Self-Consistent, Axisymmetric Two_Integral Models of Elliptical Galaxies with embedded Nuclear Discs

Recently, observations with the Hubble Space Telescope have revealed small stellar discs embedded in the nuclei of a number of ellipticals and S0s. In this paper we construct two-integral axisymmetric models for such systems. We calculate the even part of the phase-space distribution function, and specify the odd part by means of a simple parameterization. We investigate the photometric as well as the kinematic signatures of nuclear discs, including their velocity profiles (VPs), and study the influence of seeing convolution. The rotation curve of a nuclear disc gives an excellent measure of the central mass-to-light ratio whenever the VPs clearly reveal the narrow, rapidly rotating component associated with the nuclear disc. Steep cusps and seeing convolution both result in central VPs that are dominated by the bulge light, and these VPs barely show the presence of the nuclear disc, impeding measurements of the central rotation velocities of the disc stars. However, if a massive BH is present, the disc component of the VP can be seen in the wing of the bulge part, and measurements of its mean rotation provide a clear signature of the presence of the BH. This signature is insensitive to the uncertainties in the velocity anisotropy, which often lead to ambiguity in the interpretation of a central rise in velocity dispersion as due to a central BH.Comment: 13 pages, uses mn.tex (included). MNRAS accepted. The 17 PS figures (not enclosed) can be retrieved from ftp://strw.leidenuniv.nl/pub/vdbosch/diskpaper

Kinemetry: a generalisation of photometry to the higher moments of the line-of-sight velocity distribution

We present a generalisation of surface photometry to the higher-order moments of the line-of-sight velocity distribution of galaxies observed with integral-field spectrographs. The generalisation follows the approach of surface photometry by determining the best fitting ellipses along which the profiles of the moments can be extracted and analysed by means of harmonic expansion. The assumption for the odd moments (e.g. mean velocity) is that the profile along an ellipse satisfies a simple cosine law. The assumption for the even moments (e.g velocity dispersion) is that the profile is constant, as it is used in surface photometry. We find that velocity profiles extracted along ellipses of early-type galaxies are well represented by the simple cosine law (with 2% accuracy), while possible deviations are carried in the fifth harmonic term which is sensitive to the existence of multiple kinematic components, and has some analogy to the shape parameter of photometry. We compare the properties of the kinematic and photometric ellipses and find that they are often very similar. Finally, we offer a characterisation of the main velocity structures based only on the kinemetric parameters which can be used to quantify the features in velocity maps (abridged).Comment: 17 pages, 11 figures. MNRAS in press. High resolution version of the paper is available at http://www.strw.leidenuniv.nl/sauron/papers/krajnovic2005_kinemetry.pdf and software implementation of the method is freely available at http://www-astro.physics.ox.ac.uk/~dxk/idl

The curvature condition for self-consistent scale-free galaxies

We modify the curvature condition for the existence of self-consistent scale-free discs, introduced by Zhao, Carollo & de Zeeuw. We survey the parameter space of the power-law discs, and show that the modified curvature condition is in harmony with the results of Schwarzschild's numerical orbit superposition method. We study the orbital structure of the power-law discs, and find a correlation between the population of centrophobic banana orbits and the non-self-consistency index. We apply the curvature condition to other families of scale-free elongated discs and find that it rules out a large range of power-law slopes and axis ratios. We generalize the condition, and apply it, to three-dimensional scale-free axisymmetric galaxy models.Comment: 14 pages, 14 figures, Accepted for publication in MNRA

Self-consistent axisymmetric Sridhar-Touma models

We construct phase-space distribution functions for the oblate, cuspy mass models of Sridhar & Touma, which may contain a central point mass (black hole) and have potentials of St\"ackel form in parabolic coordinates. The density in the ST models is proportional to a power $r^{-\gamma}$ of the radius, with $0<\gamma<1$. We derive distribution functions $f(E, L_z)$ for the scale-free ST models (no black hole) using a power series of the energy $E$ and the component $L_z$ of the angular momentum parallel to the symmetry axis. We use the contour integral method of Hunter & Qian to construct $f(E, L_z)$ for ST models with central black holes, and employ the scheme introduced by Dejonghe & de Zeeuw to derive more general distribution functions which depend on $E$, $L_z$ and the exact third integral $I_3$. We find that self-consistent two- and three-integral distribution functions exist for all values $0 < \gamma < 1$.Comment: 10 pages, 11 Figures, Accepted for publication in MNRA

Scale-free dynamical models for galaxies: flattened densities in spherical potentials

This paper presents two families of phase-space distribution functions (DFs) that generate scale-free spheroidal mass densities in scale-free spherical potentials. The `case I' DFs are anisotropic generalizations of the flattened f(E,L_z) model, which they include as a special case. The `case II' DFs generate flattened constant-anisotropy models, in which the constant ratio of rms tangential to radial motion is characterized by Binney's parameter beta. The models can describe the outer parts of galaxies and the density cusp structure near a central black hole, but also provide general insight into the dynamical properties of flattened systems. The dependence of the intrinsic and projected properties on the model parameters and the inclination is described. The observed ratio of the rms projected line-of-sight velocities on the projected major and minor axes of elliptical galaxies is best fit by the case II models with beta > 0. These models also predict non-Gaussian velocity profile shapes consistent with existing observations. The distribution functions are used to model the galaxies NGC 2434 (E1) and NGC 3706 (E4), for which stellar kinematical measurements out to two effective radii indicate the presence of dark halos (Carollo et al.). The velocity profile shapes of both galaxies can be well fit by radially anisotropic case II models with a spherical logarithmic potential. This contrasts with the f(E,L_z) models studied previously, which require flattened dark halos to fit the data.Comment: LaTeX file, uses standard macros mn.sty, epsf.sty. 17 pages including 7 figure

Separable triaxial potential-density pairs in MOND

We study mass models that correspond to MOND (triaxial) potentials for which the Hamilton-Jacobi equation separates in ellipsoidal coordinates. The problem is first discussed in the simpler case of deep-MOND systems, and then generalized to the full MOND regime. We prove that the Kuzmin property for Newtonian gravity still holds, i.e., that the density distribution of separable potentials is fully determined once the density profile along the minor axis is assigned. At variance with the Newtonian case, the fact that a positive density along the minor axis leads to a positive density everywhere remains unproven. We also prove that (i) all regular separable models in MOND have a vanishing density at the origin, so that they would correspond to centrally dark-matter dominated systems in Newtonian gravity; (ii) triaxial separable potentials regular at large radii and associated with finite total mass leads to density distributions that at large radii are not spherical and decline as ln(r)/r^5; (iii) when the triaxial potentials admit a genuine Frobenius expansion with exponent 0<epsilon<1, the density distributions become spherical at large radii, with the profile ln(r)/r^(3+2epsilon). After presenting a suite of positive density distributions associated with MOND separable potentials, we also consider the important family of (non-separable) triaxial potentials V_1 introduced by de Zeeuw and Pfenniger, and we show that, as already known for Newtonian gravity, they obey the Kuzmin property also in MOND. The ordinary differential equation relating their potential and density along the z-axis is an Abel equation of the second kind that, in the oblate case, can be explicitly reduced to canonical form.Comment: 17 pages, 4 figures (low resolution), accepted by MNRA

The Planetary Nebulae Population in the Nuclear Regions of M31: the SAURON view

Following a first study of the central regions of M32 that illustrated the power of integral-field spectroscopy (IFS) in detecting and measuring the [O III]{\lambda}5007 emission of PNe against a strong stellar background, we turn to the very nuclear PN population of M31, within 80 pc of its centre. We show that PNe can also be found in the presence of emission from diffuse gas and further illustrate the excellent sensitivity of IFS in detecting extragalactic PNe through a comparison with narrowband images obtained with the Hubble Space Telescope. We find the nuclear PNe population of M31 is only marginally consistent with the generally adopted form of the PNe luminosity function (PNLF). In particular, this is due to a lack of PNe with absolute magnitude M5007 brighter than -3, which would only result from a rather unfortunate draw from such a model PNLF. We suggest that the observed lack of bright PNe in the nuclear regions of M31 is due to a horizontal-branch population that is more tilted toward less massive and hotter He-burning stars, so that its progeny consists mostly of UV-bright stars that fail to climb back up the asymptotic giant branch (AGB) and only of few, if any, bright PNe powered by central post-AGB stars. These results are also consistent with recent reports on a dearth of bright post-AGB stars towards the nucleus of M31, and lend further support to the idea that the metallicity of a stellar population has an impact on the way the horizontal branch is populated and to the loose anticorrelation between the strength of the UV-upturn and the specific number of PNe that is observed in early-type galaxies. Finally, our investigation also serves to stress the importance of considering the same spatial scales when comparing the PNe population of galaxies with the properties of their stellar populations.Comment: 11 pages, 10 figures, accepted for publication on Monthly Notices of the Royal Astronomical Societ

Dynamical modelling of stars and gas in NGC2974: determination of mass-to-light ratio, inclination and orbital structure by Schwarzschild's method

We study the large-scale stellar and gaseous kinematics of the E4 galaxy NGC2974, based on panoramic integral-field data obtained with SAURON. We quantify the velocity fields with Fourier methods (kinemetry), and show that the large-scale kinematics is largely consistent with axisymmetry. We construct general axisymmetric dynamical models for the stellar motions using Schwarzschild's orbit-superposition method, and compare the inferred inclination and mass-to-light ratio with the values obtained by modelling the gas kinematics. Both approaches give consistent results. However we find that the stellar models provide fairly weak constraints on the inclination. The intrinsic orbital distribution of NGC2974, which we infer from our model, is characterised by a large-scale stellar component of high angular momentum. We create semi-analytic test models, resembling NGC2974, to study the ability of Schwarzschild's modelling technique to recover the given input parameters (mass-to-light ratio and inclination) and the distribution function. We also test the influence of a limited spatial coverage on the recovery of the distribution function (i.e. the orbital structure). We find that the models can accurately recover the input mass-to-light ratio, but we confirm that even with perfect input kinematics the inclination is only marginally constrained. This suggests a possible degeneracy in the determination of the inclination, but further investigations are needed to clarify this issue. For a given potential, we find that the analytic distribution function of our test model is well recovered by the three-integral model within the spatial region constrained by integral-field kinematics.Comment: 22 pages, 24 figures. Accepted for publication in MNRAS. Version with full resolution images available at http://www.strw.leidenuniv.nl/sauron/papers/krajnovic2004_ngc2974.pd