Non-parametric reconstruction of distribution functions from observed galactic discs

A general inversion technique for the recovery of the underlying distribution function for observed galactic discs is presented and illustrated. Under the assumption that these discs are axisymmetric and thin, the proposed method yields a unique distribution compatible with all the observables available. The derivation may be carried out from the measurement of the azimuthal velocity distribution arising from positioning the slit of a spectrograph along the major axis of the galaxy. More generally, it may account for the simultaneous measurements of velocity distributions corresponding to slits presenting arbitrary orientations with respect to the major axis. The approach is non-parametric, i.e. it does not rely on a particular algebraic model for the distribution function. Special care is taken to account for the fraction of counter-rotating stars, which strongly affects the stability of the disc. An optimization algorithm is devised — generalizing the work of Skilling &38; Bryan — to carry this truly two-dimensional ill-conditioned inversion efficiently. The performance of the overall inversion technique with respect to the noise level and truncation in the data set is investigated with simulated data. Reliable results are obtained up to a mean signal-to-noise ratio of 5, and when measurements are available up to 4  R _e. A discussion of the residual biases involved in non-parametric inversions is presented. The prospects of application of the algorithm to observed galaxies and other inversion problems are discussed.     

Mass models of the Milky Way

A parametrized model of the mass distribution within the Milky Way is fitted to the available observational constraints. The most important single parameter is the ratio of the scalelength R _d* of the stellar disc to R ₀. The disc and bulge dominate v _c( R ) at R ≲ R ₀ only for R _d,*/ R ₀≲0.3. Since the only knowledge we have of the halo derives from studies like the present one, we allow it to contribute to the density at all radii. When allowed this freedom, however, the halo causes changes in assumptions relating to R  ≪  R ₀ to affect profoundly the structure of the best-fitting model at R  ≫  R ₀. For example, changing the disc slightly from an exponential surface-density profile significantly changes the form of v _c( R ) at R  ≫  R ₀, where the disc makes a negligible contribution to v _c. Moreover, minor changes in the constraints can cause the halo to develop a deep hole at its centre that is not physically plausible. These problems call into question the proposition that flat rotation curves arise because galaxies have physically distinct haloes rather than outwards-increasing mass-to-light ratios.   The mass distribution of the Galaxy and the relative importance of its various components will remain very uncertain until more observational data can be used to constrain mass models. Data that constrain the Galactic force field at z ≳ R and at R  >  R ₀ are especially important.     

On the global structure of self-gravitating discs for softened gravity

The effects of gravitational softening on the global structure of self-gravitating discs in centrifugal equilibrium are examined in relation to hydrodynamical/gravitational simulations. The one-parameter spline softening proposed by Hernquist & Katz is used.
It is found that if the characteristic size of a disc, r , is comparable to or less than the gravitational softening length, ε, then the cross-section of the simulated disc is significantly larger than that of a no-softening (Newtonian) disc with the same mass and angular momentum.
We demonstrate, furthermore, that if r ≲ε/2 then the scaling relation r ∝ε^3/4 holds for a given mass and specific angular momentum distribution with mass. Finally, we compare some of the theoretical results obtained in this paper and a previous one with the results of numerical Tree-SPH simulations and find qualitative agreement.     

Dynamical stability and evolution of the discs of Sc galaxies

We examine the local stability of galactic discs against axisymmetric density perturbations with special attention to the different dynamics of the stellar and gaseous components. In particular, the discs of the Milky Way and of NGC 6946 are studied. The Milky Way is shown to be stable, whereas the inner parts of NGC 6946, a typical Sc galaxy from the Kennicutt sample, are dynamically unstable. The ensuing dynamical evolution of the composite disc is studied by numerical simulations. The evolution is so fierce that the stellar disc heats up dynamically on a short time-scale to such a high degree, which seems to contradict the morphological appearance of the galaxy. The star formation rate required to cool the disc dynamically is estimated. Even if the star formation rate in NGC 6946 is at present high enough to meet this requirement, it is argued that the discs of Sc galaxies cannot sustain such a high star formation rate for extended periods.     

Simulations of spheroidal systems with substructure: trees in fields

We present a hybrid technique of N -body simulation to deal with collisionless stellar systems having an inhomogeneous global structure. We combine a treecode and a self-consistent field code such that each of the codes models a different component of the system being investigated. The treecode is suited to treatment of dynamically cold or clumpy components, which may undergo significant evolution within a dynamically hot system. The hot system is appropriately evolved by the self-consistent field code. This combined code is particularly suited to a number of problems in galactic dynamics. Applications of the code to these problems are briefly discussed.     

Linear response of galactic haloes to adiabatic gravitational perturbations

We determine the response of a self-similar isothermal stellar system to small adiabatic gravitational perturbations. For odd spherical harmonics the response is identical to the response of the analogous isothermal fluid system. For even spherical harmonics the response can be regarded as an infinite series of wavetrains in log r , implying alternating compression and rarefaction in equal logarithmic radius intervals. Partly because of the oscillatory nature of the solutions, tidal fields from external sources are not strongly amplified by an intervening isothermal stellar system, except at radii ≲10^−3.5 times the satellite radius; at some radii the stellar system can even screen the external tidal field in a manner analogous to Debye screening. As Weinberg has pointed out, individual resonances in a stellar system can strongly amplify external tidal fields over a limited radial range, but we cannot address this possibility because we examine only adiabatic perturbations. We also discuss the application of our method to the halo response caused by the slow growth of an embedded thin disc.     

The kinematics of the ionized gas in the Circinus galaxy

The TAURUS-2 Fabry–Perot interferometer, mounted on the 3.9-m Anglo-Australian Telescope, has been used to observe the Circinus galaxy. We have mapped the intensity and velocity distribution of the ionized hydrogen in the galaxy using the Balmer series Hα spectral line.
The semiresolved core (observed with a seeing disc of 30 pc) appears amorphous in shape, which is commonly observed in Seyfert 2 galaxies. Its peak coincides with the core position measured in the radio continuum, suggesting that ionized gas surrounds a non-thermal source.
A circumnuclear ring or spiral of radius 220 pc and a rotational velocity of 350 km s⁻¹ (assuming circular motions) surrounds the core. The inclination angle of this feature, i =40°±10°, is less than that of the previously observed radio continuum disc. The velocity channel maps obtained for the Hα ring show that the kinematics resemble those of a rotating ring and the intensity displays a complex structure indicative of several, unresolved, H II regions. We believe the ring to represent a circumnuclear starburst.
Our Hα data also show the presence of the previously detected [O III ] ionization cone to the north-west of the core, measuring more than 400 pc in length. We suggest that the ionization cone lies in a different plane from that of the starburst ring and is directed away from us. Several kinematic components of the core are derived and we calculate an outflow velocity in excess of 150–200 km s⁻¹ for gas above the core of Circinus. We also present evidence for inflowing ionized gas at the centre of Circinus.
The correlation of the Hα and radio continuum features is discussed, as well as the possible presence of a starburst-driven superwind in the Circinus galaxy.     

Orbit classification in arbitrary 2D and 3D potentials

A method of classifying generic orbits in arbitrary 2D and 3D potentials is presented. It is based on the concept of spectral dynamics introduced by Binney &38; Spergel that uses the Fourier transform of the time series of each coordinate. The method is tested using a number of potentials previously studied in the literature and is shown to distinguish correctly between regular and irregular orbits, to identify the various families of regular orbits (boxes, loops, tubes, boxlets, etc.), and to recognize the second-rank resonances that bifurcate from them. The method returns the position of the potential centre and, for 2D potentials, the orientation of the principal axes as well, should this be unknown. A further advantage of the method is that it has been encoded in a FORTRAN program that does not require user intervention, except for 'fine tuning' of search parameters that define the numerical limits of the code. The automatic character makes the program suitable for classifying large numbers of orbits.     

Frequency map analysis of the orbital structure in elliptical galaxies

We present an application of the frequency map analysis to an elliptical galaxy which is represented by a generalization of a double-power-law spherical mass model. The density distribution of this model varies as r ^−γ close to the centre and as r ⁻⁴ at large radii. We study the case with γ = 1, which is known as the 'weak-cusp' model and which represents well the density profile of the 'core' galaxies observed by the Hubble Space Telescope . The final objective of our work is to improve our understanding of the dynamics of elliptical galaxies in a similar way to Merritt &38; Fridman, finding the regions of stochasticity, looking for resonances that might play an important role in sustaining the triaxial morphology, and analysing the diffusion of orbits. To this end, we use the frequency map analysis of Laskar, which has been applied widely in the field of celestial mechanics but which is a relatively new technique in the area of galactic dynamics. Finally, we show some useful features of this method in understanding the global dynamical structure of the system.