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Relativistic Dynamics and Extreme Mass Ratio Inspirals Living Rev Relativ (2018) 21:4 https://doi.org/10.1007/s41114-018-0013-8 REVIEW ARTICLE Relativistic dynamics and extreme mass ratio inspirals Pau Amaro-Seoane1,2,3,4 Received: 13 July 2017 / Accepted: 16 February 2018 © The Author(s) 2018 Abstract It is now well-established that a dark, compact object, very likely a massive black hole (MBH) of around four million solar masses is lurking at the centre of the Milky Way. While a consensus is emerging about the origin and growth of supermas- sive black holes (with masses larger than a billion solar masses), MBHs with smaller masses, such as the one in our galactic centre, remain understudied and enigmatic. The key to understanding these holes—how some of them grow by orders of magnitude in mass—lies in understanding the dynamics of the stars in the galactic neighbourhood. Stars interact with the central MBH primarily through their gradual inspiral due to the emission of gravitational radiation. Also stars produce gases which will subsequently be accreted by the MBH through collisions and disruptions brought about by the strong central tidal field. Such processes can contribute significantly to the mass of the MBH and progress in understanding them requires theoretical work in preparation for future gravitational radiation millihertz missions and X-ray observatories. In particular, a unique probe of these regions is the gravitational radiation that is emitted by some compact stars very close to the black holes and which could be surveyed by a milli- hertz gravitational-wave interferometer scrutinizing the range of masses fundamental to understanding the origin and growth of supermassive black holes. By extracting B Pau Amaro-Seoane [email protected] 1 Institute of Space Sciences (ICE, CSIC), Institut d’Estudis Espacials de Catalunya (IEEC) at Campus UAB, Carrer de Can Magrans s/n, 08193 Barcelona, Spain 2 Institute of Applied Mathematics, Academy of Mathematics and Systems Science, CAS, Beijing 100190, China 3 Kavli Institute for Astronomy and Astrophysics, Beijing 100871, China 4 Zentrum für Astronomie und Astrophysik, TU Berlin, Hardenbergstraße 36, 10623 Berlin, Germany 123 4 Page 2 of 150 P. Amaro-Seoane the information carried by the gravitational radiation, we can determine the mass and spin of the central MBH with unprecedented precision and we can determine how the holes “eat” stars that happen to be near them. Keywords Black holes · Gravitational waves · Stellar dynamics Glossary 1 M 1 solar mass = 1.99 × 1030 kg M• Mass of super- or massive black hole 1 pc 1 parsec ≈ 3.09 × 1016 m 1 Myr/Gyr One million/billion years AGN Active galactic nucleus BH Black hole CO Compact object (a white dwarf or a neutron star), or a stellar- mass black hole. In general, a collapsed star with a mass ∈ [1.4, 10] M in this work DCO Dark compact object DF Dynamical friction EMRI Extreme mass ratio inspiral GC Galactic centre GPU Graphics processing unit GW/GWs Gravitational wave/s HB Giant stars in the horizontal branch HST Hubble space telescope IMBH Intermediate-mass black hole (M ∈[102, 105] M) IMF Initial mass function IMRI Intermediate mass ratio inspiral LISA Laser interferometer space antenna LSO Last stable orbit MBH Massive black hole (M ≈ 106 M) MC Monte carlo MW Milky Way NB6 Direct-summation N-body6 NS Neutron star PN Post-Newtonian RG Red giant RMS Root mean square SMBH Super massive black hole (M > 106 M) SNR Signal-to-noise ratio SPH Smoothed particle hydronamics TDE Tidal disruption event UCD Ultra-compact dwarf galaxy z Redshift 123 Relativistic dynamics and extreme mass ratio inspirals Page 3 of 150 4 Contents Foreword .................................................. 1 Massive dark objects in galactic nuclei ................................. 1.1 Active galactic nuclei ........................................ 1.2 Massive black holes and their possible progenitors ......................... 1.3 Tidal disruptions ........................................... 1.4 Extreme mass ratio inspirals .................................... 2 GWs as a probe to stellar dynamics and the cosmic growth of SMBHs ................. 2.1 GWs and stellar dynamics ...................................... 2.2 The mystery of the growth of MBHs ................................ 2.3 A magnifying glass ......................................... 2.3.1 A problem of ∼ 10 orders of magnitude ........................... 2.4 How stars distribute around MBHs in galactic nuclei ........................ 3 A taxonomy of orbits in galactic nuclei ................................. 3.1 Spherical potentials ......................................... 3.2 Non-spherical potentials ...................................... 4 Two-body relaxation in galactic nuclei ................................. 4.1 Introduction ............................................. 4.2 Two-body relaxation ........................................ 4.3 Dynamical friction ......................................... 4.4 The difussion and loss-cone angles ................................. 5 “Standard” mass segregation ....................................... 5.1 Introduction ............................................. 5.2 Single-mass clusters ......................................... 5.3 Mass segregation in two mass-component clusters ......................... 5.4 Clusters with a broader mass spectrum with no MBH ....................... 5.5 Core-collapse evolution ....................................... 5.6 Clusters with a broader mass spectrum with a MBH ........................ 6 Two-body extreme mass ratio inspirals ................................. 6.1 A hidden stellar population in galactic nuclei ............................ 6.2 Fundamentals of EMRIs ...................................... 6.3 Orbital evolution due to emission of gravitational waves ...................... 6.4 Decoupling from dynamics into the relativistic regime ....................... 7 Beyond the standard model of two-body relaxation ........................... 7.1 The standard picture ......................................... 7.2 Coherent or resonant relaxation ................................... 7.3 Strong mass segregation ....................................... 7.4 The cusp at the Galactic Centre ................................... 7.5 Tidal separation of binaries ..................................... 7.6 A barrier for captures ignored by rotating MBHs .......................... 7.7 Extended stars EMRIs ........................................ 7.8 The butterfly effect ......................................... 7.9 Role of the gas ............................................ 8 Integration of dense stellar systems and EMRIs ............................. 8.1 Introduction ............................................. 8.2 The Fokker–Planck approach .................................... 8.3 Moment models ........................................... 8.3.1 Equation of continuity .................................... 8.3.2 Momentum balance equation ................................. 8.3.3 Radial energy equation .................................... 8.3.4 Tangential energy equation .................................. 8.4 Solving conducting, self-gravitating gas spheres .......................... 8.5 The local approximation ...................................... 8.6 Monte Carlo codes ......................................... 8.7 Applications of Monte Carlo and Fokker–Planck simulations to the EMRI problem ....... 8.8 Direct-summation N-body codes .................................. 123 4 Page 4 of 150 P. Amaro-Seoane 8.8.1 Relativistic corrections: the post-Newtonian approach .................... 8.8.2 Relativistic corrections: a geodesic solver .......................... 8.8.3 N-body units and conversion ................................. A Comment on the use of the Ancient Greek word barathron and bathron to refer to a black hole .... References .................................................. Foreword The volume where capture orbits are produced is so small in comparison to other typical length scales of interest in astrodynamics that it has usually been seen as unimportant and irrelevant to the global dynamical evolution of the system. The only exception has been the tidal disruption of stars by massive black holes. Only when it transpired that the slow, adiabatic inspiral of compact objects onto massive black holes provides us with valuable information, did astrophysicists start to address the question in more detail. Since the problem of EMRIs (extreme mass ratio inspiral) started to draw our attention, there has been a notable progress in answering fundamental questions of stellar dynamics. The discoveries have been numerous and some of them remain puzzling. The field is developing very quickly and we are making important breakthroughs even before a millihertz mission flies. When I was approached and asked to write this review, I was glad to accept it without realising the dimensions of the task. I was told that it should be similar to a plenary talk for a wide audience. I have a personal problem with instructions like this. I remember that when I was nine years old, our Spanish teacher asked us to summarise a story we had read together in class. I asked her to define “summarise”, because I could easily produce a summary of one, two or fifty pages, depending on what she was actually expecting from us. She was confused and I never got a clear answer. She replied that “A summary is a summary and that’s it”. On this occasion, I am afraid that I have run into the same snag and I have gone for the many-pages
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