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We present a comprehensive description of the population synthesis code StarTrack. The original code has been significantly modified and updated. Special emphasis is placed here on processes leading to the formation and further evolution of compact objects (white dwarfs, neutron stars, and black holes). Both single and binary star populations are considered. The code now incorporates detailed calculations of all mass transfer phases, a full implementation of orbital evolution due to tides, as well as the most recent estimates of magnetic braking. This updated version of StarTrack can be used for a wide variety of problems, with relevance to observations with many current and planned observatories, e.g., studies of X-ray binaries (Chandra, XMM-Newton), gravitational radiation sources (LIGO, LISA), and gamma-ray burst progenitors (HETE-II, Swift). The code has already been used in studies of Galactic and extragalactic X-ray binary populations, black holes in young star clusters, Type Ia supernova progenitors, and double compact object populations. Here we describe in detail the input physics, we present the code calibration and tests, and we outline our current studies in the context of X-ray binary populations.

Observations in the past decade have revealed extrasolar planets with a wide range of orbital semimajor axes and eccentricities. Based on the present understanding of planet formation via core accretion and oligarchic growth, we expect that giant planets often form in closely packed configurations. While the protoplanets are embedded in a protoplanetary gas disk, dissipation can prevent eccentricity growth and suppress instabilities from becoming manifest. However, once the disk dissipates, eccentricities can grow rapidly, leading to close encounters between planets. Strong planet-planet gravitational scattering could produce both high eccentricities and, after tidal circularization, very short-period planets, as observed in the exoplanet population. We present new results for this scenario based on extensive dynamical integrations of systems containing three giant planets, both with and without residual gas disks. We assign the initial planetary masses and orbits in a realistic manner following the core accretion model of planet formation. We show that, with realistic initial conditions, planet-planet scattering can reproduce quite well the observed eccentricity distribution. Our results also make testable predictions for the orbital inclinations of short-period giant planets formed via strong planet scattering followed by tidal circularization.

The existence of a dominant massive planet, Jupiter, in our solar system, although perhaps essential for long-term dynamical stability and the development of life, may not be typical of planetary systems that form around other stars. In a system containing two Jupiter-like planets, the possibility exists that a dynamical instability will develop. Computer simulations suggest that in many cases this instability leads to the ejection of one planet while the other is left in a smaller, eccentric orbit. In extreme cases, the eccentric orbit has a small enough periastron distance that it may circularize at an orbital period as short as a few days through tidal dissipation. This may explain the recently detected Jupiter-mass planets in very tight circular orbits and wider eccentric orbits around nearby stars.

About 25 per cent of hot Jupiters (extrasolar Jovian-mass planets with close-in orbits) are actually orbiting counter to the spin direction of the star 1 . Perturbations from a distant binary star companion 2, 3 can produce high inclinations, but cannot explain orbits that are retrograde with respect to the total angular momentum of the system. Such orbits in a stellar context can be produced through secular (that is, long term) perturbations in hierarchical triple-star systems. Here we report a similar application to planetary bodies, including both the key octupole-order effects and tidal friction, and find that it can produce hot Jupiters in orbits that are retrograde with respect to the total angular momentum. With distant stellar mass perturbers such an outcome is not possible 2, 3 . With planetary perturbers the inner orbit's angular momentum component parallel to the total angular momentum need not be constant 4 . In fact, as we show here, it can even change sign, leading to a retrograde orbit. A brief excursion to very high eccentricity during the chaotic evolution of the inner orbit can then lead to rapid capture, forming a retrograde hot Jupiter.Despite many attempts 2,3,[5][6][7][8][9][10][11] , there is no model that can account for all the properties of the known hot Jupiter (HJ) systems. One model suggests that HJs formed far away from the star and slowly spiraled in, losing angular momentum and orbital energy to the protoplanetary disk 12, 13 . This "migration" process should produce planets with low orbital inclinations and eccentricities. However, many HJs are observed to be on orbits with high eccentricities, and misaligned with the spin direction of the star (as measured through the Rossiter-McLaughlin effect 14 ) and some of these (8 out of 32) even appear to be orbiting counter to the spin of the star. In a second model, secular perturbations from a distant binary star companion can produce increases in the eccentricity and inclination of a planetary orbit 15 . During the evolution to high eccentricity, tidal dissipation near pericenter can force the planet's orbit to decay, potentially forming a misaligned HJ 2, 3 . Recently, secular chaos involving several planets has also been proposed as a way to form HJs on eccentric and misaligned orbits 11 . A different class of models to produce a tilted orbit is via planet-planet scattering 5 , possibly combined with other perturbers and tidal friction 7 . In such models the initial configuration is a densly-packed system of planets and the final tilted orbit is a result of dynamical scattering among the planets, in contrast to the secular interactions we study here.In our general treatment of secular interactions between two orbiting bodies we allow for the magnitude and orientation of both orbital angular momenta to change (see Figure 1). The outer body (here either a planet or a brown-dwarf) gravitationally perturbs the inner planet on time scales long compared to the orbital period (i.e., we consider the secular evolution of the 1 system). We de...

We consider the formation of binary black hole mergers through the evolution of field massive triple stars. In this scenario, favorable conditions for the inspiral of a black hole binary are initiated by its gravitational interaction with a distant companion, rather than by a common-envelope phase invoked in standard binary evolution models. We use a code that follows self-consistently the evolution of massive triple stars, combining the secular triple dynamics (Lidov-Kozai cycles) with stellar evolution. After a black hole triple is formed, its dynamical evolution is computed using either the orbit-averaged equations of motion, or a high-precision direct integrator for triples with weaker hierarchies for which the secular perturbation theory breaks down. Most black hole mergers in our models are produced in the latter non-secular dynamical regime. We derive the properties of the merging binaries and compute a black hole merger rate in the range (0.3 − 1.3) Gpc −3 yr −1 , or up to ≈ 2.5 Gpc −3 yr −1 if the black hole orbital planes have initially random orientation. Finally, we show that black hole mergers from the triple channel have significantly higher eccentricities than those formed through the evolution of massive binaries or in dense star clusters. Measured eccentricities could therefore be used to uniquely identify binary mergers formed through the evolution of triple stars. While our results suggest up to ≈ 10 detections per year with Advanced-LIGO, the high eccentricities could render the merging binaries harder to detect with planned space based interferometers such as LISA.

We derive octupole-level secular perturbation equations for hierarchical triple systems, using classical Hamiltonian perturbation techniques. Our equations describe the secular evolution of the orbital eccentricities and inclinations over timescales long compared to the orbital periods. By extending previous work done to leading (quadrupole) order to octupole level (i.e., including terms of order α 3 , where α ≡ a 1 /a 2 < 1 is the ratio of semimajor axes) we obtain expressions that are applicable to a much wider range of parameters. In particular, our results can be applied to high-inclination as well as coplanar systems, and our expressions are valid for almost all mass ratios for which the system is in a stable hierarchical configuration. In contrast, the standard quadrupole-level theory of Kozai gives a vanishing result in the limit of zero relative inclination. The classical planetary perturbation theory, while valid to all orders in α, applies only to orbits of low-mass objects orbiting a common central mass, with low eccentricities and low relative inclination. For triple systems containing a close inner binary, we also discuss the possible interaction between the classical Newtonian perturbations and the general relativistic precession of the inner orbit. In some cases we show that this interaction can lead to resonances and a significant increase in the maximum amplitude of eccentricity perturbations. We establish the validity of our analytic expressions by providing detailed comparisons with the results of direct numerical integrations of the three-body problem obtained for a large number of representative cases. In addition, we show that our expressions reduce correctly to previously published analytic results obtained in various limiting regimes. We also discuss applications of the theory in the context of several observed triple systems of current interest, including the millisecond pulsar PSR B1620−26 in M4, the giant planet in 16 Cygni, and the protostellar binary TMR-1.

Physical collisions between stars occur frequently in dense star clusters, either via close encounters between two single stars, or during strong dynamical interactions involving binary stars. Here we study stellar collisions that occur during binary–single and binary–binary interactions, by performing numerical scattering experiments. Our results include cross‐sections, branching ratios and sample distributions of parameters for various outcomes. For interactions of hard binaries containing main‐sequence stars, we find that the normalized cross‐section for at least one collision to occur (between any two of the four stars involved) is essentially unity, and that the probability of collisions involving more than two stars is significant. Hydrodynamic calculations have shown that the effective radius of a collision product can be 2–30 times larger than the normal main‐sequence radius for a star of the same total mass. We study the effect of this expansion, and find that it increases the probability of further collisions considerably. We discuss these results in the context of recent observations of blue stragglers in globular clusters with masses exceeding twice the main‐sequence turn‐off mass. We also present Fewbody, a new, freely available numerical toolkit for simulating small‐N gravitational dynamics that is particularly suited to performing scattering experiments.

We study the early dynamical evolution of young, dense star clusters using Monte Carlo simulations for systems with up to N = 10 7 stars. Rapid mass segregation of massive main-sequence stars and the development of the Spitzer instability can drive these systems to core collapse in a small fraction of the initial half-mass relaxation time. If the core collapse time is less than the lifetime of the massive stars, all stars in the collapsing core may then undergo a runaway collision process leading to the formation of a massive black hole. Here we study in detail the first step in this process, up to the occurrence of core collapse. We have performed about 100 simulations for clusters with a wide variety of initial conditions, varying systematically the cluster density profile, stellar IMF, and number of stars. We also considered the effects of initial mass segregation and stellar evolution mass loss. Our results show that, for clusters with a moderate initial central concentration and any realistic IMF, the ratio of core collapse time to initial half-mass relaxation time is typically ∼ 0.1, in agreement with the value previously found by direct N -body simulations for much smaller systems. Models with even higher central concentration initially, or with initial mass segregation (from star formation) have even shorter core-collapse times. Remarkably, we find that, for all realistic initial conditions, the mass of the collapsing core is always close to ∼ 10 −3 of the total cluster mass, very similar to the observed correlation between central black hole mass and total cluster mass in a variety of environments. We discuss the implications of our results for the formation of intermediate-mass black holes in globular clusters and super star clusters, ultraluminous X-ray sources, and seed black holes in proto-galactic nuclei.

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