Secular Stellar Dynamics near Massive Black Holes Ann-Marie Madigan Secular Stellar Dynamics near Massive Black Holes PROEFSCHRIFT ter verkrijging van de graad van Doctor aan de Universiteit Leiden, op gezag van de Rector Magnificus prof. mr. P. F. van der Heijden, volgens besluit van het College voor Promoties te verdedigen op donderdag 16 februari 2012 klokke 11.15 uur door Ann-Marie Madigan geboren te Dublin in 1982 Promotiecommissie Promotor: Prof. dr. K. H. Kuijken Co-Promotors: Dr. Y. Levin (Monash University) Dr. C. Hopman Overige leden: Prof. dr. A. Ghez (University of California, Los Angeles) Prof. dr. R. Genzel (Max Planck Institute for Extraterrestrial Physics) Prof. dr. S. F. Portegies Zwart Prof. dr. S. Tremaine (Institute for Advanced Studies, Princeton) This research has been supported by a TopTalent fellowship from the Nether- lands Organisation for Scientific Research (NWO). for Ev and Dave ISBN 978-94-6191-145-2 Cover design by Dave Madigan vii Contents Chapter 1. Introduction 1 1.1 Massiveblackholes ......................... 2 1.2 MysteriesattheGalacticcenter. 2 1.3 Stellar dynamics near massive black holes . 4 1.4 Thesissummary ........................... 7 Chapter 2. The Eccentric Disk Instability 11 2.1 Introduction ............................. 12 2.2 Theeccentricityinstability . 12 2.3 N-bodysimulations ......................... 14 2.3.1 Initialconditions . 14 2.4 Results ................................ 15 2.5 ApplicationtoS-stars . 16 2.5.1 Bimodal eccentricity distribution in disk . 18 2.6 Criticaldiscussion . 19 Chapter 3. Secular Stellar Dynamics 23 3.1 Introduction ............................. 24 3.2 N-bodysimulations ......................... 25 3.2.1 Model of Galactic nucleus . 26 3.2.2 Illustrative simulations . 28 3.3 Statistical description of resonant relaxation . ..... 30 3.3.1 The autoregressive moving average model ARMA(1, 1) . 32 3.4 Interpretation of the ARMA parameters and extension ofparameterspace ......................... 35 3.4.1 Non-resonant relaxation (NR): the parameter σ ...... 36 3.4.2 Persistence of coherent torques: the parameter φ ..... 37 3.4.3 Magnitude of resonant relaxation steps: the parameter θ . 39 3.5 Results: ARMA analysis of the N-body simulations . 40 3.6 Monte Carlo simulations and applications . 44 3.6.1 Angularmomentum . 46 3.6.2 Energy............................. 48 3.6.3 Initial conditions and boundary conditions . 48 3.6.4 Timesteps .......................... 48 3.6.5 Treatmentofthelosscone. 48 viii 3.7Results ................................ 49 3.7.1 Evolutiontosteady-state. 49 3.7.2 A depression in the Galactic center . 50 3.7.3 Dynamical evolution of the S-stars . 56 3.8 Summary............................... 62 3.A Description of N-bodycode..................... 65 3.B Energyevolutionandcuspformation . 67 3.C Precession due to power-law stellar cusp . 71 3.D EquationsofARMA(1,1)model . 72 3.D.1Variance............................ 73 3.D.2 Autocorrelationfunction. 73 3.D.3 Variance at coherence time tφ ................ 73 Chapter 4. Build-up of Stars in the Galactic Center 79 4.1 Introduction ............................. 80 4.1.1 Nuclear star cluster formation . 80 4.1.2 Observational constraints from extra-galactic nuclei.... 80 4.1.3 Evidence from the Galactic center . 81 4.2 Numericalmethodsandscenarios . 84 4.3Results ................................ 88 4.3.1 Evolution of orbital eccentricities . 88 4.3.2 The high-eccentricity statistic . 90 4.4 Summary............................... 94 4.A Statistical constraints on eccentricity from the h-statistic . 96 Chapter 5. Secular Dynamical Anti-Friction 103 5.1 Introduction ............................. 104 5.2 Secular dynamical anti-friction . 105 5.2.1 Increase in orbital eccentricity . 109 5.2.2 Comparison withtheoriesintheliterature . 109 5.3 Numerical method and simulations . 110 5.3.1 Secularhorseshoeorbits . 112 5.3.2 Separation of potential into pro- and retro- grade orbits . 113 5.3.3 Increase in orbital eccentricity of IMBH . 114 5.4 Discussion .............................. 115 Nederlandse samenvatting (Dutch Summary) 119 Curriculum Vitae 123 Acknowledgments 125 1 Introduction Stars near massive black holes move on elliptical orbits which precess slowly, exerting persistent gravitational torques on each other. In this the- sis we present four important consequences of these gravitational torques: 1) A new instability which exposes the inherently unstable nature of ec- centric stellar disks in galactic nuclei, 2) A density depression of stars near massive black holes due to enhanced angular momentum relaxation and tidal disruptions, 3) A signature of high-eccentricity orbits for stars formed by Hills’ mechanism at large radii from the massive black hole, and a directly-observable statistic that can highlight these populations, and 4) A new dynamical process which we call "secular dynamical anti-friction" which boosts the orbital eccentricity of hypothesized intermediate-mass black holes as they spiral into massive black holes. Ann-Marie Madigan 2011 2 Introduction Figure 1.1 – Composite image of the Galactic center (2280′′ 840′′) with the Hubble Space Tele- scope (near-infrared, in yellow), Spitzer Space Telescope (infrared,× in red) and Chandra X-ray Ob- servatory (X-ray, in blue/violet). The bright white cluster at lower middle-right is the location of the MBH SgrA*. Credit: X-ray: NASA/CXC/UMass/D. Wang et al.; Optical: NASA/ESA/STScI/D.Wang et al.; IR: NASA/JPL-Caltech/SSC/S.Stolovy. 1.1 Massive black holes ASSIVE black holes (MBHs) are found in the nuclei of most, if not all, Mgalaxies (Ferrarese & Ford 2005). Embedded in dense star clusters, they dominate the gravitational potential out to their dynamical radius where the mass in stars becomes comparable to that of the MBH. If one understands the dynamics of stars within this radius, one can provide a theoretical framework for some of the most energetic phenomena in the universe such as the tidal disruption of stars and launching of relativistic jets. Compact stellar remnants on high-eccentricity orbits within these clusters, such as neutron stars, white dwarfs and stellar-mass black holes, interact with the MBH producing gravita- tional waves potentially detectable by future space observatories. 1.2 Mysteries at the Galactic center In the center of our Galaxy, at a distance of 8 kpc (Eisenhauer et al. 2003; Ghez et al. 2008; Gillessen et al. 2009), there∼ quietly lives a MBH1. Though highly under-luminous with respect to its Eddington luminosity (Melia & Falcke 2001), its presence is inferred by the effect it has on the orbits of stars that live in the surrounding dense nuclear star cluster. Just as (exo-)planets revolve around their sun, so too do the stars in the Galactic center orbit the MBH on near-Kepler ellipses. 1At this distance 1′′ 0.038 pc. ∼ Secular Stellar Dynamics 3 Figure 1.2 – Left - The central 20′′ of our Galaxy imaged in near-infrared. Right - The S-star orbits: the result of 16 years of tracking∼ the motions of stars around the MBH in the Galactic center. Credit: ESO/Gillessen et al. (2009). Obscuration from interstellar dust in the plane of the Galaxy results in a high optical extinction of 30 magnitudes towards the Galactic center. For this reason, observations of∼ the stars near the MBH are taken at longer, near- infrared wavelengths. The development of near-infrared instruments, adap- tive optics and integral field spectroscopy has lead to dramatic progress in our knowledge of the Galactic center in the past two decades. Angular resolution has improved by more than an order of magnitude to 5 10−4 arcseconds, ∼ × and sensitivity by three to five magnitudes to Ks 16 18, for spectroscopy and imaging respectively (Genzel et al. 2010). Using∼ astrometr− ic tracking of the or- bits of luminous B-stars, astronomers have revealed that the MBH (known as 6 SgrA*) has a mass of M• 4 10 M⊙. The most intensely-studied of these stars is called S0/S0-2; with≃ an× orbital period of just 15.2 years (Schödel et al. 2003) its orbit has been fully traced and shown to be a bound, high-eccentricity Kepler ellipse. The proximity of the Galactic center makes it unique among galactic nuclei in that one can resolve individual stars of different ages and potentially test decades-old predictions of the dynamics of stars near MBHs. One such predic- tion is a lack of young stars near the MBH; its immense tidal force prohibits star formation from the collapse and fragmentation of molecular gas clouds (Sanders 1992; Morris 1993). We do however observe many ( 200) early type stars within the central parsec. They can be separated roughly∼ into two groups. The first is a population of O and Wolf-Rayet (WR) stars, 6 2Myr old, with ∼ ± masses & 20M⊙ (Paumard et al. 2006). The majority form a well-defined, warped disk with projected radii 0.8′′ 12′′, which rotates clockwise on the sky (Levin & Beloborodov 2003; Genzel− et al. 2003; Lu et al. 2009; Bartko et al. 2009). The second population is an apparently isotropic distribution of B-stars 4 Introduction (< 10 100 Myr). Those that lie within the central 0.8′′ are often collectively referred∼ to as the “S-stars”; see Figure 1.2. Their kinematics reveal randomly- inclined and highly-eccentric orbits (Ghez et al. 2005; Gillessen et al. 2009). Their proximity to the MBH, combined with their young ages, are indicative of a “paradox of youth” (Ghez et al. 2003; Eisenhauer et al. 2005; Martins et al. 2008). They could not have formed in-situ, but due to their young ages they have not had much time to dynamically migrate to their current locations. Another prediction is that we would expect, due to gravitational interac- tions, that the population of late type stars in the Galactic nuclear star cluster is dynamically relaxed and forms a steep density power-law cusp around the MBH (e.g., Bahcall & Wolf 1976). However, the late type k-giants, 1 10 Gyr old, do not show a central concentration as is theoretically expected for− a steady- state stellar cusp (Do et al.