Stellar streams formed by tidal stripping of progenitors orbiting around the Milky Way are expected to be perturbed by encounters with dark matter subhalos. Recent studies have shown that they are an excellent proxy to infer properties of the perturbers, such as their mass. Here we present two different methodologies that make use of the fully non-Gaussian density distribution of stellar streams: a Bayesian model selection based on the probability density function (PDF) of stellar density, and a likelihood-free gradient boosting classifier. While the schemes do not assume a specific dark matter model, we are mainly interested in discerning the primordial black holes cold dark matter (PBH CDM) hypothesis form the standard particle dark matter one. Therefore, as an application we forecast model selection strength of evidence for cold dark matter clusters of masses $10^3$ - $10^5 M_{\odot}$ and $10^5$ - $10^9 M_{\odot}$, based on a GD-1-like stellar stream and including realistic observational errors. Evidence for the smaller mass range, so far under-explored, is particularly interesting for PBH CDM. We expect weak to strong evidence for model selection based on the PDF analysis, depending on the fiducial model. Instead, the gradient boosting model is a highly efficient classifier (99\% accuracy) for all mass ranges here considered. As a further test of the robustness of the method, we reach similar conclusions when performing forecasts further dividing the largest mass range into $10^5$ - $10^7 M_{\odot}$ and $10^7$ - $10^9 M_{\odot}$ ranges.
We extend our analysis of the observed disturbances on the outskirts of the HI disk of the Milky Way. We employ the additional constraints of the phase of the modes of the observed HI image and asymmetry in the radial velocity field to derive the azimuth of the perturber inferred to be responsible for the disturbances in the HI disk. We carry out a modal analysis of the phase of the disturbances in the HI image and in SPH simulations of a Milky Way-like galaxy tidally interacting with dark perturbers, the relative offset of which we utilize to derive the perturber azimuth. To make a direct con…
Three Birds with One Stone: Core-Collapsed SIDM Halos as the Common Origin of Dense Perturbers in Lenses, Streams, and Satellites
Center for Experimental Cosmology & Instrumentation, Department of Physics and Astronomy, University of California, Riverside（加州大学河滨分校物理与天文学系实验宇宙学与仪器中心）
We present a new analysis of the observed perturbations of the HI disk of the Milky Way to infer the existence of a dark sub-halo that tidally interacted with the Milky Way disk. We examine tidal interactions between perturbing dark sub-halos and the gas disk of the Milky Way using high resolution Smoothed Particle Hydrodynamics (SPH) simulations. We compare our results to the observed HI map of the Milky Way to find that the Fourier amplitudes of the planar disturbances are best-fit by a perturbing dark sub-halo with a mass that is one-hundredth of the Milky Way with a pericentric distance of…
The $100^\circ$-long thin stellar stream in the Milky Way halo, GD-1, has an ensemble of features that may be due to dynamical interactions. Using high-resolution MMT/Hectochelle spectroscopy we show that a spur of GD-1-like stars outside of the main stream are kinematically and chemically consistent with the main stream. In the spur, as in the main stream, GD-1 has a low intrinsic radial velocity dispersion, $σ_{V_r}\lesssim1\,\rm km\,s^{-1}$, is metal-poor, $\rm [Fe/H]\approx-2.3$, with little $\rm [Fe/H]$ spread and some variation in $\rm [α/Fe]$ abundances, which point to a common globular…
An extended gravitational object embedded in a parent system comprised of gas and collisionless particles may undergo both dynamical friction (DF) and mass loss by tidal forces. If the object is compact enough, it can increase its mass through accretion of material from the surrounding medium. We extend the classical linear analysis of DF on a constant-mass body in a gaseous medium to the case where its mass changes with time. We show that the structure of the wake may differ significantly from the constant-mass case. For instance, the front-back symmetry of density about subsonic constant-mas…
Formation of Parallel Stellar Streams through Encounters with Dark Matter Subhalos and Intermediate-Mass Black Holes
The University of Tokyo（东京大学）；University of Tsukuba（筑波大学）；Kitami Institute of Technology（北见工业大学）；University Observatory Munich（慕尼黑大学天文台）；Max-Planck-Institut für extraterrestrische Physik（马克斯·普朗克地外物理研究所）
We explore the stability of tidal streams to perturbations, motivated by recent claims that the clumpy structure of the stellar streams surrounding the globular cluster Palomar 5 are the result of gravitational instability. We calculate the Jeans length of tidal streams by treating them as a thin expanding cylinder of collisionless matter. We also find a general relation between the density and the velocity dispersion inside a stream, which is used to determine the longitudinal Jeans criterion. Our analytic results are checked by following the time evolution of the phase space density within s…
We present a model for the interaction of the GD-1 stellar stream with a massive perturber that naturally explains many of the observed stream features, including a gap and an off-stream spur of stars. The model involves an impulse by a fast encounter, after which the stream grows a loop of stars at different orbital energies. At specific viewing angles, this loop appears offset from the stream track. A quantitative comparison of the spur and gap features prefers models where the perturber is in the mass range of $10^6\,\rm M_\odot$ to $10^8\,\rm M_\odot$. Orbit integrations back in time show …
Tidal interaction between a gaseous disk and a massive orbiting perturber is known to result in angular momentum exchange between them. Understanding astrophysical manifestations of this coupling such as gap opening by planets in protoplanetary disks or clearing of gas by binary supermassive black holes (SMBHs) embedded in accretion disks requires knowledge of the spatial distribution of the torque exerted on the disk by a perturber. Recent hydrodynamical simulations by Dong et al (2011) have shown evidence for the tidal torque density produced in a uniform disk to change sign at the radial se…
I present a new framework for modeling the dynamics of tidal streams. The framework consists of simple models for the initial action-angle distribution of tidal debris, which can be straightforwardly evolved forward in time. Taking advantage of the essentially one-dimensional nature of tidal streams, the transformation to position-velocity coordinates can be linearized and interpolated near a small number of points along the stream, thus allowing for efficient computations of a stream's properties in observable quantities. I illustrate how to calculate the stream's average location (its "track…
The unique dynamics of the tidal disruption of satellite galaxies is an extremely sensitive probe of long-range interactions between dark-matter particles. Dark-matter forces that are several percent the strength of gravity will lead to order unity changes in the ratio of the number of stars in the leading and trailing tidal streams of a satellite galaxy. The approximate symmetry of the stellar tidal streams of the Sagittarius dwarf galaxy would thus exclude attractive dark-matter forces greater than 10% the strength of gravity which would entirely eliminate the leading stream. However, recent…
We have explored the hypothesis that the total mass-ratio of the two main galaxies of the Local Group: Andromeda Galaxy(M31) and the Milky Way (MW) can be constrained measuring the tidal force induced by the surrounding mass distribution, M31 included, on the MW. We argue that the total mass-ratio between the two groups can be approximated, at least qualitatively, finding the tidal radius where the internal binding force of the MW balances the external tidal force acting on it. Since M31 is the massive tidal perturber of the local environment, we have used a wide range of M31 to MW mass-ratio …
A few dark matter substructures have recently been detected in strong gravitational lenses though their perturbations of highly magnified images. We derive a characteristic scale for lensing perturbations and show that this is significantly larger than the perturber's Einstein radius. We show that the perturber's projected mass enclosed within this radius, scaled by the log-slope of the host galaxy's density profile, can be robustly inferred even if the inferred density profile and tidal radius of the perturber are biased. We demonstrate the validity of our analytic derivation by using several…
We present an analysis of the mechanics of thin streams, which are formed following the tidal disruption of cold, low-mass clusters in the potential of a massive host galaxy. The analysis makes extensive use of action-angle variables, in which the physics of stream formation and evolution is expressed in a particularly simple form. We demonstrate the formation of streams by considering examples in both spherical and flattened potentials, and we find that the action-space structures formed in each take on a consistent and characteristic shape. We demonstrate that tidal streams formed in realist…
Impulsive encounters between astrophysical objects are usually treated using the distant tide approximation (DTA) for which the impact parameter, $b$, is assumed to be significantly larger than the characteristic radii of the subject, $r_{\mathrm{S}}$, and the perturber, $r_{\mathrm{P}}$. The perturber potential is then expanded as a multipole series and truncated at the quadrupole term. When the perturber is more extended than the subject, this standard approach can be extended to the case where $r_{\mathrm{S}} \ll b < r_{\mathrm{P}}$. However, for encounters with $b$ of order $r_{\mathrm{S}}…
Using N-body simulations, we have modeled the production and evolution of substructures in the intracluster light (ICL) of a simulated galaxy cluster. We use a density-based definition of ICL, where ICL consists of luminous particles which are at low densities, to identify ICL particles and track their evolution. We have implemented a friends-of-friends-type clustering algorithm which finds groups of particles correlated in both position and velocity space to identify substructures in the ICL, hereafter referred to as ``streams''. We find that ~40% of the cluster's ICL is generated in the form…
Tidal streams in the Milky Way are sensitive probes of the population of dark-matter subhalos predicted in cold-dark-matter (CDM) simulations. We present a new calculus for computing the effect of subhalo fly-bys on cold tidal streams based on the action-angle representation of streams. The heart of this calculus is a line-of-parallel-angle approach that calculates the perturbed distribution function of a given stream segment by undoing the effect of all impacts. This approach allows one to compute the perturbed stream density and track in any coordinate system in minutes for realizations of t…
We report the discovery of diffuse far-infrared and far-ultraviolet emission projected near the interacting pair NGC4435/4438, in the Virgo cluster. This feature spatially coincides with a well known low surface-brightness optical plume, usually interpreted as tidal debris. If extragalactic, this stream would represent not only one of the clearest examples of intracluster dust, but also a rare case of intracluster molecular hydrogen and large-scale intracluster star formation. However, the ultraviolet, far-infrared, HI and CO emission as well as the dynamics of this feature are extremely unusu…
The dark matter halos that surround Milky Way-like galaxies in cosmological simulations are, to first order, triaxial. Nearly 30 years ago it was predicted that such triaxial dark matter halos should exhibit steady figure rotation or tumbling motions for durations of several gigayears. The angular frequency of figure rotation predicted by cosmological simulations is described by a log-normal distribution of pattern speed with a median value 0.15hkm/s/kpc (~ 0.15h rad/Gyr ~ 9h deg/Gyr) and a width of 0.83km/s/kpc. These pattern speeds are so small that they have generally been considered both u…
We present surface photometry of a giant, low surface brightness stellar arc in the halo of the nearby spiral galaxy M63 (NGC 5055) that is consistent with being a part of a stellar stream resulting from the disruption of a dwarf satellite galaxy. Using the stream's "great-circle" morphology and its photometric properties, we estimate that the stream originates from the accretion of a 10^8 M_sun satellite in the last few Gyr. The B-R color of the stream's stars is consistent with Local Group dwarfs and is also similar to the outer regions of M63's disk and stellar halo within our measurement u…
A method to search for tidal streams and to fit their orbits based on maximum likelihood is presented and applied to the Gaia data. Tests of the method are performed showing how a simulated stream produced by tidal stripping of a star cluster is recovered when added to a simulation of the Gaia catalogue. The method can be applied to search for streams associated with known progenitors or to do blind searches in a general catalogue. As the first example, we apply the method to the globular cluster M68 and detect its clear tidal stream stretching over the whole North Galactic hemisphere, and pas…
Observations of the lengthy tidal streams produced by the destruction of the Sagittarius dwarf spheroidal (Sgr dSph) are capable of providing strong constraints on the shape of the Galactic gravitational potential. However, previous work, based on modeling different stream properties in axisymmetric Galactic models has yielded conflicting results: while the angular precession of the Sgr leading arm is most consistent with a spherical or slightly oblate halo, the radial velocities of stars in this arm are only reproduced by prolate halo models. We demonstrate that this apparent paradox can be r…
We present a generative probabilistic model for a tidal stream and demonstrate how this model is used to constrain the Galactic potential. The model takes advantage of the simple structure of a stream in angle and frequency space for the correct potential. We investigate how the method performs on full 6D mock stream data, and mock data with outliers included. As currently formulated the technique is computationally costly when applied to data with large observational errors, but we describe several modifications that promise to make the technique computationally tractable.
We present a brief history of Galactic astrophysics, and explain the origin of halo substructure in the Galaxy. We motivate our study of tidal streams by highlighting the tight constraints that analysis of the trajectories of tidal streams can place on the form of the Galactic potential. We address the reconstruction of orbits from observations of tidal streams. We upgrade the scheme reported by Binney (2008) and Jin & Lynden-Bell (2007), which reconstructs orbits from streams using radial-velocity measurements, to allow it to work with erroneous input data. The upgraded algorithm can correct …
Context. Models of hierarchical structure formation predict the accretion of smaller satellite galaxies onto more massive systems and this process should be accompanied by a disintegration of the smaller companions visible, e.g., in tidal streams. Aims. In order to verify and quantify this scenario we have developed a search strategy for low surface brightness tidal structures around a sample of 474 galaxies using the Sloan Digital Sky Survey DR7 archive. Methods. Calibrated images taken from the SDSS archive were processed in an automated manner and visually inspected for possible tidal strea…
We present candidate members of the Pal 5, GD-1, Cetus Polar, and Orphan tidal stellar streams found in LAMOST DR3, SDSS DR9 and APOGEE catalogs. In LAMOST DR3, we find 20, 4, 24 high confidence candidates of tidal streams GD-1, Cetus Polar and Orphan respectively. We also list from the SDSS DR9 spectroscopic catalog 59, 118, 10 high confidence candidates of tidal streams Cetus Polar, Orphan and Pal 5, respectively. Furthermore, we find 7 high confidence candidates of the Pal 5 tidal stream in APOGEE data. Compared with SDSS, the new candidates from LAMOST DR3 are brighter, so that together, m…
Mergers and tidal interactions between massive galaxies and their dwarf satellites are a fundamental prediction of the Lambda-Cold Dark Matter cosmology. These events are thought to influence galaxy evolution throughout cosmic history and to provide important observational diagnostics of structure formation. Stellar streams in the Local Group are spectacular evidence for satellite disruption at the present day. However, constructing a significant sample of tidal streams beyond our immediate cosmic neighborhood has proven a daunting observational challenge and their potential for deepening our …
Long Tidal Tails of NGC 5024 Hidden in LMS-1 and NGC 5053 Tidal Streams
Inner Mongolia University（内蒙古大学）；National Astronomical Observatories, Chinese Academy of Sciences（中国科学院国家天文台）；Instituto de Astrofísica de Canarias（加那利天体物理研究所）；Universidad de La Laguna（拉古纳大学）；Institute for Frontiers in Astronomy and Astrophysics, Beijing Normal University（北京师范大学前沿天文与天体物理研究院）
Tidal streams from existing and destroyed satellite galaxies populate the outer regions of the Andromeda galaxy (M31). This inhomogeneous debris can be studied without many of the obstacles that plague Milky Way research. We review the history of tidal stream research in M31, and in its main satellite galaxies. We highlight the numerous tidal streams observed around M31, some of which reside at projected distances of up to 120 kpc from the center of this galaxy. Most notable is the Giant Stellar Stream, a signature of the most recent significant accretion event in the M31 system. This event in…
The dynamics of stellar streams in rotating barred potentials is explained for the first time. Naturally, neighbouring stream stars reach pericentre at slightly different times. In the presence of a rotating bar, these neighbouring stream stars experience different bar orientations during pericentric passage and hence each star receives a different torque from the bar. These differing torques reshape the angular momentum and energy distribution of stars in the stream, which in turn changes the growth rate of the stream. For a progenitor orbiting in the same sense as the bar's rotation and sati…
We simulate tidal streams in the presence and absence of substructures inside the zero redshift snapshot of the Via Lactea II (VL-2) simulation. A halo finder is used to remove and isolate the subhalos found inside the high resolution dark matter halo of VL-2, and the potentials for both the main halo and all the subhalos are constructed individually using the self-consistent field (SCF) method. This allows us to make direct comparison of tidal streams between a smooth halo and a lumpy halo without assuming idealized profiles or triaxial fits. We simulate the kinematics of a star cluster start…
By mining the data from $Gaia$ EDR3, SDSS/SEGUE DR16 and LAMOST DR8, 11 member stars of the NGC 5466 tidal stream are detected and 7 of them are newly identified. To reject contaminators, a variety of cuts are applied in sky position, color-magnitude diagram, metallicity, proper motion and radial velocity. We compare our data to a mock stream generated by modeling the cluster's disruption under a smooth Galactic potential plus the Large Magellanic Cloud (LMC). The concordant trends in phase-space between the model and observations imply that the stream might have been perturbed by LMC. The two…
In this work, we explore the idea that substructures like stellar clusters could be formed from the tidal stream produced in galactic minor mergers. We use $N$-body and SPH simulations of satellite galaxies interacting with a larger galaxy. We study the distribution of mass in streams to identify overdensity regions in which a substructure could be formed. We found that without gas, no substructure formed as none of the overdensities shows a definite morphology nor dynamical stability. Including gas we found that several clumps appear and proved to be real long standing physical structures ($t…
We report the discovery of a giant stellar tidal stream in the halo of NGC 4631, a nearby edge-on spiral galaxy interacting with the spiral NGC 4656, in deep images taken with a 40-cm aperture robotic telescope. The stream has two components: a bridge-like feature extended between NGC 4631 and NGC 4656 (stream_SE) and an overdensity with extended features on the opposite side of the NGC 4631 disk (stream_NW). Together, these features extend more than 85 kpc and display a clear (g-r) colour gradient. The orientation of stream_SE relative to the orientations of NGC 4631 and NGC 4656 is not consi…
We describe a technique that finds orbits through the Galaxy that are consistent with measurements of a tidal stream, taking into account the extent that tidal streams do not precisely delineate orbits. We show that if accurate line-of-sight velocities are measured along a well defined stream, the technique recovers the underlying orbit through the Galaxy and predicts the distances and proper motions along the stream to high precision. As the error bars on the location and velocities of the stream grow, the technique is able to find more and more orbits that are consistent with the data and th…
In this paper we discuss a method for the generation of mock tidal streams. Using an ensemble of simulations in an isochrone potential where the actions and frequencies are known, we derive an empirical recipe for the evolving satellite mass and the corresponding mass loss rate, and the ejection conditions of the stream material. The resulting stream can then be quickly generated either with direct orbital integration, or by using the action-angle formalism. The model naturally produces streaky features within the stream. These are formed due to the radial oscillation of the progenitor and the…
Dwarf galaxies that come too close to larger galaxies suffer tidal disruption; the differential gravitational force between one side of the galaxy and the other serves to rip the stars from the dwarf galaxy so that they instead orbit the larger galaxy. This process produces "tidal streams" of stars, which can be found in the stellar halo of the Milky Way, as well as in halos of other galaxies. This chapter provides a general introduction to tidal streams, including the mechanism through which the streams are created, the history of how they were discovered, and the observational techniques by …
Cold or Warm, the Dark Matter substructure spectrum must extend to objects with masses as low as $10^7 M_\odot$, according to the most recent Lyman-$α$ measurements. Around a Milky Way-like galaxy, more than a thousand of these subhaloes will not be able to form stars but are dense enough to survive even deep down in the potential well of their host. There, within the stellar halo, these dark pellets will bombard tidal streams as they travel around the Galaxy, causing small but recognizable damage to the stream density distribution. The detection and characterization of these stream ruptures w…
The Galaxy's stellar halo seems to be a tangle of disrupted systems that have been tidally stretched out into streams. Each stream approximately delineates an orbit in the Galactic force-field. In the first paper in this series we showed that all six phase-space coordinates of each point on an orbit can be reconstructed from the orbit's path across the sky and measurements of the line-of-sight velocity along the orbit. In this paper we complement this finding by showing that the orbit can also be reconstructed if we know proper motions along the orbit rather than the radial velocities. We also…
A large fraction of galactic nuclei is expected to host supermassive black hole binaries (BHB), likely formed during the early phase of galaxies assembly and merging. In this paper, we use a large set of state-of-art numerical models to investigate the interplay between a BHB and a massive star cluster (GCs) driven toward the galactic centre by dynamical friction. Varying the BHB mass and mass ratio and the GC orbit, we show that the reciprocal feedback exerted between GCs and the BHB shapes their global properties. We show that, at GC-to-BHB mass ratios above 0.1, the GC affects notably the B…
This paper presents an example where the morphology of a single stellar stream can be used to rule out a specific galactic potential form without the need for velocity information. We investigate the globular cluster Palomar5 (Pal 5), which is tidally disrupting into a cold, thin stream mapped over 22 degrees on the sky with a typical width of 0.7 degrees. We generate models of this stream by fixing Pal 5's present-day position, distance and radial velocity via observations, while allowing its proper motion to vary. In a spherical dark matter halo we easily find models that fit the observed mo…
We present two newly-discovered, compact elliptical (cE) galaxies, which exhibit clear evidence of tidal steams, found in a search of SDSS DR7. The structural parameters of the cEs are derived using GALFIT and give effective radii < 400 pc. They also possess young to intermediate-age stellar populations. These two cEs provide direct evidence, a "smoking gun", for the process of tidal stripping that is believed to be the origin of M32-type galaxies. Both are found in small group environments with many late-type galaxies, suggesting that we may be seeing the formation of such galaxies in dynamic…
Using photometry and proper motions from Gaia Data Release 2, we detect a 50 degree-long stream of about 70 stars extending westward from the halo globular cluster M5. Based on the similarities in distance, proper motions, inferred color-magnitude distribution, and trajectory, we identify this stream as the trailing tidal tail of M5. While the surface density of stars is very low (~1.5 stars per square degree, or approximately 35 magnitudes per square arcsecond), selecting only stars having proper motions consistent with the orbit of the cluster yield a detection significance ~10 sigma. While …
We present new near-infrared (NIR) observations of M63 from the Extended Disk Galaxy Exploration Science (EDGES) Survey. The extremely deep 3.6 $μ$m mosaic reaches 29 AB mag arcsec$^{-2}$ at the outer reaches of the azimuthally-averaged surface brightness profile. At this depth the consequences of galactic accretion are found within a nearby tidal stream and an up-bending break in the slope of the surface brightness profile. This break occurs at a semi-major axis length of $\sim$8', and is evidence of either an enhanced outer disc or an inner stellar halo. Simulations of galaxy evolution, alon…
We have probed a section (l ~ 150, b ~ -60) of the trailing tidal arm of the Sagittarius dwarf spheroidal galaxy by identifying a sample of Red Clump stream stars. Red Clump stars are not generally found in the halo field, but are found in significant numbers in both the Sagittarius galaxy and its tidal streams, making them excellent probes of stream characteristics. Our target sample was selected using photometric data from the Sloan Digital Sky Survey, Data Release 6, which was constrained in color to match the Sagittarius Red Clump stars. Spectroscopic observations of the target stars were …
During the past 20 years, numerous stellar streams have been discovered in both the Milky Way and the Local Group. These streams have been tidally torn from orbiting systems, which suggests that most of them should roughly trace the orbit of their progenitors around the Galaxy. As a consequence, they play a fundamental role in understanding the formation and evolution of our Galaxy. This project is based on the possibility of applying a technique developed by Binney in 2008 to various tidal streams and overdensities in the Galaxy. The aim is to develop an efficient method to constrain the Gala…
Dancing Streams In Merging Halos: Stellar Streams in a MW--LMC-like merger
Carnegie Science Observatories（卡内基科学观测站）；Northwestern University（西北大学）；Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA)（天体物理交叉探索与研究中心）；NSF-Simons AI Institute for the Sky (SkAI)（NSF-西蒙斯天空人工智能研究所）；Columbia University（哥伦比亚大学）；Boston University（波士顿大学）；Flatiron Institute（平顿研究所）
Context. Stellar tidal streams are the result of tidal interactions between a central galaxy and lower mass systems like satellite galaxies or globular clusters. For the Local Group, many diffuse substructures have been identified and their link to the galaxy evolution has been traced. However it cannot be assumed that the Milky Way or M31 are representative of their galaxy class, and a larger sample of analogue galaxies beyond the Local Group is required to be able to generalise the underlying theory. Aims. We want to characterise photometrically the stellar streams around Milky Way analogues…
The Gaia mission promises to deliver precision astrometry at an unprecedented level, heralding a new era for discerning the kinematic and spatial coordinates of stars in our Galaxy. Here, we present a new technique for estimating the age of tidally disrupted globular cluster streams using the proper motions and parallaxes of tracer stars. We evolve the collisional dynamics of globular clusters within the evolving potential of a Milky Way-like halo extracted from a cosmological $Λ$CDM simulation and analyze the resultant streams as they would be observed by Gaia. The simulations sample a variet…
I present an organic description of the regimes of collisionless tidal streams and define the orderings between the physical quantities that shape their morphology. Three fundamental dichotomies are identified in the form of dimensionless inequalities. These govern i) the speed of the stream's growth, ii) its internal coherence, iii) its thickness or opening angles. The mechanisms that regulate such main properties are analysed. The slope of the host's density profile influences the speed of the stream's growth, in both length and width, as steeper profiles enhance differential streaming. Inte…
