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Probabilistic model for constraining the Galactic potential using tidal streams
Abstract
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 when viewed in angle
and frequency space in 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.
... Section 6.3.3.2 of Chapter 6 outlines other predictive models for streams that also start from the orbit of the satellite, but do not rely on additional particles to represent the debris. Instead, these methods calculate the phase-space structure of stream populations offset from the satellite path given the orbital-phase and time since the material was lost using analytic approximations (Johnston, 1998; Bovy, 2014; Sanders, 2014). These methods can also be used to search through trial potentials to find a good fit to stream data (Johnston et al., 1999b; Sanders, 2014). ...
... Instead, these methods calculate the phase-space structure of stream populations offset from the satellite path given the orbital-phase and time since the material was lost using analytic approximations (Johnston, 1998; Bovy, 2014; Sanders, 2014). These methods can also be used to search through trial potentials to find a good fit to stream data (Johnston et al., 1999b; Sanders, 2014). However, they are limited by the extent to which the adopted analytic approximations apply, or are at least accurate enough for the purposes of recovering the potential. ...
... For example the mass distribution could be defined by a spatial grid of values or by the coefficients of a basis function expansion. @BULLET Methods that have been formulated in action-space (Peñarrubia, Koposov, & Walker, 2012; Sanders & Binney, 2013b; Bovy, 2014; Sanders, 2014; Sanderson et al., 2014) rely on being able to represent the Milky Way with an integrable potential in which actions can be found. Recent work has suggested ways of approximately recovering actions for any potential (Sanders & Binney, 2015; Bovy, 2014). ...
Tidal debris streams from galaxy satellites can provide insight into the dark matter distribution in halos. This is because we have more information about stars in a debris structure than about a purely random population of stars: we know that in the past they were all bound to the same dwarf galaxy; and we know that they form a dynamically cold population moving on similar orbits. They also probe a different region of the matter distribution in a galaxy than many other methods of mass determination, as their orbits take them far beyond the typical extent of those for the bulk of stars. Although conclusive results from this information have yet to be obtained, significant progress has been made in developing the methodologies for determining both the global mass distribution of the Milky Way’s dark matter halo and the amount of dark matter substructure within it. Methods for measuring the halo shape are divided into “predictive methods,” which predict the tidal debris properties from the progenitor satellite’s mass and orbit, given an assumed parent galaxy mass distribution; and “fundamental methods,” which exploit properties fundamental to the nature of tidal debris as global potential constraints. Methods for quantifying the prevalence of dark matter subhalos within halos through the analysis of the gaps left in tidal streams after these substructures pass through them are reviewed.
... If these orbits are calculated using an accurate MW potential, then the member stars should become bound to the progenitor at some point in their orbital history. Another common technique is the forward modeling method, such as the one described in Bonaca et al. (2014), which has several related variations (Varghese et al. 2011;Sanders 2014;Fardal et al. 2015). This method uses a Markov Chain Monte Carlo algorithm that compares simulated streams to observed streams in 6D phase space. ...
We analyse stellar streams in action-angle coordinates combined with recent local direct acceleration measurements to provide joint constraints on the potential of our galaxy. Our stream analysis uses the Kullback–Leibler divergence with a likelihood analysis based on the two-point correlation function. We provide joint constraints from pulsar accelerations and stellar streams for local and global parameters that describe the potential of the Milky Way (MW). Our goal is to build an “acceleration ladder,” where direct acceleration measurements that are currently limited in dynamic range are combined with indirect techniques that can access a much larger volume of the MW. To constrain the MW potential with stellar streams, we consider the Palomar 5, Orphan, Nyx, Helmi, and GD1 streams. Of the potential models that we have considered here, the preferred potential for the streams is a two-component Staeckel potential. We also compare the vertical accelerations from stellar streams and pulsar timing, defining the function f ( z ) = α 1 pulsar z − ∂ Φ ∂ z , where Φ is the MW potential determined from stellar streams and α 1 pulsar z is the vertical acceleration determined from pulsar timing observations. Our analysis indicates that the Oort limit determined from streams is consistently (regardless of the choice of potential) lower than that determined from pulsar timing observations. The calibration we have derived here may be used to correct the estimate of the acceleration from stellar streams.
... The stream DF approach of Bovy (2014) and similar (e.g. Sanders (2014)) are some of the few analytic means to simulate stellar streams. However the method is subject to a number of limitations. ...
Stellar streams are sensitive probes of the Galactic potential. The likelihood of a stream model given stream data is often assessed using simulations. However, comparing to simulations is challenging when even the stream paths can be hard to quantify. Here we present a novel application of Self-Organizing Maps and first-order Kalman Filters to reconstruct a stream's path, propagating measurement errors and data sparsity into the stream path uncertainty. The technique is Galactic-model independent, non-parametric, and works on phase-wrapped streams. With this technique, we can uniformly analyze and compare data with simulations, enabling both comparison of simulation techniques and ensemble analysis with stream tracks of many stellar streams. Our method is implemented in the public Python package TrackStream, available at https://github.com/nstarman/trackstream.
... Kinematically cold, metal-poor streams are particularly useful for studies of the galactic gravitational potential, as they provide a snapshot of an extended stellar orbit in phasespace. There exist several independent methods for constraining the galactic potential: fitting orbits directly to stream measurements (e.g., Johnston et al. 1999;Fardal et al. 2006;Koposov et al. 2010;Varghese et al. 2011), particle-ejection methods (Kupper et al. 2012;Bonaca et al. 2014;Gibbons et al. 2014;Fardal et al. 2015), action-angle tracks (Bovy 2014;Sanders 2014), clustering in integrals-of-motion space (Sanders & Binney 2013a;Sanderson et al. 2015;Reino et al. 2022), or, finally, direct N-body simulations (e.g., Dehnen et al. 2004;Law & Majewski 2010). While N-body simulations provide the most accurate kinematic depiction of stellar streams, they are too costly to search the parameter space efficiently for even a semirealistic galactic potential model. ...
We present a data-driven method for reconstructing the galactic acceleration field from phase-space (position and velocity) measurements of stellar streams. Our approach is based on a flexible and differentiable fit to the stream in phase-space, enabling a direct estimate of the acceleration vector along the stream. Reconstruction of the local acceleration field can be applied independently to each of several streams, allowing us to sample the acceleration field due to the underlying galactic potential across a range of scales. Our approach is methodologically different from previous works, as a model for the gravitational potential does not need to be adopted beforehand. Instead, our flexible neural-network-based model treats the stream as a collection of orbits with a locally similar mixture of energies, rather than assuming that the stream delineates a single stellar orbit. Accordingly, our approach allows for distinct regions of the stream to have different mean energies, as is the case for real stellar streams. Once the acceleration vector is sampled along the stream, standard analytic models for the galactic potential can then be rapidly constrained. We find our method recovers the correct parameters for a ground-truth triaxial logarithmic halo potential when applied to simulated stellar streams. Alternatively, we demonstrate that a flexible potential can be constrained with a neural network, and standard multipole expansions can also be constrained. Our approach is applicable to simple and complicated gravitational potentials alike and enables potential reconstruction from a fully data-driven standpoint using measurements of slowly phase-mixing tidal debris.
... There exist several independent methods for constraining the galactic potential: fitting orbits directly to stream measurements (e.g. Johnston et al. 1999;Fardal et al. 2006;Koposov et al. 2010;Varghese et al. 2011), particle ejection methods (Küpper et al. 2012;Fardal et al. 2015;Bonaca et al. 2014;Gibbons et al. 2014), action-angle tracks (Sanders 2014;Bovy 2014), clustering in integrals-of-motion space (Sanders & Binney 2013a;Sanderson et al. 2015;Reino et al. 2022), or, finally, direct N -body simulations (e.g. Dehnen et al. 2004;Law & Majewski 2010). ...
We present a data-driven method for reconstructing the galactic acceleration field from phase-space measurements of stellar streams. Our approach is based on a flexible and differentiable fit to the stream in phase-space, enabling a direct estimate of the acceleration vector along the stream. Reconstruction of the local acceleration field can be applied independently to each of several streams, allowing us to sample the acceleration field due to the underlying galactic potential across a range of scales. Our approach is methodologically different from previous works, since a model for the gravitational potential does not need to be adopted beforehand. Instead, our flexible neural-network-based model treats the stream as a collection of orbits with a locally similar mixture of energies, rather than assuming that the stream delineates a single stellar orbit. Accordingly, our approach allows for distinct regions of the stream to have different mean energies, as is the case for real stellar streams. Once the acceleration vector is sampled along the stream, standard analytic models for the galactic potential can then be rapidly constrained. We find our method recovers the correct parameters for a ground-truth triaxial logarithmic halo potential when applied to simulated stellar streams. Alternatively, we demonstrate that a flexible potential can be constrained with a neural network, though standard multipole expansions can also be constrained. Our approach is applicable to simple and complicated gravitational potentials alike, and enables potential reconstruction from a fully data-driven standpoint using measurements of slowly phase-mixing tidal debris.
... Classic VEGAS is an algorithm for adaptive multidimensional Monte Carlo integration. 1 It is widely used in particle physics: for example, in Monte Carlo event generators, 2 and to evaluate low-order 3 and high-order 4 Feynman diagrams and cross sections numerically. It has also been used in other fields for a variety of applications including, for example, path integrals for chemical physics 5 and option pricing in applied finance, 6 Bayesian statistics for astrophysics [7][8][9] and medical statistics, 10 models of neuronal networks 11 , wavefunction overlaps for atomic physics, 12 topological integrals for condensed matter physics, 13 and so on. ...
We describe a new algorithm, VEGAS+, for adaptive multidimensional Monte Carlo integration. The new algorithm adds a second adaptive strategy, adaptive stratified sampling, to the adaptive importance sampling that is the basis for its widely used predecessor VEGAS. Both VEGAS and VEGAS+ are effective for integrands with large peaks, but VEGAS+ can be much more effective for integrands with multiple peaks or other significant structures aligned with diagonals of the integration volume. We give examples where VEGAS+ is 2-17 times more accurate than VEGAS. We also show how to combine VEGAS+ with other integrators, such as the widely available MISER algorithm, to make new hybrid integrators. For a different kind of hybrid, we show how to use integrand samples, generated using MCMC or other methods, to optimize VEGAS+ before integrating. We give an example where preconditioned VEGAS+ is more than 100 times as efficient as VEGAS+ without preconditio ing. Finally, we give examples where VEGAS+ is more than 10 times as efficient as MCMC for Bayesian integrals with D = 3 and 21 parameters. We explain why VEGAS+ will often outperform MCMC for small and moderate sized problems.
... A number of methods have been developed to model observed tidal streams. These include orbit fitting [136,327,328] to the tidal stream and the remnant of the progenitor, if the progenitor still survives and the association can be identified, N-body simulations [329][330][331], approaches of particle releasing/spraying [321,[332][333][334], semi-analytic approaches [331,335] and action angle distribution of tidal debries [291,[336][337][338]. Relatively few studies had specifically constrained the mass of our MW, among which only two measurements are selected into Figure 1 that provided virial mass estimates with statistical errors. ...
We perform an extensive review of the numerous studies and methods used to determine the total mass of the Milky Way. We group the various studies into seven broad classes according to their modeling approaches. The classes include: i) estimating Galactic escape velocity using high velocity objects; ii) measuring the rotation curve through terminal and circular velocities; iii) modeling halo stars, globular clusters and satellite galaxies with the spherical Jeans equation and iv) with phase-space distribution functions; v) simulating and modeling the dynamics of stellar streams and their progenitors; vi) modeling the motion of the Milky Way, M31 and other distant satellites under the framework of Local Group timing argument; and vii) measurements made by linking the brightest Galactic satellites to their counterparts in simulations. For each class of methods, we introduce their theoretical and observational background, the method itself, the sample of available tracer objects, model assumptions, uncertainties, limits and the corresponding measurements that have been achieved in the past. Both the measured total masses within the radial range probed by tracer objects and the extrapolated virial masses are discussed and quoted. We also discuss the role of modern numerical simulations in terms of helping to validate model assumptions, understanding systematic uncertainties and calibrating the measurements. While measurements in the last two decades show a factor of two scatters, recent measurements using Gaia DR2 data are approaching a higher precision. We end with a detailed discussion of future developments in the field, especially as the size and quality of the observational data will increase tremendously with current and future surveys. In such cases, the systematic uncertainties will be dominant and thus will necessitate a much more rigorous testing and characterization of the various mass determination methods.
... 207,267,368] to the tidal stream and the remnant of the progenitor, if the progenitor still survives and the association can be identified, N-body simulations [101,215,216], approaches of particle releasing/spraying [31,140,212,213], semi-analytic approaches [100,101] and action angle distribution of tidal debries [e.g. 38,172,312,314]. Relatively fewer studies had specifically constrained the mass of our MW, among which only two measurements are selected into Fig. 1 that provided virial mass estimates with statistical errors. ...
We perform an extensive review of the numerous studies and methods used to determine the total mass of the Milky Way. We group the various methods into seven broad classes, including: i) estimating Galactic escape velocity using high velocity objects; ii) measuring the rotation curve through terminal and circular velocities; iii) modeling halo stars, globular clusters and satellite galaxies with the Spherical Jeans equation and iv) with phase-space distribution functions; v) simulating and modeling the dynamics of stellar streams and their progenitors; vi) modeling the motion of the Milky Way, M31 and other distant satellites under the framework of Local Group timing argument; and vii) measurements made by linking the brightest Galactic satellites to their counterparts in simulations. For each class of methods, we introduce their theoretical and observational background, the method itself, the sample of available tracer objects, model assumptions, uncertainties, limits and the corresponding measurements that have been achieved in the past. Both the measured total masses within the radial range probed by tracer objects and the extrapolated virial masses are discussed and quoted. We also discuss the role of modern numerical simulations in terms of helping to validate model assumptions, understanding systematic uncertainties and calibrating the measurements. While measurements in the last two decades show a factor of two scatters, recent measurements using \textit{Gaia} DR2 data are approaching a higher precision. We end with a detailed discussion of future developments, especially as the size and quality of the observational data will increase tremendously with current and future surveys. In such cases, the systematic uncertainties will be dominant and thus will necessitate a much more rigorous testing and characterization of the various mass determination methods.
... Stellar streams produced by the tidal disruption of globular clusters and dwarf galaxies are a prevalent feature of the Milky Way environs (see Newberg & Carlin 2016, for a recent review). Observations of stellar streams can provide important constraints on the formation of the Milky Way stellar halo (e.g., Johnston 1998;Bullock & Johnston 2005;Bell et al. 2008), the shape of the Galactic gravitational field (e.g., Johnston et al. 2005;Koposov et al. 2010;Law & Majewski 2010;Bovy 2014;Bonaca et al. 2014;Gibbons et al. 2014;Price-Whelan et al. 2014;Sanders 2014;Bowden et al. 2015;Küpper et al. 2015b;Erkal et al. 2016b;Bovy et al. 2016), and the abundance of low-mass dark matter substructure (e.g., Ibata et al. 2002;Johnston et al. 2002;Carlberg 2009;Yoon et al. 2011;Carlberg 2012;Ngan & Carlberg 2014;Erkal & Belokurov 2015a;Carlberg 2016;Sanderson et al. 2016;Sanders et al. 2016;Bovy et al. 2017;Erkal et al. 2017;Sandford et al. 2017). In addition, stellar streams are a direct snapshot of hierarchical structure formation (Peebles 1965;Press & Schechter 1974;Blumenthal et al. 1984) and support the standard ΛCDM cosmological model (Diemand et al. 2008;Springel et al. 2008). ...
We perform a search for stellar streams around the Milky Way using the first three years of multi-band optical imaging data from the Dark Energy Survey (DES). We use DES data covering $\sim 5000$ sq. deg. to a depth of $g > 23.5$ with a relative photometric calibration uncertainty of $< 1 \%$. This data set yields unprecedented sensitivity to the stellar density field in the southern celestial hemisphere, enabling the detection of faint stellar streams to a heliocentric distance of $\sim 50$ kpc. We search for stellar streams using a matched-filter in color-magnitude space derived from a synthetic isochrone of an old, metal-poor stellar population. Our detection technique recovers four previously known thin stellar streams: Phoenix, ATLAS, Tucana III, and a possible extension of Molonglo. In addition, we report the discovery of eleven new stellar streams. In general, the new streams detected by DES are fainter, more distant, and lower surface brightness than streams detected by similar techniques in previous photometric surveys. As a by-product of our stellar stream search, we find evidence for extra-tidal stellar structure associated with four globular clusters: NGC 288, NGC 1261, NGC 1851, and NGC 1904. The ever-growing sample of stellar streams will provide insight into the formation of the Galactic stellar halo, the Milky Way gravitational potential, as well as the large- and small-scale distribution of dark matter around the Milky Way.
... Pioneered by Tremaine (1999) and , many studies have used them to this purpose (e.g. Eyre & Binney 2011;Sanders & Binney 2013a,b;Bovy 2014;Sanders 2014) as we now summarize. ...
Tidal debris from Galactic satellites generally forms one-dimensional elongated streams, since nearby Galactic orbits have almost identical frequency ratios. We show that the situation is different for orbits close to the Galactic disc, whose vertical frequency Ωz is strongly amplitude dependent. As a consequence, stars stripped from a satellite obtain a range of values for Ωz and hence of frequency ratios, and spread into two dimensions, forming a ribbon-like structure with vertical extent comparable to that of the progenitor orbit. In integrals-of-motion space, tidal ribbons are clumps, which offers the best chance of detection and allows the determination of the Galactic potential vertically across the disc.
... For many considerations in galaxy dynamics, the canonical actionangle phase-space coordinates provide the simplest and clearest insight and tidal debris is no exception. Pioneered by Tremaine (1999), many studies have used them to this purpose (Helmi & White 1999;Eyre & Binney 2011;Sanders & Binney 2013a,b;Sanders 2014;Bovy 2014) as we now summarize. ...
Tidal debris from Galactic satellites generally forms one-dimensional elongated streams, even though the initial phase-space offsets from the progenitor fill a six-dimensional volume. This reduction in dimension is due to the fact that the orbital frequency ratios hardly vary between Galactic orbits. We show here that the situation is quite different for orbits close to the Galactic disc, whose vertical motion is anharmonic with a strongly amplitude dependent frequency $\Omega_z$. As a consequence, stars stripped from a satellite obtain a range of values for $\Omega_z$ and hence of frequency ratios, such that they spread into two dimensions, forming a ribbon-like structure with vertical extent comparable to that of the progenitor orbit. This has implications for the detectability of and search for tidal debris from Galactic satellites on disc orbits.
... The kinematical coldness of tidal streams makes them sensitive to the influence of subhalos with masses 10 8 M . Dynamical modeling of the smooth stream itself [9,10] and of the impact of subhalos [11,12] has been shown to be able to detect and characterize subhalos with masses down to 10 7 M with Gaia and LSST [13]. ...
Narrow stellar streams in the Milky Way halo are uniquely sensitive to
dark-matter subhalos, but many of these may be tidally disrupted. I calculate
the interaction between stellar and dark-matter streams using analytical and
N-body calculations, showing that disrupting objects can be detected as
low-concentration subhalos. Through this effect, we can constrain the
streaminess of the halo as well as the orbit and present position of individual
dark-matter streams. This will have profound implications for the formation of
halos and for direct and indirect-detection dark-matter searches.
Context. Flat rotation curves, v ( r ), are naturally explained by elongated (prolate) dark matter (DM) distributions, and we have provided competitive fits to the SPARC database. To further probe the geometry of the halo, or the equivalent source of gravity in other formulations, one needs observables outside the galactic plane. Stellar streams, poetically analogous to airplane contrails, but caused by tidal dispersion of massive substructures such as satellite dwarf galaxies, would lie on their own plane (consistently with angular momentum conservation) should the DM-halo gravitational field be spherically symmetric. Tracks resembling entire orbits are seldom available because their periods are commensurable with Hubble time, with streams often presenting themselves as short segments.
Aims. Therefore, we aim to establish stellar stream torsion, a local observable that measures the deviation from planarity in differential curve geometry, as a diagnostic providing sensitivity to aspherical DM distributions and ensuring the use of even relatively short streams.
Methods. We performed small-scale simulations of tidally distorted star clusters to check that indeed a central force center produces negligible torsion, while distorted halos can generate it. Turning to observational data, we identified among the known streams those that are at the largest distance from the Galactic center, and that are likely not affected by the Magellanic clouds, as the most promising for the study, and by means of polynomial fits we extracted their differential torsion.
Results. We find that the torsion of the few known streams that should be sensitive to most of the Milky Way’s DM halo is much larger than expected for a central spherical bulb alone. This is consistent with the nonsphericity of the halo.
Conclusions. Future studies of stellar stream torsion with larger samples and further out of the galactic plane should be able to extract the ellipticity of the halo to see whether it is just a slight distortion of a spherical shape or whether it rather resembles a more elongated cigar.
Stellar streams are sensitive probes of the Galactic potential. The likelihood of a stream model given stream data is often assessed using simulations. However, comparing to simulations is challenging when even the stream paths can be hard to quantify. Here we present a novel application of Self-Organizing Maps and first-order Kalman Filters to reconstruct a stream’s path, propagating measurement errors and data sparsity into the stream path uncertainty. The technique is Galactic-model independent, non-parametric, and works on phase-wrapped streams. With this technique, we can uniformly analyze and compare data with simulations, enabling both comparison of simulation techniques and ensemble analysis with stream tracks of many stellar streams. Our method is implemented in the public Python package TrackStream , available at https://github.com/nstarman/trackstream.
We present an approach to measure the Milky Way (MW) potential using the angular accelerations of stars in aggregate as measured by astrometric surveys like Gaia. Accelerations directly probe the gradient of the MW potential, as opposed to indirect methods using, e.g., stellar velocities. We show that end-of-mission Gaia stellar acceleration data may be used to measure the potential of the MW disk at approximately 3σ significance and, if recent measurements of the solar acceleration are included, the local dark matter density at ∼2σ significance. Since the significance of detection scales steeply as t5/2 for observing time t, future surveys that include angular accelerations in the astrometric solutions may be combined with Gaia to precisely measure the local dark matter density and shape of the density profile.
We describe a new algorithm, vegas+, for adaptive multidimensional Monte Carlo integration. The new algorithm adds a second adaptive strategy, adaptive stratified sampling, to the adaptive importance sampling that is the basis for its widely used predecessor vegas. Both vegas and vegas+ are effective for integrands with large peaks, but vegas+ can be much more effective for integrands with multiple peaks or other significant structures aligned with diagonals of the integration volume. We give examples where vegas+ is 2–19× more accurate than vegas. We also show how to combine vegas+ with other integrators, such as the widely available miser algorithm, to make new hybrid integrators. For a different kind of hybrid, we show how to use integrand samples, generated using MCMC or other methods, to optimize vegas+ before integrating. We give an example where preconditioned vegas+ is more than 100× as efficient as vegas+ without preconditioning. Finally, we give examples where vegas+ is more than 10× as efficient as MCMC for Bayesian integrals with D=3 and 21 parameters. We explain why vegas+ will often outperform MCMC for small and moderate sized problems.
Many astrophysical and galaxy-scale cosmological problems require a well-determined gravitational potential. Globular clusters (GCs) surrounding galaxies can be used as dynamical tracers of the luminous and dark matter distribution at large (kpc) scales. This M.Sc. project investigates - by means of the Auriga galaxy simulations and in anticipation of high-resolution IFU data of external galaxies - whether a novel action-based approach could provide a constraint for an axisymmetric approximation of the gravitational potential. In an axisymmetric potential, actions (radial JR, vertical Jz and angular momentum Lz) are integrals of motion and can be used to characterize and label orbits. In the Milky Way (MW), the assumption that stars in cold streams are on similar orbits was found to be a useful first-order constraint of its gravitational potential. In external galaxies, no individual stars but only GCs can be resolved. One could expect GCs from the same dwarf galaxy (DG) merger event to move at the present time on similar orbits in the host galaxy, analogously to stellar streams in the MW, and should therefore have similar actions in the true (axisymmetric) potential. We investigate this idea in one galaxy of the cosmological N-body simulation suite Auriga (Grand et al., 2017). As a first step, we present an effective strategy to fit analytic, axisymmetric, time-dependent potential models with slowly varying parameters to the simulation that are good enough to estimate actions. Then, we select stellar particles born in dwarf galaxies as proxies for GCs and follow the evolution of their orbital actions during the process of merging with a more massive galaxy. These actions show a significant variation over time. As a result, at z = 0, the stellar particles accreted in the same merger event show a very extended distribution in action space. We find that minimizing this distribution, however, cannot constrain the true potential since actions and their evolution are affected by complex physical processes during mergers. In local observations, we confirm this result in the stars of Gaia-Enceladus, one of the few DG mergers of our MW that we know of. Their action distribution is smeared out extensively. Based on these results, we propose that modellers need to find and develop more realistic distribution functions for GCs of a single DG merger event in simulations before being able to constrain the gravitational potential of external galaxies using action-based dynamical modelling of GCs.
We constrain the shape of the Milky Way's halo by dynamical modeling of the observed phase-space tracks of the Pal 5 and GD-1 tidal streams. We find that the only information about the potential gleaned from the tracks of these streams are precise measurements of the shape of the gravitational potential---the ratio of vertical to radial acceleration---at the location of the streams, with weaker constraints on the radial and vertical accelerations separately. The latter will improve significantly with precise proper-motion measurements from Gaia. We measure that the overall potential flattening is 0.95 +/- 0.04 at the location of GD-1 ([R,z] ~ [12.5,6.7] kpc) and 0.94 +/- 0.05 at the position of Pal 5 ([R,z] ~ [8.4,16.8] kpc). Combined with constraints on the force field near the Galactic disk, we determine that the axis ratio of the dark-matter halo's density distribution is 1.05 +/- 0.14 within the inner 20 kpc, with a hint that the halo becomes more flattened near the edge of this volume. The halo mass within 20 kpc is 1.1 +/- 0.1 x 10^{11} M_sun. A dark-matter halo this close to spherical is in tension with the predictions from numerical simulations of the formation of dark-matter halos.
Stellar streams result from the tidal disruption of satellites and star clusters as they orbit a host galaxy, and can be very sensitive probes of the gravitational potential of the host system. We select and study narrow stellar streams formed in a Milky-Way-like dark matter halo of the Aquarius suite of cosmological simulations, to determine if these streams can be used to constrain the present day characteristic parameters of the halo's gravitational potential. We find that orbits integrated in static spherical and triaxial NFW potentials both reproduce the locations and kinematics of the various streams reasonably well. To quantify this further, we determine the best-fit potential parameters by maximizing the amount of clustering of the stream stars in the space of their actions. We show that using our set of Aquarius streams, we recover a mass profile that is consistent with the spherically-averaged dark matter profile of the host halo, although we ignored both triaxiality and time evolution in the fit. This gives us confidence that such methods can be applied to the many streams that will be discovered by the Gaia mission to determine the gravitational potential of our Galaxy.
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 the subhalo distribution down to 10^5 Msun, accounting for the stream's internal dispersion and overlapping impacts. We study the properties of density and track fluctuations with suites of simulations. The one-dimensional density and track power spectra along the stream trace the subhalo mass function, with higher-mass subhalos producing power only on large scales, while lower mass subhalos cause structure on smaller scales. The time-dependence of impacts and of the evolution of the stream after an impact gives rise to bispectra. We further find that tidal streams are essentially corrugated sheets in the presence of subhalo perturbations: different projections of the track all reflect the same pattern of perturbations, facilitating their observational measurement. We apply this formalism to density data for the Pal 5 stream and make a first rigorous determination of 10^{+11}_{-6} dark-matter subhalos with masses between 3x10^6 and 10^9 Msun within 20 kpc from the Galactic center (corresponding to 1.4^{+1.6}_{-0.9} times the number predicted by CDM-only simulations or to f_{sub}(r<20 kpc) ~ 0.2%). Improved data will allow measurements of the subhalo mass function down to 10^5 Msun, thus definitively testing whether dark matter clumps on the smallest scales relevant for galaxy formation.
I present a brief overview of how stellar halos may be used to constrain the process of galaxy formation. In particular, streams and substructure in stellar halos trace merger events but can also be used to determine the mass distribution of the host galaxy and hence put constraints on the nature of dark matter. Much of the focus of this contribution is on the Milky Way, but I also present an attempt to understand the kinematics of the globular cluster system of M31.
Ample observational capabilities exist today to detect the small density perturbations that low-mass dark matter subhaloes impart on stellar streams from disrupting Galactic satellites. In anticipation of these observations, we investigate the expected number and size of gaps by combining an analytic prescription for gap evolution on circular orbits with the flux of subhaloes near the stream. We explore the distribution of gap sizes and depths for a typical cold stream around the Milky Way and find that for a given stream age and gap depth, each subhalo mass produces a characteristic gap size. For a stream with an age of a few Gyr, orbiting at a distance of 10-20 kpc from the Galactic center, even modest subhaloes with a mass of $10^6-10^7 M_\odot$ produce gaps with sizes that are on the order of several degrees. We consider the number and distribution of gap sizes created by subhaloes with masses $10^5-10^9 M_\odot$ and present predictions for six cold streams around the Milky Way. For Pal 5, we forecast 0.7 gaps with a density depletion of at least 25\% and a typical gap size of $8^\circ$. Thus, there appears to be {\it no tension} between the recent non-detection of density depletions in the Pal 5 tidal tails and $\Lambda$CDM expectations. These predictions can be used to guide the scale of future gap searches.
In aspherical potentials orbital planes continuously evolve. The gravitational torques impel the angular momentum vector to precess, that is to slowly stray around the symmetry axis, and nutate, i.e. swing up and down periodically in the perpendicular direction. This familiar orbital pole motion - if detected and measured - can reveal the shape of the underlying gravitational potential, the quantity only crudely gauged in the Galaxy so far. Here we demonstrate that the debris poles of stellar tidal streams show a very similar straying and swinging behavior, and give analytic expressions to link the amplitude and the frequency of the pole evolution to the flattening of the dark matter distribution. Most importantly, we explain how the differential orbital plane precession leads to the broadening of the stream and show that streams on polar orbits ought to scatter faster. We provide expressions for the stream width evolution as a function of the axisymmetric potential flattening and the angle from the symmetry plane and prove that our models are in good agreement with streams produced in N-body simulations. Interestingly, the same intuition applies to streams whose progenitors are on short or long-axis loops in a triaxial potential. Finally, we present a compilation of the Galactic cold stream data, and discuss how the simple picture developed here can be practically used to constrain the symmetry axes and flattening of the Milky Way.
The formation of tidal streams reflects the underlying Galactic potential, such that the resulting structures may be used to constrain the potential. One way of using streams to constrain the Galactic potential is to assume that a stream delineates an orbit. This assumption was discussed in Chap. 5, and the results indicate that an improvement over orbit-fitting is required, which accounts for the stream-orbit misalignment.
In the coming decade the Gaia satellite will precisely measure the positions
and velocities of millions of stars in the Galactic halo, including stars in
many tidal streams. These streams, the products of hierarchical accretion of
satellite galaxies by the Milky Way (MW), can be used to infer the Galactic
gravitational potential thanks to their initial compactness in phase space.
Plans for observations to extend Gaia's radial velocity (RV) measurements to
faint stars, and to determine precise distances to RR Lyrae (RRLe) in streams,
would further extend the power of Gaia's kinematic catalog to characterize the
MW's potential at large Galactocentric distances. In this work I explore the
impact of these extra data on the ability to fit the potential using the method
of action clustering, which statistically maximizes the information content
(clumpiness) of the action space of tidal streams, eliminating the need to
determine stream membership for individual stars. Using a mock halo in a toy
spherical potential, updated post-launch error models for Gaia, and estimates
for RV and distance errors for the tracers to be followed up, I show that
combining either form of additional information with the Gaia catalog greatly
reduces the bias in determining the scale radius and total mass of the Galaxy,
compared to the use of Gaia data alone.
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 satisfying a resonance
condition, the resultant stream can be substantially shorter or longer than
expected, depending on whether the pericentric passages of the progenitor occur
along the bar's minor or major axis respectively. We present a full discussion
of this phenomenon focusing mainly on streams confined to the Galactic plane.
In stark contrast with the evolution in static potentials, which give rise to
streams that grow steadily in time, rotating barred potentials can produce
dynamically old, short streams. This challenges the traditional viewpoint that
the inner halo consists of well phase-mixed material whilst the
tidally-disrupted structures in the outer halo are more spatially coherent. We
argue that this mechanism plays an important role in explaining the
mysteriously short Ophiuchus stream that was recently discovered near the bulge
region of the Milky Way.
We review the available methods for estimating actions, angles and
frequencies of orbits in both axisymmetric and triaxial potentials. The methods
are separated into two classes. Unless an orbit has been trapped by a
resonance, convergent, or iterative, methods are able to recover the actions to
arbitrarily high accuracy given sufficient computing time. Faster
non-convergent methods rely on the potential being sufficiently close to a
separable potential and the accuracy of the action estimate cannot be improved
through further computation. We critically compare the accuracy of the methods
and the required computation time for a range of orbits in an axisymmetric
multi-component Galactic potential. We introduce a new method for estimating
actions that builds on the adiabatic approximation of Sch\"onich & Binney
(2012) and discuss the accuracy required for the actions, angles and
frequencies using suitable distribution functions for the thin and thick discs,
the stellar halo and a star stream. We conclude that for studies of the disc
and smooth halo component of the Milky Way the most suitable compromise between
speed and accuracy is the St\"ackel Fudge, whilst when studying streams the
non-convergent methods do not offer sufficient accuracy and the most suitable
method is computing the actions from an orbit integration via a generating
function. All the software used in this study can be downloaded from
https://github.com/jls713/tact.
We present a freely downloadable software package for modelling the dynamics of galaxies, which we call the Torus Mapper (tm). The package is based around ‘torus mapping’, which is a non-perturbative technique for creating orbital tori for specified
values of the action integrals. Given an orbital torus and a star's position at a reference time, one can compute its position
at any other time, no matter how remote. One can also compute the velocities with which the star will pass through any given
point and the contribution it will make to the time-averaged density there. A system of angle-action coordinates for the given
potential can be created by foliating phase space with orbital tori. Such a foliation is facilitated by the ability of tm to create tori by interpolating on a grid of tori. We summarize the advantages of using tm rather than a standard time-stepper to create orbits, and give segments of code that illustrate applications of tm in several contexts, including setting up initial conditions for an N-body simulation. We examine the precision of the orbital tori created by tm and the behaviour of the code when orbits become trapped by a resonance.
We develop a formalism for modelling the impact of dark matter subhaloes on cold thin streams. Our formalism models the formation
of a gap in a stream in angle-frequency space and is able to handle general stream and impact geometry. We analyse an N-body simulation of a cold stream formed from a progenitor on an eccentric orbit in an axisymmetric potential, which is perturbed
by a direct impact from a 108M⊙ subhalo, and produce a complete generative model of the perturbed stream that matches the simulation well at a range of times.
We show how the results in angle-frequency space can be related to physical properties of the gaps and that previous results
for more constrained simulations are recovered. We demonstrate how our results are dependent upon the mass of the subhalo
and the location of the impact along the stream. We find that gaps formed far downstream grow more rapidly than those closer
to the progenitor due to the more ordered nature of the stream members far from the progenitor. Additionally, we show that
the minimum gap density plateaus in time at a value that decreases with increasing subhalo mass.
How do galaxies move relative to one another? While we can examine the motion
of dark matter subhalos around their hosts in simulations of structure
formation, determining the orbits of satellites around their parent galaxies
from observations is impossible except for a small number of nearby cases. In
this work we outline a novel approach to probing the orbital distributions of
infalling satellite galaxies using the morphology of tidal debris structures.
It has long been understood that the destruction of satellites on near-radial
orbits tends to lead to the formation of shells of debris, while those on less
eccentric orbits produce tidal streams. We combine an understanding of the
scaling relations governing the orbital properties of debris with a simple
model of how these orbits phase-mix over time to produce a `morphology metric'
that more rigorously quantifies the conditions required for shells to be
apparent in debris structures as a function of the satellite's mass and orbit
and the interaction time. Using this metric we demonstrate how differences in
orbit distributions can alter the relative frequency of shells and stream
structures observed around galaxies. These experiments suggest that more
detailed modeling and careful comparisons with current and future surveys of
low surface brightness features around nearby galaxies should be capable of
actually constraining orbital distributions and provide new insights into our
understanding of structure formation.
Palomar 5 (Pal 5) is a faint halo globular cluster associated with narrow
tidal tails. It is a useful system to understand the process of tidal
dissolution, as well as to constrain the potential of the Milky Way. A
well-determined orbit for Pal 5 would enable detailed study of these open
questions. We present here the first CCD-based proper motion measurement of Pal
5 obtained using SDSS as a first epoch and new LBT/LBC images as a second,
giving a baseline of 15 years. We perform relative astrometry, using SDSS as a
distortion-free reference, and images of the cluster and also of the Pal 5
stream for the derivation of the distortion correction for LBC. The reference
frame is made up of background galaxies. We correct for differential chromatic
refraction using relations obtained from SDSS colors as well as from
flux-calibrated spectra, finding that the correction relations for stars and
for galaxies are different. We obtain mu_alpha=-2.296+/-0.186 mas/yr and
mu_delta=-2.257+/-0.181 mas/yr for the proper motion of Pal 5. We use this
motion, and the publicly available code galpy, to model the disruption of Pal 5
in different Milky Way models consisting of a bulge, a disk and a spherical
dark matter halo. Our fits to the observed stream properties (streak and radial
velocity gradient) result in a preference for a relatively large Pal 5 distance
of around 24 kpc. A slightly larger absolute proper motion than what we measure
also results in better matches but the best solutions need a change in
distance. We find that a spherical Milky Way model, with V_0=220 km/s and V_(20
kpc), i.e., approximately at the apocenter of Pal 5, of 218 km/s, can match the
data well, at least for our choice of disk and bulge parametrization.
Several long, dynamically cold stellar streams have been observed around the
Milky Way Galaxy, presumably formed from the tidal disruption of globular
clusters. In integrable potentials---where all orbits are dynamically
regular---tidal debris phase-mixes close to the orbit of the progenitor system.
However, cosmological simulations of structure formation suggest that the Milky
Way's dark matter halo is expected not to be fully integrable; an appreciable
fraction of orbits will be chaotic. This paper examines the influence of chaos
on the phase-space morphology of cold tidal streams. We find very stark
results: Streams in chaotic regions look very different from those in regular
regions. We find that streams (simulated using test particle ensembles of
nearby orbits) can be sensitive to chaos on a much shorter time-scale than any
standard prediction (from the Lyapunov or frequency-diffusion times). For
example, on a weakly chaotic orbit with a chaotic timescale predicted to be
>1000 orbital periods (>1000 Gyr), the resulting stellar stream is, after just
a few 10's of orbits, substantially more diffuse than any formed on a nearby
but regular orbit. We find that the enhanced diffusion of the stream stars can
be understood by looking at the variance in orbital frequencies of orbit
ensembles centered around the parent (progenitor) orbit. Our results suggest
that long, cold streams around our Galaxy must exist only on regular (or very
nearly regular) orbits; they potentially provide a map of the regular regions
of the Milky Way potential. This suggests a promising new direction for the use
of tidal streams to constrain the distribution of dark matter around our
Galaxy.
Using the example of the tidal stream of the Milky Way globular cluster
Palomar 5 (Pal 5), we demonstrate how observational data on streams can be
efficiently reduced in dimensionality and modeled in a Bayesian framework. Our
approach combines detection of stream overdensities by a
Difference-of-Gaussians process with fast streakline models, a continuous
likelihood function built from these models, and inference with MCMC. By
generating $\approx10^7$ model streams, we show that the geometry of the Pal 5
debris yields powerful constraints on the solar position and motion, the Milky
Way and Pal 5 itself. All 10 model parameters were allowed to vary over large
ranges without additional prior information. Using only SDSS data and a few
radial velocities from the literature, we find that the distance of the Sun
from the Galactic Center is $8.30\pm0.25$ kpc, and the transverse velocity is
$253\pm16$ km/s. Both estimates are in excellent agreement with independent
measurements of these quantities. Assuming a standard disk and bulge model, we
determine the Galactic mass within Pal 5's apogalactic radius of 19 kpc to be
$(2.1\pm0.4)\times10^{11}$ M$_\odot$. Moreover, we find the potential of the
dark halo with a flattening of $q_z = 0.95^{+0.16}_{-0.12}$ to be essentially
spherical within the radial range that is effectively probed by Pal 5. We also
determine Pal 5's mass, distance and proper motion independently from other
methods, which enables us to perform vital cross-checks. We conclude that with
more observational data and by using additional prior information, the
precision of this method can be significantly increased.
We present an approach to approximating rapidly the actions in a general triaxial potential. The method is an extension of
the axisymmetric approach presented by Binney, and operates by assuming that the true potential is locally sufficiently close
to some Stäckel potential. The choice of Stäckel potential and associated ellipsoidal coordinates is tailored to each individual
input phase-space point. We investigate the accuracy of the method when computing actions in a triaxial Navarro–Frenk–White
potential. The speed of the algorithm comes at the expense of large errors in the actions, particularly for the box orbits.
However, we show that the method can be used to recover the observables of triaxial systems from given distribution functions
to sufficient accuracy for the Jeans equations to be satisfied. Consequently, such models could be used to build models of
external galaxies as well as triaxial components of our own Galaxy. When more accurate actions are required, this procedure
can be combined with torus mapping to produce a fast convergent scheme for action estimation.
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 bursts of stars emitted
near pericentre, rather than clumping at particular oscillation phases as sometimes suggested. When detectable, these streaky
features are a reliable diagnostic for the stream's direction of motion and encode other information on the progenitor and
its orbit. We show several tests of the recipe in alternate potentials, including a case with a chaotic progenitor orbit which
displays a marked effect on the width of the stream. Although the specific ejection recipe may need adjusting when elements
such as the orbit or satellite density profile are changed significantly, our examples suggest that model tidal streams can
be quickly and accurately generated by models of this general type for use in Bayesian sampling.
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. Internal
coherence is the requirement for the appearance of substructure in tidal
debris, and I concentrate on the `feathering' typical of GC streams.
Overdensities are associated with minima in the relative streaming velocity of
the stream members. For streams with high circularity, these are caused by the
epicyclic oscillations of stars; however, for highly non-circular progenitor's
orbits, substructure is caused by the oscillating differences in energy and
actions with which material is shed at different orbital phases of the
progenitor. This modulation results in different streaming speeds: the
streakline of material shed between two successive apocentric passages is
folded along its length, pulled at its centre by the faster streaming of
particles released near pericenter, which are therefore more widely scattered.
When the stream is coherent enough, this mechanism is potentially capable of
generating a bimodal profile in the density distributions of the longer wraps
of more massive progenitors, which I dub `bifurcations'. The conditions for
internal coherence are explored and I comment on the cases of Palomar 5,
Willman 1, the Anticenter and Sagittarius' streams. Analytical methods are
accompanied by numerical experiments, performed using a purposely built
generative model, also presented here.
We present a new method for determining the Galactic gravitational potential
based on forward modeling of tidal stellar streams. We use this method to test
the performance of smooth and static analytic potentials in representing
realistic dark matter halos, which have substructure and are continually
evolving by accretion. Our FAST-FORWARD method uses a Markov Chain Monte Carlo
algorithm to compare, in 6D phase space, an "observed" stream to models created
in trial analytic potentials. We analyze a large sample of streams evolved in
the Via Lactea II (VL2) simulation, which represents a realistic Galactic halo
potential. The recovered potential parameters are in agreement with the best
fit to the global, present-day VL2 potential. However, merely assuming an
analytic potential limits the dark matter halo mass measurement to an accuracy
of 5 to 20%, depending on the choice of analytic parametrization. Collectively,
mass estimates using streams from our sample reach this fundamental limit, but
individually they can be highly biased. Individual streams can both under- and
overestimate the mass, and the bias is progressively worse for those with
smaller perigalacticons, motivating the search for tidal streams at
galactocentric distances larger than 70 kpc. We estimate that the assumption of
a static and smooth dark matter potential in modeling of the GD-1 and Pal5-like
streams introduces an error of up to 50% in the Milky Way mass estimates.
Motivated by recent observations of the Sagittarius stream, we devise a rapid algorithm to generate faithful representations
of the centroids of stellar tidal streams formed in a disruption of a progenitor of an arbitrary mass in an arbitrary potential.
Our method works by releasing swarms of test particles at the Lagrange points around the satellite and subsequently evolving
them in a combined potential of the host and the progenitor. We stress that the action of the progenitor's gravity is crucial
to making streams that look almost indistinguishable from the N-body realizations, as indeed ours do. The method is tested on mock stream data in three different Milky Way potentials with
increasing complexity, and is shown to deliver unbiased inference on the Galactic mass distribution out to large radii. When
applied to the observations of the Sagittarius stream, our model gives a natural explanation of the stream's apocentric distances
and the differential orbital precession. We, therefore, provide a new independent measurement of the Galactic mass distribution
beyond 50 kpc. The Sagittarius stream model favours a light Milky Way with the mass 4.1 ± 0.4 × 1011 M⊙ at 100 kpc, which can be extrapolated to 5.6 ± 1.2 × 1011 M⊙ at 200 kpc. Such a low mass for the Milky Way Galaxy is in good agreement with estimates from the kinematics of halo stars
and from the satellite galaxies (once Leo I is removed from the sample). It entirely removes the ‘Too Big To Fail Problem’.
The dark matter halo of the Milky Way is expected to be triaxial and filled
with substructure. It is hoped that streams or shells of stars produced by
tidal disruption of stellar systems will provide precise measures of the
gravitational potential to test these predictions. We develop a method for
inferring the Galactic potential with tidal streams based on the idea that the
stream stars were once close in phase space. Our method can flexibly adapt to
any form for the Galactic potential: it works in phase-space rather than
action-space and hence relies neither on our ability to derive actions nor on
the integrability of the potential. Our model is probabilistic, with a
likelihood function and priors on the parameters. The method can properly
account for finite observational uncertainties and missing data dimensions. We
test our method on synthetic datasets generated from N-body simulations of
satellite disruption in a static, multi-component Milky Way including a
triaxial dark matter halo with observational uncertainties chosen to mimic
current and near-future surveys of various stars. We find that with just four
well-measured stream stars, we can infer properties of a triaxial potential
with precisions of order 5-7 percent. Without proper motions we obtain 15
percent constraints on potential parameters and precisions around 25 percent
for recovering missing phase-space coordinates. These results are encouraging
for the eventual goal of using flexible, time-dependent potential models
combined with larger data sets to unravel the detailed shape of the dark matter
distribution around the Milky Way.
We report the discovery of a narrow stellar stream crossing the
constellations of Sculptor and Fornax in the Southern celestial hemisphere. The
portion of the stream detected in the Data Release 1 photometry of the ATLAS
survey is at least 12 degrees long, while its width is $\approx$ 0.25 deg. The
Color Magnitude Diagram of this halo sub-structure is consistent with a
metal-poor [Fe/H] $\lesssim$-1.4 stellar population located at a heliocentric
distance of 20 $\pm$ 5 kpc. There are three globular clusters that could
tentatively be associated with the stream: NGC 7078, NGC 7006 (M15) and Pyxis.
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 progenitor orbit, 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') in different coordinate systems, how to quickly estimate the
dispersion around its track, and how to draw mock stream data. As a generative
model, this framework allows one to compute the full probability distribution
function and marginalize over or condition it on certain phase-space dimensions
as well as convolve it with observational uncertainties. This will be
instrumental in proper data analysis of stream data. In addition to providing a
computationally-efficient practical tool for modeling the dynamics of tidal
streams, the action-angle nature of the framework helps elucidate in exactly
what manner streams do not follow single orbits, how the observed width of the
stream relates to the velocity dispersion or mass of the progenitor, and how
the progenitors of 'orphan' streams could be located.
The practical usefulness of the proposed framework crucially depends on the
ability to calculate action-angle variables for any orbit in any gravitational
potential. A novel method for calculating actions, frequencies, and angles in
any static potential using a single orbit integration is described in an
Appendix.
emcee is an extensible, pure-Python implementation of Goodman &
Weare's Affine Invariant Markov chain Monte Carlo (MCMC) Ensemble
sampler. It's designed for Bayesian parameter estimation. The algorithm
behind emcee has several advantages over traditional MCMC sampling
methods and has excellent performance as measured by the autocorrelation
time (or function calls per independent sample). One advantage of the
algorithm is that it requires hand-tuning of only 1 or 2 parameters
compared to ˜ N^2 for a traditional algorithm in an N-dimensional
parameter space. Exploiting the parallelism of the ensemble method,
emcee permits any user to take advantage of multiple CPU cores without
extra effort.
Near-future data from ESA's Gaia mission will provide precise, full
phase-space information for hundreds of millions of stars out to heliocentric
distances of ~10 kpc. This "horizon" for full phase-space measurements is
imposed by the Gaia parallax errors degrading to worse than 10%, and could be
significantly extended by an accurate distance indicator. Recent work has
demonstrated how Spitzer observations of RR Lyrae stars can be used to make
distance estimates accurate to 2%, effectively extending the Gaia, precise-data
horizon by a factor of ten in distance and a factor of 1000 in volume. This
Letter presents one approach to exploit data of such accuracy to measure the
Galactic potential using small samples of stars associated with debris from
satellite destruction. The method is tested with synthetic observations of 100
stars from the end point of a simulation of satellite destruction: the shape,
orientation, and depth of the potential used in the simulation are recovered to
within a few percent. The success of this simple test with such a small sample
in a single debris stream suggests that constraints from multiple streams could
be combined to examine the Galaxy's dark matter halo in even more detail --- a
truly unique opportunity that is enabled by the combination of Spitzer and Gaia
with our intimate perspective on the Galaxy.
We introduce the Minimum Entropy Method, a simple statistical technique for constraining the Milky Way gravitational potential and simultaneously testing different gravity theories directly from 6D phase-space surveys and without adopting dynamical models. We demonstrate that orbital energy distributions that are separable (i.e., independent of position) have an associated entropy that increases under wrong assumptions about the gravitational potential and/or gravity theory. Of known objects, 'cold' tidal streams from low-mass progenitors follow orbital distributions that most nearly satisfy the condition of separability. Although the orbits of tidally stripped stars are perturbed by the progenitor's self-gravity, systematic variations of the energy distribution can be quantified in terms of the cross-entropy of individual tails, giving further sensitivity to theoretical biases in the host potential. The feasibility of using the Minimum Entropy Method to test a wide range of gravity theories is illustrated by evolving restricted N-body models in a Newtonian potential and examining the changes in entropy introduced by Dirac, MONDian, and f(R) gravity modifications.
To constrain the Galactic gravitational potential near the Sun ($\sim$1.5
kpc), we derive and model the spatial and velocity distribution for a sample of
9000 K-dwarfs that have spectra from SDSS/SEGUE, which yield radial velocities
and abundances ([Fe/H] & [$\alpha$/Fe]). We first derive the spatial density
distribution for stars of three abundance-selected sub-populations by
accounting for the survey's selection function. The vertical profile of these
sub-populations are simple exponentials and their vertical dispersion profile
is nearly isothermal. To model these data, we apply the `vertical' Jeans
Equation, which relates the observable tracer number density and vertical
velocity dispersion to the gravitational potential or vertical force. We
explore a number of functional forms for the vertical force law, and fit the
dispersion and density profiles of all abundance selected sub-populations
simultaneously in the same potential, and explore all parameter co-variances
using MCMC. Our fits constrain a disk {\it mass} scale height $\lesssim$ 300 pc
and the total surface mass density to be $67 \pm 6 M_{\odot} {\rm pc^{-2}}$ at
$|z| = 1.0$ kpc of which the contribution from all stars is $42 \pm 5 M_{\odot}
{\rm pc^{-2}}$ (presuming a contribution from cold gas of $13 M_{\odot} {\rm
pc^{-2}}$). We find significant constraints on the local dark matter density of
$0.0065\pm0.0023 M_{\odot} {\rm pc^{-3}}$ ($0.25\pm0.09 {\rm GeV cm^{-3}} $).
Together with recent experiments this firms up the best estimate of
$0.0075\pm0.0021 M_{\odot} {\rm pc^{-3}}$ ($0.28\pm0.08 {\rm GeV cm^{-3}} $),
consistent with global fits of approximately round dark matter halos to
kinematic data in the outskirts of the Galaxy.
We apply a new method to determine the local disc matter and dark halo matter density to kinematic and position data for ∼2000
K dwarf stars taken from the literature. Our method assumes only that the disc is locally in dynamical equilibrium, and that
the ‘tilt’ term in the Jeans equations is small up to ∼1 kpc above the plane. We present a new calculation of the photometric
distances to the K dwarf stars, and use a Monte Carlo Markov chain to marginalize over uncertainties in both the baryonic
mass distribution, and the velocity and distance errors for each individual star. We perform a series of tests to demonstrate
that our results are insensitive to plausible systematic errors in our distance calibration, and we show that our method recovers
the correct answer from a dynamically evolved N-body simulation of the Milky Way. We find a local dark matter density of M⊙ pc−3 ( GeV cm−3) at 90 per cent confidence assuming no correction for the non-flatness of the local rotation curve, and M⊙ pc−3 ( GeV cm−3) if the correction is included. Our 90 per cent lower bound on ρdm is larger than the canonical value typically assumed in the literature, and is at mild tension with extrapolations from the
rotation curve that assume a spherical halo. Our result can be explained by a larger normalization for the local Milky Way
rotation curve, an oblate dark matter halo, a local disc of dark matter or some combination of these.
The narrow GD-1 stream of stars, spanning 60ð on the sky at a distance of ~10 kpc from the Sun and ~15 kpc from the Galactic center, is presumed to be debris from a tidally disrupted star cluster that traces out a test-particle orbit in the Milky Way halo. We combine Sloan Digital Sky Survey (SDSS) photometry, USNO-B astrometry, and SDSS and Calar Alto spectroscopy to construct a complete, empirical six-dimensional (6D) phase-space map of the stream. We find that an eccentric orbit in a flattened isothermal potential describes this phase-space map well. Even after marginalizing over the stream orbital parameters and the distance from the Sun to the Galactic center, the orbital fit to GD-1 places strong constraints on the circular velocity at the Sun's radius Vc = 224 ñ 13 km s-1 and total potential flattening q æ = 0.87+0.07 -0.04. When we drop any informative priors on Vc , the GD-1 constraint becomes Vc = 221 ñ 18 km s-1. Our 6D map of
The tree code for the approximate evaluation of gravitational forces is extended and substantially accelerated by including mutual cell-cell interactions. These are computed by a Taylor series in Cartesian coordinates and in a completely symmetric fashion, such that Newton's third law is satisfied by construction and that therefore momentum is exactly conserved. The computational effort is further reduced by exploiting the mutual symmetry of the interactions. For typical astrophysical problems with N=105 and at the same level of accuracy, the new code is about 4 times faster than the tree code. For large N, the computational costs are found to scale almost linearly with N, which can also be supported by a theoretical argument, and the advantage over the tree code increases with ever larger N.
We develop a framework for modelling the Milky Way using stellar streams and a wide range of photometric and kinematic observations.
Through the use of mock data we demonstrate that a standard suite of Galactic observations leads to degeneracies in the inferred
halo parameters. We then incorporate a GD-1-like stream into this suite using the orbit-fitting technique and show that the
streams reduce the uncertainties in these parameters provided all observations are fit simultaneously. We also explore how
the assumption of a disc–halo alignment can lead unphysical models. Our results may explain why some studies based on the
Sagittarius stream find that the halo's intermediate axis is parallel to the disc spin axis even though such a configuration
is highly unstable. Finally we show that both longer streams and multiple streams lead to improvements in our ability to infer
the shape of our dark halo.
Tidal streams are a powerful probe of the Milky Way (MW) potential shape. In
this paper, we introduce a simple test particle method to fit stream data,
using a Markov Chain Monte Carlo technique to marginalise over uncertainties in
the progenitor's orbit and the Milky Way halo shape parameters. Applying it to
mock data of thin streams in the MW halo, we show that, even for very cold
streams, stream-orbit offsets - not modelled in our simple method - introduce
systematic biases in the recovered shape parameters. For the streams that we
consider, and our particular choice of potential parameterisation, these errors
are of order ~20% on the halo flattening parameters. However, larger systematic
errors can arise for more general streams and potentials; such offsets need to
be correctly modelled in order to obtain an unbiased recovery of the underlying
potential.
Assessing which of the known Milky Way streams are most constraining, we find
NGC 5466 and Pal 5 are the most promising candidates. These form an interesting
pair as their orbital planes are both approximately perpendicular to each other
and to the disc, giving optimal constraints on the MW halo shape. We show that
- while with current data their constraints on potential parameters are poor -
good radial velocity data along the Pal 5 stream will provide constraints on qz
- the flattening perpendicular to the disc. Furthermore, as discussed in a
companion paper, NGC 5466 can provide rather strong constraints on the MW halo
shape parameters, if the tentative evidence for a departure from the smooth
orbit towards its western edge is confirmed.
In the first of these two papers, we demonstrated that assuming streams delineate orbits can lead to order one errors in potential
parameters for realistic Galactic potentials. Motivated by the need for an improvement on orbit-fitting, we now present an
algorithm for constraining the Galactic potential using tidal streams without assuming that streams delineate orbits. This
approach is independent of the progenitor mass so is valid for all observed tidal streams. The method makes heavy use of angle–action
variables and seeks the potential which recovers the expected correlations in angle space. We demonstrate that the method
can correctly recover the parameters of a simple two-parameter logarithmic potential by analysing an N-body simulation of a stream. We investigate the magnitude of the errors in observational data for which the method can still
recover the correct potential and compare this to current and future errors in data. The errors in the observables of individual
stars for current and near future data are shown to be too large for the direct use of this method, but when the data are
averaged in bins on the sky, the resulting averaged data are accurate enough to constrain correctly the potential parameters
for achievable observational errors. From pseudo-data with errors comparable to those that will be furnished in the era of
Gaia (20 per cent distance errors, 1.2 mas yr−1 proper motion errors, and 10 km s−1 line-of-sight velocity errors) we recover the circular velocity, Vc = 220 km s− 1, and the flattening of the potential, q = 0.9, to be Vc = 223 ± 10 km s− 1 and q = 0.91 ± 0.09.
Tidal streams do not, in general, delineate orbits. A stream–orbit misalignment is expected to lead to biases when using orbit-fitting
to constrain models for the Galactic potential. In this first of two papers, we discuss the expected magnitude of the misalignment
and the resulting dangers of using orbit-fitting algorithms to constrain the potential. We summarize data for known streams
which should prove useful for constraining the Galactic potential, and compute their actions in a realistic Galactic potential.
We go on to discuss the formation of tidal streams in angle–action space, and explain why, in general, streams do not delineate
orbits. The magnitude of the stream–orbit misalignment is quantified for a logarithmic potential and a multicomponent Galactic
potential. Specifically, we focus on the expected misalignment for the known streams. By introducing a two-parameter family
of realistic Galactic potentials we demonstrate that assuming that these streams delineate orbits can lead to order one errors
in the halo flattening and halo-to-disc force ratio at the Sun. We present a discussion of the dependence of these results
on the progenitor mass and demonstrate that the misalignment is mass independent for the range of masses of observed streams.
Hence, orbit-fitting does not yield better constraints on the potential if one uses narrower, lower mass streams.
The usefulness of angle-action variables in galaxy dynamics is well known,
but their use is limited due to the difficulty of their calculation in
realistic galaxy potentials. Here we present a method for estimating
angle-action variables in a realistic Milky Way axisymmetric potential by
locally fitting a St\"ackel potential over the region an orbit probes. The
quality of the method is assessed by comparison with other known methods for
estimating angle-action variables of a range of disc and halo-type orbits. We
conclude by projecting the Geneva-Copenhagen survey into angle-action space.
Tidal streams provide a powerful tool by means of which the matter distribution of the dark matter halos of their host galaxies can be studied. However, the analysis is not straightforward because streams do not delineate orbits, and for most streams, especially those in external galaxies, kinematic information is absent. We present a method wherein streams are fit with simple corrections made to possible orbits of the progenitor, using a Bayesian technique known as Parallel Tempering to efficiently explore the parameter space. We show that it is possible to constrain the shape of the host halo potential or its density distribution using only the projection of tidal streams on the sky, if the host halo is considered to be axisymmetric. By adding kinematic data or the circular velocity curve of the host to the fitting data, we are able to recover other parameters of the matter distribution such as its mass and profile. We test our method on several simulated low mass stellar streams and also explore the cases for which additional data are required.
Recent years have seen the discovery of many tidal streams through the Galaxy. Relatively straightforward observations of
a stream allow one to deduce three phase-space coordinates of an orbit. An algorithm is presented that reconstructs the missing
phase-space coordinates from these data. The reconstruction starts from assumed values of the Galactic potential and a distance
to one point on the orbit, but with noise-free data the condition that energy be conserved on the orbit enables one to reject
incorrect assumptions. The performance of the algorithm is investigated when errors are added to the input data that are comparable
to those in published data for the streams of Pal 5. It is found that the algorithm returns distances and proper motions that
are accurate to of the order of 1 per cent, and enables one to reject quite resonable but incorrect trial potentials. In practical
applications, it will be important to minimize errors in the imput data, and there is considerable scope for doing this.
Partially phase-mixed structures in galaxies occupy a complex surface of dimension D in six-dimensional phase space. The appearance of such structures to observers is determined by their projection into a space
the dimensionality K of which is determined by the number of observables (e.g. sky position, distance, radial velocity, etc.). We discuss the
expected dimensionality of phase-space structures and suggest that the most prominent features in surveys with KD will be stable singularities (catastrophes). The simplest of these are the shells seen in the outer parts of elliptical galaxies.
An analysis of the kinematics of 412 stars at 1-4 kpc from the Galactic
mid-plane by Moni Bidin et al. (2012) has claimed to derive a local density of
dark matter that is an order of magnitude below standard expectations. We show
that this result is incorrect and that it arises from the assumption that the
mean azimuthal velocity of the stellar tracers is independent of Galactocentric
radius at all heights. We substitute the assumption, supported by data, that
the circular speed is independent of radius in the mid-plane. We demonstrate
that the assumption of constant mean azimuthal velocity is implausible by
showing that it requires the circular velocity to drop more steeply than
allowed by any plausible mass model, with or without dark matter, at large
heights above the mid-plane. Using the approximation that the circular velocity
curve is flat in the mid-plane, we find that the data imply a local dark-matter
density of 0.008 +/- 0.003 Msun/pc^3 = 0.3 +/- 0.1 GeV/cm3, fully consistent
with standard estimates of this quantity. This is the most robust direct
measurement of the local dark-matter density to date.
A novel code for the approximate computation of long-range forces between N mutually interacting bodies is presented. The code is based on a hierarchical tree of cubic cells and features mutual cell–cell interactions which are calculated via a Cartesian Taylor expansion in a symmetric way, such that total momentum is conserved. The code benefits from an improved and simple multipole acceptance criterion that reduces the force error and the computational effort. For N≳104, the computational costs are found empirically to rise sublinearly with N. For applications in stellar dynamics, this is the first competitive code with complexity (N); it is faster than the standard tree code by a factor of 10 or more.
We investigate the epicyclic motion of stars escaping from star clusters.
Using streaklines, we visualise the path of escaping stars and show how
epicyclic motion leads to over- and underdensities in tidal tails of star
clusters moving on circular and eccentric orbits about a galaxy. Additionally,
we investigate the effect of the cluster mass on the tidal tails, by showing
that their structure is better matched when the perturbing effect of the
cluster mass is included. By adjusting streaklines to results of N-body
computations we can accurately and quickly reproduce all observed substructure,
especially the streaky features often found in simulations which may be
interpreted in observations as multiple tidal tails. Hence, we can rule out
tidal shocks as the origin of such substructures. Finally, from the adjusted
streakline parameters we can verify that for the star clusters we studied
escape mainly happens from the tidal radius of the cluster, given by x_L =
(GM/(\Omega^2-\partial^2\Phi/\partial R^2))^{1/3}. We find, however, that there
is another limiting radius, the "edge" radius, which gives the smallest radius
from which a star can escape during one cluster orbit about the galaxy. For
eccentric cluster orbits the edge radius shrinks with increasing orbital
eccentricity (for fixed apocentric distance) but is always significantly larger
than the respective perigalactic tidal radius. In fact, the edge radii of the
clusters we investigated, which are extended and tidally filling, agree well
with their (fitted) King radii, which may indicate a fundamental connection
between these two quantities.
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 realistic
galaxy potentials are poorly represented by single orbits, contrary to what is
often assumed. We further demonstrate that attempting to constrain the
parameters of the Galactic potential by fitting orbits to such streams can lead
to significant systematic error. However, we show that it is possible to
predict accurately the track of streams from simple models of the action-space
distribution of the disrupted cluster.
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 show that accurate proper
motions of stream stars would enable distances to be determined to points on the stream that are independent of any assumption
about the Galaxy's gravitational potential. Such ‘Galactic parallaxes’ would be as fundamental as conventional trigonometric
parallaxes, but measurable to distances ∼70 times further.
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 the uncertainties in the predicted distances and proper motions increase. With radial-velocity data along a stream ~40deg long and <0.3deg wide on the sky accurate to ~1 km/s the precisions of the distances and tangential velocities along the stream are 4 percent and 5 km/s, respectively. The technique can be used to diagnose the Galactic potential: if circular-speed curve is actually flat, both a Keplerian potential and Phi(r) proportional to r are readily excluded. Given the correct radial density profile for the dark halo, the halo's mass can be determined to a precision of 5 percent. Comment: 13 pages, 16 figures, submitted to MNRAS
We study numerical simulations of satellite galaxy disruption in a potential resembling that of the Milky Way. Our goal is to assess whether a merger origin for the stellar halo would leave observable fossil structure in the phase-space distribution of nearby stars. We show how mixing of disrupted satellites can be quantified using a coarsegrained entropy. Although after 10 billion years few obvious asymmetries remain in the distribution of particles in configuration space, strong correlations are still present in velocity space. We give a simple analytic description of these effects, based on a linearised treatment in action-angle variables, which shows how the kinematic and density structure of the debris stream changes with time. By applying this description we find that a single dwarf elliptical-like satellite of current luminosity 10 8 L fi disrupted 10 Gyr ago from an orbit circulating in the inner halo (mean apocentre 12 kpc) would contribute about 30 kinematically cold stre...
Angle-action coordinates are used to study the relic of an N-body simulation of a self-gravitating satellite galaxy that was released on a short-period orbit within the disc of the Galaxy. Satellite stars that lie within 1.5 kpc of the Sun are confined to a grid of patches in action space. As the relic phase mixes for longer, the patches become smaller and more numerous. These patches can be seen even when the angle-action coordinates of an erroneous Galactic potential are used, but using the wrong potential displaces them. Diagnostic quantities constructed from the angle coordinates both allow the true potential to be identified, and the relic to be dated. Hence, when the full phase space coordinates of large numbers of solar-neighbourhood stars are known, it should be possible to identify members of particular relics from the distribution of stars in an approximate action space. This would then open up the possibility of determining the time since the relic was disrupted and gaining better knowledge of the Galactic potential.
The availability of angle-action coordinates for arbitrary potentials is the key to these developments. The paper includes a brief introduction to the torus technique used to generate them.
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