusters containing up to 106stars to core collapse and beyond. We quantify several aspects of core collapse over multiple realizations and different parameters while always resolving the cluster core with a realistic number of particles.Methods.We run a large set ofN-body simulations with our new code MPCDSS. The cluster mass function is a pure power law with no stellar evolution, allowing us to clearly measure the effects of the mass spectrum on core collapse.Results.Leading up to core collapse, we find a power-law relation between the size of the core and the time left to core collapse. Our simulations thus confirm the theoretical self-similar contraction picture but with a dependence on the slope of the mass function. The time of core collapse has a non-monotonic dependence on the slope, which is well fitted by a parabola. This also holds for the depth of core collapse and for the dynamical friction timescale of heavy particles. Cluster density profiles at core collapse show a broken-power-law structure, suggesting that central cusps are a genuine feature of collapsed cores. The core bounces back after collapse, with visible fluctuations, and the inner density slope evolves to an asymptotic value. The presence of an intermediate-mass black hole inhibits core collapse, making it much shallower, irrespective of the mass-function slope.Conclusions.We confirm and expand on several predictions of star cluster evolution before, during, and after core collapse. Such predictions were based on theoretical calculations or small-size directN-body simulations. Here we put them to the test in MPC simulations with a much larger number of particles, allowing us to resolve the collapsing core.

The spatial distribution of globular clusters in dwarf spheroidal galaxies and the timing problem

ABSTRACT The dynamical friction time-scale of massive globular clusters (GCs) in the inner regions of cuspy dark haloes in dwarf spheroidal (dSph) galaxies can be much shorter than the Hubble time. This implies that a small fraction of the GCs is expected to be caught close to the centre of these galaxies. We compare the radial distribution of GCs predicted in simple Monte Carlo models with that of a sample of 38 spectroscopically confirmed GCs plus 17 GC candidates associated mainly with low-luminosity dSph galaxies. If dark matter haloes follow an NFW profile, the observed number of off-centre GCs at projected distances less than one-half the galaxy effective radius is significantly higher than the models predict. This timing problem can be viewed as a fine-tuning of the starting GC distances. As a result of the short sinking time-scale for GCs in the central regions, the radial distribution of GCs is expected to evolve significantly during the next 1–2 Gyr. However, dark matter haloes with cores of size comparable to the galaxy effective radii can lead to a slow orbital inspiral of GCs in the central regions of these galaxies, providing a simple solution to the timing problem. We also examine any indication of mass segregation in the summed distribution of our sample of GCs.

General Circulation Model Errors Are Variable across Exoclimate Parameter Spaces

Abstract General circulation models (GCMs) are often used to explore exoclimate parameter spaces and classify atmospheric circulation regimes. Models are tuned to give reasonable climate states for standard test cases, such as the Held–Suarez test, and then used to simulate diverse exoclimates by varying input parameters such as rotation rates, instellation, atmospheric optical properties, frictional timescales, and so on. In such studies, there is an implicit assumption that the model works reasonably well for the standard test case will be credible at all points in an arbitrarily wide parameter space. Here, we test this assumption using the open-source GCM THOR to simulate atmospheric circulation on tidally locked Earth-like planets with rotation periods of 0.1–100 days. We find that the model error, as quantified by the ratio between physical and spurious numerical contributions to the angular momentum balance, is extremely variable across this range of rotation periods with some cases where numerical errors are the dominant component. Increasing model grid resolution does improve errors, but using a higher-order numerical diffusion scheme can sometimes magnify errors for finite-volume dynamical solvers. We further show that to minimize error and make the angular momentum balance more physical within our model, the surface friction timescale must be smaller than the rotational timescale.

Radiative-dynamical Simulation of Jupiter’s Stratosphere and Upper Troposphere

Abstract We present a two-dimensional radiative-dynamical model of the combined stratosphere and upper troposphere of Jupiter to understand its temperature distribution and meridional circulation pattern. Our study highlights the importance of radiative and mechanical forcing for driving the middle atmospheric circulation on Jupiter. Our model adopts a state-of-the-art radiative transfer scheme with recent observations of Jovian gas abundances and haze distribution. Assuming local radiative equilibrium, latitudinal variation of hydrocarbon abundances is not able to explain the observed latitudinal temperature variations in the mid-latitudes. With mechanical forcing parameterized as a frictional drag on zonal wind, our model produces ∼2 K latitudinal temperature variations observed in low to mid-latitudes in the troposphere and lower stratosphere, but cannot reproduce the observed 5 K temperature variations in the middle stratosphere. In the high latitudes, temperature and meridional circulation depend strongly on polar haze radiation. The simulated residual mean circulation shows either two broad equator-to-pole cells or multi-cell patterns, depending on the frictional drag timescale and polar haze properties. A more realistic wave parameterization and a better observational characterization of haze distribution and optical properties are needed to better understand latitudinal temperature distributions and circulation patterns in the middle atmosphere of Jupiter.

Formation of GW190521 via Gas Accretion onto Population III Stellar Black Hole Remnants Born in High-redshift Minihalos

Abstract The recent gravitational-wave merger event, GW190521, has challenged our understanding of stellar-mass black hole (BH) formation. The primary and secondary BHs are both inferred to fall inside the pair-instability (PI) mass gap. Here we propose that the formation of such binaries is possible through gas accretion onto the BH remnants of Population III stars born in high-redshift (z > 10) minihalos. Once the parent halo has grown to the atomic-cooling limit, even brief episodes of gas accretion in the dense central regions of the halo can increase the masses of Population III remnant BHs above the PI limit. Starting with a binary black hole (BBH) with an initial mass of O(100) M ⊙ we find that it would only need to spend about 100 Myr in the inner few parsecs of an atomic-cooling halo to accrete about 50 M ⊙ of material and resemble a system similar to GW190521. The dynamical friction timescale for the binary to sink to the dense inner region of its parent halo is comparable or shorter than the accretion timescale required to increase their mass above the PI limit. Once in the core of the halo, the binary can enter a phase of hyper-Eddington accretion, where it would only take a few thousand years to exceed the PI limit through accretion. Even more massive BBHs could form through this channel, and be detectable by detectors with improved low-frequency sensitivity. Single Population III BH remnants would also grow through accretion and could later form binaries dynamically. As little as a few percent of Population III BH remnants may be sufficient to match the rate of massive BBH mergers inferred from GW190521 of .

FURTHER PROPERTIES OF THE DYNAMICAL CLOCK A+ INDICATOR IN A TOY MODEL OF PURE DYNAMICAL FRICTION

The Alessandrini A+ indicator is a measure of star cluster dynamical evolution based on the mass-segregation of blue straggler stars. A+ is defined as the integral of the cumulative distribution of blue stragglers over log radius, minus a term related to the reference population used. In a companion paper I introduced a model of dynamical friction and calculated the A+ indicator analytically. Here I show further properties of the time evolution of A+, focusing on the physical interpretation of its time derivative dA+ /dt. I find that dA+ /dt is the mean of the reciprocal dyamical friction timescale, weighted by the density of blue stragglers. I show that it is non-negative (as expected based on monotonicity) due to the density of blue-stragglers being non-negative and that, for a radially non-decreasing dynamical friction timescale, dA+ /dt is also non-decreasing with time, making A+ a convex function.

Dynamics of Disordered Proteins under Confinement: Memory Effects and Internal Friction.

Many proteins are disordered under physiological conditions. How efficiently they can search for their cellular targets and how fast they can fold upon target binding is determined by their intrinsic dynamics, which have thus attracted much recent attention. Experiments and molecular simulations show that the inherent reconfiguration timescale for unfolded proteins has a solvent friction component and an internal friction component, and the microscopic origin of the latter, along with its proper mathematical description, has been a topic of considerable debate. Internal friction varies across different proteins of comparable length and increases with decreasing denaturant concentration, showing that it depends on how compact the protein is. Here we report on a systematic atomistic simulation study of how confinement, which induces a more compact unfolded state, affects dynamics and friction in disordered peptides. We find that the average reconfiguration timescales increase exponentially as the peptide's spatial dimensions are reduced; at the same time, confinement broadens the spectrum of relaxation timescales exhibited by the peptide. There are two important implications of this broadening: First, it limits applicability of the common Rouse and Zimm models with internal friction, as those models attempt to capture internal friction effects by introducing a single internal friction timescale. Second, the long-tailed distribution of relaxation times leads to anomalous diffusion effects in the dynamics of intramolecular distances. Analysis and interpretation of experimental signals from various measurements that probe intramolecular protein dynamics (such as single-molecule fluorescence correlation spectroscopy and single-molecule force spectroscopy) rely on the assumption of diffusive dynamics along the distances being probed; hence, our results suggest the need for more general models allowing for anomalous diffusion effects.

Timescales of major mergers from simulations of isolated binary galaxy collisions

A six-dimensional parameter space based on high-resolution numerical simulations of isolated binary galaxy collisions has been constructed to investigate the dynamical friction timescales, τmer, for major mergers. Our experiments follow the gravitational encounters between ∼600 pairs of similarly massive late- and early-type galaxies with orbital parameters that meet the predictions of the Λ-cold dark matter (ΛCDM) cosmology. We analyse the performance of different schemes for tracking the secular evolution of mergers, finding that the product of the intergalactic distance and velocity is best suited to identify the time of coalescence. In contrast, a widely used merger-time estimator such as the exhaustion of the orbital spin is shown to systematically underpredict τmer, resulting in relative errors that can reach 60% for nearly radial encounters. We find that the internal spins of the progenitors can lead to total variations in the merger times larger than 30% in highly circular encounters, whereas only the spin of the principal halo is capable of modulating the strength of the interaction prevailing throughout a merger. The comparison of our simulated merger times with predictions from different variants of a well-known fitting formula has revealed an only partially satisfactory agreement, which has led us to recalculate the values of the coefficients of these expressions to obtain relations that fit major mergers perfectly. The observed biases between data and predictions, which do not only apply to the present work, are inconsistent with expectations from differences in the degree of idealisation of the collisions, their metric, spin-related biases, or the simulation set-up. This indicates a certain lack of accuracy of the dynamical friction modelling, arising perhaps from a still incomplete identification of the parameters governing orbital decay.

Wave-frequency flows within a near-bed vegetation canopy

We study water flows and wave dissipation within near-bed pneumatophore canopies at the wave-exposed fringe of a mangrove forest on Cù Lao Dung Island, in the Mekong Delta. To evaluate canopy drag, the three-dimensional geometry of pneumatophore stems growing upward from the buried lateral roots of Sonneratia caseolaris mangroves was reconstructed from photogrammetric surveys. In cases where hydrodynamic measurements were obtained, up to 84 stems per square meter were observed, with stem heights < 0.6m, and basal diameters 0.01–0.02m. The parameter a= (frontal area of pneumatophores blocking the flow)/(canopy volume) ranged from zero to 1.8m−1. Within-canopy water velocity displayed a phase lead and slight attenuation relative to above-canopy flows. The phase lead was frequency-dependent, ranging from 0 to 30 degrees at the frequencies of energetic waves (> 0.1Hz), and up to 90 degrees at lower frequencies. A model is developed for wave-induced flows within the vertically variable canopy. Scaling suggests that acceleration-induced forces and vertical mixing were negligible at wave frequencies. Consistent with theory, drag-induced vertical variability in velocity scaled with Λ=Tw/(2πTf), where Tw= wave period, Tf=2/(CDa|u|) is the frictional time scale, CD≈2 is the drag coefficient, and |u| is a typical flow speed. For fixed wave conditions (|u| and Tw), theory predicts increasing dissipation with increasing vegetation density (i.e. increasing a), until a maximum is reached for order-one Λ. For larger Λ, within-canopy flow is so inhibited by drag that further increases in a reduce within-canopy dissipation. For observed cases, Λ⩽0.38 at energetic wave frequencies, so wave dissipation near the forest edge is expected to increase with increasing pneumatophore canopy density. However, under different wave conditions, the most dense canopies may occasionally approach the dissipation maximum (Λ≈1). Predicted dissipation by the pneumatophore canopy was sufficient to attenuate most wave energy over distances slightly less (more) than 100m into the marsh in 1m (2m) water depth.

A rumble in the dark: signatures of self-interacting dark matter in supermassive black hole dynamics and galaxy density profiles

We explore for the first time the effect of self-interacting dark matter (SIDM) on the dark matter (DM) and baryonic distribution in massive galaxies formed in hydrodynamical cosmological simulations, including explicit baryonic physics treatment. A novel implementation of Super-Massive Black Hole (SMBH) formation and evolution is used, as in Tremmel et al.(2015, 2016), allowing to explicitly follow SMBH dynamics at the center of galaxies. A high SIDM constant cross-section is chosen, $\sigma$=10 $\rm cm^2/gr$, to amplify differences from CDM models. Milky Way-like galaxies form a shallower DM density profile in SIDM than they do in CDM, with differences already at 20 kpc scales. This demonstrates that even for the most massive spirals the effect of SIDM dominates over the adiabatic contraction due to baryons. Strikingly, the dynamics of SMBHs differs in the SIDM and reference CDM case. SMBHs in massive spirals have sunk to the centre of their host galaxy in both the SIDM and CDM run, while in less massive galaxies about 80$\%$ of the SMBH population is off-centered in the SIDM case, as opposed to the CDM case in which $\sim$90$\%$ of SMBHs have reached their host's centre. SMBHs are found as far as $\sim$9 kpc away from the centre of their host SIDM galaxy. This difference is due to the increased dynamical friction timescale caused by the lower DM density in SIDM galaxies compared to CDM, resulting in 'core stalling'. This pilot work highlights the importance of simulating in a full hydrodynamical context different DM models combined to SMBH physics to study their influence on galaxy formation.

Dynamical Friction and the Evolution of Supermassive Black Hole Binaries: The Final Hundred-parsec Problem

Abstract The supermassive black holes originally in the nuclei of two merging galaxies will form a binary in the remnant core. The early evolution of the massive binary is driven by dynamical friction before the binary becomes “hard” and eventually reaches coalescence through gravitational-wave emission. We consider the dynamical friction evolution of massive binaries consisting of a secondary hole orbiting inside a stellar cusp dominated by a more massive central black hole. In our treatment, we include the frictional force from stars moving faster than the inspiralling object, which is neglected in the standard Chandrasekhar treatment. We show that the binary eccentricity increases if the stellar cusp density profile rises less steeply than . In cusps shallower than , the frictional timescale can become very long due to the deficit of stars moving slower than the massive body. Although including fast stars increases the decay rate, low mass-ratio binaries ( ) in sufficiently massive galaxies have decay timescales longer than one Hubble time. During such minor mergers, the secondary hole stalls on an eccentric orbit at a distance of order one-tenth the influence radius of the primary hole (i.e., for massive ellipticals). We calculate the expected number of stalled satellites as a function of the host galaxy mass and show that the brightest cluster galaxies should have of such satellites orbiting within their cores. Our results could provide an explanation for a number of observations, which include multiple nuclei in core ellipticals, off-center AGNs, and eccentric nuclear disks.

In the cold light of day: the potential importance of under-ice convective mixed layers to primary producers

Temperate lakes are ice covered for much of the year; however, winter lake conditions have not been well studied and are undergoing rapid change. Using data collected during ice-on periods from 4 north-temperate water bodies, we report observations of stable surface layers, solar-induced convective mixed layers, and their potential impacts on phytoplankton. The convective mixed layer is defined as the region where the convective Richardson number (Ri) is ≤1. In the absence of a convective mixed layer, peaks in chlorophyll a were near the ice–water interface. Light conditions here seemed sufficient to support phytoplankton biomass accrual in the short-term in 50% of our measurements, although snow depths >13.5 cm may lead to light limitation. When a convective mixed layer was present, light conditions were sufficient for biomass accrual in 37.5% of cases. The frictional timescale for damping averaged 15 minutes, indicative of a lack of mixing at night. Convective mixing depths and velocity increased as snow declined, and results demonstrated the potential for rapid convective mixed layer deepening (up to 6.6 m h−1), underscoring the highly dynamic physical environment under ice. Although declining periods of ice cover have been subject to much attention, changes in snow cover may have equally important implications for primary producers and the potential for under-ice blooms. This link between physics and biology must be further explored to better understand how changing winters will affect water bodies.

Theoretical and computational validation of the Kuhn barrier friction mechanism in unfolded proteins

A long time ago, Kuhn predicted that long polymers should approach a limit where their global motion is controlled by solvent friction alone, with ruggedness of their energy landscapes having no consequences for their dynamics. In contrast, internal friction effects are important for polymers of modest length. Internal friction in proteins, in particular, affects how fast they fold or find their binding targets and, as such, has attracted much recent attention. Here we explore the molecular origins of internal friction in unfolded proteins using atomistic simulations, coarse-grained models and analytic theory. We show that the characteristic internal friction timescale is directly proportional to the timescale of hindered dihedral rotations within the polypeptide chain, with a proportionality coefficient b that is independent of the chain length. Such chain length independence of b provides experimentally testable evidence that internal friction arises from concerted, crankshaft-like dihedral rearrangements. In accord with phenomenological models of internal friction, we find the global reconfiguration timescale of a polypeptide to be the sum of solvent friction and internal friction timescales. At the same time, the time evolution of inter-monomer distances within polypeptides deviates both from the predictions of those models and from a simple, one-dimensional diffusion model.

Theoretical lower limits on sizes of ultrafaint dwarf galaxies from dynamical friction

Dwarf spheroidal galaxies are the smallest known stellar systems where under Newtonian interpretations, a significant amount of dark matter is required to explain observed kinematics. In fact, they are in this sense the most heavily dark matter dominated objects known. That, plus the increasingly small sizes of the newly discovered ultra faint dwarfs, puts these systems in the regime where dynamical friction on individual stars starts to become relevant. We calculate the dynamical friction timescales for pressure supported isotropic spherical dark matter dominated stellar systems, yielding $\tau_{DF} =0.93 (r_{h}/10 pc)^{2} (\sigma/ kms^{-1}) Gyr$, { where $r_{h}$ is the half-light radius}. For a stellar velocity dispersion value of $3 km/s$, as typical for the smallest of the recently detected ultra faint dwarf spheroidals, dynamical friction timescales becomes smaller than the $10 Gyr$ typical of the stellar ages for these systems, for $r_{h}<19 pc$. Thus, this becomes a theoretical lower limit below which dark matter dominated stellar systems become unstable to dynamical friction. We present a comparison with structural parameters of the smallest ultra faint dwarf spheroidals known, showing that these are already close to the stability limit derived, any future detection of yet smaller such systems would be inconsistent with a particle dark matter hypothesis.

Equilibration of a Baroclinic Planetary Atmosphere toward the Limit of Vanishing Bottom Friction

Abstract This paper discusses whether and how a baroclinic atmosphere can equilibrate with very small bottom friction in a dry primitive equation general circulation model. The model is forced by a Newtonian relaxation of temperature to a prescribed temperature profile, and it is damped by a linear friction near the lower boundary. When friction is decreased by four orders of magnitude, kinetic energy dissipation by friction gradually becomes negligible, while “energy recycling” becomes dominant. In this limit kinetic energy is converted back into potential energy at the largest scales, thus closing the energy cycle without significant frictional dissipation. The momentum fluxes are of opposite sign in the upper and lower atmosphere: in the upper atmosphere, eddies converge momentum into the westerly jets; however, in the lower atmosphere, the eddies diverge momentum out of the westerly jets. The secondary circulation driven by the meridional eddy momentum fluxes thus acts to increase the baroclinicity of the westerly jet. This regime may be relevant for the Jovian atmosphere, where the frictional time scale may be much larger than the radiative damping time scale.