Actions for axisymmetric potentials

We give an algorithm for the economical calculation of angles and actions for stars in axisymmetric potentials. We test the algorithm by integrating orbits in a realistic model of the Galactic potential, and find that, even for orbits characteristic of thick-disc stars, the errors in the actions are typically smaller than 2 percent. We describe a scheme for obtaining actions by interpolation on tabulated values that significantly accelerates the process of calculating observables quantities, such as density and velocity moments, from a distribution function.Comment: 5 pages accepted for publication in MNRA

Probing a regular orbit with spectral dynamics

We have extended the spectral dynamics formalism introduced by Binney & Spergel, and have implemented a semi-analytic method to represent regular orbits in any potential, making full use of their regularity. We use the spectral analysis code of Carpintero & Aguilar to determine the nature of an orbit (irregular, regular, resonant, periodic) from a short-time numerical integration. If the orbit is regular, we approximate it by a truncated Fourier time series of a few tens of terms per coordinate. Switching to a description in action-angle variables, this corresponds to a reconstruction of the underlying invariant torus. We then relate the uniform distribution of a regular orbit on its torus to the non-uniform distribution in the space of observables by a simple Jacobian transformation between the two sets of coordinates. This allows us to compute, in a cell-independent way, all the physical quantities needed in the study of the orbit, including the density and in the line-of-sight velocity distribution, with much increased accuracy. The resulting flexibility in the determination of the orbital properties, and the drastic reduction of storage space for the orbit library, provide a significant improvement in the practical application of Schwarzschild's orbit superposition method for constructing galaxy models. We test and apply our method to two-dimensional orbits in elongated discs, and to the meridional motion in axisymmetric potentials, and show that for a given accuracy, the spectral dynamics formalism requires an order of magnitude fewer computations than the more traditional approaches.Comment: 13 pages, 18 eps figures, submitted to MNRA

Origin and possible birthplace of the extreme runaway star HIP 60350

Using the recently determined spatial velocity components of the extreme runaway star HIP 60350 and a gravitation potential model of the Galaxy, we integrate the orbit of HIP 60350 back to the plane of the Galaxy. In this way, a possible location of the formation of the star is determined. We estimate the uncertainty of the result due to the uncertainties of the gravitational potential model and the errors in the spatial velocity components. The place of birth lies (within the errors) near the position of the open cluster NGC 3603. However, the ejection event which occured about 20 Myr ago is in contradiction with the cluster mean age of 3-4 Myr. We suggest that it occured at an earlier phase in sequential star formation in that region. We discuss also ejection mechanisms. Due to the rather high mass of the star (about 5 \Msol), the most probable model is that of dynamical ejection.Comment: Astron. Astrophys. submitted, 7 pages, 5 eps figure

Difficulties with Recovering The Masses of Supermassive Black Holes from Stellar Kinematical Data

We investigate the ability of three-integral, axisymmetric, orbit-based modeling algorithms to recover the parameters defining the gravitational potential (M/L ratio and black hole mass Mh) in spheroidal stellar systems using stellar kinematical data. We show that the potential estimation problem is generically under-determined when applied to long-slit kinematical data of the kind used in most black hole mass determinations to date. A range of parameters (M/L, Mh) can provide equally good fits to the data, making it impossible to assign best-fit values. We illustrate the indeterminacy using a variety of data sets derived from realistic models as well as published observations of the galaxy M32. In the case of M32, our reanalysis demonstrates that data published prior to 2000 are equally consistent with Mh in the range 1.5x10^6-5x10^6 solar masses, with no preferred value in that range. While the HST/STIS data for this galaxy may overcome the degeneracy in Mh, HST data for most galaxies do not resolve the black hole's sphere of influence and in these galaxies the degree of degeneracy allowed by the data may be substantial. We investigate the effect on the degeneracy of enforcing smoothness (regularization) constraints. However we find no indication that the true potential can be recovered simply by enforcing smoothness. For a given smoothing level, all solutions in the minimum-chisquare valley exhibit similar levels of noise. These experiments affirm that the indeterminacy is real and not an artifact associated with non-smooth solutions. (Abridged)Comment: Accepted for publication in The Astrophysical Journal. Changes include discussion of regularizatio

ON the CONSERVATION of the VERTICAL ACTION in GALACTIC DISKS

We employ high-resolution N-body simulations of isolated spiral galaxy models, from low-amplitude, multi-armed galaxies to Milky Way-like disks, to estimate the vertical action of ensembles of stars in an axisymmetrical potential. In the multi-armed galaxy the low-amplitude arms represent tiny perturbations of the potential, hence the vertical action for a set of stars is conserved, although after several orbital periods of revolution the conservation degrades significantly. For a Milky Way-like galaxy with vigorous spiral activity and the formation of a bar, our results show that the potential is far from steady, implying that the action is not a constant of motion. Furthermore, because of the presence of high-amplitude arms and the bar, considerable in-plane and vertical heating occurs that forces stars to deviate from near-circular orbits, reducing the degree at which the actions are conserved for individual stars, in agreement with previous results, but also for ensembles of stars. If confirmed, this result has several implications, including the assertion that the thick disk of our Galaxy forms by radial migration of stars, under the assumption of the conservation of the action describing the vertical motion of stars. © 2016. The American Astronomical Society. All rights reserved

Relativistic and Newtonian core-shell models: analytical and numerical results

We make a detailed analysis of Newtonian as well as relativistic core-shell models recently proposed to describe a black hole or neutron star surrounded by shells of matter, and in a seminal sense also galaxies, supernovae and star remnants since there are massive shell-like structures surrounding many of them and also evidences for many galactic nuclei hiding black holes. We discuss the unicity of the models in relation to their analyticity at the black hole horizon and also to the full elimination of conical singularities. Secondly, we study the role played by the presence/lack of discrete reflection symmetries about equatorial planes in the chaotic behavior of the orbits, which is to be contrasted with the almost universal acceptance of reflection symmetries as default assumptions in galactic modeling. We also compare the related effects if we change a true central black hole by a Newtonian central mass. The numerical findings are: 1- The breakdown of the reflection symmetry about the equatorial plane in both Newtonian and relativistic core-shell models does i) enhance in a significant way the chaoticity of orbits in reflection symmetric oblate shell models and ii) inhibit significantly also the occurrence of chaos in reflection symmetric prolate shell models. In particular, in the prolate case the lack of the reflection symmetry provides the phase space with a robust family of regular orbits that is otherwise not found at higher energies. 2- The relative extents of the chaotic regions in the relativistic cases (i. e. with a true central black hole) are significantly larger than in the corresponding Newtonian ones (which have just a $-1/r$ central potential).Comment: AASTEX, 22 pages plus 28 postscript figures, to appear in Ap.

Galactic kinematics with modified Newtonian dynamics

We look for observational signatures that could discriminate between Newtonian and modified Newtonian (MOND) dynamics in the Milky Way, in view of the advent of large astrometric and spectroscopic surveys. Indeed, a typical signature of MOND is an apparent disk of "phantom" dark matter, which is uniquely correlated with the visible disk-density distribution. Due to this phantom dark disk, Newtonian models with a spherical halo have different signatures from MOND models close to the Galactic plane. The models can thus be differentiated by measuring dynamically (within Newtonian dynamics) the disk surface density at the solar radius, the radial mass gradient within the disk, or the velocity ellipsoid tilt angle above the Galactic plane. Using the most realistic possible baryonic mass model for the Milky Way, we predict that, if MOND applies, the local surface density measured by a Newtonist will be approximately 78 Msun/pc2 within 1.1 kpc of the Galactic plane, the dynamically measured disk scale-length will be enhanced by a factor of 1.25 with respect to the visible disk scale-length, and the local vertical tilt of the velocity ellipsoid at 1 kpc above the plane will be approximately 6 degrees. None of these tests can be conclusive for the present-day accuracy of Milky Way data, but they will be of prime interest with the advent of large surveys such as GAIA.Comment: 5 page

Kinematical and chemical vertical structure of the Galactic thick disk I. Thick disk kinematics

The variation of the kinematical properties of the Galactic thick disk with Galactic height Z are studied by means of 412 red giants observed in the direction of the south Galactic pole up to 4.5 kpc from the plane. We confirm the non-null mean radial motion toward the Galactic anticenter found by other authors, but we find that it changes sign at |Z|=3 kpc, and the proposed inward motion of the LSR alone cannot explain these observations. The rotational velocity decreases with |Z| by -30 km/s/kpc, but the data are better represented by a power-law with index 1.25, similar to that proposed from the analysis of SDSS data. All the velocity dispersions increase with |Z|, but the vertical gradients are small. The dispersions grow proportionally, with no significant variation of the anisotropy. The ratio sigma_U/sigma_W=2 suggests that the thick disk could have formed from a low-latitude merging event. The vertex deviation increases with Galactic height, reaching ~20 degrees at |Z|=3.5 kpc. The tilt angle also increases, and the orientation of the ellipsoid in the radial-vertical plane is constantly intermediate between the alignment with the cylindrical and the spherical coordinate systems. The tilt angle at |Z|=2 kpc coincides with the expectations of MOND, but an extension of the calculations to higher |Z| is required to perform a conclusive test. Finally, between 2.5 and 3.5 kpc we detect deviations from the linear trend of many kinematical quantities, suggesting that some kinematical substructure could be present.Comment: Accepted for publication in Ap

Axisymmetric Three-Integral Models for Galaxies

We describe an improved, practical method for constructing galaxy models that match an arbitrary set of observational constraints, without prior assumptions about the phase-space distribution function (DF). Our method is an extension of Schwarzschild's orbit superposition technique. As in Schwarzschild's original implementation, we compute a representative library of orbits in a given potential. We then project each orbit onto the space of observables, consisting of position on the sky and line-of-sight velocity, while properly taking into account seeing convolution and pixel binning. We find the combination of orbits that produces a dynamical model that best fits the observed photometry and kinematics of the galaxy. A key new element of this work is the ability to predict and match to the data the full line-of-sight velocity profile shapes. A dark component (such as a black hole and/or a dark halo) can easily be included in the models. We have tested our method, by using it to reconstruct the properties of a two-integral model built with independent software. The test model is reproduced satisfactorily, either with the regular orbits, or with the two-integral components. This paper mainly deals with the technical aspects of the method, while applications to the galaxies M32 and NGC 4342 are described elsewhere (van der Marel et al., Cretton & van den Bosch). (abridged)Comment: minor changes, accepted for publication in the Astrophysical Journal Supplement

Triaxial orbit based galaxy models with an application to the (apparent) decoupled core galaxy NGC 4365

We present a flexible and efficient method to construct triaxial dynamical models of galaxies with a central black hole, using Schwarzschild's orbital superposition approach. Our method is general and can deal with realistic luminosity distributions, which project to surface brightness distributions that may show position angle twists and ellipticity variations. The models are fit to measurements of the full line-of-sight velocity distribution (wherever available). We verify that our method is able to reproduce theoretical predictions of a three-integral triaxial Abel model. In a companion paper (van de Ven, de Zeeuw & van den Bosch), we demonstrate that the method recovers the phase-space distribution function. We apply our method to two-dimensional observations of the E3 galaxy NGC 4365, obtained with the integral-field spectrograph SAURON, and study its internal structure, showing that the observed kinematically decoupled core is not physically distinct from the main body and the inner region is close to oblate axisymmetric.Comment: 21 Pages, 14 (Colour) Figures, Companion paper is arXiv:0712.0309 Accepted to MNRAS. Full resolution version at http://www.strw.leidenuniv.nl/~bosch/papers/RvdBosch_triaxmethod.pd