Self-consistent Dynamical Model Research Articles

Self-consistent dynamical models with a finite extent – IV. Wendland models based on compactly supported radial basis functions

ABSTRACT We present a new step in our systematic effort to develop self-consistent dynamical models with a finite radial extent. The focus is on models with simple analytical density profiles allowing for analytical calculations of many dynamical properties. In this paper, we introduce a family of models, termed Wendland models, based on compactly supported radial basis functions. The family of models is characterized by a parameter k that controls the smoothness of the transition at the truncation radius. In the limit $k\rightarrow \infty$, the Wendland model reduces to a non-truncated model with a Gaussian density profile. For each Wendland model, the density, mass and gravitational potential are simple truncated polynomial functions of radius. Via the sphecow tool we demonstrate that all Wendland models can be supported by isotropic distribution functions. Surprisingly, the isotropic distribution function exhibits varied behaviour across different Wendland models. Additionally, each model can be supported by a continuum of Osipkov–Merritt orbital structures, ranging from radially anisotropic to completely tangential at the truncation radius. To the best of our knowledge, the Wendland models presented here are the first family of models accommodating both radial and tangential Osipkov–Merritt distribution functions. Using linear superposition, these models can easily be combined to generate Wendland models with even more diverse orbital structures. While the Wendland models are not fully representative of real dynamical systems due to their Gaussian-like density profile, this study lays important groundwork for constructing more realistic models with truncated density profiles that can be supported by a range of orbital structures.

The most stringent upper limit set on the mass of a central black hole in 47 Tucanae using dynamical models

Globular clusters (GCs) have been proposed as promising sites for discovering intermediate-mass black holes (IMBHs), offering the possibility to gain crucial insights into the formation and evolution of these elusive objects. The Galactic GC 47 Tucanae (also known as NGC 104) has been suggested as a potential IMBH host, yet previous studies have yielded conflicting results. Therefore, we present a set of self-consistent dynamical models based on distribution functions (DFs) that depend on action integrals to assess the presence (or absence) of an IMBH in 47 Tucanae. Leveraging the state-of-the-art Multi Unit Spectroscopic Explorer (MUSE) and Hubble Space Telescope (HST) data, we analyzed the three-dimensional (3D) kinematics of the cluster’s central regions, fitting individual star velocities down to the sub-arcsec scale (approximately 10−2 pc). According to our analysis, the inner kinematics of 47 Tucanae is incompatible with a central BH more massive than 578 M⊙ (at 3σ). This is the most stringent upper limit placed thus far on the mass of a putative IMBH in 47 Tucanae via a dynamical study.

Self-consistent dynamical models with a finite extent – III. Truncated power-law spheres

ABSTRACT Fully analytical dynamical models usually have an infinite extent, while real star clusters, galaxies, and dark matter haloes have a finite extent. The standard method for generating dynamical models with a finite extent consists of taking a model with an infinite extent and applying a truncation in binding energy. This method, however, cannot be used to generate models with a preset analytical mass density profile. We investigate the self-consistency and dynamical properties of a family of power-law spheres with a general tangential Cuddeford (TC) orbital structure. By varying the density power-law slope γ and the central anisotropy β0, these models cover a wide parameter space in density and anisotropy profiles. We explicitly calculate the phase–space distribution function for various parameter combinations, and interpret our results in terms of the energy distribution of bound orbits. We find that truncated power-law spheres can be supported by a TC orbital structure if, and only if, γ ≥ 2β0, which means that the central density slope–anisotropy inequality is both a sufficient and a necessary condition for this family. We provide closed expressions for structural and dynamical properties such as the radial and tangential velocity dispersion profiles, which can be compared against more complex numerical modelling results. This work significantly adds to the available suite of self-consistent dynamical models with a finite extent and an analytical description.

Testing quasi-linear modified Newtonian dynamics theory with Galactic globular clusters in a weak external field

ABSTRACT We developed self-consistent dynamical models of stellar systems in the framework of quasi-linear modified Newtonian dynamics (QUMOND). The models are constructed from the anisotropic distribution function of Gunn and Griffin, combined with the modified Poisson equation defining this gravitation theory and take into account the external field effect. We have used these models, and their Newtonian analogues, to fit the projected density and the velocity dispersion profiles of a sample of 18 Galactic globular clusters, using the most updated data sets of radial velocities and Gaia proper motions. We have thus obtained, for each cluster, estimates of the dynamical mass-to-light ratio (M/L) for each theory of gravity. The selected clusters have accurate proper motions and a well-sampled mass function down to the very low-mass regime. This allows us to constrain the degree of anisotropy and to provide, from comparison with stellar evolution isochrones, a dynamics-independent estimate of the minimum mass-to-light ratio (M/L)min. Comparing the best-fitting dynamical M/L with (M/L)min, we find that for none of the analysed clusters the two gravity theories are significantly incompatible with the observational data, although for one of them (NGC 5024) the dynamical M/L predicted by QUMOND lies at 2.8σ below (M/L)min. Though the proposed approach suffers from some limitations (in particular the lack of a treatment of mass segregation), the obtained results suggest that the kinematics of globular clusters in a relatively weak external field can be a powerful tool to prove alternative theories of gravitation.

Self-consistent dynamical models with a finite extent – II. Radially truncated models

ABSTRACT Galaxies, dark matter haloes, and star clusters have a finite extent, yet most simple dynamical models have an infinite extent. The default method to generate dynamical models with a finite extent is to apply an energy truncation to the distribution function, but this approach is not suited to construct models with a preset density profile and it imposes unphysical constraints on the orbit population. We investigate whether it is possible to construct simple dynamical models for spherical systems with a preset density profile with a finite extent, and ideally with a different range of orbital structures. We systematically investigate the consistency of radially truncated dynamical models, and demonstrate that no spherical models with a discontinuous density truncation can be supported by an ergodic orbital structure. On the other hand, we argue that many radially truncated models can be supported by a tangential Osipkov–Merritt orbital structure that becomes completely tangential at the truncation radius. We formulate a consistency hypothesis for radially truncated models with such an orbital structure, and test it using an analytical example and the numerical exploration of a large model parameter space using the sphecow code. We physically interpret our results in terms of the occupancy of bound orbits, and we discuss possible extensions of the tangential Osipkov–Merritt orbital structure that can support radially truncated models.

A self-consistent dynamical model of the Milky Way disc adjusted to Gaia data

Context. Accurate astrometry achieved by Gaia for many stars in the Milky Way provides an opportunity to reanalyse the Galactic stellar populations from a large and homogeneous sample and to revisit the Galaxy gravitational potential. Aims. This paper shows how a self-consistent dynamical model can be obtained by fitting the gravitational potential of the Milky Way to the stellar kinematics and densities from Gaia data. Methods. We derived a gravitational potential using the Besancon Galaxy Model, and computed the disc stellar distribution functions based on three integrals of motion (E, Lz, I3) to model stationary stellar discs. The gravitational potential and the stellar distribution functions are built self-consistently, and are then adjusted to be in agreement with the kinematics and the density distributions obtained from Gaia observations. A Markov chain Monte Carlo (MCMC) is used to fit the free parameters of the dynamical model to Gaia parallax and proper motion distributions. The fit is done on several sets of Gaia data, mainly a subsample of the GCNS (Gaia catalogue of nearby stars to 100 pc) with G < 17, together with 26 deep fields selected from eDR3, widely spread in longitudes and latitudes. Results. We are able to determine the velocity dispersion ellipsoid and its tilt for subcomponents of different ages, both varying with R and z. The density laws and their radial scale lengths for the thin and thick disc populations are also obtained self-consistently. This new model has some interesting characteristics that come naturally from the process, such as a flaring thin disc. The thick disc is found to present very distinctive characteristics from the old thin disc, both in density and kinematics. This lends significant support to the idea that thin and thick discs were formed in distinct scenarios, as the density and kinematics transition between them is found to be abrupt. The dark matter halo is shown to be nearly spherical. We also derive the solar motion with regards to the Local Standard of Rest (LSR), finding U⊙ = 10.79 ± 0.56 km s−1, V⊙ = 11.06 ± 0.94 km s−1, and W⊙ = 7.66 ± 0.43 km s−1, in close agreement with recent studies. Conclusions. The resulting fully self-consistent gravitational potential, still axisymmetric, is a good approximation of a smooth mass distribution in the Milky Way and can be used for further studies, including finding streams, substructures, and to compute orbits for real stars in our Galaxy.

Self-consistent modelling of the Milky Way’s nuclear stellar disc

Abstract The nuclear stellar disc (NSD) is a flattened high-density stellar structure that dominates the gravitational field of the Milky Way at Galactocentric radius $30\, {\rm pc}\lesssim R\lesssim 300\, {\rm pc}$. We construct axisymmetric self-consistent equilibrium dynamical models of the NSD in which the distribution function is an analytic function of the action variables. We fit the models to the normalized kinematic distributions (line-of-sight velocities + VIRAC2 proper motions) of stars in the NSD survey of Fritz et al., taking the foreground contamination due to the Galactic Bar explicitly into account using an N-body model. The posterior marginalized probability distributions give a total mass of $M_{\rm NSD} = 10.5^{+1.1}_{-1.0} \times 10^8 \, \, \rm M_\odot$, roughly exponential radial and vertical scale lengths of $R_{\rm disc} = 88.6^{+9.2}_{-6.9} \, {\rm pc}$ and $H_{\rm disc}=28.4^{+5.5}_{-5.5} \, {\rm pc}$, respectively, and a velocity dispersion $\sigma \simeq 70\, {\rm km\, s^{-1}}$ that decreases with radius. We find that the assumption that the NSD is axisymmetric provides a good representation of the data. We quantify contamination from the Galactic Bar in the sample, which is substantial in most observed fields. Our models provide the full 6D (position + velocity) distribution function of the NSD, which can be used to generate predictions for future surveys. We make the models publicly available as part of the software package agama.

Self-consistent dynamical models with a finite extent – I. The uniform density sphere

ABSTRACT The standard method to generate dynamical models with a finite extent is to apply a truncation in binding energy to the distribution function (DF). This approach has the disadvantages that one cannot choose the density to start with, that the important dynamical quantities cannot be calculated analytically, and that a fraction of the possible bound orbits are excluded a priori. We explore another route and start from a truncation in radius rather than a truncation in binding energy. We focus on the simplest truncated density profile, the uniform density sphere. We explore the most common inversion techniques to generate DFs for the uniform density sphere, corresponding to a large range of possible anisotropy profiles. We find that the uniform density sphere cannot be supported by the standard isotropic, constant anisotropy or Osipkov–Merritt models, as all these models are characterized by negative DFs. We generalize the Cuddeford inversion method to models with a tangential anisotropy and present a one-parameter family of dynamical models for the uniform density sphere. Each member of this family is characterized by an anisotropy profile that smoothly decreases from an arbitrary value β0 ≤ 0 at the centre to completely tangential at the outer radius. All models have a positive DF over the entire phase space, and a non-zero occupancy of all possible bound orbits. This shows that one can generate non-trivial self-consistent dynamical models based on preset density profile with a finite extent.

The Fornax3D project: Assembly histories of lenticular galaxies from a combined dynamical and population orbital analysis

In order to assess the impact of the environment on the formation and evolution of galaxies, accurate assembly histories of such galaxies are needed. However, these measurements are observationally difficult owing to the diversity of formation paths that lead to the same present-day state of a galaxy. In this work, we apply a powerful new technique in order to observationally derive accurate assembly histories through a self-consistent combined stellar dynamical and population galaxy model. We present this approach for three edge-on lenticular galaxies from the Fornax3D project – FCC 153, FCC 170, and FCC 177 – in order to infer their mass assembly histories individually and in the context of the Fornax cluster. The method was tested on mock data from simulations to quantify its reliability. We find that the galaxies studied here have all been able to form dynamically-cold (intrinsic vertical velocity dispersion σz ≲ 50 km s−1) stellar disks after cluster infall. Moreover, the pre-existing (old) high angular momentum components have retained their angular momentum (orbital circularity λz > 0.8) through to the present day. Comparing the derived assembly histories with a comparable galaxy in a low-density environment – NGC 3115 – we find evidence for cluster-driven suppression of stellar accretion and merging. We measured the intrinsic stellar age–velocity-dispersion relation and find that the shape of the relation is consistent with galaxies in the literature across redshift. There is tentative evidence for enhancement in the luminosity-weighted intrinsic vertical velocity dispersion due to the cluster environment. But importantly, there is an indication that metallicity may be a key driver of this relation. We finally speculate that the cluster environment is responsible for the S0 morphology of these galaxies via the gradual external perturbations, or ‘harassment’, generated within the cluster.

Modeling the Chronologies and Size Distributions of Ceres and Vesta Craters

Abstract We infer the crater chronologies of Ceres and Vesta from a self-consistent dynamical model of asteroid impactors. The model accounts for planetary migration/instability early in the history of our solar system and tracks asteroid orbits over 4.56 Gyr. It is calibrated on the current population of the asteroid belt. The model provides the number of asteroid impacts on different worlds at any time throughout the solar system's history. We combine the results with an impactor-crater scaling relationship to determine the crater distribution of Ceres and Vesta and compare these theoretical predictions with observations. We find that: (i) The Ceres and Vesta chronologies are similar, whereas they significantly differ from the lunar chronology. Therefore, using the lunar chronology for main belt asteroids, as often done in previous publications, is incorrect. (ii) The model results match the number and size distribution of large (diameter >90 km) craters observed on Vesta, but overestimate the number of large craters on Ceres. This implies that large crater erasure is required for Ceres. (iii) In a model where planetary migration/instability happens early, the probability to form the Rheasilvia basin on Vesta during the last 1 Gyr is 10%, a factor of ∼1.5 higher than for the late instability case and ∼2.5 times higher than found in previous studies. Thus, while the formation of the Rheasilvia at ∼1 Gyr ago (Ga) would be somewhat unusual, it cannot be ruled out at more than ≃1.5σ. In broader context, our work provides a self-consistent framework for modeling asteroid crater records.

The CARMENES search for exoplanets around M dwarfs

Context. GJ 1148 is an M-dwarf star hosting a planetary system composed of two Saturn-mass planets in eccentric orbits with periods of 41.38 and 532.02 days. Aims. We reanalyze the orbital configuration and dynamics of the GJ 1148 multi-planetary system based on new precise radial velocity measurements taken with CARMENES. Methods. We combined new and archival precise Doppler measurements from CARMENES with those available from HIRES for GJ 1148 and modeled these data with a self-consistent dynamical model. We studied the orbital dynamics of the system using the secular theory and direct N-body integrations. The prospects of potentially habitable moons around GJ 1148 b were examined. Results. The refined dynamical analyses show that the GJ 1148 system is long-term stable in a large phase-space of orbital parameters with an orbital configuration suggesting apsidal alignment, but not in any particular high-order mean-motion resonant commensurability. GJ 1148 b orbits inside the optimistic habitable zone (HZ). We find only a narrow stability region around the planet where exomoons can exist. However, in this stable region exomoons exhibit quick orbital decay due to tidal interaction with the planet. Conclusions. The GJ 1148 planetary system is a very rare M-dwarf planetary system consisting of a pair of gas giants, the inner of which resides in the HZ. We conclude that habitable exomoons around GJ 1148 b are very unlikely to exist.

Investigating the spectral age problem with powerful radio galaxies

ABSTRACTThe ‘spectral age problem’ is our systematic inability to reconcile the maximum cooling time of radiating electrons in the lobes of a radio galaxy with its age as modelled by the dynamical evolution of the lobes. While there are known uncertainties in the models that produce both age estimates, ‘spectral’ ages are commonly underestimated relative to dynamical ages, consequently leading to unreliable estimates of the time-averaged kinetic feedback of a powerful radio galaxy. In this work, we attempt to solve the spectral age problem by observing two cluster-centre powerful radio galaxies; 3C 320 and 3C 444. With high-resolution broad-band Karl G. Jansky Very Large Array observations of the radio sources and deep XMM–Newton and Chandra observations of their hot intracluster media, coupled with the use of an analytic model, we robustly determine their spectral and dynamical ages. After finding self-consistent dynamical models that agree with our observational constraints, and accounting for sub-equipartition magnetic fields, we find that our spectral ages are still underestimated by a factor of two at least. Equipartition magnetic fields will underestimate the spectral age by factors of up to ∼20. The turbulent mixing of electron populations in the radio lobes is likely to be the main remaining factor in the spectral age/dynamical age discrepancy, and must be accounted for in the study of large samples of powerful radio galaxies.

Direct evidence of AGN feedback: a post-starburst galaxy stripped of its gas by AGN-driven winds

Post starburst E+A galaxies show indications of a powerful starburst that was quenched abruptly. Their disturbed, bulge-dominated morphologies suggest that they are merger remnants. The more massive E+A galaxies are suggested to be quenched by AGN feedback, yet little is known about AGN-driven winds in this short-lived phase. We present spatially-resolved IFU spectroscopy by the Keck Cosmic Web Imager of SDSS J003443.68+251020.9, at z=0.118. The system consists of two galaxies, the larger of which is a post starburst E+A galaxy hosting an AGN. Our modelling suggests a 400 Myrs starburst, with a peak star formation rate of 120 Msun/yr. The observations reveal stationary and outflowing gas, photoionized by the central AGN. We detect gas outflows to a distance of 17 kpc from the central galaxy, far beyond the region of the stars (about 3 kpc), inside a conic structure with an opening angle of 70 degrees. We construct self-consistent photoionization and dynamical models for the different gas components and show that the gas outside the galaxy forms a continuous flow, with a mass outflow rate of about 24 Msun/yr. The gas mass in the flow, roughly $10^9$ Msun, is larger than the total gas mass within the galaxy, some of which is outflowing too. The continuity of the flow puts a lower limit of 60 Myrs on the duration of the AGN feedback. Such AGN are capable of removing, in a single episode, most of the gas from their host galaxies and expelling enriched material into the surrounding CGM.

Influence of molecular dissociation on blob-induced atom density perturbations

The effects of enhanced electron and ion pressure perturbations mediated in filamentary structures (blobs) on the densities of neutral atoms and molecules are investigated through a self-consistent dynamical fluid model for plasma and neutral fields. The electron and ion densities and pressures, and the generalized vorticity, are simulated by a 2D drift-fluid model in an edge and scrape-off layer slab domain of a toroidally magnetically confined plasma. The plasma dynamics are coupled with a diffusion model for densities of neutral atoms and molecules. The combined model allows for determining the response of the density of neutrals with various temperatures to blobs. It is found that blobs locally deplete densities of molecules and atoms that do not originate from dissociation of molecules, whereas the density of atoms created by dissociation may increase during blob events. The neutral species, their temperature, and origin should thus be taken into consideration when estimating the effect of blobs on neutral density perturbations when calculating emission rates, e.g. for gas puff imaging.

The CARMENES search for exoplanets around M dwarfs

Context. The main goal of the CARMENES survey is to find Earth-mass planets around nearby M-dwarf stars. Seven M dwarfs included in the CARMENES sample had been observed before with HIRES and HARPS and either were reported to have one short period planetary companion (GJ 15 A, GJ 176, GJ 436, GJ 536 and GJ 1148) or are multiple planetary systems (GJ 581 and GJ 876). Aims. We aim to report new precise optical radial velocity measurements for these planet hosts and test the overall capabilities of CARMENES. Methods. We combined our CARMENES precise Doppler measurements with those available from HIRES and HARPS and derived new orbital parameters for the systems. Bona-fide single planet systems were fitted with a Keplerian model. The multiple planet systems were analyzed using a self-consistent dynamical model and their best fit orbits were tested for long-term stability. Results. We confirm or provide supportive arguments for planets around all the investigated stars except for GJ 15 A, for which we find that the post-discovery HIRES data and our CARMENES data do not show a signal at 11.4 days. Although we cannot confirm the super-Earth planet GJ 15 Ab, we show evidence for a possible long-period (Pc = 7030-630+970 d) Saturn-mass (mcsini = 51.8-5.8+5.5M⊕) planet around GJ 15 A. In addition, based on our CARMENES and HIRES data we discover a second planet around GJ 1148, for which we estimate a period Pc = 532.6-2.5+4.1 days, eccentricity ec = 0.342-0.062+0.050 and minimum mass mcsini = 68.1-2.2+4.9M⊕. Conclusions. The CARMENES optical radial velocities have similar precision and overall scatter when compared to the Doppler measurements conducted with HARPS and HIRES. We conclude that CARMENES is an instrument that is up to the challenge of discovering rocky planets around low-mass stars.

Self-Consistent Dynamical Model of the Broad Line Region

We develope a self-consistent description of the Broad Line Region based on the concept of the failed wind powered by the radiation pressure acting on dusty accretion disk atmosphere in Keplerian motion. The material raised high above the disk is illuminated, dust evaportes, and the matter falls back towards the disk. This material is the source of emission lines. The model predicts the inner and outer radius of the region, the cloud dynamics under the dust radiation pressure and, subsequently, just the gravitational field of the central black hole, which results in assymetry between the rise and fall. Knowledge of the dynamics allows to predict the shapes of the emission lines as functions of the basic parameters of an active nucleus: black hole mass, accretion rate, black hole spin (or accretion efficiency) and the viewing angle with respect to the symmetry axis. Here we show preliminary results based on analytical approximations to the cloud motion.

Three planets around HD 27894

Aims. Our new program with HARPS aims to detect mean motion resonant planetary systems around stars which were previously reported to have a single bona fide planet, often based only on sparse radial velocity data. Methods. Archival and new HARPS radial velocities for the K2V star HD 27894 were combined and fitted with a three-planet self-consistent dynamical model. The best-fit orbit was tested for long-term stability. Results. We find clear evidence that HD 27894 is hosting at least three massive planets. In addition to the already known Jovian planet with a period $P_{\rm b}$ $\approx$ 18 days we discover a Saturn-mass planet with $P_{\rm c}$ $\approx$ 36 days, likely in a 2:1 mean motion resonance with the first planet, and a cold massive planet ($\approx$ 5.3 $M_{\mathrm{Jup}}$) with a period $P_{\rm d}$ $\approx$ 5170 days on a moderately eccentric orbit ($e_{\rm d}$ = 0.39). Conclusions. HD 27894 is hosting a massive, eccentric giant planet orbiting around a tightly packed inner pair of massive planets likely involved in an asymmetric 2:1 mean motion resonance. HD 27894 may be an important milestone for probing planetary formation and evolution scenarios.

Towards a self-consistent dynamical nuclear model

Density functional theory (DFT) is a powerful and accurate tool, exploited in nuclear physics to investigate the ground-state and some of the collective properties of nuclei along the whole nuclear chart. Models based on DFT are not, however, suitable for the description of single-particle dynamics in nuclei. Following the field theoretical approach by A Bohr and B R Mottelson to describe nuclear interactions between single-particle and vibrational degrees of freedom, we have taken important steps towards the building of a microscopic dynamic nuclear model. In connection with this, one important issue that needs to be better understood is the renormalization of the effective interaction in the particle-vibration approach. One possible way to renormalize the interaction is by the so-called subtraction method. In this contribution, we will implement the subtraction method in our model for the first time and study its consequences.

Mixed Diffusive-Convective Relaxation of a Warm Beam of Energetic Particles in Cold Plasma

This work addresses the features of fast particle transport in the bump-on-tail problem for varying the width of the fluctuation spectrum, in the view of possible applications to studies of energetic particle transport in fusion plasmas. Our analysis is built around the idea that strongly-shaped beams do not relax through diffusion only and that there exists an intermediate time scale where the relaxations are convective (ballistic-like). We cast this idea in the form of a self-consistent nonlinear dynamical model, which extends the classic equations of the quasi-linear theory to “broad” beams with internal structure. We also present numerical simulation results of the relaxation of a broad beam of energetic particles in cold plasma. These generally demonstrate the mixed diffusive-convective features of supra-thermal particle transport essentially depending on nonlinear wave-particle interactions and phase-space structures. Taking into account the modes of the stable linear spectrum is crucial for the self-consistent evolution of the distribution function and the fluctuation intensity spectrum.

The distribution function of the Galaxy's dark halo

Starting from the hypothesis that the Galaxy's dark halo responded adiabatically to the infall of baryons, we have constructed a self-consistent dynamical model of the Galaxy that satisfies a large number of observations, including measurements of gas terminal velocities and masers, the kinematics of 180 000 giant stars from the RAVE (RAdial Velocity Experiment) survey and star-count data from the Sloan Digital Sky Survey. The stellar disc and the dark halo are both specified by distribution functions of the action integrals. The model is obtained by extending the work of Piffl et al. from the construction of a single model to a systematic search of model space. Whereas the model of Piffl et al. violated constraints on the terminal-velocity curve, our model respects these constraints by adopting a long scalelength Rd = 3.66 kpc for the thin and thick discs. The model is, however, inconsistent with the measured optical depth for microlensing of bulge stars because it attributes too large a fraction of the density at R ≲ 3 kpc to dark matter rather than stars. Moreover, it now seems likely that the thick disc's scalelength is significantly shorter than the model implies. Shortening this scalelength would cause the constraints from the rotation curve to be violated anew. We conclude that we can now rule out adiabatic compression of our Galaxy's dark halo.