The secular evolution of an infinitely thin tepid isolated galactic disc made of a finite number of particles is investigated using the inhomogeneous Balescu-Lenard equation expressed in terms of angle-action variables. The matrix method is implement
ed numerically in order to model the induced gravitational polarization. Special care is taken to account for the amplification of potential fluctuations of mutually resonant orbits and the unwinding of the induced swing amplified transients. Quantitative comparisons with ${N-}$body simulations yield consistent scalings with the number of particles and with the self-gravity of the disc: the fewer particles and the colder the disc, the faster the secular evolution. Secular evolution is driven by resonances, but does not depend on the initial phases of the disc. For a Mestel disc with ${Q sim 1.5}$, the polarization cloud around each star boosts up its secular effect by a factor of the order of a thousand or more, promoting accordingly the dynamical relevance of self-induced collisional secular evolution. The position and shape of the induced resonant ridge are found to be in very good agreement with the prediction of the Balescu-Lenard equation, which scales with the square of the susceptibility of the disc. In astrophysics, the inhomogeneous Balescu-Lenard equation may describe the secular diffusion of giant molecular clouds in galactic discs, the secular migration and segregation of planetesimals in proto-planetary discs, or even the long-term evolution of population of stars within the Galactic centre. It could be used as a valuable check of the accuracy of ${N-}$body integrators over secular timescales.
We run hydrodynamical simulations of a 2D isothermal non self-gravitating inviscid gas flowing in a rigidly rotating externally imposed potential formed by only two components: a monopole and a quadrupole. We explore systematically the effects of var
ying the quadrupole while keeping fixed the monopole and discuss the consequences for the interpretation of longitude-velocity diagrams in the Milky Way. We find that the gas flow can constrain the quadrupole of the potential and the characteristics of the bar that generates it. The exponential scale length of the bar must be at least $1.5rm, kpc$. The strength of the bar is also constrained. Our global interpretation favours a pattern speed of $Omega=40,rm km s^{-1} {kpc}^{-1}$. We find that for most observational features, there exist a value of the parameters that matches each individual feature well, but is difficult to reproduce all the important features at once. Due to the intractably high number of parameters involved in the general problem, quantitative fitting methods that can run automatic searches in parameter space are necessary.
Spiral arms that emerge from the ends of a galactic bar are important in interpreting observations of our and external galaxies. It is therefore important to understand the physical mechanism that causes them. We find that these spiral arms can be un
derstood as kinematic density waves generated by librations around underlying ballistic closed orbits. This is even true in the case of a strong bar, provided the librations are around the appropriate closed orbits and not around the circular orbits that form the basis of the epicycle approximation. An important consequence is that it is a potentials orbital structure that determines whether a bar should be classified as weak or strong, and not crude estimates of the potentials deviation from axisymmetry.
We use a Cartesian grid to simulate the flow of gas in a barred Galactic potential and investigate the effects of varying the sound speed in the gas and the resolution of the grid. For all sound speeds and resolutions, streamlines closely follow clos
ed orbits at large and small radii. At intermediate radii shocks arise and the streamlines shift between two families of closed orbits. The point at which the shocks appear and the streamlines shift between orbit families depends strongly on sound speed and resolution. For sufficiently large values of these two parameters, the transfer happens at the cusped orbit as hypothesised by Binney et al. over two decades ago. For sufficiently high resolutions the flow downstream of the shocks becomes unsteady. If this unsteadiness is physical, as appears to be the case, it provides a promising explanation for the asymmetry in the observed distribution of CO.
We present a new method for fitting simple hydrodynamical models to the (l,v) distribution of atomic and molecular gas observed in the Milky Way. The method works by matching features found in models and observations. It is based on the assumption th
at the large-scale features seen in (l,v) plots, such as ridgelines and the terminal velocity curve, are influenced primarily by the underlying large-scale Galactic potential and are only weakly dependent on local ISM heating and cooling processes. In our scheme one first identifies by hand the features in the observations: this only has to be done once. We describe a procedure for automatically extracting similar features from simple hydrodynamical models and quantifying the distance between each models features and the observations. Application to models of the Galactic Bar region (|l|<30deg) shows that our feature-fitting method performs better than chi^2 or envelope distances at identifying the correct underlying galaxy model.
We present a new framework for estimating a galaxys gravitational potential, Phi(x), from its stellar kinematics by adopting a fully non-parametric model for the galaxys unknown action-space distribution function, f(J). Having an expression for the j
oint likelihood of Phi and f, the likelihood of Phi is calculated by using a Dirichlet process mixture to represent the prior on f and marginalising. We demonstrate that modelling machinery constructed using this framework is successful at recovering the potentials of some simple systems from perfect discrete kinematical data, a situation handled effortlessly by traditional moment-based methods, such as the virial theorem, but in which other, more modern, methods are less than satisfactory. We show how to generalise the machinery to account for realistic observational errors and selection functions. A practical implementation is likely to raise some interesting algorithmic and computational challenges.
We present three-dimensional eccentric disc models of the nucleus of M31, modelling the disc as a linear combination of thick rings of massless stars orbiting in the potential of a central black hole. Our models are nonparametric generalisations of t
he parametric models of Peiris & Tremaine. The models reproduce well the observed WFPC2 photometry, the detailed line-of-sight velocity distributions from STIS observations along P1 and P2, together with the qualitative features of the OASIS kinematic maps. We confirm Peiris & Tremaines finding that nuclear discs aligned with the larger disc of M31 are strongly ruled out. Our optimal model is inclined at 57 degrees with respect to the line of sight of M31 and has a position angle of 55 degrees. It has a central black hole of mass 10^8 solar masses, and, when viewed in three dimensions, shows a clear enhancement in the density of stars around the black hole. The distribution of orbit eccentricities in our models is similar to Peiris & Tremaines model, but we find significantly different inclination distributions, which might provide valuable clues to the origin of the disc.
We present a general scheme for constructing Monte Carlo realizations of equilibrium, collisionless galaxy models with known distribution function (DF) f_0. Our method uses importance sampling to find the sampling DF f_s that minimizes the mean-squar
e formal errors in a given set of projections of the DF f_0. The result is a multi-mass N-body realization of the galaxy model in which ``interesting regions of phase-space are densely populated by lots of low-mass particles, increasing the effective N there, and less interesting regions by fewer, higher-mass particles. As a simple application, we consider the case of minimizing the shot noise in estimates of the acceleration field for an N-body model of a spherical Hernquist model. Models constructed using our scheme easily yield a factor ~100 reduction in the variance in the central acceleration field when compared to a traditional equal-mass model with the same number of particles. When evolving both models with a real N-body code, the diffusion coefficients in our model are reduced by a similar factor. Therefore, for certain types of problems, our scheme is a practical method for reducing the two-body relaxation effects, thereby bringing the N-body simulations closer to the collisionless ideal.
