A. Just

Supermassive Black Hole Binary Evolution in Axisymmetric Galaxies: The Final Parsec Problem is Not a Problem

During a galaxy merger, the supermassive black hole (SMBH) in each galaxy is thought to sink to the center of the potential and form an SMBH binary; this binary can eject stars via three-body scattering, bringing the SMBHs ever closer. In a static spherical galaxy model, the binary stalls at a separation of about a parsec after ejecting all the stars in its loss cone—this is the well-known final parsec problem. Earlier work has shown that the centrophilic orbits in triaxial galaxy models are key in refilling the loss cone at a high enough rate to prevent the black holes from stalling. However, the evolution of binary SMBHs has never been explored in axisymmetric galaxies, so it is not clear if the final parsec problem persists in these systems. Here we use a suite of direct N-body simulations to follow SMBH binary evolution in galaxy models with a range of ellipticity. For the first time, we show that mere axisymmetry can solve the final parsec problem; we find the SMBH evolution is independent of N for an axis ratio of c/a = 0.8, and that the SMBH binary separation reaches the gravitational radiation regime for c/a = 0.75.

The properties of the local spiral arms from RAVE data: two-dimensional density wave approach

Using the Radial Velocity Experiment (RAVE) survey, we recently brought to light a gradient in the mean galactocentric radial velocity of stars in the extended solar neighbourhood. This gradient likely originates from non-axisymmetric perturbations of the potential, among which a perturbation by spiral arms is a possible explanation. Here, we apply the traditional density wave theory and analytically model the radial component of the two-dimensional velocity field. Provided that the radial velocity gradient is caused by relatively long-lived spiral arms that can affect stars substantially above the plane, this analytic model provides new independent estimates for the parameters of the Milky Way spiral structure. Our analysis favours a two-armed perturbation with the Sun close to the inner ultra-harmonic 4:1 resonance, with a pattern speed Ωp=18.6-0.2+0.3 km s-1 kpc-1 and a small amplitude A=0.55-0.02+0.02 per cent of the background potential (14 per cent of the background density). This model can serve as a basis for numerical simulations in three dimensions, additionally including a possible influence of the Galactic bar and/or other non-axisymmetric modes.

Towards a fully consistent Milky Way disc model – I. The local model based on kinematic and photometric data

We present a fully consistent evolutionary disc model of the solar cylinder. The model is based on a sequence of stellar subpopulations described by the star formation history (SFR) and the dynamical heating law [given by the age-velocity dispersion relation (AVR)]. The stellar subpopulations are in dynamical equilibrium and the gravitational potential is calculated self-consistently including the influence of the dark matter halo and the gas component. The combination of kinematic data from Hipparcos and the finite lifetimes of main-sequence (MS) stars enables us to determine the detailed vertical disc structure independent of individual stellar ages and only weakly dependent on the initial mass function (IMF). The disc parameters are determined by applying a sophisticated best-fitting algorithm to the MS star velocity distribution functions in magnitude bins. We find that the AVR is well constrained by the local kinematics, whereas for the SFR the allowed range is larger. The model is consistent with the local kinematics of MS stars and fulfils the known constraints on scaleheights, surface densities and mass ratios. A simple chemical enrichment model is included in order to fit the local metallicity distribution of G dwarfs. In our favoured Model A, the power-law index of the AVR is 0.375 with a minimum and maximum velocity dispersion of 5.1 and 25.0 km s -1 , respectively. The SFR shows a maximum 10 Gyr ago and declines by a factor of four to the present-day value of 1.5 M ⊙ pc -2 Gyr -1 . A best fit of the IMF leads to power-law indices of -1.46 below and -4.16 above 1.72M ⊙ avoiding a kink at 1 M ⊙ . An isothermal thick-disc component with local density of ~6 per cent of the stellar density is included. A thick disc containing more than 10 per cent of local stellar mass is inconsistent with the local kinematics of K and M dwarfs. Neglecting the thick-disc component results in slight variations of the thin-disc properties, but has a negligible influence on the AVR and the normalized SFR. The model allows detailed predictions of the density, age, metallicity and velocity distribution functions of MS stars as a function of height above the mid-plane. The complexity of the model does not allow to rule out other star formation scenarios using the local data alone. The incorporation of multiband star count and kinematic data of larger samples in the near future will improve the determination of the disc structure and evolution significantly.

Large scale inhomogeneity and local dynamical friction

We investigate the effect of a density gradient on Chandrasekhars dynamical friction formula based on the method of 2-body encounters in the local approximation. We apply these generalizations to the orbit evolution of satellite galaxies in Dark Matter haloes. We find from the analysis that the main influence occurs through a position-dependent maximum impact parameter in the Coulomb logarithm, which is determined by the local scale-length of the density distribution. We also show that for eccentric orbits the explicit dependence of the Coulomb logarithm on position yields significant differences for the standard homogeneous force. Including the velocity dependence of the Coulomb logarithm yields ambigous results. The orbital fits in the first few periods are further improved, but the deviations at later times are much larger. The additional force induced by the density gradient, the inhomogeneous force, is not antiparallel to the satellite motion and can exceed 10% of the homogeneous friction force in magnitude. However, due to the symmetry properties of the inhomogeneous force, there is a deformation and no secular effect on the orbit at the first order. Therefore the inhomogeneous force can be safely neglected for the orbital evolution of satellite galaxies. For the homogeneous force we compare numerical N-body calculations with semi-analytical orbits to determine quantitatively the accuracy of the generalized formulae of the Coulomb logarithm in the Chandrasekhar approach. With the local scale-length as the maximum impact parameter we find a significant improvement of the orbital fits and a better interpretation of the quantitative value of the Coulomb logarithm.

Quantitative analysis of clumps in the tidal tails of star clusters

Tidal tails of star clusters are not homogeneous but show well-defined clumps in observations as well as in numerical simulations. Recently, an epicyclic theory for the formation of these clumps was presented. A quantitative analysis was still missing. We present a quantitative derivation of the angular momentum and energy distribution of escaping stars from a star cluster in the tidal field of the Milky Way and derive the connection to the position and width of the clumps. For the numerical realization we use star-by-star N-body simulations. We find a very good agreement of theory and models. We show that the radial offset of the tidal arms scales with the tidal radius, which is a function of cluster mass and the rotation curve at the cluster orbit. The mean radial offset is 2.77 times the tidal radius in the outer disc. Near the Galactic Centre the circumstances are more complicated, but to lowest order the theory still applies. We have also measured the Jacobi energy distribution of bound stars and showed that there is a large fraction of stars (about 35 per cent) above the critical Jacobi energy at all times, which can potentially leave the cluster. This is a hint that the mass loss is dominated by a self-regulating process of increasing Jacobi energy due to the weakening of the potential well of the star cluster, which is induced by the mass loss itself.

Dynamical friction of massive objects in galactic centres

Dynamical friction leads to an orbital decay of massive objects like young compact star clusters or massive black holes in central regions of galaxies. The dynamical friction force can be well approximated by Chandrasekhars standard formula, but recent investigations show that corrections to the Coulomb logarithm are necessary. With a large set of N-body simulations we show that the improved formula for the Coulomb logarithm fits the orbital decay very well for circular and eccentric orbits. The local scalelength of the background density distribution serves as the maximum impact parameter for a wide range of power-law indices of -1 ... -5. For each type of code the numerical resolution must be compared to the effective minimum impact parameter in order to determine the Coulomb logarithm. We also quantify the correction factors by using self-consistent velocity distribution functions instead of the standard Maxwellian often used. These factors enter directly the decay time-scale and cover a range of 0.5 ... 3 for typical orbits. The new Coulomb logarithm combined with self-consistent velocity distribution functions in the Chandrasekhar formula provides a significant improvement of orbital decay times with correction up to one order of magnitude compared to the standard case. We suggest the general use of the improved formula in parameter studies as well as in special applications.

Dynamical friction in flattened systems: a numerical test of Binney's approach

We carry out a set of self-consistent N-body calculations to investigate how important the velocity anisotropy in non-spherical dark matter haloes is for dynamical friction. For this purpose, we allow satellite galaxies to orbit within flattened and live dark matter haloes (DMHs) and compare the resulting orbit evolution with a semi-analytic code. This code solves the equation of motion of the same satellite orbits with mass loss and assumes the same DMH, but either employs Chandrasekhars dynamical friction formula, which does not incorporate the velocity anisotropy, or Binneys description of dynamical friction in anisotropic systems. In the numerical and the two semi-analytic models, the satellites are given different initial orbital inclinations and orbital eccentricities, whereas the parent galaxy is composed of a DMH with aspect ratio qh= 0.6. We find that Binneys approach successfully describes the overall satellite decay and orbital inclination decrease for the whole set of orbits, with an averaged discrepancy of less than 4 per cent in orbital radius during the first three orbits. If Chandrasekhars expression is used instead, the discrepancy increases to 20 per cent. Binneys treatment therefore appears to provide a significantly improved treatment of dynamical friction in anisotropic systems. The velocity anisotropy of the DMH velocity distribution function leads to a significant decrease with time of the inclination of non-polar satellite orbits. But, at the same time, it reduces the difference in decay times between polar and coplanar orbits evident in a flattened DMH when the anisotropic DMH velocity distribution function is not taken into account explicitly. Our N-body calculations furthermore indicate that polar orbits survive about 1.6 times longer than coplanar orbits and that the orbital eccentricity e remains close to its initial value if satellites decay slowly towards the galaxy centre. However, orbits of rapidly decaying satellites modelled with the semi-analytic code show a strong orbital circularization () not present in the N-body computations.

Chemical gradients in the Milky Way from the RAVE data

Aims. We provide new constraints on the chemo-dynamical models of the Milky Way by measuring the radial and vertical chemical gradients for the elements Mg, Al, Si, Ti, and Fe in the Galactic disc and the gradient variations as a function of the distance from the Galactic plane (Z). Methods. We selected a sample of giant stars from the RAVE database using the gravity criterium 1.7 < log g< 2.8. We created a RAVE mock sample with the Galaxia code based on the Besancon model and selected a corresponding mock sample to compare the model with the observed data. We measured the radial gradients and the vertical gradients as a function of the distance from the Galactic plane Z to study their variation across the Galactic disc. Results. The RAVE sample exhibits a negative radial gradient of d[Fe/H]/dR = −0.054 dex kpc −1 close to the Galactic plane (|Z| < 0.4 kpc) that becomes flatter for larger |Z|. Other elements follow the same trend although with some variations from element to element. The mock sample has radial gradients in fair agreement with the observed data. The variation of the gradients with Z shows that the Fe radial gradient of the RAVE sample has little change in the range |Z| 0.6 kpc and then flattens. The iron vertical gradient of the RAVE sample is slightly negative close to the Galactic plane and steepens with |Z|. The mock sample exhibits an iron vertical gradient that is always steeper than the RAVE sample. The mock sample also shows an excess of metal-poor stars in the [Fe/H] distributions with respect to the observed data. These discrepancies can be reduced by decreasing the number of thick disc stars and increasing their average metallicity in the Besancon model.

N‐body models of rotating globular clusters

In this paper we examine the dynamical evolution of rotating globular clusters with direct N-body models. Our initial models are rotating King models, and we obtain results both for equal-mass systems and for systems composed of two mass components. Previous investigations using a Fokker–Planck solver have shown that rotation has a noticeable influence on stellar systems such as globular clusters that evolve by two-body relaxation. In particular, it accelerates their dynamical evolution through the gravogyro instability. We have validated the occurrence of the gravogyro instability with direct N-body models. In the case of systems composed of two mass components, mass segregation takes place, a process that competes with the rotation in the acceleration of the core collapse. The ‘accelerating’ effect of rotation was detected in our isolated two-mass N-body models. Finally, we look at rotating N-body models in a tidal field within the tidal approximation. It turns out that rotation increases the escape rate significantly. A difference between retrograde- and prograde-rotating stellar clusters, with respect to the orbit of the cluster around the Galaxy, occurs. This difference is the result of the presence of a ‘third integral’ and chaotic scattering, respectively.

Chemical gradients in the Milky Way from the RAVE data : II. Giant stars

Aims. We aim at measuring the chemical gradients of the elements Mg, Al, Si, and Fe along the Galactic radius to provide new constraints on the chemical evolution models of the Galaxy and Galaxy models such as the Besancon model. Thanks to the large number of stars of our RAVE sample we can study how the gradients vary as function of the distance from the Galactic plane. Methods. We analysed three different samples selected from three independent datasets: a sample of 19 962 dwarf stars selected from the RAVE database, a sample of 10 616 dwarf stars selected from the Geneva-Copenhagen Survey (GCS) dataset, and a mock sample (equivalent to the RAVE sample) created by using the GALAXIA code, which is based on the Besancon model. The three samples were analysed by using the very same method for comparison purposes. We integrated the Galactic orbits and obtained the guiding radii (Rg) and the maximum distances from the Galactic plane reached by the stars along their orbits (Zmax). We measured the chemical gradients as functions of Rg at different Zmax. Results. We found that the chemical gradients of the RAVE and GCS samples are negative and show consistent trends, although they are not equal: at Zmax < 0.4 kpc and 4.5 < Rg(kpc) < 9.5, the iron gradient for the RAVE sample is d[Fe/H]/dRg = −0.065 dex kpc −1 , whereas for the GCS sample it is d[Fe/H]/dRg = −0.043 dex kpc −1 with internal errors of ±0.002 and ±0.004 dex kpc −1 , respectively. The gradients of the RAVE and GCS samples become flatter at larger Zmax. Conversely, the mock sample has a positive iron gradient of d[Fe/H]/dRg =+ 0.053 ± 0.003 dex kpc −1 at Zmax < 0.4 kpc and remains positive at any Zmax. These positive and unrealistic values originate from the lack of correlation between metallicity and tangential velocity in the Besancon model. In addition, the low metallicity and asymmetric drift of the thick disc causes a shift of the stars towards lower Rg and metallicity which, together with the thin-disc stars with a higher metallicity and Rg, generates a fictitious positive gradient of the full sample. The flatter gradient at larger Zmax found in the RAVE and the GCS samples may therefore be due to the superposition of thin- and thick-disc stars, which mimicks a flatter or positive gradient. This does not exclude the possibility that the thick disc has no chemical gradient. The discrepancies between the observational samples and the mock sample can be reduced by i) decreasing the density; ii) decreasing the vertical velocity; and iii) increasing the metallicity of the thick disc in the Besancon model.