Three-integral models for axisymmetric galactic discs

We present new equilibrium component distribution functions that depend on three analytic integrals in a Stäckel potential, and that can be used to model stellar discs of galaxies. These components are generalizations of two-integral ones and can thus provide thin discs in the two-integral approximation. Their most important properties are the partly analytical expression for their moments, the disc-like features of their configuration space densities (exponential decline in the galactic plane and finite extent in the vertical direction) and the anisotropy of their velocity dispersions. We further show that a linear combination of such components can fit a van der Kruit disc.     

Integrable models of galactic discs with double nuclei

We introduce a new class of 2D mass models, whose potentials are of Stäckel form in elliptic coordinates. Our model galaxies have two separate strong cusps that form double nuclei. The potential and surface density distributions are locally axisymmetric near the nuclei and become highly non-axisymmetric outside the nucleus. The surface density diverges toward the cuspy nuclei with the law     Our model is sustained by four general types of regular orbits: butterfly , nucleophilic banana , horseshoe and aligned loop orbits. Horseshoes and nucleophilic bananas support the existence of cuspy regions. Butterflies and aligned loops control the non-axisymmetric shape of outer regions. Without any need for central black holes, our distributed mass models resemble the nuclei of M31 and NGC 4486B. It is also shown that the self-gravity of the stellar disc can prevent the double nucleus to collapse.     

Exploring disc galaxy dynamics using integral field unit data

In order to test the basic equations believed to dictate the dynamics of disc galaxies, we present and analyse deep two-dimensional spectral data obtained using the PPAK integral field unit for the early-type spiral systems NGC 2273, NGC 2985, NGC 3898 and NGC 5533. We describe the care needed to obtain and process such data to a point where reliable kinematic measurements can be obtained from these observations, and a new more optimal method for deriving the rotational motion and velocity dispersions in such disc systems. The data from NGC 2273 and NGC 2985 show systematic variations in velocity dispersion with azimuth, as one would expect if the shapes of their velocity ellipsoids are significantly anisotropic, while the hotter discs in NGC 3898 and NGC 5533 appear to have fairly isotropic velocity dispersions. Correcting the rotational motion for asymmetric drift using the derived velocity dispersions reproduces the rotation curves inferred from emission lines reasonably well, implying that this correction is quite robust, and that the use of the asymmetric drift equation is valid. NGC 2985 is sufficiently close to face on for the data, combined with the asymmetric drift equation, to determine all three components of the velocity ellipsoid. The principal axes of this velocity ellipsoid are found to be in the ratio  σ _z : σ_φ : σ _R ≈ 0.7 : 0.7 : 1  , which shows unequivocally that this disc distribution function respects a third integral of motion. The ratio is also consistent with the predictions of epicyclic theory, giving some confidence in the application of this approximation to even fairly early-type disc galaxies.     

The distribution function of dark matter in massive haloes

We study the distribution function (DF) of dark matter particles in haloes of mass range  10¹⁴–10¹⁵ M_⊙  . In the numerical part of this work we measure the DF for a sample of relaxed haloes formed in the simulation of a standard Λ cold dark matter (ΛCDM) model. The DF is expressed as a function of energy E and the absolute value of the angular momentum L , a form suitable for comparison with theoretical models. By proper scaling we obtain the results that do not depend on the virial mass of the haloes. We demonstrate that the DF can be separated into energy and angular momentum components and propose a phenomenological model of the DF in the form     . This formulation involves three parameters describing the anisotropy profile in terms of its asymptotic values (β₀ and  β_∞  ) and the scale of transition between them ( L ₀). The energy part   f _E ( E )  is obtained via inversion of the integral for spatial density. We provide a straightforward numerical scheme for this procedure as well as a simple analytical approximation for a typical halo formed in the simulation. The DF model is extensively compared with the simulations: using the model parameters obtained from fitting the anisotropy profile, we recover the DF from the simulation as well as the profiles of the dispersion and kurtosis of radial and tangential velocities. Finally, we show that our DF model reproduces the power-law behaviour of phase-space density   Q =ρ( r )/σ³( r )  .     

On the potential–density relation for Stäckel discs

The general form of the surface density of an infinitely thin disc is given that generates a Sta¨ckel potential in the disc only, using formulae for the potential of elliptic and hyperbolic strings. This is useful for problems in which a simple form for the potential is important, while the corresponding surface density need only be known to check (numerically) that it is positive. A simple potential with a positive surface density is given. Also, formulae are given to calculate the surface density of such a Sta¨ckel disc, in the case in which the rotation curve is given and all the mass is concentrated in the disc.     

The haloes of merger remnants

We perform collisionless N -body simulations of 1:1 galaxy mergers, using models which include a galaxy halo, disc and bulge, focusing on the behaviour of the halo component. The galaxy models are constructed without recourse to a Maxwellian approximation. We investigate the effect of varying the galaxies' orientation, their mutual orbit and the initial velocity anisotropy or cusp strength of the haloes upon the remnant halo density profiles and shape, as well as on the kinematics. We observe that the halo density profile (determined as a spherical average, an approximation we find appropriate) is exceptionally robust in mergers, and that the velocity anisotropy of our remnant haloes is nearly independent of the orbits or initial anisotropy of the haloes. The remnants follow the halo anisotropy – local density slope (β–γ) relation suggested by Hansen & Moore in the inner parts of the halo, but β is systematically lower than this relation predicts in the outer parts. Remnant halo axis ratios are strongly dependent on the initial parameters of the haloes and on their orbits. We also find that the remnant haloes are significantly less spherical than those described in studies of simulations which include gas cooling.     

Measuring galaxy potentials using shell kinematics

We show that the kinematics of the shells seen around some elliptical galaxies provide a new, independent means for measuring the gravitational potentials of elliptical galaxies out to large radii. A numerical simulation of a set of shells formed in the merger between an elliptical and a smaller galaxy reveals that the shells have a characteristic observable kinematic structure, with the maximum line-of-sight velocity increasing linearly as one moves inward from a shell edge. A simple analytic calculation shows that this structure provides a direct measure of the gradient of the gravitational potential at the shell radius. In order to extract this information from attainable data, we have also derived a complete distribution of line-of-sight velocities for material within a shell; comparing the observed spectra of a shell to a stellar template convolved with this distribution will enable us to measure the gradient of the potential at this radius. Repeating the analysis for a whole series of nested shells in a galaxy allows the complete form of the gravitational potential as a function of radius to be mapped out. The requisite observations lie within reach of the up-coming generation of large telescopes.     

New biorthogonal potential–density basis functions

We use the weighted integral form of spherical Bessel functions and introduce a new analytical set of complete and biorthogonal potential–density basis functions. The potential and density functions of the new set have finite central values and they fall off, respectively, similar to   r ^{−(1+ l )}  and   r ^{−(4+ l )}  at large radii, where l is the latitudinal quantum number of spherical harmonics. The lowest order term associated with   l = 0  is the perfect sphere of de Zeeuw. Our basis functions are intrinsically suitable for the modelling of three-dimensional, soft-centred stellar systems and they complement the basis sets of Clutton-Brock, Hernquist & Ostriker and Zhao. We test the performance of our functions by expanding the density and potential profiles of some spherical and oblate galaxy models.     

Dark matter halo response to the disc growth

We consider the sensitivity of the circular-orbit adiabatic contraction approximation to the baryon condensation rate and the orbital structure of dark matter haloes in the Λ cold dark matter (ΛCDM) paradigm. Using one-dimensional hydrodynamic simulations including the dark matter halo mass accretion history and gas cooling, we demonstrate that the adiabatic approximation is approximately valid even though haloes and discs may assemble simultaneously. We further demonstrate the validity of the simple approximation for ΛCDM haloes with isotropic velocity distributions using three-dimensional N -body simulations. This result is easily understood: an isotropic velocity distribution in a cuspy halo requires more circular orbits than radial orbits. Conversely, the approximation is poor in the extreme case of a radial orbit halo. It overestimates the response of a core dark matter halo, where radial orbit fraction is larger. Because no astronomically relevant models are dominated by low angular momentum orbits in the vicinity of the disc and the growth time-scale is never shorter than a dynamical time, we conclude that the adiabatic contraction approximation is useful in modelling the response of dark matter haloes to the growth of a disc.     

Dynamics of the boxy elliptical galaxy NGC 1600

We use three-integral models to infer the distribution function (DF) of the boxy E3–E4 galaxy NGC 1600 from surface brightness and line-profile data on the minor and major axes. We assume axisymmetry and that the mass-to-light ratio is constant in the central ∼1 R _e. Stars in the resulting gravitational potential move mainly on regular orbits. We use an approximate third integral K from perturbation theory and write the DF as a sum of basis functions in the three integrals E , L _z and K . We then fit the projected moments of these basis functions to the kinematic observables and deprojected density, using a non-parametric algorithm. The deduced dynamical structure is radially anisotropic, with σ _θ σ _r ≈ σ _φ σ _r ≈0.7 on the major axis. Both on the minor axis and near the centre the velocity distribution is more isotropic; thus the model is flattened by equatorial radial orbits. The kinematic data are fitted without the need for a central black hole; the central mass determined previously from ground-based data therefore overestimates the actual black-hole mass. The mass-to-light ratio of the stars is M L _V =6  h ₅₀. The anisotropy structure of NGC 1600 with a radially anisotropic main body and more nearly isotropic centre is similar to that found recently in NGC 1399, 2434, 3379 and 6703, suggesting that this pattern may be common amongst massive elliptical galaxies. We discuss a possible merger origin of NGC 1600 in the light of these results.     

Tidal tails of star clusters

Based on recent findings of a formation mechanism of substructure in tidal tails by Küpper et al., we investigate a more comprehensive set of N -body models of star clusters on orbits about a Milky Way like potential. We find that the predicted epicyclic overdensities arise in any tidal tail no matter which orbit the cluster follows as long as the cluster lives long enough for the overdensities to build up.
The distance of the overdensities along the tidal tail from the cluster centre depends for circular orbits only on the mass of the cluster and the strength of the tidal field, and therefore decreases monotonically with time, while for eccentric orbits the orbital motion influences the distance, causing a periodic compression and stretching of the tails and making the distance oscillate with time. We provide an approximation for estimating the distance of the overdensities in this case.
We describe an additional type of overdensity which arises in extended tidal tails of clusters on eccentric orbits, when the acceleration of the tidal field on the stellar stream is no longer homogeneous. Moreover, we conclude that a pericentre passage or a disc shock is not the direct origin of an overdensity within a tidal tail. Escape due to such tidal perturbations does not take place immediately after the perturbation but is rather delayed and spread over the orbit of the cluster. All observable overdensities are therefore of the mentioned two types. In particular, we note that substructured tidal tails do not imply the existence of dark matter substructures in the haloes of galaxies.     

A numerical comparison of theories of violent relaxation

Using N -body simulations with a large set of massless test particles, we compare the predictions of two theories of violent relaxation, the well-known Lynden-Bell theory and the more recent theory by Nakamura. We derive 'weakened' versions of both the theories in which we use the whole equilibrium coarse-grained distribution function     as a constraint instead of the total energy constraint. We use these weakened theories to construct expressions for the conditional probability   K_i (τ)  that a test particle initially at the phase-space coordinate τ would end-up in the i th macro-cell at equilibrium. We show that the logarithm of the ratio   R_ij (τ) ≡ K_i (τ)/ K_j (τ)  is directly proportional to the initial phase-space density   f ₀(τ)  for the Lynden-Bell theory and inversely proportional to   f ₀(τ)  for the Nakamura theory. We then measure   R_ij (τ)  using a set of N -body simulations of a system undergoing a gravitational collapse to check the validity of the two theories of violent relaxation. We find that both the theories are at odds with the numerical results, both qualitatively and quantitatively.     

The tidal stripping of satellites

We present an improved analytic calculation for the tidal radius of satellites and test our results against N -body simulations.
The tidal radius in general depends upon four factors: the potential of the host galaxy, the potential of the satellite, the orbit of the satellite and the orbit of the star within the satellite . We demonstrate that this last point is critical and suggest using three tidal radii to cover the range of orbits of stars within the satellite. In this way we show explicitly that prograde star orbits will be more easily stripped than radial orbits; while radial orbits are more easily stripped than retrograde ones. This result has previously been established by several authors numerically, but can now be understood analytically. For point mass, power-law (which includes the isothermal sphere), and a restricted class of split power-law potentials our solution is fully analytic. For more general potentials, we provide an equation which may be rapidly solved numerically.
Over short times (≲1–2 Gyr ∼1 satellite orbit), we find excellent agreement between our analytic and numerical models. Over longer times, star orbits within the satellite are transformed by the tidal field of the host galaxy. In a Hubble time, this causes a convergence of the three limiting tidal radii towards the prograde stripping radius. Beyond the prograde stripping radius, the velocity dispersion will be tangentially anisotropic.     

Morphology, photometry and kinematics of N-body bars – I. Three models with different halo central concentrations

We discuss the morphology, photometry and kinematics of the bars which have formed in three N -body simulations. These have initially the same disc and the same halo-to-disc mass ratio, but their haloes have very different central concentrations. The third model includes a bulge. The bar in the model with the centrally concentrated halo (model MH) is much stronger, longer and thinner than the bar in the model with the less centrally concentrated halo (model MD). Its shape, when viewed side-on, evolves from boxy to peanut and then to 'X'-shaped, as opposed to that of model MD, which stays boxy. The projected density profiles obtained from cuts along the bar major axis, for both the face-on and the edge-on views, show a flat part, as opposed to those of model MD which are falling rapidly. A Fourier analysis of the face-on density distribution of model MH shows very large  m=2  , 4, 6 and 8 components. Contrary to this, for model MD the components  m=6  and 8 are negligible. The velocity field of model MH shows strong deviations from axial symmetry, and in particular has wavy isovelocities near the end of the bar when viewed along the bar minor axis. When viewed edge-on, it shows cylindrical rotation, which the MD model does not. The properties of the bar of the model with a bulge and a non-centrally concentrated halo (MDB) are intermediate between those of the bars of the other two models. All three models exhibit a lot of inflow of the disc material during their evolution, so that by the end of the simulations the disc dominates over the halo in the inner parts, even for model MH, for which the halo and disc contributions were initially comparable in that region.     

Invariant distributions and collisionless equilibria

This paper discusses the possibility of constructing time-independent solutions to the collisionless Boltzmann equation which depend on quantities other than global isolating integrals such as energy and angular momentum. The key point is that, at least in principle, a self-consistent equilibrium can be constructed from any set of time-independent phase-space building blocks which, when combined, generate the mass distribution associated with an assumed time-independent potential. This approach provides a way to justify Schwarzschild's method for the numerical construction of self-consistent equilibria with arbitrary time-independent potentials, generalizing thereby an approach developed by Vandervoort for integrable potentials. As a simple illustration, Schwarzschild's method is reformulated to allow for a straightforward computation of equilibria which depend only on one or two global integrals and no other quantities, as is reasonable, for example, for modelling axisymmetric configurations characterized by a non-integrable potential.     

Radio emission at 1.4 GHz from luminous compact galaxies

We analyse the properties of the 1.4 GHz continuum emission in 52 star‐forming compact luminous galaxies (LCGs). The fluxes of the 1.4 GHz thermal (free‐free) component are derived from the extinction‐ and aperture‐corrected fluxes of the Hα emission line. The fraction of the thermal 1.4 GHz emission is shown tobe in the range of 2–52 % with a median value of about 15–17 %, and its distribution is close to a log‐normal one. Both the thermal and non‐thermal radio continuum luminosities are shown to be proportional to the masses of the young stellar population of the LCGs. We discuss the approximation of the ratio of the 1.4 GHz luminosity‐to‐the mass of the young stellar population as a function of starburst age. The derived approximation is shown to be similar but less clearly defined compared to ones obtained previously for LCGs from their Hα and UV continuum luminosities. (© 2015 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

BHαBAR: big Hα kinematical sample of barred spiral galaxies – I. Fabry–Perot observations of 21 galaxies

We present the Hα gas kinematics of 21 representative barred spiral galaxies belonging to the BHαBAR sample. The galaxies were observed with FaNTOmM, a Fabry–Perot integral-field spectrometer, on three different telescopes. The three-dimensional data cubes were processed through a robust pipeline with the aim of providing the most homogeneous and accurate data set possible useful for further analysis. The data cubes were spatially binned to a constant signal-to-noise ratio, typically around 7. Maps of the monochromatic Hα emission line and of the velocity field were generated and the kinematical parameters were derived for the whole sample using tilted-ring models. The photometrical and kinematical parameters (position angle of the major axis, inclination, systemic velocity and kinematical centre) are in relative good agreement, except perhaps for the later-type spirals.     

nmagic: a fast parallel implementation of a χ2-made-to-measure algorithm for modelling observational data

We describe a made-to-measure (M2M) algorithm for constructing N -particle models of stellar systems from observational data (χ²M2M), extending earlier ideas by Syer & Tremaine. The algorithm properly accounts for observational errors, is flexible, and can be applied to various systems and geometries. We implement this algorithm in a parallel code nmagic and carry out a sequence of tests to illustrate its power and performance. (i) We reconstruct an isotropic Hernquist model from density moments and projected kinematics and recover the correct differential energy distribution and intrinsic kinematics. (ii) We build a self-consistent oblate three-integral maximum rotator model and compare how the distribution function is recovered from integral field and slit kinematic data. (iii) We create a non-rotating and a figure rotating triaxial stellar particle model, reproduce the projected kinematics of the figure rotating system by a non-rotating system of the same intrinsic shape, and illustrate the signature of pattern rotation in this model. From these tests, we comment on the dependence of the results from χ²M2M on the initial model, the geometry, and the amount of available data.