On the distribution of stellar remnants around massive black holes: slow mass segregation, star cluster inspirals and correlated orbits

We study the long term dynamical evolution of stellar mass black holes (BHs) at the Galactic center (GC) and put constraints on their number and central mass distribution. Models of the GC are considered that have not yet achieved a steady state under the influence of random gravitational encounters. Contrary to some recent claims that mass-segregation can rapidly rebuild a density cusp in the stars, we find that time scales associated with cusp regrowth are longer than the Hubble time. These results cast doubts on standard models that postulate high densities of BHs near the GC and motivate studies that start from initial conditions which correspond to well-defined physical models. For the first time, we consider the distribution of BHs in a dissipationless formation model for the Milky Way nuclear cluster (NC), in which massive stellar clusters merge in the GC to form a nucleus. We simulate the successive inspiral of massive clusters containing an inner dense cluster of BHs. The pre-existing mass segregation is not completely erased as the clusters are disrupted by the massive black hole tidal field. As a result, after 12 inspiral events a NC forms in which the BHs have higher central densities than the stars. After evolving the model for 5-10 Gyr, the BHs do form a steep central cusp, while the stellar distribution maintains properties that resemble those of the Milky Way NC. Finally, we investigate the effect of BH perturbations on the motion of the GC S-stars, as a means of constraining the number of the perturbers. We find that reproducing the S-star orbital distribution requires >~1000 BHs within 0.1 pc of Sgr A*. A dissipationless formation scenario for the Milky Way NC is consistent with this lower limit and therefore could reconcile the need for high central densities of BHs (to explain the orbits of the S-stars), with the missing-cusp problem of the GC giant star population.Comment: 23 pages, 21 Figures. Accepted for publication in Ap

Generalized Dirac operators on Lorentzian manifolds and propagation of singularities

We survey the correct definition of a generalized Dirac operator on a Space--Time and the classical result about propagation of singularities. This says that light travels along light--like geodesics. Finally we show this is also true for generalized Dirac operators

The Calderon projection over C* algebras

We construct the Calderon projection on the space of Cauchy datas for a twisted Dirac operator in the Mischenko--Fomenko pseudodifferential calculus for operators acting on bundles of finitely generated $C^*$--Hilbert modules on a compact manifold with boundary. In particular an invertible double is constructed generalizing the classical result

Dynamical friction and the evolution of Supermassive Black hole Binaries: the final hundred-parsec problem

The supermassive black holes originally in the nuclei of two merging galaxies will form a binary in the remnant core. The early evolution of the massive binary is driven by dynamical friction before the binary becomes "hard" and eventually reaches coalescence through gravitational wave emission. { We consider the dynamical friction evolution of massive binaries consisting of a secondary hole orbiting inside a stellar cusp dominated by a more massive central black hole.} In our treatment we include the frictional force from stars moving faster than the inspiralling object which is neglected in the standard Chandrasekhar's treatment. We show that the binary eccentricity increases if the stellar cusp density profile rises less steeply than $\rho\propto r^{-2}$. In cusps shallower than $\rho\propto r^{-1}$ the frictional timescale can become very long due to the deficit of stars moving slower than the massive body. Although including the fast stars increases the decay rate, low mass-ratio binaries ($q\lesssim 10^{-3}$) in sufficiently massive galaxies have decay timescales longer than one Hubble time. During such minor mergers the secondary hole stalls on an eccentric orbit at a distance of order one tenth the influence radius of the primary hole (i.e., $\approx 10-100\rm pc$ for massive ellipticals). We calculate the expected number of stalled satellites as a function of the host galaxy mass, and show that the brightest cluster galaxies should have $\gtrsim 1$ of such satellites orbiting within their cores. Our results could provide an explanation to a number of observations, which include multiple nuclei in core ellipticals, off-center AGNs and eccentric nuclear disks.Comment: 18 pages, 13 Figures. Accepted for publication in Ap

Massive binary star mergers in galactic nuclei: implications for blue stragglers, binary S-stars and gravitational waves

Galactic nuclei are often found to contain young stellar populations and, in most cases, a central supermassive black hole (SMBH). Most known massive stars are found in binaries or higher-multiplicity systems, and in a galactic nucleus the gravitational interaction with the SMBH can affect their long-term evolution. In this paper, we study the orbital evolution of stellar binaries near SMBHs using high precision $N$-body simulations, and including tidal forces and Post-Newtonian corrections to the motion. We focus on the Lidov-Kozai (LK) effect induced by the SMBH on massive star binaries. We investigate how the properties of the merging binaries change with varying the SMBH mass, the slope of the initial mass function, the distributions of the binary orbital parameters and the efficiency in energy dissipation in dissipative tides. We find that the fraction of merging massive binary stars is in the range $\sim 4\%$--$15\%$ regardless of the details of the initial distributions of masses and orbital elements. For a Milky Way-like nucleus, we find a typical rate of binary mergers $\Gamma\approx 1.4\times 10^{-7}$ yr$^{-1}$. The merger products of massive binaries can be rejuvenated blue-straggler stars, more massive than each of their original progenitors, and G2-like objects. Binary systems that survive the LK cycles can be source of X-rays and gravitational waves, observable with present and upcoming instruments.Comment: 13 pages, 7 figures, 1 table, accepted by MNRA

From conjecture generation by maintaining dragging to proof

In this paper we propose a hypothesis about how different uses of maintaining dragging, either as a physical tool in a dynamic geometry environment or as a psychological tool for generating conjectures can influence subsequent processes of proving. Through two examples we support the hypothesis that using maintaining dragging as a physical tool may foster cognitive rupture between the conjecturing phase and the proof, while using it as a psychological tool may foster cognitive unity between them.Comment: Research report at the 40th PME conference, Hungar

Maintaining dragging and the pivot invariant in processes of conjecture generation

In this paper, we analyze processes of conjecture generation in the context of open problems proposed in a dynamic geometry environment, when a particular dragging modality, maintaining dragging, is used. This involves dragging points while maintaining certain properties, controlling the movement of the figures. Our results suggest that the pragmatic need of physically controlling the simultaneous movements of the different parts of figures can foster the production of two chains of successive properties, hinged together by an invariant that we will call pivot invariant. Moreover, we show how the production of these chains is tied to the production of conjectures and to the processes of argumentation through which they are generated.Comment: Research report at the 40th PME Conference, Hungar