A Contraction Analysis of the Convergence of Risk-Sensitive Filters

A contraction analysis of risk-sensitive Riccati equations is proposed. When the state-space model is reachable and observable, a block-update implementation of the risk-sensitive filter is used to show that the N-fold composition of the Riccati map is strictly contractive with respect to the Riemannian metric of positive definite matrices, when N is larger than the number of states. The range of values of the risk-sensitivity parameter for which the map remains contractive can be estimated a priori. It is also found that a second condition must be imposed on the risk-sensitivity parameter and on the initial error variance to ensure that the solution of the risk-sensitive Riccati equation remains positive definite at all times. The two conditions obtained can be viewed as extending to the multivariable case an earlier analysis of Whittle for the scalar case.Comment: 22 pages, 6 figure

Robust Kalman Filtering: Asymptotic Analysis of the Least Favorable Model

We consider a robust filtering problem where the robust filter is designed according to the least favorable model belonging to a ball about the nominal model. In this approach, the ball radius specifies the modeling error tolerance and the least favorable model is computed by performing a Riccati-like backward recursion. We show that this recursion converges provided that the tolerance is sufficiently small

Recognizing the fingerprints of the Galactic bar: a quantitative approach to comparing model (l,v) distributions to observation

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 that 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 model's 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.Comment: Accepted for publication in MNRA

Gas flow in barred potentials

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 closed 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.Comment: Accepted for publication in MNRA

Gas flow in barred potentials II. Bar Driven Spiral Arms

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 understood 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 potential's orbital structure that determines whether a bar should be classified as weak or strong, and not crude estimates of the potential's deviation from axisymmetry.Comment: Accepted for publication in MNRA

Models of rotating coronae

Fitting equilibrium dynamical models to observational data is an essential step in understanding the structure of the gaseous hot haloes that surround our own and other galaxies. However, the two main categories of models that are used in the literature are poorly suited for this task: (i) simple barotropic models are analytic and can therefore be adjusted to match the observations, but are clearly unrealistic because the rotational velocity $v_\phi(R,z)$ does not depend on the distance $z$ from the galactic plane, while (ii) models obtained as a result of cosmological galaxy formation simulations are more realistic, but are impractical to fit to observations due to high computational cost. Here we bridge this gap by presenting a general method to construct axisymmetric baroclinic equilibrium models of rotating galactic coronae in arbitrary external potentials. We consider in particular a family of models whose equipressure surfaces in the $(R,z)$ plane are ellipses of varying axis ratio. These models are defined by two one-dimensional functions, the axial ratio of pressure $q_{\rm axis}(z)$ and the value of the pressure $P_{\rm axis}(z)$ along the galaxy's symmetry axis. These models can have a rotation speed $v_\phi(R,z)$ that realistically decreases as one moves away from the galactic plane, and can reproduce the angular momentum distribution found in cosmological simulations. The models are computationally cheap to construct and can thus be used in fitting algorithms. We provide a python code that given $q_{\rm axis}(z)$, $P_{\rm axis}(z)$ and $\Phi(R,z)$ returns $\rho(R,z)$, $T(R,z)$, $P(R,z)$, $v_\phi(R,z)$. We show a few examples of these models using the Milky Way as a case study.Comment: Accepted for publication in MNRA

Nuclear Spirals in the inner Milky Way

We use hydrodynamical simulations to construct a new coherent picture for the gas flow in the Central Molecular Zone (CMZ), the region of our Galaxy within $R\leq 500\, \mathrm{pc}$. We relate connected structures observed in $(l,b,v)$ data cubes of molecular tracers to nuclear spiral arms. These arise naturally in hydrodynamical simulations of barred galaxies, and are similar to those that can be seen in external galaxies such as NGC4303 or NGC1097. We discuss a face-on view of the CMZ including the position of several prominent molecular clouds, such as Sgr B2, the $20\,{\rm km\, s^{-1}}$ and $50\,{\rm km\, s^{-1}}$ clouds, the polar arc, Bania Clump 2 and Sgr C. Our model is also consistent with the larger scale gas flow, up to $R\simeq 3\,\rm kpc$, thus providing a consistent picture of the entire Galactic bar region.Comment: Accepted for publication in MNRAS, 12 pages, 12 figure

Periodicity makes galactic shocks unstable - I. Linear analysis

We study the dynamical stability of stationary galactic spiral shocks. The steady-state equilibrium flow contains a shock of the type derived by Roberts in the tightly wound approximation. We find that boundary conditions are critical in determining whether the solutions are stable or not. Shocks are unstable if periodic boundary conditions are imposed. For intermediate strengths of the spiral potential, the instability disappears if boundary conditions are imposed such that the upstream flow is left unperturbed as in the classic analysis of D'yakov and Kontorovich. This reconciles apparently contradictory findings of previous authors regarding the stability of spiral shocks. This also shows that the instability is distinct from the Kelvin-Helmholtz instability, confirming the findings of Kim et al. We suggest that instability is a general characteristics of periodic shocks, regardless of the presence of shear, and provide a physical picture as to why this is the case. For strong spiral potentials, high post-shock shear makes the system unstable also to parasitic Kelvin-Helmholtz instability regardless of the boundary conditions. Our analysis is performed in the context of a simplified problem that, while preserving all the important characteristics of the original problem, strips it from unnecessary complications, and assumes that the gas is isothermal, non self-gravitating, non-magnetised.Comment: Accepted for publication in MNRA