N-Body models of star clusters

Most star clusters (SCs, both young SCs and globular clusters) are collisional environments: their two-body relaxation timescale is shorter than their lifetime. Thus, dynamical interactions between stars drive the structural evolution of SCs. Particular importance have three-body encounters, that is gravitational encounters between a binary and a star. In fact, each binary has a energy reservoir (its internal energy) that can be exchanged with single stars during each dynamical interaction, altering the energetic budget of the SC. Direct N-Body codes are used to study the structural evolution of SCs, by accounting for both two-body and three-body (or few-body) encounters.

As the dynamical evolution of a SC is deeply connected with the stellar evolution of its members, a number of direct N-body codes also include recipes for stellar evolution. Each particle in the simulation corresponds to a star, in the sense that it is associated not only with a mass, a position and a velocity, but also with a stellar type (main sequence, helium star, giant star, etc), a stellar radius, an effective temperature and luminosity. All these properties are evolved in time according to stellar and/or binary evolution recipes.

The public software environment STARLAB (Portegies Zwart et al. 2001, http://www.sns.ias.edu/~starlab/index.html) includes recipes for both dynamics (through a fourth-order predictor-corrector Hermitian scheme) and stellar evolution. STARLAB is optimized to run on graphics processing units (GPUs), through the CUDA parallel computing architecture. The version of STARLAB which is being developed at OAPd includes accurate metallicity-dependent recipes for stellar evolution, stellar winds and remnant formation (Mapelli et al. 2013; Mapelli & Bressan 2013).

This code is currently used to study: (i) the structural evolution of SCs with different metallicity; (ii) the evolution of exotic binaries in SCs (e.g., X-ray binaries, black hole-black hole binaries, etc.).

(i) Structural evolution of SCs with different metallicity (see Mapelli & Bressan 2013):

The evaporation of stars from the core of a SC removes part of its kinetic energy (Spitzer 1987). Since a SC  has negative heat capacity, this leads to gravothermal instability and to the collapse of the core (Binney & Tremaine 1987). Core collapse in SCs is reversed mostly by the kinetic energy injected by three-body encounters (i.e. close encounters between stars and binaries).
It has long been debated whether mass-loss by stellar winds and/or supernovae (SNe) is efficient in affecting core collapse. In fact, stellar winds and SNe eject mass from a SC, making the central potential well shallower and quenching the onset of gravothermal instability.

The timescale for core collapse in young dense SCs is expected to be of the same order of magnitude as the lifetime of massive stars. Thus, mass-loss by stellar winds and SNe peaks during the epochs of core collapse and post-core collapse. Actually, mass-loss by stellar evolution is expected to delay the core collapse (quenching the gravothermal instability) and/or to reverse more rapidly the core collapse, depending on the interplay between core collapse timescale and massive star lifetime.

Two further ingredients of this scenario are the dependence of mass-loss on stellar metallicity and the formation of stellar remnants. Stellar winds are suppressed at low metallicity (e.g. Kudritzki, Pauldrach & Puls 1987; Vink, de Koter & Lamers 2001). Thus, metal-rich SCs are expected to lose more mass by stellar winds than metal-poor ones.
Massive stars that end their life with mass higher than ~40 Msun are expected to collapse directly into black holes (BHs), with no or faint SN explosion (e.g. Fryer 1999). Massive metal-poor stars lose less mass by stellar winds, and thus are more likely to collapse directly into BHs. This mechanism allows to form BHs with mass higher than 25 Msun (e.g. Mapelli et al. 2009). If retained inside the SC, these BHs become the most massive objects in the SC after a few tens Myr, dominating the energy budget of three-body encounters.
At OAPd, we investigate the importance of these effects through direct N-Body simulations. Figure 1 shows the SIMULATED behaviour of the core radius as a function of time for three different metallicities. The effect of metallicity appears immediately after the collapse, during the first phase of re-expansion: the core radius in metal-rich SCs expands more than in metal-poor SCs, as mass-loss by stellar winds in metal-rich SCs removes more matter from the core potential well.


FIGURE 1: Core radius as a function of time for 3 metallicities: Z=0.01 Zsun (solid red line), Z=0.1 Zsun (dashed black line) and Z=1 Zsun (dotted blue line). Each line in this Figure is the median value of 100 simulated SCs. In the insert: zoom of the first 7.5 Myr.

(ii) Exotic binaries (see Mapelli et al. 2013):
The first data by second-generation ground-based gravitational wave detectors are upcoming: Advanced LIGO and VIRGO will start the scientific runs in 2015-2016. Thus, the first direct detection of gravitational waves from the merger of compact-object binaries might occur in the immediate future.
Despite this, the available theoretical predictions and models to help interpreting these new data present many open issues. Firstly, most of the available predictions for the merger rate do not account for the environment of compact-object binaries. It is established that a large fraction of stars (~80%) form in SCs, i.e. in systems where dynamical interactions are essential. Most predictions for the merger of compact-object binaries consider only isolated field binaries, and miss the contribution of dynamics to the evolution of binaries. Secondly, the mass spectrum of compact objects with stellar origin, and in particular that of BHs is highly uncertain: while it is commonly believed that the mass of a stellar BH does not exceed 20 Msun, recent observations indicate the existence of at least one stellar BH in the Local Group with mass >20 Msun. Dynamical processes and stellar metallicity are two key ingredients to shape the mass spectrum of BHs. Dynamical interactions influence the mass of BHs, as they trigger mass transfer and mergers between stars and between stars and BHs. The metallicity of the progenitor star strongly influences the mass of the remnant, as metal-poor stars lose less mass by stellar winds than metal-rich stars.

At OAPd, we study the formation and evolution of compact-object binaries in young SCs, by means of direct N-body simulations. The results of such simulations are used to study the population of BH-BH, BH-NS, and NS-NS binaries. Software development is done in collaboration with Dr. Alessia Gualandris at the University of Leicester. The implications of the study of compact-object binaries for gravitational wave emission are addressed in the frame of the FIRB project `New perspectives on the violent Universe: unveiling the physics of compact objects with joint observations of gravitational waves and electromagnetic radiation' (2013-2016, PI: Branchesi, University of Urbino; CO-PIs: Mapelli, INAF; Razzano, University of Pisa). A PhD student is currently working on this project and other Master/PhD theses are available on topics related to the FIRB project.



FIGURE 2: Mass of the BHs versus zero age main sequence mass of the progenitor star, as derived by our code. The solid red line: 0.01 Zsun; the dotted black line: 0.1 Zsun; the dot–dashed green line: 0.3 Zsun and the dashed blue line: 1 Zsun.

People: Michela Mapelli, Luca Zampieri

Collaborations: Brunetto Ziosi (PhD student at UniPd), Marica Branchesi (University of Urbino, LIGO/Virgo collaboration), Massimiliano Razzano (University of Pisa), Alessandro Bressan (SISSA/ISAS), Alessia Gualandris (University of Leicester), Emanuele Ripamonti (University of Milano Bicocca), Andrea Possenti and Marta Burgay (INAF-Cagliari).

Recent pubblications: Mapelli et al. (2009, MNRAS, 395, 71); Mapelli et al. (2010, MNRAS, 408, 234);  Mapelli et al. (2011, MNRAS, 416, 1756); Mapelli et al. (2013, MNRAS, in press); Mapelli & Bressan (2013, MNRAS, submitted).


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