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Register a new userMiyamoto-Nagai discs embedded in the Binney logarithmic potential: analytical solution of the two-integrals Jeans equations
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We present the analytical solution of the two-integrals Jeans equations for Miyamoto-Nagai discs embedded in Binney logarithmic dark matter haloes. The equations can be solved (both with standard methods and with the Residue Theorem) for arbitrary choices of the parameters, thus providing a very flexible two-component galaxy model, ranging from flattened discs to spherical systems. A particularly interesting case is obtained when the dark matter halo reduces to the Singular Isothermal Sphere. Azimuthal motions are separated in the ordered and velocity dispersion components by using the Satoh decomposition. The obtained formulae can be used in numerical simulations of galactic gas flows, for testing codes of stellar dynamics, and to study the dependence of the stellar velocity dispersion and of the asymmetric drift in the equatorial plane as a function of disc and halo flattenings. Here, we estimate the inflow radial velocities of the interstellar medium, expected by the mixing of the stellar mass losses of the lagging stars in the disc with a pre-existing gas in circular orbit.
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I present a flexible solution for the axisymmetric Jeans equations of stellar hydrodynamics under the assumption of an anisotropic (three-integral) velocity ellipsoid aligned with the spherical polar coordinate system. I describe and test a robust and efficient algorithm for its numerical computation. I outline the evaluation of the intrinsic velocity moments and the projection of all first and second velocity moments, including both the line-of-sight velocities and the proper motions. This spherically-aligned Jeans Anisotropic Modelling (JAM_sph) method can describe in detail the photometry and kinematics of real galaxies. It allows for a spatially-varying anisotropy, or stellar mass-to-light ratios gradients, as well as for the inclusion of general dark matter distributions and supermassive black holes. The JAM_sph method complements my previously derived cylindrically-aligned JAM_cyl and spherical Jeans solutions, which I also summarize in this paper. Comparisons between results obtained with either JAM_sph or JAM_cyl can be used to asses the robustness of inferred dynamical quantities. As an illustration, I modelled the Atlas3D sample of 260 early-type galaxies with high-quality integral-field spectroscopy, using both methods. I found that they provide statistically indistinguishable total-density logarithmic slopes. This may explain the previously-reported success of the JAM method in recovering density profiles of real or simulated galaxies. A reference software implementation of JAM_sph is included in the publicly-available JAM software package.
We investigate the diffusion of particles in an attractive one-dimensional potential that grows logarithmically for large $|x|$ using the Fokker-Planck equation. An eigenfunction expansion shows that the Boltzmann equilibrium density does not fully describe the long time limit of this problem. Instead this limit is characterized by an infinite covariant density. This non-normalizable density yields the mean square displacement of the particles, which for a certain range of parameters exhibits anomalous diffusion. In a symmetric potential with an asymmetric initial condition, the average position decays anomalously slowly. This problem also has applications outside the thermal context, as in the diffusion of the momenta of atoms in optical molasses.
Analytic methods to investigate periodic orbits in galactic potentials. To evaluate the quality of the approximation of periodic orbits in the logarithmic potential constructed using perturbation theory based on Hamiltonian normal forms. The solutions of the equations of motion corresponding to periodic orbits are obtained as series expansions computed by inverting the normalizing canonical transformation. To improve the convergence of the series a resummation based on a continued fraction may be performed. This method is analogous to that looking for approximate rational solutions (Prendergast method). It is shown that with a normal form truncated at the lowest order incorporating the relevant resonance it is possible to construct quite accurate solutions both for normal modes and periodic orbits in general position.
Although N-body studies of dark matter halos show that the density profiles, rho(r), are not simple power-laws, the quantity rho/sigma^3, where sigma(r) is the velocity dispersion, is in fact a featureless power-law over ~3 decades in radius. In the first part of the paper we demonstrate, using the semi-analytic Extended Secondary Infall Model (ESIM), that the nearly scale-free nature of rho/sigma^3 is a robust feature of virialized halos in equilibrium. By examining the processes in common between numerical N-body and semi-analytic approaches, we argue that the scale-free nature of rho/sigma^3 cannot be the result of hierarchical merging, rather it must be an outcome of violent relaxation. The empirical results of the first part of the paper motivate the analytical work of the second part of the paper, where we use rho/sigma^3 proportional to r^{-alpha} as an additional constraint in the isotropic Jeans equation of hydrostatic equilibrium. Our analysis shows that the constrained Jeans equation has different types of solutions, and in particular, it admits a unique ``periodic solution with alpha=1.9444. We derive the analytic expression for this density profile, which asymptotes to inner and outer profiles of rho ~ r^{-0.78}, and rho ~ r^{-3.44}, respectively.
We present an analytical model to identify thin discs in galaxies, and apply this model to a sample of SDSS MaNGA galaxies. This model fits the velocity and velocity dispersion fields of galaxies with regular kinematics. By introducing two parameters $zeta$ related to the comparison of the models asymmetric drift correction to the observed gas kinematics and $eta$ related to the dominant component of a galaxy, we classify the galaxies in the sample as disc-dominated, non-disc-dominated, or disc-free indicating galaxies with a dominating thin disc, a non-dominating thin disc, or no thin disc detection with our method, respectively. The dynamical mass resulting from our model correlates with stellar mass, and we investigate discrepancies by including gas mass and variation of the initial mass function. As expected, most spiral galaxies in the sample are disc-dominated, while ellipticals are predominantly disc-free. Lenticular galaxies show a dichotomy in their kinematic classification, which is related to their different star formation rates and gas fractions. We propose two possible scenarios to explain these results. In the first scenario, disc-free lenticulars formed in more violent processes than disc-dominated ones, while in the second scenario, the quenching processes in lenticulars lead to a change in their kinematic structures as disc-dominated lenticulars evolve to disc-free ones.
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