{
  "id": "1509.02397",
  "version": "v1",
  "published": "2015-09-08T14:56:51.000Z",
  "updated": "2015-09-08T14:56:51.000Z",
  "title": "Stellar Dynamics around a Massive Black Hole I: Secular Collisionless Theory",
  "authors": [
    "S. Sridhar",
    "Jihad R. Touma"
  ],
  "comment": "Submitted to MNRAS; 27 preprint pages",
  "categories": [
    "astro-ph.GA",
    "gr-qc"
  ],
  "abstract": "We present a theory in 3 parts, of the long-term (or secular) evolution of stellar systems orbiting within the sphere of influence of massive black holes in galactic nuclei. Here we describe the secular collisionless dynamics of a (Keplerian) stellar system of mass $M$ orbiting a black hole of mass $M_\\bullet \\gg M$. The stellar distribution function (DF) $f$ obeys the collisionless Boltzmann equation (CBE) in 6-dim phase space. The small mass ratio, $\\varepsilon = M/M_\\bullet \\ll 1$, implies a separation of time scales in the motions of stars: the fast Kepler orbital periods and the secular time scale which is longer by a factor $\\varepsilon^{-1}$. We orbit-average the CBE over the fast Keplerian orbital phase using the Method of Multiple Scales. Then $f$ is expressed as the sum of a secular DF $F$ in a 5-dim (Gaussian Ring) space, and small fluctuations that remain of $O(\\varepsilon)$ over secular times. $F$ obeys a secular CBE that includes stellar self-gravity, general relativistic corrections up to 1.5 post-Newtonian order, and external sources. Secular dynamics conserves the semi-major axis of every star. This additional integral of motion promotes extra regularity of the stellar orbits, and enables the construction of secular equilibrium DFs ($F_0$) through a Secular Jeans theorem. Secular equilibria allow for varied spatial geometries including figure rotation. A linearized secular CBE determines the linear response and stability of $F_0$. Spherical, non-rotating equilibria may support small-amplitude, long-lived, warp-like distortions. We also prove that an axisymmetric, zero-thickness, flat disc is secularly stable to all in-plane perturbations, when its DF $F_0$ is a monotonic function of the angular momentum at fixed energy.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-09-08T14:56:51.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "massive black hole",
      "secular collisionless theory",
      "stellar dynamics",
      "stellar system",
      "fast kepler orbital periods"
    ],
    "publication": {
      "doi": "10.1093/mnras/stw542"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 1392429
    }
  }
}