Anisotropy ansatz for the Jeans equations: oblate galaxies

ABSTRACTWe present the theoretical framework to efficiently solve the Jeans equations for multicomponent axisymmetric stellar systems, focusing on the scaling of all quantities entering them. The models may include an arbitrary number of stellar distributions, a dark matter halo, and a central supermassive black hole; each stellar distribution is implicitly described by a two- or three-integral distribution function, and the stellar components can have different structural (density profile, flattening, mass, scale length), dynamical (rotation, velocity dispersion anisotropy), and population (age, metallicity, initial mass function, mass-to-light ratio) properties. In order to determine the ordered rotational velocity and the azimuthal velocity dispersion fields of each component, we introduce a decomposition that can be used when the commonly adopted Satoh decomposition cannot be applied. The scheme developed is particularly suitable for a numerical implementation; we describe its realization within our code JASMINE2, optimized to maximally exploit the scalings allowed by the Poisson and the Jeans equations, also in the post-processing procedures. As applications, we illustrate the building of three multicomponent galaxy models with two distinct stellar populations, a central black hole, and a dark matter halo; we also study the solution of the Jeans equations for an exponential thick disc, and for its multicomponent representation as the superposition of three Miyamoto–Nagai discs. A useful general formula for the numerical evaluation of the gravitational potential of factorized thick discs is finally given.

Studies of strong gravitational lensing in current and upcoming wide and deep photometric surveys, and of stellar kinematics from (integral-field) spectroscopy at increasing redshifts, promise to provide valuable constraints on galaxy density profiles and shapes. However, both methods are affected by various selection and modelling biases, whch we aim to investigate in a consistent way. In this first paper in a series we develop a flexible but efficient pipeline to simulate lensing by realistic galaxy models. These galaxy models have separate stellar and dark matter components, each with a range of density profiles and shapes representative of early-type, central galaxies without significant contributions from other nearby galaxies. We use Fourier methods to calculate the lensing properties of galaxies with arbitrary surface density distributions, and Monte Carlo methods to compute lensing statistics such as point-source lensing cross-sections. Incorporating a variety of magnification bias modes lets us examine different survey limitations in image resolution and flux. We rigorously test the numerical methods for systematic errors and sensitivity to basic assumptions. We also determine the minimum number of viewing angles that must be sampled in order to recover accurate orientation-averaged lensing quantities. We find that for a range of non-isothermal stellar and dark matter density profiles typical of elliptical galaxies, the combined density profile and corresponding lensing properties are surprisingly close to isothermal around the Einstein radius. The converse implication is that constraints from strong lensing and/or stellar kinematics, which are indeed consistent with isothermal models near the Einstein radius, cannot trivially be extrapolated to smaller and larger radii.

The Jeans equations do not form a closed system, and to solve them a parametrization relating the velocity moments is often adopted. For axisymmetric models, a phenomenological choice (the “b-ansatz”) is widely used for the relation between the vertical ($\sigma _z^2$) and radial ($\sigma _R^2$) components of the velocity dispersion tensor, thus breaking their identity present in two-integral systems. However, the way in which the ansatz affects the resulting kinematical fields can be quite complicated, so that the analysis of these fields is usually performed only after numerically computing them. We present here a general procedure to study the properties of the ansatz-dependent fields $\overline{v_{\varphi }^2}$, $\Delta =\overline{v_{\varphi }^2}- \sigma _z^2$ and $\Delta _R= \overline{v_{\varphi }^2}- \sigma _R^2$. Specifically, the effects of the b-ansatz can be determined before solving the Jeans equations once the behaviour over the (R, z)-plane of three easy-to-build ansatz-independent functions is known. The procedure also constrains the ansatz to exclude unphysical results (as a negative $\overline{v_{\varphi }^2}$). The method is illustrated by discussing the cases of three well-known galaxy models: the Miyamoto & Nagai and Satoh disks, and the Binney logarithmic halo, for which the regions and the constraints on the ansatz values can be determined analytically; a two-component (Miyamoto & Nagai plus logarithmic halo) model is also discussed.

Made-to-measure dynamical particle models of elliptical galaxies

We present a detailed diagnostic study of the observed temperatures of the hot X-ray coronae of early-type galaxies. By extending the investigation carried out in Pellegrini (2011) with spherical models, we focus on the dependence of the energy budget and temperature of the hot gas on the galaxy structure and internal stellar kinematics. By solving the Jeans equations we construct realistic axisymmetric three-component galaxy models (stars, dark matter halo, central black hole) with different degrees of flattening and rotational support. The kinematical fields are projected along different lines of sight, and the aperture velocity dispersion is computed within a fraction of the circularized effective radius. The model parameters are chosen so that the models resemble real ETGs and lie on the Faber-Jackson and Size-Luminosity relations. For these models we compute T_* (the stellar heating contribution to the gas injection temperature) and T_gm (the temperature equivalent of the energy required for the gas escape). In particular, different degrees of thermalisation of the ordered rotational field of the galaxy are considered. We find that T_* and T_gm can vary only mildly due to a pure change of shape. Galaxy rotation instead, when not thermalised, can lead to a large decrease of T_*; this effect can be larger in flatter galaxies that can be more rotationally supported. Recent temperature measurements T_x, obtained with Chandra, are larger than, but close to, the T_* values of the models, and show a possible trend for a lower T_x in flatter and more rotationally supported galaxies; this trend can be explained by the lack of thermalisation of the whole stellar kinetic energy. Flat and rotating galaxies also show lower L_x values, and then a lower gas content, but this is unlikely to be due to the small variation of T_gm found here for them.

By solving Poisson's equation for logarithmic density disturbance in three-dimensional (3D) spiral galaxies, the disturbed potential can be given on the disk. Combining it with the corresponding dispersion relation, the local gravitational stability of our galaxy is discussed here. We have generalized the Toomre's stability criterion to the more realistic 3D galaxy models. The generalized Toomre's parameter Q near the sun is 2.08 or 2.20 for two or four armed spiral galaxy models, respectively.

Galaxies are self-gravitating structures composed by several components encompassing spherical, axial, and triaxial symmetry. Although real systems feature heterogeneous components whose properties are intimately connected, semi-analytical approaches often exploit the linearity of the Poisson’s equation to represent the potential and mass distribution of a multicomponent galaxy as the sum of the individual components. In this work, we expand the semi-analytical framework developed in Bonetti et al. (2020) by including both a detailed implementation of the gravitational potential of exponential disc (modelled with a sech2 and an exponential vertical profile) and an accurate prescription for the dynamical friction experienced by massive perturbers (MP) in composite galaxy models featuring rotating disc structures. Such improvements allow us to evolve arbitrary orbits either within or outside the galactic disc plane. We validate the results obtained by our numerical model against public semi-analytical codes as well as full N-body simulations, finding that our model is in excellent agreement to the codes it is compared with. The ability to reproduce the relevant physical processes responsible for the evolution of MP orbits and its computational efficiency make our framework perfectly suited for large parameter-space exploration studies.

We report on the results of a radio interferometric observation of NGC7027 in the CO J=2-1 and 13CO J=2-1 lines. The results are analyzed with morpho-kinematic models developed from the software tool Shape. Our goal is to reveal the morpho-kinematic properties of the central region of the nebula, and to explore the nature of unseen high-velocity jets that may have created the characteristic structure of the central region consisting of molecular and ionized components. A simple ellipsoidal shell model explains the intensity distribution around the systemic velocity, but the high velocity features deviate from the ellipsoidal model. Through the Shape automatic reconstruction model, we found a possible trail of a jet only in one direction, but no other possible holes were created by the passage of a jet.

ABSTRACTRecently, two-component spherical galaxy models have been presented, where the stellar profile is described by a Jaffe law, and the total density by another Jaffe law, or by an r−3 law at large radii. We extend these two families to their ellipsoidal axisymmetric counterparts: the JJe and J3e models. The total and stellar density distributions can have different flattenings and scale lengths, and the dark matter halo is defined by difference. First, the analytical conditions required to have a nowhere negative dark matter halo density are derived. The Jeans equations for the stellar component are then solved analytically, in the limit of small flattenings, also in the presence of a central BH. The azimuthal velocity dispersion anisotropy is described by the Satoh k-decomposition. Finally, we present the analytical formulae for velocity fields near the centre and at large radii, together with the various terms entering the virial theorem. The JJe and J3e models can be useful in a number of theoretical applications, e.g. to explore the role of the various parameters (flattening, relative scale lengths, mass ratios, rotational support) in determining the behaviour of the stellar kinematical fields before performing more time-expensive integrations with specific galaxy models, to test codes of stellar dynamics and in numerical simulations of gas flows in galaxies.

A human face provides a variety of different communicative functions such as identification, the perception of emotional expression, and lip-reading. For these reasons, many applications in robotics require tracking and recognizing a human face. A novel face recognition system should be able to deal with various changes in face images, such as pose, illumination, and expression, among which pose variation is the most difficult one to deal with. Therefore, face registration (alignment) is the key of robust face recognition. If we can register face images into frontal views, the recognition task would be much easier. To align a face image into a canonical frontal view, we need to know the pose information of a human head. Therefore, in this paper, we propose a novel method for modeling a human head as a simple 3D ellipsoid. And also, we present 3D head tracking and pose estimation methods using the proposed ellipsoidal model. After recovering full motion of the head, we can register face images with pose variations into stabilized view images which are suitable for frontal face recognition. By doing so, simple and efficient frontal face recognition can be easily carried out in the stabilized texture map space instead of the original input image space. To evaluate the feasibility of the proposed approach using a simple ellipsoid model, 3D head tracking experiments are carried out on 45 image sequences with ground truth from Boston University, and several face recognition experiments are conducted on our laboratory database and the Yale Face Database B by using subspace-based face recognition methods such as PCA, PCA+LAD, and DCV.

Context. Dynamically self-consistent galactic models are necessary for analysing and interpreting star counts, stellar density distributions, and stellar kinematics in order to understand the formation and the evolution of our Galaxy. Aims. We modify and improve the dynamical self-consistency of the Besançon Galaxy model in the case of a stationary and axisymmetric gravitational potential. Methods. Each stellar orbit is modelled by determining a Stäckel approximate integral of motion. Generalised Shu distribution functions (DFs) with three integrals of motion are used to model the stellar distribution functions. Results. This new version of the Besançon model is compared with the previous axisymmetric BGM2014 version and we find that the two versions have similar densities for each stellar component. The dynamically self-consistency is improved and can be tested by recovering the forces and the potential through the Jeans equations applied to each stellar distribution function. Forces are recovered with an accuracy better than one per cent over most of the volume of the Galaxy.

Spherical self-consistent elliptical galaxy models, embedded in a dark matter halo, are constructed. Two different stellar distributions are considered : the King law, and another one reproducing the R 1/4 luminosity law; the stellar mass-to-light ratio is constant. The concentration and the amount of the stellar and dark matter distributions are determined by five parameters; two characteristic radii determine the radial trend of the anisotropy for the stellar and dark halo velocity dispersion tensors. The multicomponent decomposition of dynamical models of galaxies and a theorem concerning their consistency are introduced; then, the region of the parameter space where the models are self-consistently generated are determined

Recent orbital studies of 3D bar structure in various numerical and analytical models show that X-structures that reside in boxy/peanut-shaped (B/PS) bulges are not delineated by some specific type of orbits, but are natural parts of them and formed by the same orbits that constitute such bulges. This implies that to accurately account for B/PS bulges and their X-structures in photometric studies, one needs the photometric model of B/PS bulge that includes an X-structure as its natural part. To find such a model, we considered a self-consistent numerical galaxy model where a typical B/PS bulge arises. Using spectral characteristics of particle-‘stars’, we decomposed the galaxy model on to the bar and non-bar components. We used the extracted 3D bar component to find an appropriate B/PS bulge photometric model, which can account for X-structures residing in such bulges. The resulted B/PS bulge photometric model has a truncated 2D Sersic profile with truncations introduced above (in the upper half-plane) and below (in the bottom half-plane) the rays of X-structures. We applied this model to represent B/PS bulges of various numerical models and some real galaxies. The comparison with previous works revealed that there are systematic shifts between the X-structure parameters of the same galaxies measured within the different approaches. We found that the geometric parameters of X-structures of real and modelled galaxies are consistent with each other if we measure them using our new model.

Secondary evolution of galaxies investigated by N-body simulations

The presence of a dark matter core in the central kiloparsec of many dwarf galaxies has been a long standing problem in galaxy formation theories based on the standard cold dark matter paradigm. Recent cosmological simulations, based on Smooth Particle Hydrodynamics and rather strong feedback recipes have shown that it was indeed possible to form extended dark matter cores using baryonic processes related to a more realistic treatment of the interstellar medium. Using adaptive mesh refinement, together with a new, stronger supernovae feedback scheme that we have recently implemented in the RAMSES code, we show that it is also possible to form a prominent dark matter core within the well-controlled framework of an isolated, initially cuspy, 10 billion solar masses dark matter halo. Although our numerical experiment is idealized, it allows a clean and unambiguous identification of the dark matter core formation process. Our dark matter inner profile is well fitted by a pseudo-isothermal profile with a core radius of 800 pc. The core formation mechanism is consistent with the one proposed recently by Pontzen & Governato. We highlight two key observational predictions of all simulations that find cusp-core transformations: (i) a bursty star formation history with peak to trough ratio of 5 to 10 and a duty cycle comparable to the local dynamical time; and (ii) a stellar distribution that is hot with v/sigma=1. We compare the observational properties of our model galaxy with recent measurements of the isolated dwarf WLM. We show that the spatial and kinematical distribution of stars and HI gas are in striking agreement with observations, supporting the fundamental role played by stellar feedback in shaping both the stellar and dark matter distribution.