Corotation Resonance Research Articles

Effects of Halo Spin on the Formation and Evolution of Bars in Disk Galaxies

The spin of dark halos has been shown to significantly affect bar formation and evolution in disk galaxies. To understand the physical role of halo spin in bar formation, we run N-body simulations of isolated, Milky Way–sized galaxies by varying the halo spin parameter in the range −0.16 ≤ λ ≤ 0.16 and the bulge mass. We find that our adopted halo alone is subject to swing amplification of an m = 2 nonaxisymmetric mode rotating in the same sense as the halo, which assists or inhibits the bar formation in a disk depending on its sense of rotation. The m = 2 mode in the disk, growing via swing amplification, interacts constructively (destructively) with the m = 2 mode in the prograde (retrograde) halo, promoting (delaying) bar formation. A bar grows by losing its angular momentum primarily to a halo. Since the halo particles inside (outside) the corotation resonance with the bar can emit (absorb) angular momentum to (from) the bar, the bar pattern speed decays more slowly for larger λ > 0, while it decreases relatively fast almost independent of λ ≤ 0. Models with a strong bar develop a boxy peanut-shaped bulge. In models without a bulge, this occurs rapidly via buckling instability, while bars with a bulge thicken gradually without undergoing buckling instability. Among the models considered in the present work, the bar in the λ = 0.06 model with a bulge of 10% of the disk mass best describes the Milky Way in terms of its length and pattern speed.

Trojan Globular Clusters: Radial Migration via Trapping in Bar Resonances

We search for globular clusters (GCs) trapped in resonances with the bar of the Milky Way (MW). By integrating their orbits in a potential with a decelerating bar, we select 10 whose orbits are significantly changed by its presence. Most of these are trapped in the corotation resonance, including M22 and 47 Tuc. The decelerating bar is capable of transporting these GCs to their current positions from much lower energies, angular momenta, and radii. Our results indicate that the bar is likely to have reshaped the MW’s GC system via its resonances. We also discuss implications for the origins of specific GCs, including the possible nuclear star cluster M22. Finally, we consider the effects of the bar on the tidal tails of a trapped GC by running simulations of stars stripped from 47 Tuc. Instead of forming narrow tails, the stripped stars make up a diffuse extended halo around the cluster, consistent with observations of 47 Tuc.

Radial halo substructure in harmony with the Galactic bar

ABSTRACT Overdensities in the radial phase space $(r,v_{\rm r})$ of the Milky Way’s halo have previously been associated with the phase-mixed debris of a highly radial merger event such as Gaia Sausage–Enceladus. We present and test an alternative theory in which the overdense ‘chevrons’ are instead composed of stars trapped in resonances with the Galactic bar. We develop an analytic model of resonant orbits in the isochrone potential, and complement this with a test particle simulation of a stellar halo in a realistic barred Milky Way potential. These models are used to predict the appearance of action space $(J_\phi ,J_{\rm r})$ and radial phase space in the Solar neighbourhood. They are able to reproduce almost all salient features of the observed chevrons. In particular, both the analytic model and simulation predict that the chevrons are more prominent at $v_{\rm r}\ \lt\ 0$ when viewed near the Sun, as is observed by Gaia. This is inconsistent with formation by an ancient merger event. We also associate individual chevrons with specific resonances. At a bar pattern speed of $\Omega _\mathrm{b}=35$ km s$^{-1}$ kpc$^{-1}$, the two most prominent prograde chevrons align very closely with the corotation and outer Lindblad resonances. The former can be viewed as a highly eccentric extension of the Hercules stream. Finally, our model predicts that the $v_{\rm r}$ asymmetry changes sign as a function of Galactic radius and azimuth, and we find evidence that this is indeed the case in the Milky Way.

Secular evolution of resonant planets in the coplanar case

Aims. We study the secular evolution of two planets in mutual deep mean-motion resonance (MMR) in the planar elliptic three-body problem framework for different mass ratios. We do not consider any restriction in the eccentricity of the inner planet e1 or in the eccentricity of the outer planet e2. Methods. The method we used is based on a semi-analytical model that consists of calculating the averaged resonant disturbing function numerically. It is assumed for this that all the orbital elements (except for the mean longitudes) of both planets are constant on the resonant timescale. In order to obtain the secular evolution inside the MMR, we make use of the adiabatic invariance principle, assuming a zero-amplitude resonant libration. We constructed two phase portraits, called the ℋ1 and ℋ2 surfaces, in the three-dimensional spaces (e1, Δϖ, σ) and (e2, Δϖ, σ), where Δϖ is the difference between the planetary longitude of perihelia and σ is the critical angle. These surfaces, which are related through the angular moment conservation, allow us to find the apsidal corotation resonances (ACRs) and to predict the secular evolution of e1, e2, Δϖ, and σ (libration center). Results. While studying the 1:1, 2:1, 3:1, and 3:2 MMR, we found that large excursions in eccentricity can exist in some particular cases. We compared the secular variations of e1, e2, Δϖ, and σ predicted by the model with a numerical integration of the exact equations of motion for different mass ratios. We obtained good matches. Finally, the model was applied to study the secular evolution of the resonant exoplanet systems HD 73526 and HD 31527. They both have a pair of planets and are very close to the deep MMR condition. In the first system, we found that the pair of planets that constitutes the system evolves in a symmetrical ACR, whereas in the second system, we found that planets c and d, which are in an unusual 16:3 MMR, are close to an ACR, but outside its dynamical region, where Δϖ circulates.

The Milky Way bar pattern speed using Hercules and Gaia DR3

ABSTRACT The distribution of moving groups in the solar neighbourhood has been used to constrain dynamical properties of the Milky Way for decades. The kinematic bimodality between the main mode (Hyades, Pleiades, Coma Berenices, and Sirius) and Hercules can be explained by two different bar models – via the outer Lindblad resonance of a bar with a high pattern speed (∼55 km s−1 kpc−1), or via the corotation resonance of a bar with a low pattern speed (∼40 km s−1 kpc−1). Recent works directly studying the kinematics of bar stars and gas flows near the centre of the Galaxy have converged on the low pattern speed model. In this paper, we independently confirm this result by using Gaia DR3 to directly study the variation of Hercules across Galactic azimuth. We find that Hercules increases in Vϕ and becomes stronger as we move towards the minor axis of the bar, and decreases in Vϕ and becomes weaker as we move towards the major axis of the bar. This is in direct agreement with theoretical predictions of a low pattern speed bar model in which Hercules is formed by the corotation resonance with stars orbiting the bar’s L4/L5 Lagrange points.

Impacts of the Local arm on the local circular velocity inferred from the Gaia DR3 young stars in the Milky Way

ABSTRACT A simple one-dimensional axisymmetric disc model is applied to the kinematics of O type and B type stars (OB stars) near the Sun obtained from Gaia Data Release 3 catalogue. The model determines the ‘local centrifugal speed’ Vc(R0) – defined as the circular velocity in the Galactocentric rest frame, where the star would move in a near-circular orbit if the potential is axisymmetric with the local potential of the Galaxy. We find that the Vc(R0) values and their gradient vary across the selected region of stars within the solar neighbourhood. By comparing with an N-body/hydrodynamic simulation of a Milky Way-like galaxy, we find that the kinematics of the young stars in the solar neighbourhood is affected by the Local arm, which makes it difficult to measure Vc(R0). However, from the resemblance between the observational data and the simulation, we suggest that the known rotational velocity gap between the Coma Bernices and Hyades-Pleiades moving groups could be driven by the co-rotation resonance of the Local arm, which can be used to infer the azimuthally averaged circular velocity. We find that Vc(R) obtained from the D < 2 kpc sample is well matched with this gap at the position of the Local arm. Hence, we argue that our results from the D < 2 kpc sample, Vc(R0) = 234 ± 2 km s−1, are close to the azimuthally averaged circular velocity rather than the local centrifugal speed, which is influenced by the presence of the Local arm.

Saturation of spiral instabilities in disc galaxies

ABSTRACT Spiral density waves can arise in galactic discs as linear instabilities of the underlying stellar distribution function. Such an instability grows exponentially in amplitude at some fixed growth rate β before saturating non-linearly. However, the mechanisms behind saturation, and the resulting saturated spiral amplitude, have received little attention. Here, we argue that one important saturation mechanism is the non-linear trapping of stars near the spiral’s corotation resonance. Under this mechanism, we show analytically that an m-armed spiral instability will saturate when the libration frequency of resonantly trapped orbits reaches $\omega _\mathrm{lib} \sim \mathrm{a\, \, few}\times m^{1/2} \beta$. For a galaxy with a flat rotation curve, this implies a maximum relative spiral surface density $\vert \delta \Sigma /\Sigma _0\vert \sim \mathrm{a\, \, few} \times (\beta /\Omega _\mathrm{p})^2 \cot \alpha$, where Ωp is the spiral pattern speed and α is its pitch angle. This result is in reasonable agreement with recent N-body simulations, and suggests that spirals driven by internally-generated instabilities reach relative amplitudes of at most a few tens of per cent; higher amplitude spirals, like in M51 and NGC 1300, are likely caused by very strong bars and/or tidal perturbations.

Origin of reduced dynamical friction by dark matter haloes with net prograde rotation

ABSTRACT We provide an explanation for the reduced dynamical friction on galactic bars in spinning dark matter haloes. Earlier work based on linear theory predicted an increase in dynamical friction when dark haloes have a net forward rotation because prograde orbits couple to bars with greater strength than retrograde orbits. Subsequent numerical studies, however, found the opposite trend: dynamical friction weakens with increasing spin of the halo. We revisit this problem and demonstrate that linear theory in fact correctly predicts a reduced torque in forward-rotating haloes. We show that shifting the halo mass from retrograde to prograde phase space generates a positive gradient in the distribution function near the origin of the z-angular momentum (Lz = 0), which results in a resonant transfer of Lz to the bar, making the net dynamical friction weaker. While this effect is subdominant for the major resonances, including the corotation resonance, it leads to a significant positive torque on the bar for the series of direct radial resonances as these resonances are strongest at Lz = 0. The overall dynamical friction from spinning haloes is shown to decrease with the halo’s spin in agreement with the secular behaviour of N-body simulations. We validate our linear calculation by computing the non-linear torque from individual resonances using the angle-averaged Hamiltonian.

On the interaction of pebble accreting embryos with the gaseous disc: importance of thermal forces

ABSTRACT A planetary embryo embedded in a gaseous disc can grow by pebble accretion while subjected to a gravitational force from the disc that changes its orbital elements. Usually, that force is considered to arise from the Lindblad and corotation resonances with the embryo. However, more important contributions exist for low-mass planets. Radiative thermal diffusion in the vicinity of embryos yields an additional contribution to the disc’s force that damps the eccentricity and inclination much more vigorously than the resonant interaction with the disc, and that in general induces fast inward migration. In addition, the irradiation of the disc by a hot embryo gives rise to an additional contribution that excites eccentricity and inclination, and induces outward migration. Which of the two contributions dominates depends on the embryo’s luminosity. We assess the importance of these contributions (termed thermal forces) on the dynamics and growth of a set of pebble-accreting embryos initially of Martian mass, by means of N-body simulations that include analytic expressions for the disc’s force. We find very different outcomes for the embryos subjected to thermal forces and those subjected only to resonant forces. Importantly, we find that the median final mass of the embryos subjected to thermal forces is nearly independent of the metallicity, whereas this mass roughly scales with the metallicity when they are subjected only to resonant forces. These results can be explained by the strong damping of eccentricity and inclination at low metallicity, which enhances the embryos’ accretion efficiency.

Evidence for Corotation Origin of Super-metal-rich Stars in LAMOST-Gaia: Multiple Ridges with a Similar Slope in the ϕ versus L z Plane

Super-metal-rich (SMR) stars in the solar neighborhood are thought to be born in the inner disk and come to their present location by radial migration, which is most intense at the corotation resonance (CR) of the Galactic bar. In this work, we show evidence for the CR origin of SMR stars in the Large Sky Area Multi-Object Fiber Spectroscopic Telescope and Gaia by detecting six ridges and undulations in the ϕ versus L z space coded by median V R, following a similar slope of −8 km s−1 kpc deg−1. The slope is predicted by Monario et al.'s model for CR of a large and slow Galactic bar. For the first time, we show the variation in the angular momentum with azimuths from −10° to 20° for two outer and broad undulations with negative V R around − 18 km s−1 following this slope. The wave-like pattern with large amplitude outside CR and a wide peak of the second undulation indicate that minor merger of the Sagittarius dwarf galaxy with the disk might play a role besides the significant impact of the CR of the Galactic bar.

Chemodynamical Signatures of Bar Resonances in the Galactic Disk: Current Data and Future Prospects

The Galactic disk exhibits complex chemical and dynamical substructure thought to be induced by the bar, spiral arms, and satellites. Here, we explore the chemical signatures of bar resonances in action and velocity space, and characterize the differences between the signatures of corotation (CR) and higher-order resonances using test particle simulations. Thanks to recent surveys, we now have large data sets containing metallicities and kinematics of stars outside the solar neighborhood. We compare the simulations to the observational data from Gaia EDR3 and LAMOST DR5 and find weak evidence for a slow bar with the “hat” moving group (250 km s−1 ≲ v ϕ ≲ 270 km s−1) associated with its outer Lindblad resonance and “Hercules” (170 km s−1 ≲ v ϕ ≲ 195 km s−1) with CR. While constraints from current data are limited by their spatial footprint, stars closer in azimuth than the Sun to the bar’s minor axis show much stronger signatures of the bar’s outer Lindblad and CR resonances in test particle simulations. Future data sets with greater azimuthal coverage, including the final Gaia data release, will allow reliable chemodynamical identification of bar resonances.

The effects of non-linearities on tidal flows in the convective envelopes of rotating stars and planets in exoplanetary systems

ABSTRACT In close exoplanetary systems, tidal interactions drive orbital and spin evolution of planets and stars over long time-scales. Tidally forced inertial waves (restored by the Coriolis acceleration) in the convective envelopes of low-mass stars and giant gaseous planets contribute greatly to the tidal dissipation when they are excited and subsequently damped (e.g. through viscous friction), especially early in the life of a system. These waves are known to be subject to non-linear effects, including triggering differential rotation in the form of zonal flows. In this study, we use a realistic tidal body forcing to excite inertial waves through the residual action of the equilibrium tide in the momentum equation for the waves. By performing 3D non-linear hydrodynamical simulations in adiabatic and incompressible convective shells, we investigate how the addition of non-linear terms affects the tidal flow properties, and the energy and angular momentum redistribution. In particular, we identify and justify the removal of terms responsible for unphysical angular momentum evolution observed in a previous numerical study. Within our new set-up, we observe the establishment of strong cylindrically sheared zonal flows, which modify the tidal dissipation rates from prior linear theoretical predictions. We demonstrate that the effects of this differential rotation on the waves neatly explains the discrepancies between linear and non-linear dissipation rates in many of our simulations. We also highlight the major role of both corotation resonances and parametric instabilities of inertial waves, which are observed for sufficiently high tidal forcing amplitudes or low viscosities, in affecting the tidal flow response.

Proximity of exoplanets to first-order mean-motion resonances

ABSTRACT Planetary formation theories and, more specifically, migration models predict that planets can be captured in mean-motion resonances (MMRs) during the disc phase. The distribution of period ratios between adjacent planets shows an accumulation in the vicinity of the resonance, which is not centred on the nominal resonance but instead presents an offset slightly exterior to it. Here, we extend on previous works by thoroughly exploring the effect of different disc and planet parameters on the resonance offset during the disc migration phase. The dynamical study is carried out for several first-order MMRs and for both low-mass Earth-like planets undergoing type-I migration and giant planets evolving under type-II migration. We find that the offset varies with time during the migration of the two-planet system along the apsidal corotation resonance family. The departure from the nominal resonance increases for higher planetary masses and stronger eccentricity damping. In the Earth to super-Earth regime, we find offset values in agreement with the observations when using a sophisticated modelling for the planet–disc interactions, where the damping time-scale depends on the eccentricity. This dependence causes a feedback that induces an increase of the resonance offsets. Regarding giant planets, the offsets of detected planet pairs are well reproduced with a classical K-factor prescription for the planet–disc interactions when the eccentricity damping rate remains low to moderate. In both regimes, eccentricities are in agreement with the observations too. As a result, planet–disc interactions provide a generic channel to generate the offsets found in the observations.

Photometric Signature of Ultraharmonic Resonances in Barred Galaxies

Bars may induce morphological features, such as rings, through their resonances. Previous studies suggested that the presence of “dark gaps,” or regions of a galaxy where the difference between the surface brightness along the bar major axis and that along the bar minor axis is maximal, can be attributed to the location of bar corotation. Here, using GALAKOS, a high-resolution N-body simulation of a barred galaxy, we test this photometric method’s ability to identify the bar corotation resonance. Contrary to previous work, our results indicate that “dark gaps” are a clear sign of the location of the 4:1 ultraharmonic resonance instead of bar corotation. Measurements of the bar corotation can indirectly be inferred using kinematic information, e.g., by measuring the shape of the rotation curve. We demonstrate our concept on a sample of 578 face-on barred galaxies with both imaging and integral field observations and find that the sample likely consists primarily of fast bars.

Secular evolution of resonant small bodies: semi-analytical approach for arbitrary eccentricities in the coplanar case

ABSTRACT We study the secular evolution of a particle in deep mean motion resonance (MMR) with a planet in the planar elliptic restricted three body problem. We do not consider any restriction neither in the planet’s eccentricity ep nor in the particle’s eccentricity e. The methodology used is based on a semi-analytical model that consists on calculating the averaged resonant disturbing function numerically, assuming for this that in the resonant scale of time all the orbital elements of the particle are constant. In order to obtain the secular evolution inside the MMR, we make use of the adiabatic invariance principle, assuming a zero-amplitude resonant libration. We construct two-dimensional surfaces (called $\mathcal {H}$ surfaces) in the three-dimensional space (σ, e, ϖ) that allow us to predict the secular evolution of these three variables. The 2:1 MMR is used as example to show some results. We found four apsidal corotation resonance (ACR) families, two symmetric and two asymmetric. One of the symmetric families exists for almost any ep value. The other one for ep > 0.3 and the asymmetric ones for ep > 0.44. We corroborate the secular variations in e and ϖ predicted by the model through numerical integrations even when the initial conditions are displaced from those ACR. Some peculiar examples are presented for the 3:1 and 3:2 MMR showing large excursions in eccentricity. As an application, the Planet 9 is investigated as a possible responsible of high eccentric distant TNOs.

Properties of Fast and Slow Bars Classified by Epicyclic Frequency Curves from Photometry of Barred Galaxies

Abstract We test the idea that bar pattern speeds decrease with time owing to angular momentum exchange with a dark matter halo. If this process actually occurs, then the radii of the corotation resonance and other resonances should generally increase with time. We therefore derive the angular velocity Ω and epicyclic frequency κ as functions of galactocentric radius for 85 barred galaxies using photometric data. Mass maps are constructed by assuming a dynamical mass-to-light ratio and then solving the Poisson equation for the gravitational potential. The locations of Lindblad resonances and the corotation resonance radius are then derived using the standard precession frequency curves in conjunction with bar pattern speeds recently estimated from the Tremaine–Weinberg method as applied to integral field spectroscopy data. Correlations between physical properties of bars and their host galaxies indicate that bar length and the corotation radius depend on the disk circular velocity while bar strength and pattern speed do not. As the bar pattern speed decreases, bar strength, length, and corotation radius increase, but when bars are subclassified into fast, medium, and slow domains, no significant change in bar length is found. Only a hint of an increase in bar strength from fast to slow bars is found. These results suggest that bar length in a galaxy undergoes little evolution, and is determined instead mainly by the size of the host galaxy.

Neptune’s ring arcs from VLT/SPHERE-IRDIS near-infrared observations

Context. Neptune’s incomplete ring arcs have been stable since their discovery in 1984 by stellar occultation. Although these structures should be destroyed within a few months through differential Keplerian motion, imaging data over the past couple of decades have shown that these structures remain stable. Aims. We present the first SPHERE near-infrared observations of Neptune’s ring arcs taken at 2.2 μm (broadband Ks) with the IRDIS camera at the Very Large Telescope (VLT) in August 2016. Methods. The images were aligned using the ephemerides of the satellite Proteus and were suitably co-added to enhance ring and satellite signals. Results. We analyse high-angular-resolution near-infrared images of Neptune’s ring arcs obtained in 2016 at the ESO VLT-UT3 with the adaptive-optics-fed camera SPHERE-IRDIS. We derive accurate mean motion values for the arcs and the nearby satellite Galatea. The trailing arcs Fraternité and Égalité have been stable since they were last observed in 2007. Furthermore, we confirm the fading away of the leading arcs Courage and Liberté. Finally, we confirm the mismatch between the arcs’ position and the 42:43 inclined and eccentric corotation resonances with Galatea, thus demonstrating that no 42:43 corotation model works to explain the azimuthal confinement of the arcs’ materiel.

The current orbit of Methone (S/2004 S 1)

The Cassini spacecraft discovered many close-in small satellites in Saturnian system, and several of them exhibit exotic orbital states due to interactions with Mimas and the oblateness of the planet. This work is devoted to Methone, which is currently involved in a 15:14 Mean-Motion Resonance with Mimas. We give an in deep study the current orbit of Methone by analyzing and identifying the short, resonant and long-term gravitational perturbations on its orbit. In addition, we perform numerical integrations of full equations of motion of ensembles of close-in small bodies orbiting the non-central field of Saturn. Spectral analyses of the orbits and interpretation of them in dynamical maps allow us to describe the orbit and the dynamics of Methone in view of resonant and long-term dynamics. We show that the current geometric orbit of Methone is aligned with Mimas' due to a forced resonant component in eccentricity, leading to simultaneous oscillations of several critical angles of the expanded disturbing function. Thus, we explain the simultaneous oscillations of four critical arguments associated to the resonance. The mapping of the Mimas-Methone resonance shows that the domains of the 15:14 Mimas-Methone resonance are dominated by regular motions associated to the Corotation resonance located at eccentricities lower than $\sim 0.015$ and osculating semi-major axis in the interval 194,660-194,730 km. Methone is currently located deeply within this site.

Dynamics of the Spiral-Arm Corotation and Its Observable Footprints in the Solar Neighborhood

This article discusses the effects of the spiral-arm corotation on the stellar dynamics in the Solar Neighborhood (SN). All our results presented here rely on: (1) observational evidence that the Sun lies near the corotation circle, where stars rotate with the same angular velocity as the spiral-arm pattern; the corotation circle establishes domains of the corotation resonance (CR) in the Galactic disk; (2) dynamical constraints that put the spiral-arm potential as the dominant perturbation in the SN, comparing with the effects of the central bar in the SN; (3) a long-lived nature of the spiral structure, promoting a state of dynamical relaxing and phase-mixing of the stellar orbits in response to the spiral perturbation. With an analytical model for the Galactic potential, composed of an axisymmetric background deduced from the observed rotation curve, and perturbed by a four-armed spiral pattern, numerical simulations of stellar orbits are performed to delineate the domains of regular and chaotic motions shaped by the resonances. Such studies show that stars can be trapped inside the stable zones of the spiral CR, and this orbital trapping mechanism could explain the dynamical origin of the Local arm of the Milky Way (MW). The spiral CR and the near high-order epicyclic resonances influence the velocity distribution in the SN, creating the observable structures such as moving groups and their radially extended counterpart known as diagonal ridges. The Sun and most of the SN stars evolve inside a stable zone of the spiral CR, never crossing the main spiral-arm structure, but oscillating in the region between the Sagittarius-Carina and Perseus arms. This orbital behavior of the Sun brings insights to our understanding of questions concerning the solar system evolution, the Earth environment changes, and the preservation of life on Earth.

Tree-ring structure of Galactic bar resonance

ABSTRACT Galaxy models have long predicted that galactic bars slow down by losing angular momentum to their postulated dark haloes. When the bar slows down, resonance sweeps radially outwards through the galactic disc while growing in volume, thereby sequentially capturing new stars at its surface/separatrix. Since trapped stars conserve their action of libration, which measures the relative distance to the resonance centre, the order of capturing is preserved: the surface of the resonance is dominated by stars captured recently at large radius, while the core of the resonance is occupied by stars trapped early at small radius. The slow down of the bar thus results in a rising mean metallicity of trapped stars from the surface towards the centre of the resonance as the Galaxy’s metallicity declines towards large radii. This argument, when applied to Solar neighbourhood stars, allows a novel precision measurement of the bar’s current pattern speed $\Omega _{\rm p}= 35.5 \pm 0.8 \, {\rm km}\, {\rm s}^{-1}\, {\rm kpc}^{-1}$, placing the corotation radius at $R_{\rm CR}= 6.6 \pm 0.2 \, {\rm kpc}$. With this pattern speed, the corotation resonance precisely fits the Hercules stream in agreement with kinematics. Beyond corroborating the slow bar theory, this measurement manifests the deceleration of the bar of more than $24{{\ \rm per\ cent}}$ since its formation and thus the angular momentum transfer to the dark halo by dynamical friction. The measurement therefore supports the existence of a standard dark-matter halo rather than alternative models of gravity.