{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Allgemeine Relativitätstheorie mit dem Computer \n",
    "# General Theory of Relativity on the Computer\n",
    "\n",
    "## Vorlesung gehalten an der J.W.Goethe-Universität in Frankfurt am Main (Wintersemester 2026/27)\n",
    "\n",
    "### von Dr.phil.nat. Dr.rer.pol. Matthias Hanauske\n",
    "\n",
    "### Frankfurt am Main 28.09.2026"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Bewegung eines Probekörpers um ein nicht-rotierendes schwarzes Loch in der Ebene\n",
    "\n",
    "Die Bewegung eines Probekörpers (eines Teilchens mit vernachlässigbarer Masse, das die Raumzeit selbst nicht deformiert) um ein nicht-rotierendes, ungeladenes schwarzes Loch wird durch die Schwarzschild-Metrik der Allgemeinen Relativitätstheorie bestimmt. In der Vorlesung <A href=\"https://itp.uni-frankfurt.de/~hanauske/VARTC/index.html\" target=blank>Allgemeine Relativit&auml;tstheorie mit dem Computer</A> behandelten wir die <A href=\"https://itp.uni-frankfurt.de/~hanauske/VARTC/2026/python/Schwarzschild_OGRePy/Schwarzschild_OGRePy.html\" target=blank>Eigenschaften der Schwarzschild-Metrik</A>. Da das System sphärisch symmetrisch ist, bleibt der Drehimpuls erhalten. Die Bewegung eines Teilchens findet daher immer in einer konstanten Bahnebene statt, die man äquatorial bei $\\theta = \\pi/2$ ansetzen kann. In dem Jupyter Notebook <A href=\"https://itp.uni-frankfurt.de/~hanauske/VARTC/2026/python/GeodSchwarzschildI_OGRePy/GeodSchwarzschildI_OGRePy.html\" target=blank>Teil I: Radialer Fall eines Probekörpers in ein schwarzes Loch</A> betrachteten wir zunächst den einfachsten Fall eines radialen Einfallens eines Testteilchens, wohingegen wir im Jupyter Notebook <A href=\"https://itp.uni-frankfurt.de/~hanauske/VARTC/2026/python/GeodSchwarzschildII_OGRePy/GeodSchwarzschildII_OGRePy.html\" target=blank>Teil II: Klassifizierung unterschiedlicher Bahnbewegungen</A> betrachteten wir allgemeine geodätische Bewegungen um ein nicht-rotierendes, ungeladenes schwarzes Loch und klassifizierten diese mittels eines effektiven Potenzials."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Teil III: Der ISCO und die Photonensphäre\n",
    "\n",
    "In diesem Jupyter Notebook werden wir im Folgenden das Konzept der <i>Innersten stabilen kreisförmige Bahnbewegung (Innermost Stable Circular Orbit (ISCO))</i> vorstellen und den Begriff der <i>Photonensphäre</i> eines nicht-rotierenden schwarzen Loches diskutieren.\n",
    "\n",
    "## a) Die innerste stabile kreisförmige Bahnbewegung eines Probekörpers (Innermost Stable Circular Orbit (ISCO)) \n",
    "\n",
    "Der ISCO markiert die innerste stabile Kreisbahn, auf der sich ein Körper mit Masse dauerhaft bewegen kann. Der ISCO befindet sich bei drei Schwarzschild-Radien $r_{\\rm ISCO} = 3 \\, R_s = 6 \\, M $. Für Radien $r > r_{\\rm ISCO}$ existieren stabile kreisförmige Bahnen. Fällt ein Probekörper unter diesen Radius, verliert das effektive Potenzial sein lokales Minimum. Es gibt dort keine stabilen Kreisbahnen mehr. Erreicht Materie (z. B. in einer Akkretionsscheibe um ein schwarzes Loch) den ISCO, beginnt sie ohne Zufuhr weiterer Energie spiralförmig und rapide in das schwarze Loch zu stürzen."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Geodätische Bewegung eines Probekörpers\n",
    "\n",
    "Um die Geodätengleichung \n",
    "\n",
    "$$\n",
    "\\frac{d^2 x^\\mu}{d\\tau^2} +  \\Gamma^\\mu_{\\nu \\rho} \\frac{d x^\\nu}{d\\tau} \\frac{d x^\\rho}{d\\tau} ~=~ 0 \n",
    "$$\n",
    "\n",
    "zu berechnen, definieren zunächst den ersten Term der Gleichung $\\frac{d^2 x^\\mu}{d\\tau^2}$:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Wir beginnen mit der Berechnung des ISCOs eines schwarzen Loches und betrachten wieder die Geodätengleichung in vorgegebener Schwarzschild Raumzeit. \n",
    "\n",
    "Zunächst wird das Python-Modul \"OGRePy\" eingebunden, welches auf dem Modul SymPy basiert und symbolische Berechnungen in der Allgemeinen Relativitätstheorie relativ einfach möglich macht. Eine ausführliche Einführung in das Modul \"OGRePy\" finden sie unter dem folgenden Link: <A href=\"https://github.com/bshoshany/OGRePy/blob/master/OGRePy/docs/OGRePy_Documentation.ipynb\" target=\"_blank\">OGRePy-Documentation</A>. Dann definieren wir die  zugrundeliegenden Koordinaten und die kovariante Form der Raumzeit-Metrik der Schwarzschildmetrik."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/markdown": [
       "<b>\n",
       "OGRePy: An <u>O</u>bject-Oriented <u>G</u>eneral <u>Re</u>lativity Package for <u>Py</u>thon<br />\n",
       "By <a href=\"https://github.com/bshoshany\">Barak Shoshany</a> (<a href=\"mailto:baraksh@gmail.com\">baraksh@gmail.com</a>) (<a href=\"https://baraksh.com/\">baraksh.com</a>)<br />\n",
       "v2.0.0 (2026-05-11)<br />\n",
       "GitHub repository: <a href=\"https://github.com/bshoshany/OGRePy\">https://github.com/bshoshany/OGRePy</a><br />\n",
       "Documentation: <a href=\"/home/hanauske/neu_2026/VARTC_2026/ART_env/lib/python3.14/site-packages/OGRePy/docs/OGRePy_Documentation.ipynb\">.ipynb</a>, <a href=\"/home/hanauske/neu_2026/VARTC_2026/ART_env/lib/python3.14/site-packages/OGRePy/docs/OGRePy_Documentation.pdf\">.pdf</a>, <a href=\"#\" onclick=\"window.open('file:///home/hanauske/neu_2026/VARTC_2026/ART_env/lib/python3.14/site-packages/OGRePy/docs/OGRePy_Documentation.html', '_blank')\">.html</a>\n",
       "</b>"
      ],
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import OGRePy as gr\n",
    "from sympy import *\n",
    "init_printing()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Um symbolische Rechnungen in einer gekrümmten Raumzeit machen zu können, müssen wir zunächst wieder das verwendete Koordinatensystem und die gekrümmte Metrik der Raumzeit (hier die Schwarzschild-Metrik) definieren."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/markdown": [
       "$$g{}_{\\mu}{}_{\\nu}\\Bigg|_{\\left(t, r, \\theta, \\phi\\right)} = \\left(\\begin{matrix}- \\frac{2 M}{r} + 1 & 0 & 0 & 0\\\\0 & - \\frac{1}{- \\frac{2 M}{r} + 1} & 0 & 0\\\\0 & 0 & - r^{2} & 0\\\\0 & 0 & 0 & - r^{2} \\sin^{2}{\\left(\\theta \\right)}\\end{matrix}\\right)$$"
      ],
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from OGRePy.abc import t, phi, theta\n",
    "r = gr.sym(\"r\", nonnegative=True)\n",
    "M = gr.sym(\"M\", nonnegative=True)\n",
    "\n",
    "Spherical = gr.Coordinates(t, r, theta, phi)\n",
    "\n",
    "g = gr.Metric(\n",
    "    coords = Spherical,\n",
    "    components = gr.diag((1-2*M/r), -1/(1-2*M/r), -r**2, -r**2*gr.s.sin(theta)**2),\n",
    ")\n",
    "g.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Wir benutzen im Folgenden die direkte Berechnung der Geodätengleichung mittels der Christoffel Symbole. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/markdown": [
       "**OGRePy**: You have the latest version of the package."
      ],
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/markdown": [
       "$$0{}^{\\mu}\\Bigg|_{\\left(t, r, \\theta, \\phi\\right)} = \\left(\\begin{matrix}\\frac{- 2 M \\dot{r} \\dot{t} + \\ddot{t} r \\left(2 M - r\\right)}{r \\left(2 M - r\\right)}\\\\\\frac{M \\dot{r}^{2} r^{2} - M \\dot{t}^{2} \\left(2 M - r\\right)^{2} + r^{3} \\left(2 M - r\\right) \\left(\\ddot{r} + \\dot{\\phi}^{2} \\left(2 M - r\\right) \\sin^{2}{\\left(\\theta \\right)} + \\dot{\\theta}^{2} \\left(2 M - r\\right)\\right)}{r^{3} \\left(2 M - r\\right)}\\\\\\ddot{\\theta} - \\frac{\\dot{\\phi}^{2} \\sin{\\left(2 \\theta \\right)}}{2} + \\frac{2 \\dot{\\theta} \\dot{r}}{r}\\\\\\ddot{\\phi} + \\frac{2 \\dot{\\phi} \\dot{\\theta}}{\\tan{\\left(\\theta \\right)}} + \\frac{2 \\dot{\\phi} \\dot{r}}{r}\\end{matrix}\\right)$$"
      ],
      "text/plain": [
       "<OGRePy._core.GeodesicFromChristoffel at 0x71044e10aa50>"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g.geodesic_from_christoffel()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Die Geodätengleichung ist somit ein System von nichtlinearen gekoppelten Differenzialgleichungen zweiter Ordnung. Die entsprechenden vier Gleichungen lauten:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/latex": [
       "$\\displaystyle \\frac{d^{2}}{d \\lambda^{2}} t{\\left(\\lambda \\right)} = \\frac{2 M \\frac{d}{d \\lambda} r{\\left(\\lambda \\right)} \\frac{d}{d \\lambda} t{\\left(\\lambda \\right)}}{\\left(2 M - r{\\left(\\lambda \\right)}\\right) r{\\left(\\lambda \\right)}}$"
      ],
      "text/plain": [
       "                               d                    d                \n",
       "    2                   2⋅M⋅────────(r(\\lambda))⋅────────(t(\\lambda))\n",
       "   d                        d\\lambda             d\\lambda            \n",
       "─────────(t(\\lambda)) = ─────────────────────────────────────────────\n",
       "        2                       (2⋅M - r(\\lambda))⋅r(\\lambda)        \n",
       "d\\lambda                                                             "
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "param = Spherical.of_param()\n",
    "dot = Spherical.of_param_dot()\n",
    "ddot = Spherical.of_param_ddot()\n",
    "\n",
    "GeodEq = g.geodesic_from_christoffel().components(indices=(1,),coords=Spherical)\n",
    "GeodEq0 = Eq(0,simplify(GeodEq[0]))\n",
    "GeodEq0_a = Eq(ddot[0],solve(GeodEq0,ddot[0])[0])\n",
    "GeodEq0_a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/latex": [
       "$\\displaystyle \\frac{d^{2}}{d \\lambda^{2}} r{\\left(\\lambda \\right)} = \\frac{- 4 M^{2} r^{3}{\\left(\\lambda \\right)} \\sin^{2}{\\left(\\theta{\\left(\\lambda \\right)} \\right)} \\left(\\frac{d}{d \\lambda} \\phi{\\left(\\lambda \\right)}\\right)^{2} - 4 M^{2} r^{3}{\\left(\\lambda \\right)} \\left(\\frac{d}{d \\lambda} \\theta{\\left(\\lambda \\right)}\\right)^{2} + M \\left(2 M - r{\\left(\\lambda \\right)}\\right)^{2} \\left(\\frac{d}{d \\lambda} t{\\left(\\lambda \\right)}\\right)^{2} + 4 M r^{4}{\\left(\\lambda \\right)} \\sin^{2}{\\left(\\theta{\\left(\\lambda \\right)} \\right)} \\left(\\frac{d}{d \\lambda} \\phi{\\left(\\lambda \\right)}\\right)^{2} + 4 M r^{4}{\\left(\\lambda \\right)} \\left(\\frac{d}{d \\lambda} \\theta{\\left(\\lambda \\right)}\\right)^{2} - M r^{2}{\\left(\\lambda \\right)} \\left(\\frac{d}{d \\lambda} r{\\left(\\lambda \\right)}\\right)^{2} - r^{5}{\\left(\\lambda \\right)} \\sin^{2}{\\left(\\theta{\\left(\\lambda \\right)} \\right)} \\left(\\frac{d}{d \\lambda} \\phi{\\left(\\lambda \\right)}\\right)^{2} - r^{5}{\\left(\\lambda \\right)} \\left(\\frac{d}{d \\lambda} \\theta{\\left(\\lambda \\right)}\\right)^{2}}{\\left(2 M - r{\\left(\\lambda \\right)}\\right) r^{3}{\\left(\\lambda \\right)}}$"
      ],
      "text/plain": [
       "                                                                               ↪\n",
       "                             2  3             2                  ⎛   d         ↪\n",
       "    2                   - 4⋅M ⋅r (\\lambda)⋅sin (\\theta(\\lambda))⋅⎜────────(\\ph ↪\n",
       "   d                                                             ⎝d\\lambda     ↪\n",
       "─────────(r(\\lambda)) = ────────────────────────────────────────────────────── ↪\n",
       "        2                                                                      ↪\n",
       "d\\lambda                                                                       ↪\n",
       "\n",
       "↪             2                                               2                ↪\n",
       "↪            ⎞       2  3          ⎛   d                     ⎞                 ↪\n",
       "↪ i(\\lambda))⎟  - 4⋅M ⋅r (\\lambda)⋅⎜────────(\\theta(\\lambda))⎟  + M⋅(2⋅M - r(\\ ↪\n",
       "↪            ⎠                     ⎝d\\lambda                 ⎠                 ↪\n",
       "↪ ──────────────────────────────────────────────────────────────────────────── ↪\n",
       "↪                                                                              ↪\n",
       "↪                                                                              ↪\n",
       "\n",
       "↪                                 2                                            ↪\n",
       "↪         2 ⎛   d                ⎞         4             2                  ⎛  ↪\n",
       "↪ lambda)) ⋅⎜────────(t(\\lambda))⎟  + 4⋅M⋅r (\\lambda)⋅sin (\\theta(\\lambda))⋅⎜─ ↪\n",
       "↪           ⎝d\\lambda            ⎠                                          ⎝d ↪\n",
       "↪ ──────────────────────────────────────────────────────────────────────────── ↪\n",
       "↪                                                                              ↪\n",
       "↪                                                                   (2⋅M - r(\\ ↪\n",
       "\n",
       "↪                        2                                              2      ↪\n",
       "↪   d                   ⎞         4          ⎛   d                     ⎞       ↪\n",
       "↪ ───────(\\phi(\\lambda))⎟  + 4⋅M⋅r (\\lambda)⋅⎜────────(\\theta(\\lambda))⎟  - M⋅ ↪\n",
       "↪ \\lambda               ⎠                    ⎝d\\lambda                 ⎠       ↪\n",
       "↪ ──────────────────────────────────────────────────────────────────────────── ↪\n",
       "↪           3                                                                  ↪\n",
       "↪ lambda))⋅r (\\lambda)                                                         ↪\n",
       "\n",
       "↪                                   2                                          ↪\n",
       "↪  2          ⎛   d                ⎞     5             2                  ⎛    ↪\n",
       "↪ r (\\lambda)⋅⎜────────(r(\\lambda))⎟  - r (\\lambda)⋅sin (\\theta(\\lambda))⋅⎜─── ↪\n",
       "↪             ⎝d\\lambda            ⎠                                      ⎝d\\l ↪\n",
       "↪ ──────────────────────────────────────────────────────────────────────────── ↪\n",
       "↪                                                                              ↪\n",
       "↪                                                                              ↪\n",
       "\n",
       "↪                      2                                          2\n",
       "↪ d                   ⎞     5          ⎛   d                     ⎞ \n",
       "↪ ─────(\\phi(\\lambda))⎟  - r (\\lambda)⋅⎜────────(\\theta(\\lambda))⎟ \n",
       "↪ ambda               ⎠                ⎝d\\lambda                 ⎠ \n",
       "↪ ─────────────────────────────────────────────────────────────────\n",
       "↪                                                                  \n",
       "↪                                                                  "
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "GeodEq1 = Eq(0,simplify(GeodEq[1]))\n",
    "GeodEq1_a = Eq(ddot[1],solve(GeodEq1,ddot[1])[0])\n",
    "GeodEq1_a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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DjzwooR7i+LzLKIz1LzaG4sdcqGJK45WH/tPfRW4zwnPLFPu75nj3dN+Mop68PXlmDaJr6vessuxBdK08GgiOQ1DWDNpbueL7szhgvH8rChrFY6FxpMGukQlVJaWzq+SYi8Xu9IAApwun3iEsVSftblnnc+mlOUCZOYOKecYudKTQ9NwyLpmnvJvumXu/lRrWiFsKT6rowYxLgPx4Bn26Blp7vGcxFNaLGbbiNWsQxQZdTb83qyw1WEGpxQmwk8+iw5ot7S7JywKuUizPrnNOyMEDXqF+Hq6dInYYEBy7n4SWqDO2l3aPFJZ1YC7d8uW+yqHwUIScZtQIZZEady/0HE4uSgXEi7wYhLR1do4q/16vCpSf9XP0GIoPc6CKqdKMMFyLxqtluDB/1fHeA6uljKIeg4hmXU2/n3z8uNiPT+wxnu2sWpAoRI40uDw0CBiFaSxHwqOjVj1zk3D2p/aUB4Flt8JGPBQ/Ingq4g/yr8UyHvXfH9oIaF7cKccv8rnEMqKYxvzhG72gqOUz73DhOFjPYV7FvFzp5xUHChhFySuH2fmsfJuk2Adefeyl0DfZmY5GqZ/Mhc2Mt9rDrvDguxcqj0E0908WdtfW763uLN9osEa3oeLAKHr83aDise6LOwcyp6S8KF6EVOtYzorAE6vJyRGYIKC5xDx6K39yAhHTgjKkoJ4x/mxnOcwpxSPUMAqDopRPXlOmrV0rWTdJsU/hjsMmG7hCo9Rnxnoz4632IDsPNlQoC485qJTnqvq9OWWpAbjTICEo8mPVZ3Hw8kHkosWw+4x5Wh7C6EWsp5UPns1LHK3Cnnb5CGgOcTrBDVYUYU4ch6EE2SmiFDmSfCm3U5wJVub4oChJi8RxLAbjORL9pO/XRtc63lfT780dw0qQcASAtc5/OBkuQyTxw9EWq1Hx6Q00opqk/AgurvIjpKrWl9IQcrybmRyz5RUoD0KvJDDzrOkzt4qvUaikGHjYEXAEHIGzQOB2g60M1rfaxe3WtHnsNrkZm+8s2Yl+SDPOhC0vR7FVZRl5olhnSW3qOdad5eMZHAFHwBFwBLaJwK0E/aPf8FEb0l9d4f0OFx2GXZfCdjTL8VROTxVhx1p52ug58uHWIUdF4btNxQ2711Hmh/eg1PuopPY9+vg8KgBe+VUioHmfyoQJBnEtczFqnzXKu9Taeg91+HqbQO0REQGUZXNSnhKpuACokluDKdmliHD5IU3oDYs3u0QU5RexDEe97Ahru8JuJRz5reJtaXxW6aAzdQQOQEDrAgN59hXJvqx9ve2L2PXk3+IxLOjn1h87wa39/8NhlmiB+TvLAQ0POAKOgCNweQhsSlliLcqNUNYz7yrZ5aEwS8Q7zOq7RZXnWJfjmvwIBgXHt0i40jeX1Jm/H1XUlFS+tjudZvYYR8ARcAQcgbNDYHVlKUXCOwW7Bm9Kjf9HyDFKiVBc4T1ELMtx6TH//9B+hGD0eYl420+RoehKypIjnnyHqygnR8ARcAQuEwHJRX4lrWuTkCNwTNmc1xafb07QKC7S8JkGjks7H+T43KNG/IYrv9xjn5AQbimtH5Sf3WOJ2I3y+UdJGZKf9BdKNyVOnNELBfJ3p5bmviPgCDgCF4VAlLk1WdrTVxQtcvsiqfmdpToOcN/LoUze6Xk4blSY3R9pO4WHm6s8p6Q0bnMOP1unZ3jxnePoO8q0zL5h8YRffsy6L5shf2zjz/K7/k3MUNADZ49AHHs78ueHMDDuMPRaBtvZ99s7cN0IaH7zuuu5fJv7AyCK65L1FFBelCWv0w6+jDlUvLHATas96jAfznMcCVijXZbi7xXX8588ULD/lluT7GbrUnUwYeDpdEUIaE5jyPEvtTiNsB+kQEn+qmdOGpwcgYtDQHMb+e7/tWlmZJvKkrKJkBi984tpKEyOOKv/JUHl81usKE9+u3AxS506xJMjAITdURR5wOviLKOjgLmOwhOjS/MAw4l5nv4HketAw3t5LQgw77nwWCWtg1lZnxSG18VtNmaVpTrN76Oi3ACrRADDfxhgG98k5eFYl3yLfx8lnhwFNwdc6T0Ej+G4uaeA57kYBNg98luvd1mPMBSZ40cbYxlff3QEtoDA15rbPZuDLlkfecEzX0db6OvBbehRlgiQya7SahQgpKFImwpG+RA0WBu8q6wpXiUfRpEnR2cHv2COZeFx0G2ww1rupTaEAHO5ZRhe1OLfEO7elEdCQLKOzUuXvFPeLlkfuwLPrx+pW6tUe5tyFRim0Ojob3KAw27wrVyLsEpeqzzW90QRRr5ckgj/xSM+7+R3DVKr4jQt8mv93muafRJW+YPLTph5xNkhoPGvXVQLNwSVvtirg7MDxxt8NghonjJfX8p9rjCXK5Hr9jqBk5N0F4lMRs73UlPWJ0zgCe+0riT5/ILDzlIAsoO0/62HYkNx2LnzHJh2/Dl5dxkHinQuTjB4DCQD90HOyRHYNAJxvpqw2XRbvXGOQETgpeYtMnYnP7xWkm8nfybTDaxnCvA1QS9VZX3GAJ6sm4uhsLMUkBwvhR8YVzjd7RHmRuxkt5ghYIqPAcl3Zyhg+I++rUwGL2Plj47AphBgXfBf4/N5valGemMcARDQPEVBmfJ7qjCXFe0TOORwvtMjzuS3grNkeUuyPi1MvstTluoU1gZHqDmQ7DZrH/QrKQwOYAdFK3/ynzzE8xPyOTkC54aA5i5WNO8wa8ez59Ylb+/lI4A8fhe7ySme7Sh3msdDOIEBhXqfPFeDKt+U9VlBNlrkvxiyd5a8iB0dtUZgsAxG31emPVce0lGUvf/JIy3uYUdgswhobvNK4an88J59sw31hjkCCQKar+95lM9GBzLF+fB04F/x21fWdyvhA5t08mI3AgHtj8uVYrjJpPSRErUWKh6rhTKvFL7HKcwu9KJuQKk/TleGgOYyNwQ/kz/sKBVe5DveK4PSu/t4CDB3Wze7rWUclzZ3gJr7h8h6eNqRrdV11v5N0vr0XSXRWNRmpfCfObAsAkXwftEDn1mEPDGJYyuOcxE2To7A2SEQ53bpZ7+Y0xe1+M9ucLzB+yAw+wotMkPuD7I9r+AIWc/OMtcpOfuzemZnyY6Q3eMAWFR2WBP2M3VcPw4dl88gcFmHHeVo1xmfyVc6G1e0kyOwXQQ0f1kDvFbA4OP29uAUh2HIWnFyBDaNQJzHzOX8tLDUbjY7z0sJ4nOMrL+4/9p0E0Fiy441zbeSXPZB4bGz5J3Na/l2XVjB8NkHgqN28Ycryy+UzmA5OQLnhADChXnL+8rcuaI8p5G87rYyhzmCHW1mKpD8oHg2RiVClh8q61G0Pcq6VO8m45r/dWSTLfZGOQKOgCPgCCyGgJQqn5r4f22aQfRmJt2THQFHwBFwBC4bAU4Tl3x1xo4UnhdFriwvaji9M46AI+AI7IeAdpZ8X7/Ibe/4+g1e+Tf7+zVqg7ldWW5wULxJjoAj4AicGAHuraR3Uw6tHh5L7lIPbcfi5fyd5eKQOkNHwBFwBM4Pgbgr5DPBg37aUeW4DMpPQ17UJyM2kv8HEjjCslLY1VkAAAAASUVORK5CYII=",
      "text/latex": [
       "$\\displaystyle \\frac{d^{2}}{d \\lambda^{2}} \\theta{\\left(\\lambda \\right)} = \\frac{\\sin{\\left(2 \\theta{\\left(\\lambda \\right)} \\right)} \\left(\\frac{d}{d \\lambda} \\phi{\\left(\\lambda \\right)}\\right)^{2}}{2} - \\frac{2 \\frac{d}{d \\lambda} \\theta{\\left(\\lambda \\right)} \\frac{d}{d \\lambda} r{\\left(\\lambda \\right)}}{r{\\left(\\lambda \\right)}}$"
      ],
      "text/plain": [
       "                                                                             2 ↪\n",
       "                                                    ⎛   d                   ⎞  ↪\n",
       "    2                        sin(2⋅\\theta(\\lambda))⋅⎜────────(\\phi(\\lambda))⎟  ↪\n",
       "   d                                                ⎝d\\lambda               ⎠  ↪\n",
       "─────────(\\theta(\\lambda)) = ───────────────────────────────────────────────── ↪\n",
       "        2                                            2                         ↪\n",
       "d\\lambda                                                                       ↪\n",
       "\n",
       "↪                                                    \n",
       "↪         d                         d                \n",
       "↪    2⋅────────(\\theta(\\lambda))⋅────────(r(\\lambda))\n",
       "↪      d\\lambda                  d\\lambda            \n",
       "↪  - ────────────────────────────────────────────────\n",
       "↪                       r(\\lambda)                   \n",
       "↪                                                    "
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "GeodEq2 = Eq(0,simplify(GeodEq[2]))\n",
    "GeodEq2_a = Eq(ddot[2],solve(GeodEq2,ddot[2])[0])\n",
    "GeodEq2_a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\frac{d^{2}}{d \\lambda^{2}} \\phi{\\left(\\lambda \\right)} = - \\frac{2 \\left(\\frac{r{\\left(\\lambda \\right)} \\frac{d}{d \\lambda} \\theta{\\left(\\lambda \\right)}}{\\tan{\\left(\\theta{\\left(\\lambda \\right)} \\right)}} + \\frac{d}{d \\lambda} r{\\left(\\lambda \\right)}\\right) \\frac{d}{d \\lambda} \\phi{\\left(\\lambda \\right)}}{r{\\left(\\lambda \\right)}}$"
      ],
      "text/plain": [
       "                              ⎛              d                                 ↪\n",
       "                              ⎜r(\\lambda)⋅────────(\\theta(\\lambda))            ↪\n",
       "                              ⎜           d\\lambda                       d     ↪\n",
       "    2                      -2⋅⎜──────────────────────────────────── + ──────── ↪\n",
       "   d                          ⎝        tan(\\theta(\\lambda))           d\\lambda ↪\n",
       "─────────(\\phi(\\lambda)) = ─────────────────────────────────────────────────── ↪\n",
       "        2                                                         r(\\lambda)   ↪\n",
       "d\\lambda                                                                       ↪\n",
       "\n",
       "↪             ⎞                         \n",
       "↪             ⎟                         \n",
       "↪             ⎟    d                    \n",
       "↪ (r(\\lambda))⎟⋅────────(\\phi(\\lambda)) \n",
       "↪             ⎠ d\\lambda                \n",
       "↪ ──────────────────────────────────────\n",
       "↪                                       \n",
       "↪                                       "
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "GeodEq3 = Eq(0,simplify(GeodEq[3]))\n",
    "GeodEq3_a = Eq(ddot[3],solve(GeodEq3,ddot[3])[0])\n",
    "GeodEq3_a"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Wir lassen im Folgenden nur ebene Bewegungen zu ($\\theta=\\frac{\\pi}{2}, {\\rm cos}(\\theta)=0, {\\rm sin}(\\theta)=1, \\frac{d \\theta}{d\\tau}=0, \\frac{d^2 \\theta}{d\\tau^2}=0$). Unter diesen Annahmen sind nur drei der vier Geodätengleichungen relevant und diese lauten dann:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/latex": [
       "$\\displaystyle \\frac{d^{2}}{d \\lambda^{2}} t{\\left(\\lambda \\right)} = \\frac{2 M \\frac{d}{d \\lambda} r{\\left(\\lambda \\right)} \\frac{d}{d \\lambda} t{\\left(\\lambda \\right)}}{\\left(2 M - r{\\left(\\lambda \\right)}\\right) r{\\left(\\lambda \\right)}}$"
      ],
      "text/plain": [
       "                               d                    d                \n",
       "    2                   2⋅M⋅────────(r(\\lambda))⋅────────(t(\\lambda))\n",
       "   d                        d\\lambda             d\\lambda            \n",
       "─────────(t(\\lambda)) = ─────────────────────────────────────────────\n",
       "        2                       (2⋅M - r(\\lambda))⋅r(\\lambda)        \n",
       "d\\lambda                                                             "
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "GeodEq0_a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/latex": [
       "$\\displaystyle \\frac{d^{2}}{d \\lambda^{2}} r{\\left(\\lambda \\right)} = \\frac{- 4 M^{2} r^{3}{\\left(\\lambda \\right)} \\left(\\frac{d}{d \\lambda} \\phi{\\left(\\lambda \\right)}\\right)^{2} + M \\left(2 M - r{\\left(\\lambda \\right)}\\right)^{2} \\left(\\frac{d}{d \\lambda} t{\\left(\\lambda \\right)}\\right)^{2} + 4 M r^{4}{\\left(\\lambda \\right)} \\left(\\frac{d}{d \\lambda} \\phi{\\left(\\lambda \\right)}\\right)^{2} - M r^{2}{\\left(\\lambda \\right)} \\left(\\frac{d}{d \\lambda} r{\\left(\\lambda \\right)}\\right)^{2} - r^{5}{\\left(\\lambda \\right)} \\left(\\frac{d}{d \\lambda} \\phi{\\left(\\lambda \\right)}\\right)^{2}}{\\left(2 M - r{\\left(\\lambda \\right)}\\right) r^{3}{\\left(\\lambda \\right)}}$"
      ],
      "text/plain": [
       "                                                                    2          ↪\n",
       "                             2  3          ⎛   d                   ⎞           ↪\n",
       "    2                   - 4⋅M ⋅r (\\lambda)⋅⎜────────(\\phi(\\lambda))⎟  + M⋅(2⋅M ↪\n",
       "   d                                       ⎝d\\lambda               ⎠           ↪\n",
       "─────────(r(\\lambda)) = ────────────────────────────────────────────────────── ↪\n",
       "        2                                                                      ↪\n",
       "d\\lambda                                                                       ↪\n",
       "\n",
       "↪                                       2                                      ↪\n",
       "↪               2 ⎛   d                ⎞         4          ⎛   d              ↪\n",
       "↪  - r(\\lambda)) ⋅⎜────────(t(\\lambda))⎟  + 4⋅M⋅r (\\lambda)⋅⎜────────(\\phi(\\la ↪\n",
       "↪                 ⎝d\\lambda            ⎠                    ⎝d\\lambda          ↪\n",
       "↪ ──────────────────────────────────────────────────────────────────────────── ↪\n",
       "↪                                                             3                ↪\n",
       "↪                                         (2⋅M - r(\\lambda))⋅r (\\lambda)       ↪\n",
       "\n",
       "↪        2                                       2                             ↪\n",
       "↪       ⎞       2          ⎛   d                ⎞     5          ⎛   d         ↪\n",
       "↪ mbda))⎟  - M⋅r (\\lambda)⋅⎜────────(r(\\lambda))⎟  - r (\\lambda)⋅⎜────────(\\ph ↪\n",
       "↪       ⎠                  ⎝d\\lambda            ⎠                ⎝d\\lambda     ↪\n",
       "↪ ──────────────────────────────────────────────────────────────────────────── ↪\n",
       "↪                                                                              ↪\n",
       "↪                                                                              ↪\n",
       "\n",
       "↪             2\n",
       "↪            ⎞ \n",
       "↪ i(\\lambda))⎟ \n",
       "↪            ⎠ \n",
       "↪ ─────────────\n",
       "↪              \n",
       "↪              "
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "GeodEq1_a = GeodEq1_a.subs({(param[2],pi/2),(dot[2],0)})\n",
    "GeodEq1_a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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caP2m7AKFA9M4IBlDTgltTzaEmIkqeJt1ohwf8ibrmeUgbhef/vKtAJuJKDSewSV5it2Ri5LmWBCPILaxyLHPU71nwAoUDszNAeTK/Q5mH+5bFWBSCwFyOlaOmRBzjFGo3c3mJW8tbpbqQljhQCYHNClxFDnbbVY7yR3yJqvzDDJZW4oVDuyOA3i6cy5b8SjAeTgoxuBwQ1o65HNAMznfzvBx0uRlq8UBrvb3OH6Tu00XY7DboSuED+DAXLdZzefPA9rdVdGyZ7Cr4SrEjuWAndVH/2yZ6rOR/kbx4Y4UHU//BdWk0kFo5YSBAAAAAElFTkSuQmCC",
      "text/latex": [
       "$\\displaystyle \\frac{d^{2}}{d \\lambda^{2}} \\phi{\\left(\\lambda \\right)} = - \\frac{2 \\frac{d}{d \\lambda} \\phi{\\left(\\lambda \\right)} \\frac{d}{d \\lambda} r{\\left(\\lambda \\right)}}{r{\\left(\\lambda \\right)}}$"
      ],
      "text/plain": [
       "                                 d                       d                 \n",
       "    2                      -2⋅────────(\\phi(\\lambda))⋅────────(r(\\lambda)) \n",
       "   d                          d\\lambda                d\\lambda             \n",
       "─────────(\\phi(\\lambda)) = ────────────────────────────────────────────────\n",
       "        2                                     r(\\lambda)                   \n",
       "d\\lambda                                                                   "
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "GeodEq3_a = GeodEq3_a.subs({(param[2],pi/2),(dot[2],0)})\n",
    "GeodEq3_a"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Um dieses System von drei gekoppelten Differentialgleichungen zweiter Ordnung mit Python nummerisch lösen zu können, müssen wir es zunächst in ein System von sechs Differenzialgleichungen erster Ordnung umschreiben. Details zum numerischen Lösung von Differenzialgleichungen finden Sie in der Vorlesung <A href=\"https://itp.uni-frankfurt.de/~hanauske/VPROG/index.html\" target=\"_blank\">Einführung in die Programmierung für Studierende der Physik</A> in den Unterpunkten <A href=\"https://itp.uni-frankfurt.de/~hanauske/VPROG/V8/DGL_1.html\" target=\"_blank\">Differenzialgleichungen: Numerische Lösung von Anfangswertproblemen</A> und \n",
    "<A href=\"https://itp.uni-frankfurt.de/~hanauske/VPROG/V9/DGL_2.html\" target=\"_blank\">Systeme von gekoppelten Differenzialgleichungen und Differenzialgleichungen zweiter Ordnung</A>.\n",
    "\n",
    "Wir definieren dazu formal\n",
    "\n",
    "$$\n",
    "t(\\tau) = y_1(\\tau) \\, , \\quad r(\\tau) = y_2(\\tau), \\quad \\phi(\\tau) = y_3(\\tau) \\, , \\quad \\frac{d t(\\tau)}{d\\tau} = y_4(\\tau) \\, , \\quad \\frac{d r(\\tau)}{d\\tau} = y_5(\\tau), \\quad \\frac{d \\phi(\\tau)}{d\\tau} = y_6\n",
    "$$\n",
    "\n",
    "und schreiben unser DGL-System wie folgt um:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "y1, y2, y3, y4, y5, y6 = symbols('y_1, y_2, y_3, y_4, y_5, y_6')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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AZ+Gu3xyaBNjf3ESDutZ96ffpSdlIUyjt4gDiVs+jIBrSPNI+PPApwuIy9rJEppZCO7n2iHa4ywuvyPurlyp5H8X+Jzl8YujC7fpBAPtz9lpz35PR/bHDGjImuxhkCH2PduxLZYHxy1iJaO5j3wXYnSwLyRGnyGdqg73x/wEcPDfhuovuGQAAAABJRU5ErkJggg==",
      "text/latex": [
       "$\\displaystyle \\frac{d^{2}}{d \\lambda^{2}} t{\\left(\\lambda \\right)} = \\frac{2 M y_{4} y_{5}}{y_{2} \\left(2 M - y_{2}\\right)}$"
      ],
      "text/plain": [
       "    2                                \n",
       "   d                      2⋅M⋅y₄⋅y₅  \n",
       "─────────(t(\\lambda)) = ─────────────\n",
       "        2               y₂⋅(2⋅M - y₂)\n",
       "d\\lambda                             "
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "SysDGL_0 = Eq(ddot[0],GeodEq0_a.rhs.subs({(param[1],y2),(param[3],y3),(dot[0],y4),(dot[1],y5),(dot[3],y6)}))\n",
    "SysDGL_0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/latex": [
       "$\\displaystyle \\frac{d^{2}}{d \\lambda^{2}} r{\\left(\\lambda \\right)} = \\frac{- 4 M^{2} y_{2}^{3} y_{6}^{2} + 4 M y_{2}^{4} y_{6}^{2} - M y_{2}^{2} y_{5}^{2} + M y_{4}^{2} \\left(2 M - y_{2}\\right)^{2} - y_{2}^{5} y_{6}^{2}}{y_{2}^{3} \\left(2 M - y_{2}\\right)}$"
      ],
      "text/plain": [
       "    2                        2   3   2         4   2       2   2       2       ↪\n",
       "   d                    - 4⋅M ⋅y₂ ⋅y₆  + 4⋅M⋅y₂ ⋅y₆  - M⋅y₂ ⋅y₅  + M⋅y₄ ⋅(2⋅M  ↪\n",
       "─────────(r(\\lambda)) = ────────────────────────────────────────────────────── ↪\n",
       "        2                                             3                        ↪\n",
       "d\\lambda                                            y₂ ⋅(2⋅M - y₂)             ↪\n",
       "\n",
       "↪      2     5   2\n",
       "↪ - y₂)  - y₂ ⋅y₆ \n",
       "↪ ────────────────\n",
       "↪                 \n",
       "↪                 "
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "SysDGL_1 = Eq(ddot[1],GeodEq1_a.rhs.subs({(param[1],y2),(param[3],y3),(dot[0],y4),(dot[1],y5),(dot[3],y6)}))\n",
    "SysDGL_1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\frac{d^{2}}{d \\lambda^{2}} \\phi{\\left(\\lambda \\right)} = - \\frac{2 y_{5} y_{6}}{y_{2}}$"
      ],
      "text/plain": [
       "    2                               \n",
       "   d                       -2⋅y₅⋅y₆ \n",
       "─────────(\\phi(\\lambda)) = ─────────\n",
       "        2                     y₂    \n",
       "d\\lambda                            "
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "SysDGL_3 = Eq(ddot[3],GeodEq3_a.rhs.subs({(param[1],y2),(param[3],y3),(dot[0],y4),(dot[1],y5),(dot[3],y6)}))\n",
    "SysDGL_3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Wir werden später die DGLs mittels Python numerisch lösen und setzen im Folgenden $M=1$. Wir benutzen hierfür das Python-Modul <A href=\"https://docs.scipy.org/doc/scipy/tutorial/index.html#user-guide\" target=\"_blank\">SciPy</A>, welches eine breite Kollektion von mathematischen Algorithmen und Funktionen bereitstellt. Im Speziellen werden wir in diesem Semester die Funktion <A href=\"https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.solve_ivp.html#scipy.integrate.solve_ivp\" target=\"_blank\">'solve_ivp(...)'</A> verwenden, die sich im Untermodul <A href=\"https://docs.scipy.org/doc/scipy/reference/integrate.html#module-scipy.integrate\" target=\"_blank\">'scipy.integrate'</A> befindet, welches Funktionen zum Lösen von gewöhnlichen Differenzialgleichungen bereitstellt. Zusätzlich berechnen wir die numerische Lösung auch mittels der älteren Methode \"integrate.odeint()\", die ebenfalls in dem Python-Modul \"scipy\" definiert ist. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt \n",
    "from matplotlib import rcParams\n",
    "from scipy import integrate"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Wir definieren das System der gekoppelten sechs Differenzialgleichungen:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "set_M = 1\n",
    "f_4 = lambdify((y1,y2,y3,y4,y5,y6), SysDGL_0.rhs.subs(M,set_M))\n",
    "f_5 = lambdify((y1,y2,y3,y4,y5,y6), SysDGL_1.rhs.subs(M,set_M))\n",
    "f_6 = lambdify((y1,y2,y3,y4,y5,y6), SysDGL_3.rhs.subs(M,set_M))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "def DGLsys(vy, time):\n",
    "    y1,y2,y3,y4,y5,y6 = vy\n",
    "    dy1 = y4\n",
    "    dy2 = y5\n",
    "    dy3 = y6\n",
    "    dy4 = f_4(y1,y2,y3,y4,y5,y6)\n",
    "    dy5 = f_5(y1,y2,y3,y4,y5,y6)\n",
    "    dy6 = f_6(y1,y2,y3,y4,y5,y6)\n",
    "    return np.array([dy1,dy2,dy3,dy4,dy5,dy6])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "def DGLsys_ivp(time, vy):\n",
    "    y1,y2,y3,y4,y5,y6 = vy\n",
    "    dy1 = y4\n",
    "    dy2 = y5\n",
    "    dy3 = y6\n",
    "    dy4 = f_4(y1,y2,y3,y4,y5,y6)\n",
    "    dy5 = f_5(y1,y2,y3,y4,y5,y6)\n",
    "    dy6 = f_6(y1,y2,y3,y4,y5,y6)\n",
    "    return np.array([dy1,dy2,dy3,dy4,dy5,dy6])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Bevor wir die Geodätengleichung numerisch lösen und die innerste stabile kreisförmige Bahnbewegung eines Probekörpers visualisieren können, benötigen wir die Anfangsbedingungen des ISCOSs. \n",
    "\n",
    "Literatur:\n",
    "\n",
    "<ol>\n",
    "   <li><i>General relativity : An introduction for physicists</i> von M. P. Hobson, G. P. Efstathiou und A. N. Lasenby</li>\n",
    "   <li><i>Gravity : An introduction to Einstein's general relativity</i> von James B. Hartle</li>\n",
    "   <li><i>Allgemeine Relativit&auml;tstheorie</i> von Torsten Flie&szlig;bach</li>\n",
    "   <li><i>Relativistic hydrodynamics</i> von Luciano Rezzolla und Olindo Zanotti</li>\n",
    "</ol>\n",
    "\n",
    "Man kann zeigen (siehe z.B. Seite 206 in <A href=\"https://doi.org/10.1017/CBO9780511790904\" target=\"_blank\">General relativity : An introduction for physicists by M. P. Hobson, G. P. Efstathiou and A. N. Lasenby</A>), dass sich die erste und vierte Gleichung des Systems der Differenzialgleichungen der Geodätengleichung in die folgenden Gleichungen umschreiben lässt:\n",
    "$$\n",
    "\\begin{eqnarray}\n",
    "\\hbox{1. Gleichung:}&& \\frac{d}{d\\tau} \\left[  \\left( 1 - \\frac{2 M}{r} \\right) \\frac{dt}{d\\tau} \\right] = 0 \\\\\n",
    "\\rightarrow  && \\left( 1 - \\frac{2 M}{r}  \\right) \\frac{dt}{d\\tau} = e = \\hbox{const} \\\\\n",
    "\\hbox{4. Gleichung:}&& \\frac{d}{d\\tau}  \\left( r^2 \\hbox{sin}^2(\\theta)  \\frac{d\\phi}{d\\tau} \\right)   = 0 \\\\\n",
    "\\rightarrow &&  r^2 \\hbox{sin}^2(\\theta)  \\frac{d\\phi}{d\\tau}  = l = \\hbox{const} \\quad ,\n",
    "\\end{eqnarray}\n",
    "$$\n",
    "wobei die w&auml;hrend der Bewegungen erhaltenen Gr&ouml;&szlig;en e (Teilchenenergie pro Masse, $e=\\frac{E}{m}$) und l (Drehimpuls pro Masse m, $l=\\frac{L}{m}$) sich aus der Definition des Viererimpulses $p_\\mu = m \\, u_\\mu = m \\, g_{\\mu\\nu} \\frac{dx^\\nu}{d\\tau} = m \\, \\left( g_{tt} \\, \\frac{dt}{d\\tau}, g_{rr} \\, \\frac{dr}{d\\tau}, g_{\\theta\\theta} \\, \\frac{d\\theta}{d\\tau}, g_{\\phi\\phi} \\, \\frac{d\\phi}{d\\tau}  \\right) $ ergeben (Voraussetzung: diagonale Form der Metrik). F&uuml;r die Schwarzschildmetrik erhält man:\n",
    "\n",
    "$$\n",
    "\\begin{eqnarray}\n",
    "&& p_t = p_0 = m \\frac{dx_0}{d\\tau} = m g_{00} \\frac{dx^0}{d\\tau} = m g_{00} \\frac{dt}{d\\tau} = m \\left( 1 - \\frac{2 M}{r} \\right) \\frac{dt}{d\\tau} = m \\, e \\\\\n",
    "&& p_\\phi = p_3 = m \\frac{dx_3}{d\\tau} = m g_{33} \\frac{dx^3}{d\\tau} = m g_{33} \\frac{d\\phi}{d\\tau} = - m \\left( r^2 \\hbox{sin}^2(\\theta) \\right) \\frac{d\\phi}{d\\tau} = - m \\, l\n",
    "\\end{eqnarray}\n",
    "$$\n",
    "\n",
    "Die Klassifikation m&ouml;glicher Bahnen von Probek&ouml;rpern in vorgegebener Schwarzschild-Raumzeit kann mittels eines definierten effektiven Potenzials illustriert werden. Dieses Potenzial h&auml;ngt von dem, bei der Bewegung erhaltenen Drehimpuls pro Masse m ab. Die im Zentralfeld m&ouml;glichen Bewegungen werden mittels zweier erhaltener Gr&ouml;&szlig;en (l: Drehimpuls pro Masse m und e: Energie pro Masse) charakterisiert. Die Definition des effektiven Potenzial erfolgt mittels der radialen, 2. Geod&auml;tengleichung. So definieren z.B. die Literaturangaben 1-3 (siehe Angaben oben) das effektive Potenzial wie folgt:\n",
    "$$\n",
    "\\begin{eqnarray}\n",
    "\\hbox{2. Gleichung:}&\\rightarrow&\\frac{1}{2} \\left( \\frac{dr}{d\\tau}  \\right)^2 + V(r,M,l) = \\frac{1}{2} \\left(  e^2 -1 \\right)  \\\\\n",
    "\\hbox{wobei:} && V(r,M,l) =   \\frac{l^2}{2 r^2} \\left(  1 - \\frac{2 M}{r} \\right) - \\frac{M}{r} \\quad .\n",
    "\\end{eqnarray}\n",
    "$$\n",
    "In der Literaturangabe 4 wird das Pozential V(r,M,l) hingegen wie folgt definiert:\n",
    "\n",
    "$$\n",
    "\\begin{eqnarray}\n",
    "\\hbox{2. Gleichung:}&\\rightarrow& \\left( \\frac{dr}{d\\tau}  \\right)^2 + \\left( V(r,M,l) \\right)^2 = e^2   \\\\\n",
    "\\hbox{wobei:} && V(r,M,l) =  \\sqrt{ \\left( 1 - \\frac{2 M}{r} \\right) \\left( 1 + \\frac{l^2}{r^2} \\right) } \\quad .\n",
    "\\end{eqnarray}\n",
    "$$  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Wir betrachten nun kreisförmige gebundene Orbits, bei denen der Abstand zum schwarzen Loch (der Radiuswert $r$) während der gesamten Trajektorie konstant ist. Solche Trajektorien sind nur bei den Extrema des effektiven Potenzial $V(r,M,l)$ möglich, wobei stabile Orbits nur im lokalen Minimum möglich sind. \n",
    "\n",
    "$$\n",
    "\\frac{d V}{dr} ~=~ 0 \\,\\, , \\quad \\frac{d^2 V}{dr^2} ~<~ 0\n",
    "$$\n",
    "\n",
    "Die Ableitungen berechnen sich zu:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "l = symbols('l')\n",
    "VeffRez = sqrt(1-2*M/r)*sqrt(1+l**2/r**2)\n",
    "VeffNewton = 1 - M/r + l**2/(2*r**2)\n",
    "VeffRezl = lambdify((r,l),VeffRez.subs({(M,set_M)}))\n",
    "VeffNewtonl = lambdify((r,l),VeffNewton.subs({(M,set_M)}))\n",
    "rval = np.linspace(2.2,30, 1001)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/latex": [
       "$\\displaystyle \\frac{M \\sqrt{\\frac{l^{2}}{r^{2}} + 1}}{r^{2} \\sqrt{- \\frac{2 M}{r} + 1}} - \\frac{l^{2} \\sqrt{- \\frac{2 M}{r} + 1}}{r^{3} \\sqrt{\\frac{l^{2}}{r^{2}} + 1}}$"
      ],
      "text/plain": [
       "         ________                      \n",
       "        ╱  2                           \n",
       "       ╱  l                 ___________\n",
       " M⋅   ╱   ── + 1      2    ╱   2⋅M     \n",
       "     ╱     2         l ⋅  ╱  - ─── + 1 \n",
       "   ╲╱     r             ╲╱      r      \n",
       "────────────────── - ──────────────────\n",
       "       ___________            ________ \n",
       " 2    ╱   2⋅M                ╱  2      \n",
       "r ⋅  ╱  - ─── + 1     3     ╱  l       \n",
       "   ╲╱      r         r ⋅   ╱   ── + 1  \n",
       "                          ╱     2      \n",
       "                        ╲╱     r       "
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d_VeffRez = diff(VeffRez,r)\n",
    "d_VeffRez"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Die Extrema von $V(r,M,l)$ liegen bei den folgenden Radiuswerten."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left[ \\frac{l^{2}}{2 M} - \\frac{l \\sqrt{- 12 M^{2} + l^{2}}}{2 M}, \\  \\frac{l^{2}}{2 M} + \\frac{l \\sqrt{- 12 M^{2} + l^{2}}}{2 M}\\right]$"
      ],
      "text/plain": [
       "⎡           ______________             ______________⎤\n",
       "⎢ 2        ╱       2    2    2        ╱       2    2 ⎥\n",
       "⎢l     l⋅╲╱  - 12⋅M  + l    l     l⋅╲╱  - 12⋅M  + l  ⎥\n",
       "⎢─── - ───────────────────, ─── + ───────────────────⎥\n",
       "⎣2⋅M           2⋅M          2⋅M           2⋅M        ⎦"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Extrem_V = solve(d_VeffRez,r)\n",
    "Extrem_V"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Setzt man die freien Parameterwerte z.B. auf $l=4.5$ und $M=1$, so erhält man für die Extrema von $V(r,M=1,l=4.5)$ die folgenden Radiuswerte."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left[ 3.66236702264472, \\  16.5876329773553\\right]$"
      ],
      "text/plain": [
       "[3.66236702264472, 16.5876329773553]"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "set_l = 4.5\n",
    "Extrem_V_set = solve(d_VeffRez.subs({(M,set_M),(l,set_l)}),r)\n",
    "Extrem_V_set"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Anhand der Werte für die zweite Ableitung sieht man welcher dieser Werte ein Maximum bzw. Minimum ist."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/latex": [
       "$\\displaystyle \\frac{- \\frac{2 M l^{2}}{r^{2} \\sqrt{- \\frac{2 M}{r} + 1} \\sqrt{\\frac{l^{2}}{r^{2}} + 1}} + \\frac{M \\sqrt{\\frac{l^{2}}{r^{2}} + 1} \\left(\\frac{M}{r \\left(\\frac{2 M}{r} - 1\\right)} - 2\\right)}{\\sqrt{- \\frac{2 M}{r} + 1}} - \\frac{l^{2} \\sqrt{- \\frac{2 M}{r} + 1} \\left(\\frac{l^{2}}{r^{2} \\left(\\frac{l^{2}}{r^{2}} + 1\\right)} - 3\\right)}{r \\sqrt{\\frac{l^{2}}{r^{2}} + 1}}}{r^{3}}$"
      ],
      "text/plain": [
       "                                                                               ↪\n",
       "                                              ________                      2  ↪\n",
       "                                             ╱  2                          l ⋅ ↪\n",
       "                                            ╱  l       ⎛     M         ⎞       ↪\n",
       "                                      M⋅   ╱   ── + 1 ⋅⎜─────────── - 2⎟       ↪\n",
       "                                          ╱     2      ⎜  ⎛2⋅M    ⎞    ⎟       ↪\n",
       "                    2                   ╲╱     r       ⎜r⋅⎜─── - 1⎟    ⎟       ↪\n",
       "               2⋅M⋅l                                   ⎝  ⎝ r     ⎠    ⎠       ↪\n",
       "- ───────────────────────────────── + ────────────────────────────────── - ─── ↪\n",
       "                           ________                ___________                 ↪\n",
       "         ___________      ╱  2                    ╱   2⋅M                      ↪\n",
       "   2    ╱   2⋅M          ╱  l                    ╱  - ─── + 1                  ↪\n",
       "  r ⋅  ╱  - ─── + 1 ⋅   ╱   ── + 1             ╲╱      r                       ↪\n",
       "     ╲╱      r         ╱     2                                                 ↪\n",
       "                     ╲╱     r                                                  ↪\n",
       "────────────────────────────────────────────────────────────────────────────── ↪\n",
       "                                                       3                       ↪\n",
       "                                                      r                        ↪\n",
       "\n",
       "↪     ___________ ⎛     2         ⎞\n",
       "↪    ╱   2⋅M      ⎜    l          ⎟\n",
       "↪   ╱  - ─── + 1 ⋅⎜─────────── - 3⎟\n",
       "↪ ╲╱      r       ⎜   ⎛ 2    ⎞    ⎟\n",
       "↪                 ⎜ 2 ⎜l     ⎟    ⎟\n",
       "↪                 ⎜r ⋅⎜── + 1⎟    ⎟\n",
       "↪                 ⎜   ⎜ 2    ⎟    ⎟\n",
       "↪                 ⎝   ⎝r     ⎠    ⎠\n",
       "↪ ─────────────────────────────────\n",
       "↪                ________          \n",
       "↪               ╱  2               \n",
       "↪              ╱  l                \n",
       "↪        r⋅   ╱   ── + 1           \n",
       "↪            ╱     2               \n",
       "↪          ╲╱     r                \n",
       "↪ ─────────────────────────────────\n",
       "↪                                  \n",
       "↪                                  "
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dd_VeffRez = diff(VeffRez,r,r)\n",
    "dd_VeffRez"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left[ -0.067312449891573, \\  0.000175703779112795\\right]$"
      ],
      "text/plain": [
       "[-0.067312449891573, 0.000175703779112795]"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[dd_VeffRez.subs({(M,set_M),(l,set_l),(r,Extrem_V_set[0])}),dd_VeffRez.subs({(M,set_M),(l,set_l),(r,Extrem_V_set[1])})]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Im Folgenden betrachten wir uns die Form des effektiven Potenzials (Literaturangabe 4) bei Variation des Parameters $l$ ($l \\in [3,4.5]$ und $M=1$). "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "rcParams.update({\n",
    "    'figure.figsize'    : [10,8],\n",
    "    'axes.titlesize' : 14,\n",
    "    'axes.labelsize' : 16,  \n",
    "    'xtick.labelsize' : 14 ,\n",
    "    'ytick.labelsize' : 14 \n",
    "})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 1000x800 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.cla()\n",
    "plt.xlim(2.3, 20)\n",
    "plt.ylim(0.8, 1.1)\n",
    "plt.ylabel(r\"$\\rm V_{eff}(r)$\")\n",
    "plt.xlabel(r\"$\\rm Radius \\, r \\,[km]$\")\n",
    "\n",
    "set_data = [\n",
    "    (4.5, \"yellow\", r\"$\\rm l=4.5$\", 1),\n",
    "    (4, \"green\", r\"$\\rm l=4$\", 1),\n",
    "    (3.7, \"red\", r\"$\\rm l=3.7$\", 1),\n",
    "    (float(2 * sqrt(3)), \"black\", r\"$\\rm l=2 \\, \\sqrt{3}$\", 0),\n",
    "    (3, \"blue\", r\"$\\rm l=3$\", None)  \n",
    "]\n",
    "\n",
    "for l_val, color, label_text, i in set_data:\n",
    "    plt.plot(rval, VeffRezl(rval, l_val), linewidth=1, color=color, label=label_text)\n",
    "    if i is not None:\n",
    "        Extrem_V_set = solve(d_VeffRez.subs({(M, set_M), (l, l_val)}), r)\n",
    "        x = Extrem_V_set[i]\n",
    "        y = float(VeffRez.subs({(M, set_M), (l, l_val), (r, x)}))\n",
    "        plt.scatter(x, y, c=color, s=60, marker=\"o\")\n",
    "\n",
    "plt.legend(loc=\"lower right\", fontsize=18)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In der oberen Abbildung erkennt man, dass bei kleinen Drehimpuls-Werten mit $l < 2 \\, \\sqrt{3} \\approx 3.4641$ keine gebundenen stabilen Orbits des Probekörpers mehr möglich sind (fehlendes Minimum in der blauen Kurve). Für $l > 2 \\, \\sqrt{3}$ sind kreisförmige und elliptische gebundene Orbits möglich, wobei der jeweilige Punkt auf den Kurven den Radiuswert der kreisförmigen Bahn des Probekörpers kennzeichnet. Der Punkt auf der schwarzen Kurve stellt die Grenze der innersten stabilen kreisförmigen Bahnbewegungen dar, der sogenannte \"Innermost Stable Circular Orbit (ISCO)\" eines Probekörpers um ein nicht-rotierendes schwarzes Loch. Er befindet sich bei $r_{\\rm ISCO}=6M$ und $l_{\\rm ISCO}=2 \\, \\sqrt{3}\\,M$, was wir weiter unten zeigen werden. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Die Existenz eines solchen ISCOs ist ein allgemein-relativistischer Effekt der in der Newtonschen Theorie nicht vorhanden ist. Um dies zu verdeutlichen, vergleichen wir das effektive Potenzial der allgemeinen Relativitätstheorie $V(r,M=1,l) = V(r,l)$ mit dem effektiven Newtonschen Potenzial $V_{Newton}(r,l)$. Für große Abstände geht das effektive Potenzial in das Newtonsche über und es gilt (für das effektive Potenzial der Literaturangabe 4):\n",
    "\n",
    "$$\n",
    "\\lim\\limits_{r \\to \\infty}V(r,l) \\approx 1 - \\frac{M}{r} + \\frac{l^2}{2 \\, r^2} = V_{Newton}(r,l)\n",
    "$$\n",
    "\n",
    "Wir betrachten uns nun die Form dieses effektiven Newtonschen Potenzials bei Variation des Parameters $l$ ($l \\in [1,4]$ und $M=1$). "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "rval = np.linspace(0.1,30, 1001)\n",
    "VeffNewton = 1-M/r + l**2/(2*r**2)\n",
    "VeffNewtonl = lambdify((r,l),VeffNewton.subs({(M,set_M)}))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 1000x800 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.cla()\n",
    "plt.xlim(0.1, 20)\n",
    "plt.ylim(0.45, 1.1)\n",
    "plt.ylabel(r\"$\\rm V_{Newton}(r)$\")\n",
    "plt.xlabel(r\"$\\rm Radius \\, r \\,[km]$\")\n",
    "\n",
    "set_data = [\n",
    "    (4, \"green\", r\"$\\rm l=4$\"),\n",
    "    (3.7, \"red\", r\"$\\rm l=3.7$\"),\n",
    "    (float(2 * sqrt(3)), \"black\", r\"$\\rm l=2 \\,\\sqrt{3}$\"),\n",
    "    (3, \"blue\", r\"$\\rm l=3$\"),\n",
    "    (2.5, \"orange\", r\"$\\rm l=2.5$\"),\n",
    "    (2, \"brown\", r\"$\\rm l=2$\"),\n",
    "    (1, \"grey\", r\"$\\rm l=1$\")\n",
    "]\n",
    "\n",
    "for l_val, color, label_text in set_data:\n",
    "    plt.plot(rval, VeffNewtonl(rval, l_val), linewidth=1, color=color, label=label_text)\n",
    "    rmin_set = solve(Eq(diff(VeffNewton.subs({(M, set_M), (l, l_val)}), r), 0), r)\n",
    "    if rmin_set:\n",
    "        rmin = rmin_set[0]\n",
    "        plt.scatter(rmin, VeffNewtonl(rmin, l_val), c=color, s=60, marker='o')\n",
    "\n",
    "plt.legend(loc='lower right', fontsize=18)\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Man erkennt, dass stabile kreisförmige Bahnen in der Newtonschen Theorie auch für kleine Abstände (kleine Drehimpulse) möglich sind und eine Grenze der innersten stabilen kreisförmigen Bahnbewegung (ein ISCO) nicht existiert. Zusätzlich sehen wir, dass Probekörper nie in das Zentrum bei $r=0$ fallen können, da es eine unendlich starke Potenzialbarriere für kleine Radien gibt."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Berechnung des ISCOs\n",
    "\n",
    "In der allgemeinen Relativitätstheorie sind jedoch nur stabile Bahnen möglich, falls der Wert des Drehimpulses l oberhalb einer gewissen Grenze ist, da nur dann ein Minimum im Potenzial vorhanden ist. Kreisförmige Bahnbewegungen sind dadurch charakterisiert, dass der Wert des Radius sich im Laufe der Zeit nicht verändert und somit sich der radiale Abstand des Probekörpers vom schwarzen Loch gerade im Minimum des effektiven Potenzials befindet. Für kreisförmige Bahnbewegungen muss somit gelten: $\\frac{dV(r,M,l)}{dr}=0$. Dies führt zu folgender Gleichung:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/latex": [
       "$\\displaystyle \\frac{M \\sqrt{\\frac{l^{2}}{r^{2}} + 1}}{r^{2} \\sqrt{- \\frac{2 M}{r} + 1}} - \\frac{l^{2} \\sqrt{- \\frac{2 M}{r} + 1}}{r^{3} \\sqrt{\\frac{l^{2}}{r^{2}} + 1}} = 0$"
      ],
      "text/plain": [
       "         ________                          \n",
       "        ╱  2                               \n",
       "       ╱  l                 ___________    \n",
       " M⋅   ╱   ── + 1      2    ╱   2⋅M         \n",
       "     ╱     2         l ⋅  ╱  - ─── + 1     \n",
       "   ╲╱     r             ╲╱      r          \n",
       "────────────────── - ────────────────── = 0\n",
       "       ___________            ________     \n",
       " 2    ╱   2⋅M                ╱  2          \n",
       "r ⋅  ╱  - ─── + 1     3     ╱  l           \n",
       "   ╲╱      r         r ⋅   ╱   ── + 1      \n",
       "                          ╱     2          \n",
       "                        ╲╱     r           "
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "EqCircular = Eq(d_VeffRez,0)\n",
    "EqCircular"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "L&ouml;st man diese Gleichung nach r auf, so erh&auml;lt man zwei L&ouml;sungen, wobei die zweite (positives Vorzeichen) dem stabilen Minimum und die erste (negatives Vorzeichen) dem instabilen Maximum entspricht:\n",
    "\n",
    "$$\n",
    "\\begin{eqnarray}\n",
    "\\frac{dV(r,M,l)}{dr}=0 \\quad \\Rightarrow \\, r_\\pm = \\frac{l}{2 M} \\left( l \\pm \\sqrt{l^2 -12 M^2}  \\right)\n",
    "\\end{eqnarray}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/latex": [
       "$\\displaystyle \\left( \\frac{l \\left(l - \\sqrt{- 12 M^{2} + l^{2}}\\right)}{2 M}, \\  \\frac{l \\left(l + \\sqrt{- 12 M^{2} + l^{2}}\\right)}{2 M}\\right)$"
      ],
      "text/plain": [
       "⎛  ⎛       ______________⎞    ⎛       ______________⎞⎞\n",
       "⎜  ⎜      ╱       2    2 ⎟    ⎜      ╱       2    2 ⎟⎟\n",
       "⎜l⋅⎝l - ╲╱  - 12⋅M  + l  ⎠  l⋅⎝l + ╲╱  - 12⋅M  + l  ⎠⎟\n",
       "⎜─────────────────────────, ─────────────────────────⎟\n",
       "⎝           2⋅M                        2⋅M           ⎠"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "LoesCircular = solve(EqCircular,r)\n",
    "LoesCircular[0].simplify(),LoesCircular[1].simplify()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Die innerste kreisförmige Bahnbewegung (ISCO) hat zusätzlich die Eigenschaft, dass das Maximum und Minimum zu einem Sattelpunkt zusammen fällt - es muss somit zusätzlich gelten: $\\frac{d^2V(r,M,l)}{d^2r}=0$. Dies führt zu folgender Gleichung:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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G+IiGa/CVUqow9sLx/Eb3HIMVUwntoKg3KLaQu4ifYP3sLZQIpasr/witxlfhrZfZixyEcTjyiCPL9lYgoJltISm4TgUgBcIi0fBc+RSTxFZbiZT99GxUPnSgRKfs6AtM8GJNqEYnlCprUl8vQcNi2+ivev/ocTC1/lzfI51yW+ukzVsCOmFJM1Cdb+oSSpoMmKouBF4qkG0BCAGOyGrsk+16oQx7rWvvnjflUxcGA9lsHlUJOdH0NlJpbJmYLNxVbudyoMLBPNAEnU8UhoAJedPfvpKP4pQ3CiC0sO0jlNvOm7Lhb4uPr9pljT1Db9YCbohB1QHL8NT71P3YQ9mtiivbw9CEalX+kV4O1rnHyi/0kcVZi+cpE8o5fZ0xrzHhVxmLx0G9O/bhbDb9Wbww1uHqmN+HZPR7lxGtJlWdUQ2PQRalhkF2t+VKg78NAqpLBCbKCxbAyYMJ7UDpGcxReC965oMDwlJzdSEXizYEZv14sd58hQm49m2LIJ9AH5Zg8ARLTnlggH/NPWFrnfJhewEeITfk4GsszdD7UeJK/slrcpuMUvDOmYhPjgM6NY+xIBVO9AU87TiaU75hQtmlBE/u6yKICSYrJZ9HIo4xoNOpjKz6s5gAR1wY7+6fMvu1hTWvCqvPFhFqUYRpXhCcktpX4iocazOHwdT3vF00sKP8BSHXJZCu+FX6gEXdusdSZLA+I0heX73YClB63kewYMEmFsWkyEPPCDX2L0ITkwauTxVeHEmje+IuesbqOolu0vc55UEZHIeGAj3k4OtwhVU0nELADQHtuEUIoPCwVWaSU3sf7Mtlf2BCyThBPw19POQ/qa+XiX/SlfK4bupEZ279mck4Ro2sdQYrrDObtSqcDlQIM12DYPmq3RD0jKAkHYKRDtnuiAouBOILXehkLAGyJ5GOcNGVcn6U54Dr4stmXeuWAARf6ktIItFuAgIsY/FhQVH3E9JflJb2sVqRmlJWhDSzhJzKQ2nr3e6guEfygwOv8On6+1pOY2iHs1+1vve3Hc9fW6q4e6f79ikDISrWlT5O3R7tTiHgjgbxhOWzqhfk3xT2evuy+lLvhLKW8ZK+zlJ+LYvh2437dCr9+YlQ2FyRH0Z6fawV1vkY0hmqD1h0j8JJx2v8/7TCsaaEJfvOGaniaUR0SKw6VZ56vuiZGSfLvUEpJrhwCqfMYpmzDPIlYwRUn9R/3Xo+yo3So7DlMmGZJeTEG9teeGfxET96f2vF8qIy6JtD+x/ZwjN3UjEqoEcbR5wEpxBwcaBwLjUEMKwgA+tH8dWim7dK19uXFTdrzGvm/M+T8hns64qnLTMJpL/2ncd8UTqMQPXJIqsryNrgcu7PH4mJXORFwPvqaoX1CpLRgK/ViOsNl0ZOWPiyuJ4BHRKlkzQNp/R0jOCDYttIowesHA2Bp/fIa2gJpZ2HnzNAQPVat55nQPEsEmcJuTLnoe0OWG3qgmUWMbESl3XWmGhGyBu+4O9odwoBdzSIZytfbR5Dyh/ii3Y/NFmrsz7Ul+vpuu5j9HXk9dh5zBelafCjZ1Y81/bvw/uzeGAFl+0A2cuYu64W4rBBBGjALHFOdXTuKyupwlgCxkTP7K+hlOo5OOIqZVbpseSmcFRRoM9XIzCKgNrtWyUKQm5Kegb5vj7B+wy8XX2KuCSdMAhbD7Cg8g9dbBXqcsQfKlhE22kEXBfADluNwHPl0Nd+G5mrLY315Ub6jocYfb1uyWXcYDxa5XLqz2IUnaVuKV7F+5Ev28I6E3011PZxMzQGZi8I5MrpmY7BXy0Sd+GZexLoWiipumLF4LnxbhlGR+edSnDrvt7xSGZnBHJBACFXrA6MEax2jkAZWr6iv3RusxnLO8SrDJQy+hOTwODqH3aFsChXlQfN+M697LVCoOlqPKjF73F7GgG3B1i3VobaMsv8P8rzfcXgvkjFj/XlMfhi9PX6BJAxaGhsGaOniBdfWfRn0Ykegm/rLZP4TC3RXWoE5USPGgMKJzPNLkWyUEpLfmjcQTlFEUUw0onqacqk1YW4owVXRYxvjMAaBNTeEWyPdJ1kmRkp66Xii/40km4o+rUiw4ddDOZMDOuCbejdLePo99WqypYFdeWt+jmVgOvi0WFREGD5/FmUnIYzidHXixLUtsO3H5URaLjo9ZZY5X9of1b5hZFAvDNxyN5ZYV1YheXATmPgHL+uxlA/348tBMEiFLYCUDJ7VPuE09X+VV6wm4+A6ofN81jU7I5FIIqQU10WEzldFyutehcFmj6GQLzomZWQQWsR6bZ0Kh9lEVqmCtQtyDmVgNsCIOdZtNHCYqe2ikK2mVP+q/s6xCmf0K75GLo6FWSIcKVr7GkdStsVp/cP7c8qn/GRrUinsK6C8cMuoB02jEDZEPnwCYF3KZ+59llosOTwL0Z07row4rmvUxD3jd7BGtulECvabiIChVV7Ylon2wgBteMf5GnTT+Tr/WBJiQgglq7xS92aj0GWljn0HmMBfB3iVCdBwK1eMj2EARe6NwJfqUCO5GuckLMBEaN9XW23d4uP4qYcn7UB2YVsP6w/i6Ef5deMj1tgsirPu1Vv3+DLavzMmlha4AtCZi8M8ggalhgLV6ZBSQ2OWeJj+bAV4FK+x7WYQYaEtWvYv2rLYA0U32aPAEJu9QcA6jdYDegjhRVjLip6j0nMWqV5brG96Us+4OdIa8jpBFwv4I5YjYDaKrILObi6Pw8RU/aJsb6O/O3c4qP3We18UPNbK9gXlcW4dFh/VvlYkVk1SmaMG6rjqXFWWKci9U86ZmtYP7kG/7UaRmEFLRsKy/xPdU86HFsC+Mu4YsmxTMPM663u+2ZgWKNodH0KraLsjEBeCKhNxxRyWAIXCUvRQd9LyZIIH4dZQ8qx5nQCLq/ekR+1ajd8RMje9K0NK4N9XeWntsXnsP4sLNA70D9SGt+iNO4H7961/9glSr7OxAgkg4A6Lh/6MNsd+0I7GZrPTojqgqVE/kVqlUVR72PJ4GvlbOtWtDMx5cvrvi1FmzYHlYuA4zD4ro9HNy3bmZ8DAbUd+jPb5DZrw8p7sK8r/o18cd7qkaiKhsP6s8pm5eh/8p/qvjCiHYlF7LKtsMZG1Pklh4A6rhXW5Gql2Pu9uZBLkO2kSDq7gEsKbBOzGQJlO+aordNZFTcDLcOMC4W1NmihnU91X+i9xnK1nm2unYqe081GQO1r7C/4WIYJ2zDq+T8qH9jn1HZ8NTo4yLldtyHz860jMNQXFResPKvkSRfG7otdqDjMCKxDYKg/r8s57tu2sMbF07kliIA6oy2sCdaLSTICRsAIGAEjMBWBh1MTOt05EZAyx76g+okGyTKayywwWQBNmBEwAkZgJQKWGSsB9OuLEbDCuhi607yI9ZEzYk91/MVpaiczRtSOOObtqTwfHtyU84Tqpqr7lpm1zLjl2j+QdyusB4KfSNEoq9l+YZ0IhibjHwQ4po2vhfv+EOOflCvvVAZKMeWxn77xL1V6RnHma9lX8tBzui9mxZedETgCAcuMI1B3mef6p6tSSNm6M7Fhl3g1PpwbelXp+aCJDynYRsDhy/9VmC2zAsKu+OofJfGiNjG5TfXhNrGt0fY4Fot/nGsorHp+LA8th51tSvl2RuBMCKg/0ccn9++J/fhMEJmXDRE4m4V1N+vOhnWyZ9Yo9y9nFIiCWpxzpyuK6xv5wS/3Z+TtpPkj8Ews0AdjuCltDaWU8jgeq3JqmyxZch7kb1Wgb4yAEYiBgGVGDBSdxyIETqOwSkhFs+5MQVLlnWE5kr+W7Vy6VThWVBSB57oP1qv6weLE57LMCp250CpS83Nle+HPGWJZ3Ke0teKvjsuyC9DKe5RVVgP4xzk7I2AE4iFwKzIjHmLOKRoCp1FYhUg0646EHsIOC+LQ0nfWy5HicXBpR/H8TSPnllbLP4TpOTgsW4Pnl4aER19Fdywl6mhWUi6fiU/n5GcJ0TPbGn97jLJM+0Sg8q9RjAecyxvVKV/GhS/lv9F9XamOWo4zMwKpIaD2fjMyIzXsTc89AndnAEIdCcUypnWH5UiEHh8jIYS7LDVhObLYKxdw1Du5LEeyt29MoD8WP3UltWBTYbzHhyxWBEPF3/BV7QAljrYSvT30tbWyzGA1Z1L1RGFMNAMN3M/aElCWpde6neIR2PxnOvzi7YzALSFgmXFLtZ0gr7srrAgVef77+1v5FzxHwCWqdUf01C0nKMNBMNZJLZYjFUB84cQL94WVR9cuJfc+YRq/wSLVSU3Jy5XAVziWVZRV/iHq5o4u6gTLgbSJ57FhGGlrTBRD+/xV93x4ddE7b+VRJl9zT9gMV/XlrneUH22eSezVJK4rvcOMwMkQsMw4WYXmxs7dAQRPsV5OJqsUTlGtO8qzLpDGlr6L5ciSYJYjsfZ8Il8tpU9mZiQhvMp/Lf/7SNLBaL3PJGFMoSZNg4eyXJTUN7rnb3g3+QJbeUfhU/TZbYwAdaUimISGfc5RSlR+tPHOtqY4VjHol0/LwgqrqsJ/kccK+iPhuo8xGS6L8MUI3C4CZV9KUmaINsuLG2maDw/gc4r1cg5Zm1h3IEAdISx9txU3hHSw3hCHwEbJrS9HfqXnyY6y5HsVQMUhiIMApvw1jr2nY/sNqaf219f1ultU/s58LqLRL81C4JlS0wejOrWT3ramOJTjSkHWM32x2E+te/pjFnurowLmzIzAtggcIjPUn/eUi9si6NxXI7C7hVUNcI71cpBB5YXiFt26Q6HKGyHct/Sd+3IkSztB4YbdLodyjPUYa1ZM98FQZirPy65DAO0YR93Lh0lSX8mk4SxUOyNgBM6LwFEyw/LivG1qNmfRLawIOVHBfjKWDx7Jo1RyyPx3iquUVd13Wi+Vbo7byrrDciQWTfbZQg90F/8GVfJHuVhTseZwfarwsBxJ3EXPKNLB4kpQEk40UT8Ny2kPYViw+MCksmT1pHPwCRFQO2E5nrZM2++0dpZpXvSxr3j60BTrK1/c18cGtptEdyqjOjNY99DdFoZsLWove7KFyQp59NpwhlsioDY7SQ5PoaHMyzJjClhOsykC0RVWUYuSSuPGY6H8Xj4ogIVQ0nOwXvKBBEphoQwqfeUURocjfkjpw7qDMhzVKc9OAU0hikt+OVI0sqenz4LKvr/R7Qp6f2zLQFTMnVk6CKjuUTRR0t6Xp3+GvdltIlE0e/uf4pjQFR9DtV8cetZ7lWI5lG5NnMq42n6jsJ/lZ9O7hg6/awQ2QmBUDtfLVbu3zKgD4vskEYiqsKrRI+hQMJ/Ic45nsEx8pXuE10XXoLxeWS+JxynNauvOfU7zf1X25tad+VRNf0P0YyEC616FU2n6lNnLWv71/ubKxnQ0nHIJAqrD0Ff58j/0xca+UKXB+t5rXV1Srt8xAkZgPQLqm6NyuF6K0i+WGXp3lbzU+5YX9crw/SACsRXWIOiwUrwMJatRFuE8677XelnGx7LukM+S5cjNO5Aw2GQ5UvmiXIA11ukrhVXxhFf1ovsrpzTR+N+KzyuiHbAJAqo/VjiYdH6ta/tIG9rXYF/ehKgDMwUDFY9llgk5eDC+/K3r1QqRwu2MwCEIqD2OyuFAmNKukhl63/IigOnr5ghEVVhr1DKgL7K+1DrbKutOmU+Sy3uibZPlSOXL9guEKtbrtoJB9bDXtmEpI3CO0/vULfuSKYflYPb4dW7bUPgmfKpMu/0QQCnDmoqCWtSn6pWJD3+s0WupV/xmTuVCD8oyqzXBsaS5qeKo/NnSdDURDAT4agQSQ2BUDtNn5JOQGaLD8iKxBpQaOXexCSobP8KjU4mZWp7eRxhi3WGfKh2q7hAaYbtBPfzm74UVQhWPUlE5haNkxnBRz9GNQZDz2A6Bsj3Rl7Gyhjb0TM/Ptyt1NOfXSvGn/IeiiXEA+or98bqucTHyWFO+3zUCURBQv0BmTpLDZR+3zIiCvDPZEoEtLKwItVhflidn3dmyMobyLgcgZqDMmseWI8H/qXzd4vSlnge3Ayh+iqsvAzMoMu4OxuIAAAaDSURBVLGI5mbyGa1cZzSIAP2QdofSimDjuLOo9T5YeitSZWPdZd9dYe0saVqtbCqfKwtPq2g/GoFcEJgrh7OUGeqzyKCpcjGXujOdPQhEV1jVgNg/s2rZOdCKIJIP1h1OHEBIYt35NKTZ+yoacliORDFtbwv4QrSv3iKhPOqKAYpMlLoO9Vjm72XXAEgCV9UJR7bRr+l71H/UOl/IIkdQFXv1Fr7v14zAaREo+8acfpqlzBCfjEeWF6dtyU3G7pqPST6hFOGw7rDMfah1R+UnvxxZDlZ05GJbgJ6ZbdcVTT2uc8qTPcpMIuZu/YhKxzou/PYMBNgCULQj1fmhdVi257ntbgarTmoEbgsB9SkmfynKjEPHmttqBelzG93CGptlFKKyMyVh3REtuSxH1pd4sApPOfi5qD7xOGhFVjyTiMLirXssufWtB4NNQGm97DqIUJqRqjfaPULtcGuG6GClZY71KE1QTZURSAuBRTJD/XFQXsCi0iySGXrP8iKtNrI5NWVboZy/5fmwu/rTqRwsrBCejHUHYuRyWI5kiYcD37GKfabrHItUrxVZ+fBlNkehvNE9Z/B5QBEIt+BU3x/L2+JxC5VtHm8RgaUyo1deAKLGDMuMW2xNC3gu28qvumIQwxCGgYQ/dGGv8uXBu3erzv0lj11cyQj7MA8VmCofBfBHXZO38IjGN6L1FRWk+1mKpdIXH7XoilXNzggYASNgBE6OgMb7RTLD8uLkDWMH9tSGsNSjqDb+PVHPWOcxvn2W/JaAgJOIrX+dHoJ3v4qOnJYjUVbDcs1crHKwIs/lyemNgBEwAkagH4GlMsPyoh9Tx0xDACNgl4HsV4Wz9fC9XLYETGPXqdoIsG+VfyvqagTttNUzDUMPc7YQVO/6xggYASNgBLJFYLbMsLzItq5TI5yjE9le0nZhVf2JFdY2NCd61kCC0jnbMq33UHKT3/JwoqoyK0bACBiBwxFYIjMsLw6vtuwJUBvCSDbmHmWzJWCME8d3I6CGEGYn3QkcagSMgBEwAkagRMAyw01hCAG1D460xBo6x/2h94aMYI/KzNhy2efes8LaB43DjYARMAJGwAgYASNgBCoEpHjO+oC7enH9zfsPVXgexwSsZ9Y5GAEjYASMgBEwAkbACHQgIH3wQUfwHkFde1dDucH6+jcK61EEBmJ8NQJGwAgYASNgBIyAEbhBBKSH8t0MnHftZQ1hf3lLwA02DrNsBIyAETACRsAIGIG5CEix3GIPK2TwkXjxBwEtmoKF9RcrrC1k/GgEjIARMAJGwAgYASNwjYAU1q32sHKkGn8S0HacdMRHW2+z+aerNgd+NgJGwAjEREADIrP7J/Lsp+L+E/nnCp91jrHesTMCRsAIGIGZCGis/VOv8G9XP/GqrmwH+D/5T3X/hy2soGJnBIyAEbif3f+lgfEHwND1W13+J/8vnu2MgBEwAkZgUwSwpn6nsRdjwd/yXAtlVdeLFVZQsDMCRsAIyJraAuF9PQ99vdpK7kcjYASMgBFYioAUVc5h7d1yYIV1KbJ+zwgYgVMhoMGyvfT/uRjs2lN1Kr7NjBEwAkYgBwSssOZQS6bRCBiB3RAotwI8VYEvdF9sD9itcBdkBIyAETACnQj4o6tOWBxoBIzALSMgRZXN/ny1+rPuv79lLMy7ETACRiAFBKywplALpsEIGIHkEJCi+pGI+l3+Y923twskR68JMgJGwAicGQFvCThz7Zo3I2AELqXiyRL/R7r/TJ4jq/4tzwb/P/X8g3zj+BSF44jHPZa3wlpA4R8jYASMwDEI3B1TrEs1AkbACOyGwFMppCioF135l5ZvdA1fohYfVekZ5ZQTAf6SD+6Jbgh/FQJ8NQJGwAgYgWMQsIX1GNxdqhEwAjsgIEUUayqHUeP4i78PFPZh8XT/v9X1j6q+UPgzxXP+H44zANkOECytRaB/jIARMAJGYH8ErLDuj7lLNAJGYD8E3lNRwULKntRgWb1IEa3uIUfPLPt76R8w7IyAETACiSHgj64SqxCTYwSMQHwEpIyyvP+z/L90b4tpfIidoxEwAkZgUwS8h3VTeJ25ETACiSDAcj9/u2plNZEKMRlGwAgYgTkIWGGdg5bTGgEjkCsCWFh/ypV4020EjIARuHUErLDeegsw/0bg5AjIqsqHV3i2BNgZASNgBIxAhghYYc2w0kyyETACsxBAWWU7wC+z3nJiI2AEjIARSAaB/weNjLiJEUoDgAAAAABJRU5ErkJggg==",
      "text/latex": [
       "$\\displaystyle \\frac{- \\frac{2 M l^{2}}{r^{2} \\sqrt{- \\frac{2 M}{r} + 1} \\sqrt{\\frac{l^{2}}{r^{2}} + 1}} + \\frac{M \\sqrt{\\frac{l^{2}}{r^{2}} + 1} \\left(\\frac{M}{r \\left(\\frac{2 M}{r} - 1\\right)} - 2\\right)}{\\sqrt{- \\frac{2 M}{r} + 1}} - \\frac{l^{2} \\sqrt{- \\frac{2 M}{r} + 1} \\left(\\frac{l^{2}}{r^{2} \\left(\\frac{l^{2}}{r^{2}} + 1\\right)} - 3\\right)}{r \\sqrt{\\frac{l^{2}}{r^{2}} + 1}}}{r^{3}} = 0$"
      ],
      "text/plain": [
       "                                                                               ↪\n",
       "                                              ________                      2  ↪\n",
       "                                             ╱  2                          l ⋅ ↪\n",
       "                                            ╱  l       ⎛     M         ⎞       ↪\n",
       "                                      M⋅   ╱   ── + 1 ⋅⎜─────────── - 2⎟       ↪\n",
       "                                          ╱     2      ⎜  ⎛2⋅M    ⎞    ⎟       ↪\n",
       "                    2                   ╲╱     r       ⎜r⋅⎜─── - 1⎟    ⎟       ↪\n",
       "               2⋅M⋅l                                   ⎝  ⎝ r     ⎠    ⎠       ↪\n",
       "- ───────────────────────────────── + ────────────────────────────────── - ─── ↪\n",
       "                           ________                ___________                 ↪\n",
       "         ___________      ╱  2                    ╱   2⋅M                      ↪\n",
       "   2    ╱   2⋅M          ╱  l                    ╱  - ─── + 1                  ↪\n",
       "  r ⋅  ╱  - ─── + 1 ⋅   ╱   ── + 1             ╲╱      r                       ↪\n",
       "     ╲╱      r         ╱     2                                                 ↪\n",
       "                     ╲╱     r                                                  ↪\n",
       "────────────────────────────────────────────────────────────────────────────── ↪\n",
       "                                                       3                       ↪\n",
       "                                                      r                        ↪\n",
       "\n",
       "↪     ___________ ⎛     2         ⎞    \n",
       "↪    ╱   2⋅M      ⎜    l          ⎟    \n",
       "↪   ╱  - ─── + 1 ⋅⎜─────────── - 3⎟    \n",
       "↪ ╲╱      r       ⎜   ⎛ 2    ⎞    ⎟    \n",
       "↪                 ⎜ 2 ⎜l     ⎟    ⎟    \n",
       "↪                 ⎜r ⋅⎜── + 1⎟    ⎟    \n",
       "↪                 ⎜   ⎜ 2    ⎟    ⎟    \n",
       "↪                 ⎝   ⎝r     ⎠    ⎠    \n",
       "↪ ─────────────────────────────────    \n",
       "↪                ________              \n",
       "↪               ╱  2                   \n",
       "↪              ╱  l                    \n",
       "↪        r⋅   ╱   ── + 1               \n",
       "↪            ╱     2                   \n",
       "↪          ╲╱     r                    \n",
       "↪ ───────────────────────────────── = 0\n",
       "↪                                      \n",
       "↪                                      "
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "EqSattel = Eq(dd_VeffRez,0)\n",
    "EqSattel"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Wir setzen in diese Gleichung für r die Lösung von \"LoesCircular\" ein:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/latex": [
       "$\\displaystyle \\frac{16 M^{4} \\sqrt{\\frac{- 4 M^{2} + l^{2} + l \\sqrt{- 12 M^{2} + l^{2}}}{l \\left(l + \\sqrt{- 12 M^{2} + l^{2}}\\right)}} \\left(4 M^{2} \\left(4 M^{2} + \\left(l + \\sqrt{- 12 M^{2} + l^{2}}\\right)^{2}\\right) \\left(4 M^{2} - l^{2} - l \\sqrt{- 12 M^{2} + l^{2}}\\right) - \\left(4 M^{2} + \\left(l + \\sqrt{- 12 M^{2} + l^{2}}\\right)^{2}\\right)^{2} \\left(- 3 M^{2} + l^{2} + l \\sqrt{- 12 M^{2} + l^{2}}\\right) + \\left(8 M^{2} + 3 \\left(l + \\sqrt{- 12 M^{2} + l^{2}}\\right)^{2}\\right) \\left(4 M^{2} - l^{2} - l \\sqrt{- 12 M^{2} + l^{2}}\\right)^{2}\\right)}{l^{2} \\sqrt{\\frac{4 M^{2} + \\left(l + \\sqrt{- 12 M^{2} + l^{2}}\\right)^{2}}{\\left(l + \\sqrt{- 12 M^{2} + l^{2}}\\right)^{2}}} \\left(4 M^{2} + \\left(l + \\sqrt{- 12 M^{2} + l^{2}}\\right)^{2}\\right) \\left(l + \\sqrt{- 12 M^{2} + l^{2}}\\right)^{4} \\left(4 M^{2} - l^{2} - l \\sqrt{- 12 M^{2} + l^{2}}\\right)^{2}} = 0$"
      ],
      "text/plain": [
       "              ___________________________________ ⎛                            ↪\n",
       "             ╱                    ______________  ⎜     ⎛                      ↪\n",
       "            ╱       2    2       ╱       2    2   ⎜     ⎜       ⎛       ______ ↪\n",
       "    4      ╱   - 4⋅M  + l  + l⋅╲╱  - 12⋅M  + l    ⎜   2 ⎜   2   ⎜      ╱       ↪\n",
       "16⋅M ⋅    ╱    ───────────────────────────────── ⋅⎝4⋅M ⋅⎝4⋅M  + ⎝l + ╲╱  - 12⋅ ↪\n",
       "         ╱           ⎛       ______________⎞                                   ↪\n",
       "        ╱            ⎜      ╱       2    2 ⎟                                   ↪\n",
       "      ╲╱           l⋅⎝l + ╲╱  - 12⋅M  + l  ⎠                                   ↪\n",
       "────────────────────────────────────────────────────────────────────────────── ↪\n",
       "                                                                               ↪\n",
       "                                                                             ╱ ↪\n",
       "                                                                            ╱  ↪\n",
       "                                                                           ╱   ↪\n",
       "                                                                  2       ╱    ↪\n",
       "                                                                 l ⋅     ╱     ↪\n",
       "                                                                        ╱      ↪\n",
       "                                                                       ╱       ↪\n",
       "                                                                      ╱        ↪\n",
       "                                                                    ╲╱         ↪\n",
       "\n",
       "↪                                                                              ↪\n",
       "↪          2⎞                                     ⎛                            ↪\n",
       "↪ ________⎞ ⎟ ⎛                 ______________⎞   ⎜       ⎛       ____________ ↪\n",
       "↪  2    2 ⎟ ⎟ ⎜   2    2       ╱       2    2 ⎟   ⎜   2   ⎜      ╱       2     ↪\n",
       "↪ M  + l  ⎠ ⎠⋅⎝4⋅M  - l  - l⋅╲╱  - 12⋅M  + l  ⎠ - ⎝4⋅M  + ⎝l + ╲╱  - 12⋅M  + l ↪\n",
       "↪                                                                              ↪\n",
       "↪                                                                              ↪\n",
       "↪                                                                              ↪\n",
       "↪ ──────────────────────────────────────────────────────────────────────────── ↪\n",
       "↪ _________________________________                                            ↪\n",
       "↪                                2                                             ↪\n",
       "↪         ⎛       ______________⎞   ⎛                              2⎞          ↪\n",
       "↪     2   ⎜      ╱       2    2 ⎟   ⎜       ⎛       ______________⎞ ⎟ ⎛        ↪\n",
       "↪  4⋅M  + ⎝l + ╲╱  - 12⋅M  + l  ⎠   ⎜   2   ⎜      ╱       2    2 ⎟ ⎟ ⎜      ╱ ↪\n",
       "↪  ─────────────────────────────── ⋅⎝4⋅M  + ⎝l + ╲╱  - 12⋅M  + l  ⎠ ⎠⋅⎝l + ╲╱  ↪\n",
       "↪                            2                                                 ↪\n",
       "↪     ⎛       ______________⎞                                                  ↪\n",
       "↪     ⎜      ╱       2    2 ⎟                                                  ↪\n",
       "↪     ⎝l + ╲╱  - 12⋅M  + l  ⎠                                                  ↪\n",
       "\n",
       "↪      2                                                                       ↪\n",
       "↪    2⎞                                        ⎛                               ↪\n",
       "↪ __⎞ ⎟  ⎛                   ______________⎞   ⎜         ⎛       _____________ ↪\n",
       "↪ 2 ⎟ ⎟  ⎜     2    2       ╱       2    2 ⎟   ⎜   2     ⎜      ╱       2    2 ↪\n",
       "↪   ⎠ ⎠ ⋅⎝- 3⋅M  + l  + l⋅╲╱  - 12⋅M  + l  ⎠ + ⎝8⋅M  + 3⋅⎝l + ╲╱  - 12⋅M  + l  ↪\n",
       "↪                                                                              ↪\n",
       "↪                                                                              ↪\n",
       "↪                                                                              ↪\n",
       "↪ ──────────────────────────────────────────────────────────────────────────── ↪\n",
       "↪                                                                              ↪\n",
       "↪                                                                              ↪\n",
       "↪                4                                  2                          ↪\n",
       "↪ ______________⎞  ⎛                 ______________⎞                           ↪\n",
       "↪        2    2 ⎟  ⎜   2    2       ╱       2    2 ⎟                           ↪\n",
       "↪  - 12⋅M  + l  ⎠ ⋅⎝4⋅M  - l  - l⋅╲╱  - 12⋅M  + l  ⎠                           ↪\n",
       "↪                                                                              ↪\n",
       "↪                                                                              ↪\n",
       "↪                                                                              ↪\n",
       "↪                                                                              ↪\n",
       "\n",
       "↪                                        ⎞    \n",
       "↪   2⎞                                  2⎟    \n",
       "↪ _⎞ ⎟ ⎛                 ______________⎞ ⎟    \n",
       "↪  ⎟ ⎟ ⎜   2    2       ╱       2    2 ⎟ ⎟    \n",
       "↪  ⎠ ⎠⋅⎝4⋅M  - l  - l⋅╲╱  - 12⋅M  + l  ⎠ ⎠    \n",
       "↪                                             \n",
       "↪                                             \n",
       "↪                                             \n",
       "↪ ──────────────────────────────────────── = 0\n",
       "↪                                             \n",
       "↪                                             \n",
       "↪                                             \n",
       "↪                                             \n",
       "↪                                             \n",
       "↪                                             \n",
       "↪                                             \n",
       "↪                                             \n",
       "↪                                             \n",
       "↪                                             "
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "EqSattel1 = EqSattel.subs({(r,LoesCircular[1])}).simplify()\n",
    "EqSattel1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Lösen wir diese Gleichung nach l auf erhalten wir zwei reellwertige Lösungen ($l_{\\rm ISCO}=+/- \\, 2\\sqrt{3}\\,M$), wobei diese einem rechts bzw. links rotierenden Probekörper am ISCO entsprechen."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left[ - 2 \\sqrt{3} M, \\  2 \\sqrt{3} M, \\  2 i M\\right]$"
      ],
      "text/plain": [
       "[-2⋅√3⋅M, 2⋅√3⋅M, 2⋅ⅈ⋅M]"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "LoesEqSattel1 = solve(EqSattel1,l)\n",
    "LoesEqSattel1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Setzen wir eine dieser Lösungen in \"LoesCircular\" ein, erhalten wir den Radiuswert der innersten kreisförmigen Bahnbewegung ($r_{\\rm ISCO}=6M$):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle 6 M$"
      ],
      "text/plain": [
       "6⋅M"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "LoesCircular[1].subs({(l,LoesEqSattel1[1])})"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Zusammenfassend erhalten wir somit:\n",
    "\n",
    "$$\n",
    "\\begin{eqnarray}\n",
    "\\frac{d V(r,M,l)}{dr}=0 \\,\\,,\\,\\,\\, \\frac{d^2 V(r,M,l)}{dr^2}=0 \\quad \\underbrace{\\Rightarrow}_{\\hbox{ISCO}} r=6 M \\,\\,,\\,\\,\\, l = 2 \\sqrt{3} M\n",
    "\\end{eqnarray}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Numerische Simulation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Wir wollen uns nun drei kreisförmige Bahnbewegung um ein nicht-rotierendes schwarzes Loch (Masse $M=1$) darstellen, wobei eine davon der ISCO ist. Wir wählen zur Zeit $t=\\tau=0$ die Anfangswerte der Radien zu $r_0=12, 9.5$ und $6$ km und starten jeweils bei einem Winkel $\\phi_0=0$. Die Anfangsgeschwindigkeit in radialer Richtung sei $\\left. \\frac{dr}{d\\tau} \\right|_{\\tau=0}=0$ und die Anfangsgeschwindigkeit in $\\phi$-Richtung ergibt sich durch die festgelegten, konstanten Drehimpuls Werte der kreisförmigen Bahnbewegungen mittels der Gleichung $\\left.  \\frac{d\\phi}{d\\tau} \\right|_{\\tau=0}   = \\frac{l}{r_0^2}$ ($l=4, 3.7$ und $2\\sqrt{3}$). Die weitere nötige Anfangsbedingung ( $\\left. \\frac{dt}{d\\tau} \\right|_{\\tau=0}$ ) erhalten wir mittels der Bedingung $\\left( \\frac{ds}{d\\tau} \\right)^2= g_{\\mu\\nu} \\frac{dx^\\mu}{d\\tau} \\frac{dx^\\nu}{d\\tau} = g_{\\mu\\nu} u^\\mu u^\\nu = 1$, die für massive Probekörper immer erfüllt sein muss ($u^\\mu$ ist die Vierergeschwindigkeit des Probekörpers). Die Gleichung für das infinitesimale Weglängenelement $ds^2=1$ in der Ebene lautet:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle - \\mathrm{d}\\phi^{2} r^{2} \\sin^{2}{\\left(\\theta \\right)} - \\mathrm{d}\\theta^{2} r^{2} - \\frac{\\mathrm{d}r^{2}}{- \\frac{2 M}{r} + 1} + \\mathrm{d}t^{2} \\left(- \\frac{2 M}{r} + 1\\right) = 1$"
      ],
      "text/plain": [
       "                                                                      2        ↪\n",
       "                2  2    2                           2  2   \\mathrm{d}r         ↪\n",
       "- \\mathrm{d}\\phi ⋅r ⋅sin (\\theta) - \\mathrm{d}\\theta ⋅r  - ──────────── + \\mat ↪\n",
       "                                                              2⋅M              ↪\n",
       "                                                            - ─── + 1          ↪\n",
       "                                                               r               ↪\n",
       "\n",
       "↪                         \n",
       "↪        2 ⎛  2⋅M    ⎞    \n",
       "↪ hrm{d}t ⋅⎜- ─── + 1⎟ = 1\n",
       "↪          ⎝   r     ⎠    \n",
       "↪                         \n",
       "↪                         "
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ds2 = g.line_element()\n",
    "Eq_ds = Eq(ds2,1)\n",
    "Eq_ds"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Umbenennung der Symbole"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [],
   "source": [
    "ds_symb = list(ds2.free_symbols)\n",
    "for s in ds_symb:\n",
    "    if (latex(s)==r\"\\mathrm{d}t\"):\n",
    "        dt = s\n",
    "    if (latex(s)==r\"\\mathrm{d}r\"):\n",
    "        dr = s\n",
    "    if (latex(s)==r\"\\mathrm{d}\\theta\"):\n",
    "        dtheta = s\n",
    "    if (latex(s)==r\"\\mathrm{d}\\phi\"):\n",
    "        dphi = s "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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pjtxVlY6PXYqE4KoC9ywO7i4NHKytX3SQzoXwfyLtzMKzy/8KqJ7J9y70nEFgk0GbRMIOJz0fsiBXfxGzXdE1n4u/fJ4Fi5XdY+0NCyWxBdqwrPFqrwrU5vCpFzW6upCbK4uDRTUOmKF3IZIoE57JSTwJ2YrMomNIGZx0H2tqyf9GrKDmaouy1sQmB3bFlTghY0xUFa/PKEms0fxYaL9MNtPDUJPs8YfOgfo4oIkIqxUl8bQ06kUbi9wcVe0EysFHtR0lSdttUT8H2t1wuLLYjdVekXNgVw4wIX2tCSolvLkHgUyWhKBKDZFtyQPzKqpa2DaGuLIwTnhaCwfa39mvhebd6dRkjHfBVtqivAuUhI7Ot8p2Z84BFarNeBUcjw+oPkuVN1mwOBLnwMYc0GDL8f8BG1NZHHompl/EO9YvqrRmi+PocoLY+MHGlBLXkGa1ypXFLDZdTiYJKwvN7PMmZswOJcIUWHp8AbjYCUW0hW9DiU6HmRwQz9hl9lzZmajoc4cDOKA+YMMBfVH1or6HoQ4QnoOrREFgpX8PHRJgXtLjcx+8JLUlYFFVa1VtyZgtccf+9bfxt2TyBG7xP4SflFZv7Bz6fxYTPPZHG3BAAotCYMJGgPkmT9iyy30d/h2oDXheCkr1L59/4W+Lq7ZuS+HnHDoYV8qHUfYfnVdvKLmymNPrF5ZHgotn8atSvAqHK+GA+pvQI+9euGGwcZ+Lx/dUBS/bXoSigF2uLODClYEE+W81mcW2a9y+eGW97c11DuThgCuLPHysBosUBPvc/1BaxBvf1TDOCXUOXDkHfIH7+gQA99jj1tfX795i58AqDvwfKcQKe+tbD7YAAAAASUVORK5CYII=",
      "text/latex": [
       "$\\displaystyle - \\mathrm{d}\\phi^{2} r^{2} - \\frac{\\mathrm{d}r^{2}}{- \\frac{2 M}{r} + 1} + \\mathrm{d}t^{2} \\left(- \\frac{2 M}{r} + 1\\right) = 1$"
      ],
      "text/plain": [
       "                                  2                               \n",
       "                2  2   \\mathrm{d}r               2 ⎛  2⋅M    ⎞    \n",
       "- \\mathrm{d}\\phi ⋅r  - ──────────── + \\mathrm{d}t ⋅⎜- ─── + 1⎟ = 1\n",
       "                          2⋅M                      ⎝   r     ⎠    \n",
       "                        - ─── + 1                                 \n",
       "                           r                                      "
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eq_ds = Eq_ds.subs({(dtheta,0),(theta,pi/2)})\n",
    "Eq_ds"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Wir lösen diese Gleichung nach $dt$ auf:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/latex": [
       "$\\displaystyle \\left[ \\frac{\\sqrt{r} \\sqrt{- 2 M \\mathrm{d}\\phi^{2} r^{2} - 2 M + \\mathrm{d}\\phi^{2} r^{3} + \\mathrm{d}r^{2} r + r}}{- 2 M + r}, \\  \\frac{\\sqrt{r} \\sqrt{- 2 M \\mathrm{d}\\phi^{2} r^{2} - 2 M + \\mathrm{d}\\phi^{2} r^{3} + \\mathrm{d}r^{2} r + r}}{2 M - r}\\right]$"
      ],
      "text/plain": [
       "⎡      _______________________________________________________________________ ↪\n",
       "⎢     ╱                     2  2                       2  3              2     ↪\n",
       "⎢√r⋅╲╱  - 2⋅M⋅\\mathrm{d}\\phi ⋅r  - 2⋅M + \\mathrm{d}\\phi ⋅r  + \\mathrm{d}r ⋅r + ↪\n",
       "⎢───────────────────────────────────────────────────────────────────────────── ↪\n",
       "⎣                                    -2⋅M + r                                  ↪\n",
       "\n",
       "↪ ___        _________________________________________________________________ ↪\n",
       "↪           ╱                     2  2                       2  3              ↪\n",
       "↪  r   √r⋅╲╱  - 2⋅M⋅\\mathrm{d}\\phi ⋅r  - 2⋅M + \\mathrm{d}\\phi ⋅r  + \\mathrm{d} ↪\n",
       "↪ ───, ─────────────────────────────────────────────────────────────────────── ↪\n",
       "↪                                          2⋅M - r                             ↪\n",
       "\n",
       "↪ _________⎤\n",
       "↪  2       ⎥\n",
       "↪ r ⋅r + r ⎥\n",
       "↪ ─────────⎥\n",
       "↪          ⎦"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "LoesEq_ds = solve(Eq_ds,dt)\n",
    "LoesEq_ds"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Wir benutzen die erste Lösung (positive Lösung) für unsere Anfangsbedingungen $\\left. \\frac{dt}{d\\tau} \\right|_{\\tau=0}$. Die Anfangsbedingungen lauten somit:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [],
   "source": [
    "t0 = 0.0\n",
    "phi0 = 0.0\n",
    "dr0 = 0.0\n",
    "\n",
    "#Grüne Kurve\n",
    "Ar0 = 12.0\n",
    "Adphi0 = 4.0/Ar0**2\n",
    "Adt0 = float(LoesEq_ds[0].subs({(r,Ar0),(dr,dr0),(dphi,Adphi0),(M,set_M)}))\n",
    "\n",
    "#Rote Kurve\n",
    "Br0 = 9.25\n",
    "Bdphi0 = 3.7/Br0**2\n",
    "Bdt0 = float(LoesEq_ds[0].subs({(r,Br0),(dr,dr0),(dphi,Bdphi0),(M,set_M)}))\n",
    "\n",
    "#Schwarze Kurve (ISCO)\n",
    "Cr0  =6.0\n",
    "Cdphi0 = float(2*sqrt(3)/Cr0**2)\n",
    "Cdt0 = float(LoesEq_ds[0].subs({(r,Cr0),(dr,dr0),(dphi,Cdphi0),(M,set_M)}))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [],
   "source": [
    "tauval = np.linspace(0, 800, 10001)\n",
    "\n",
    "Ainitialval = np.array([t0,Ar0,phi0,Adt0,dr0,Adphi0])\n",
    "Binitialval = np.array([t0,Br0,phi0,Bdt0,dr0,Bdphi0])\n",
    "Cinitialval = np.array([t0,Cr0,phi0,Cdt0,dr0,Cdphi0])\n",
    "\n",
    "ALoes = integrate.odeint(DGLsys, Ainitialval, tauval)\n",
    "BLoes = integrate.odeint(DGLsys, Binitialval, tauval)\n",
    "CLoes = integrate.odeint(DGLsys, Cinitialval, tauval)\n",
    "CLoes_ivp = integrate.solve_ivp(DGLsys_ivp, [tauval[0], tauval[-1]], Cinitialval, t_eval=tauval, rtol=10**(-10), atol=10**(-10))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [],
   "source": [
    "rcParams.update({\n",
    "    'figure.figsize'    : [8,8],\n",
    "    'axes.titlesize' : 14,\n",
    "    'axes.labelsize' : 16,  \n",
    "    'xtick.labelsize' : 14 ,\n",
    "    'ytick.labelsize' : 14 \n",
    "})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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KlrGR8+I+HMtWcHHoOLr2DCA42LXs2nPPwZSS75GQPS8XHnwc58lTpuOKeA2NBKeCRoJF/oHdTtL8RUSOnMQ71mGM31731pfy53OyPKQ1udvWIeugnpAnj8GgIp4hLg5mzoTPJyWxYGcEebhAEn7cX+IUHQbk4bHHIGtW0ylF3JOmQ6QDlWCRv7twAdb3m85D87sDMJ3H6GGZTosWrrdyW7QAPz/DIUU8lPPyFS72eYXQBTP4ydGM9vY5gOu8uccegyFNtlO0RrjWDhT5HyrB6UAlWOQPTie/7bLw4Yeu0Spr4g1OEUE4l4jMUYobm3dTsKiar0iauXqV2JPRzFhZgMmTYfduACebqElVy3Yu1mtH7oVfYA3VtokimhMsImnL6cS+4lfO1XqIKcVGU6kSTJ3q2gigUq1MHO0/luQfFpHzwh4VYJG0FhJCaJkCPPUU/PYbrF4NL9+7gRpswea0c37NYcrWCGbSJLh+3XRYEc+gkeBU0Eiw+Krr1+Hb907y2GtFsOHgDPkoZj1Om44BPPss1KplOqGID4qM5MrbE7F9+gkvOkYzOd41JSlbNujbFwZXXk6OFjUhRKPD4ls0HSIdqASLr4mOhgkTYNw4iIyE72lHO+YRE5KP6/N/IW/jMqYjikhCArFXrXz5lT8ffQTHjkFuzvM7hbEHZib+2ZfJ8c4LplOKZJiU9jW9Zykif3btGnFjp3Do253cd2IqcXGuq4sUAXunYcSX7EhY1w6E+fubzSkiLoGBhAbCM8/AgAGwcCHEPvUhQWcSICGByWMusOMMvPQSlC1rOqyI+9BIcCpoJFi83e+/g7NWLYpc2ARANbYQX7YaQ4dCp05a5UHEUzj37efC86MJ++U77rEf4hz5AGjbFl5+0U6NPCddr2xFvJBOjBORFDt2DHr2hGLFYNSFXreun959Bbt2wSOPqACLeBJLmdLk+WkqmaJO8+PWfHTo4NqTZv58GFNnHvZ7inOhZU/XK18RH6USLOKrrl4l+pX3GNgzjpIlXSs92O1wulF3Trfpj3PXbspMfRGrfkuIeK6sWalaFebMgX37oEd3J8N4G5vTTu7FU3mt/R62bjUdUsQM/XkT8UGXv/6JuFxFyfrWC4RMHU9yMjRtChs2wOLlgRSYPwFL+XKmY4pIGipVCr78NIkizz3EtYCsbKImb2xvSfXqrmkSu3aZTiiSsTQnOBU0J1g8XWQkjBkDS8YfYkdCaWw4iPYPZ89Pp6nXONB0PBHJKNHRnNx6kVdnlOCrr8DhcF39Q6kXqVvXQo7RL0KOHGYzitwhLZGWDlSCxSM5nVw9Hc17n2fj/ffh6lXX1Qty9qZ6uevknvAaltKlzGYUEWMOHIDXX4cNs09wmOIEkERMUG5id/1ORPEg0/FEUk0nxokIyStXc7FoTbYW68KIEa4CXKUKLFoErc5NIc+Kb1SARXxcqVIwaxasHr0Ri9UCwMfxvSlRIYhhwyA21nBAkXSiEizihZxO+OnHZM43606u37fQMHEp3fMs4dtvYetWaNECLDb97y8i/6/Qi53x//0I5zs/w9o6Q4iPh7ffdq0a88nHDpLXbDAdUSRN6a+giJfZsQOaNIEWrf0YnDQKgKi85ZjydWY6dnQtkyQiclsREeSZ9SGL14Yyfz6ULOk6l2Dt07Pwa1CH89Va4tx/wHRKkTShEiziDZKSiHl7An0euUrVqrB8OQQEQMTznbk69TvCT+3Ev1F90ylFxENYLNCmDezeDZM+jGe09WUA8mxbzOBOJ9m+3XBAkTSgEizi4ZI3buVSgYqEDRtA4W/exumEhx+Ggwfh3fcshHRvDzab6Zgi4oH8/aHv0wHkmPAm0VkK8Iu1KWP3NKVaNXjySbh82XRCkTunEiziwZYtgwe7hhFy8SgAz1k+YNuSKGbOhMKFzWYTES9htZK5XzeyXjhEmY1f8vDDrvMOJk2CEiVgXZfx2NdtNJ1SJNVUgkU80MmT0KGDa+7vz0eLMzHT81wsWpOAjWuo0jTcdDwR8UaZMpG/ej5mzoRVq6BcOchzaQ81Zw/CVq825zs+bTqhSKqoBIt4kMQffmJn1d6UKunk++/BaoWBA6HH76+T6/B6rDWqmY4oIj7g3ntdJ+HOqv0RftgB+Pi73PTuDRcvGg4nkkIqwSIe4nSn5who24JK27+gVfwcGjRw/REaNw6y5gpwNWIRkQzi5wflVn9C7MhxHM9RlfcYzBdfuFaU+Pxz15QJEXemv5oibu7KFXjiCRgwp8Gt68bU+YFVq6BCBXO5RETw8yN02ECKRG5h5fogKleG6Gh4/HF4o9y3XO77Mty4YTqlyG1p2+RU0LbJkpGcTvjuO3j6abhwwXXd6lJPUGVgPYL7ddOCvyLidpKTXe9OffDqZbbdKE1uLnI5RzGy7NmIf54cpuOJj9C2ySKeKimJ6CGj+Kbk63Tq5CrAJUvCr79C/f1TCH6yuwqwiLglPz94/nnYNm4t2S1XAFh5qQJVm+Zg0ybD4UT+QiPBqaCRYElvjoQkoorXJtepbSRjo7bfVloMrcTQoRAYaDqdiEjKOfft5+wjg2l2cgp7r+TDYnG9s/XWWxASYjqdeDONBIt4mBMnoEkLfz451QoAC04WDFnHiBEqwCLieSxlSpN/5yJWHcrHY4+5pnh99BF0KLOPCw/00E4bYpxKsIhhTid88QWULw8rVsAHmYZxoPpjWDZtIu/Ip0zHExG5K+HhMH06LFkCRQokMfJUN3L/PI2YQuVJ2LbHdDzxYSrBIqbY7cS+Ppa3qs+nd2+Ii4M6dWDrb/6U2jxda/6KiFdp2hR+m7Wf0gGuHS7PXA2jdtd72LbNcDDxWSrBIgY4Y+O4WO4+Qkc8T59tfcjvf5F334XVq6F4cdPpRETSR5a6FQg+tofz1VvxXPZp7DiQiVq1YMQISEoynU58jUqwSAa7cgW6PB7CugOu5YLCiWLj6z8zeDDYbIbDiYikt/z5ybP5R746WJ0OHVzLqr3+OjSpHk3kgBGQmGg6ofgIlWCRDLRmDVSsCN/OsTDANokz+WvgWLaSAkO7mY4mIpKhwsPh22/hm28gWzbo9dtAck54ncgSdXAeOWo6nvgAlWCRDJC8diPjBx6mYUM4dQqKFYMfNuYm/6mN+DW+13Q8EREjLBbo0gX2LzpGF8tsAAJPHGLgMxaio81mE++nEiySnux2ooeMgvr1qD3+YWyORLp3h+3boVo1tOmFiAiQu3ZR/LZs5HJ4CfpbJ/Px4qJUqgTr15tOJt5MJVgkHc2fFU/ke1Pxw041trG+22SmToUsWUwnExFxL9aqlcl+ehcDNzxM0aKutdMbNIC33nRg373PdDzxQirBIukgPh6efBIe6hpMZ8c3xFuCuDLgVap9/qTpaCIi7iswkBo1YMcOePRRsNshavg4nBUrEjP8fdfC6iJpxM90ABGv4nBw/FASHR4NZPt212yHZkOqYOt/nGwReUynExHxCKGh8NVX0LnsHpoNHYKfM5mwNwezLVcdqg6obTqeeAmNBIuklUuXuFijJWsqDmD7dteZzz//DKNGgb8KsIhIqj04uCRX+zwPwGhepMYztRkzRgPCkjY0EiySBuwJyVwqVZ9cUfvpBpwsUpvuv/YiIsJ0MhERD+bvT/bJo4hv/wAHZtTB8RUMGQIbNsDUqRAWZjqgeDKNBIvcpagoeOBBP56PehmAuEw5GTKxsAqwiEgaCWragC+m+zF5MgQEwPz50K/sGqIeGajNNeSOqQSL3IVNm6BKFfjlF5ib+TG2dR1LlkPb8W/WyHQ0ERGvYrFAnz6wdi1Uy3+OsWc6Ef7NeCLL3AuRkabjiQdSCRa5E2fPsuz5n2jQwLX5RYkSrkJcdcYgKFDAdDoREa9VvTosH7ONHNYrAOw9Gshzb2QlOdlwMPE4KsEiqWTfuIWYktWpO7YdFRK30LYtbNkC5cqZTiYi4htCH2mFbcM6jhVuRAe+44OP/WnZEu0yJ6miEiySCjEx8FPnLwm7epZMxPN9gWf4/jsnoaGmk4mI+BZbjaoUPb6cyd+FkzkzLF0KNWvCke2xrgWGRf6DSrBICh0+DLVqQbuTH7LOWo/IUvUpuG0+Vpu2PhYRMaV9e1i3DiIi4PihRM7XaE1knTYQG2s6mrg5lWCRFFi2zDXCcOAA5C4QQMiyH8j52zLIlct0NBERn1epkmta2jd5BlHP/is5Ny/iZPV2Wk9Y/pVKsMi/OX2aY5Ue4uFml7lyxTUSvGULVLwvu2udHhERcQu5c8ODX7TjakA24gmk46G36NMHkpJMJxN3pRIs8g8cv+0mpnQtiv42n28cnejZNYmVKyGPNn8TEXFLAQ80Jnj3Jpb0mMVWa00++wxatdLMCLk9lWCR24iPh0HDMpNw1bUIe9Xsx/l81EWCggwHExGRf2UpUZw2X7ZlwQJunTDXoAFELt6i/ZblT1SCRf7i8mVo1gw+WnQPD9l+5HzphmTbvwFLgfymo4mISAq1bAm//uo6daPMbzPJ2bIGlx57VitHyC0qwSI3OZ38ftxJ3bqwejWEhsKbS2uSZ+8KnQAnIuKBqlWDrd+fYKqlJwA5vv6IPW/OM5xK3IVKsAhAUhJRDz3BrPJvceCAa9O3tWuhUSNce3WKiIhHiqhXiMRxk0i2+DGZPlR5qz0zZ5pOJe7Az3QAEeMcDiLv7UDODQt4CYiLKEj/Dd3Ir9kPIiJeIeTpniRUKsvKcVVI+t7Co4/C2bMweLDpZGKSRoLF582db2Xc5loAJFoCGDYiUAVYRMTLBNavwcxv/XjuOdfnL7wA7w44gfPgIbPBxBiNBItPmzoVevcGh+MlKpaOp83Ye8ncvJHpWCIikg6sVnj/fciZEz54+QJtJ9zP1S9iCF6zBGvVyqbjSQazOJ1aLySlYmNjCQsLIyYmhtDQUNNx5G44nXw4zsKgQa5Pe/eGyZPBZjMbS0REMsahml0psflrAI7kqEmhsxvwD9A5IN4gpX1N0yHE5zgPH+F0gVp8Msj1Ftjzz8OUKSrAIiK+pMSSj4ksUYcTFOS+S3No38HCjRumU0lGUgkWn+LYvZe4SvUocHYzy7ifcYNP8e67WgBCRMTnZM1Kzu1LOfrZKqKCIvjxR3jgAe0u50tUgsVnOBww6J3c/H7dteZvcN4wBj7npwIsIuKrgoNp1LsIS5ZAliyuzTXub+wk7tftppNJBlAJFp9gt7vm/X40M5wmluUcrdedHHt+hbx5TUcTERHDGjSAVasgR3YnHbYOIXPD6lybNMN0LElnKsHi9W4W4KlTXWcGj5uZk3vWTIXs2U1HExERN1GlCmwdvoAXeRcbDgKf7EnMtiOmY0k6UgkWr2bftJVl5Z5l+jQHNhvMnAlduphOJSIi7qjwwNZEPTwAgP58QqM+xbh82XAoSTdaJ1i8ln3jFuIbNKFZUgyfWeLI8vUUOnbW6z4REfkHFgvhX3/EseadmT+4HpHb4f77YdkyvXnojdQIxCslJ8NHL54mMOkqAA+WPkLHB+MNpxIREbdnsVC0Wz1WroRcuWDHDmjcGC4fvWI6maQxlWDxOg4H9OoFz615iK7Wb7hQ4X5ybl4MmTObjiYiIh6ibFlYuRJy5wbrzm1YSxbj2sTppmNJGlIJFq/idMJTT8GMGeDnB52/60junUshONh0NBER8TBlysCamadYbrmfrPbLBPXvyfX5S03HkjTicyV4165dzJo1i7lz55qOImnMeew4E7puYNIk1+YXM2bAQw+hnTBEROSOFb+vAPbOjwKwjrp0GluLeM2u8woWp9PpNB0io8TGxlK6dGnOnj1LWFgY0dHRqb5/SvaiFgPOnOFK+foEXLlAG36gy5T7efxx06FERMQrOBycfH4c1ab0JfJaZlq1grlzwd/fdDC5nZT2NZ8aCR4yZAg2m40sWbKYjiJp7LfOb5HtynGCuc7M/C/weC+H6UgiIuItrFYKfjCIOYsyExQECxdCt26udejFc/lMCd6wYQOffvopEydOJCAgwHQcSUOffw41141lPm24kq0IuTYtdO2KISIikobuvff/R4C/n5XIurJ9cO7ZazqW3CGfaApJSUk88cQTdOrUiZYtW5qOI2lozhx44glIIIgNg+aQdfdayJ/fdCwREfFSDzwAs7+8zg+0pcHBKcTWaoLz+O+mY8kd8IkSPHr0aM6fP8+4ceNMR5G0Yrfz67IkunZ1rQjRpw+8874/lvz5TCcTEREv99CDyVQvHAlAwLUrzH37gOFEcie8vgQfPnyYt956i7Fjx5IrV65U3TchIYHY2Ng/fYgbcDq51PUZkh9oRUBiHO3awSefaBEIERHJIKGhhG/9mTOF69CY5XT4rDnffGM6lKSW15fgvn370qBBA7p165bq+44aNYqwsLBbHxEREemQUFIr+pX3yDFrAo2Tl7Ip5H6+mmbHZjOdSkREfEqOHOQ7upaaz9YBoHt3WLHCcCZJFa8uwV9++SWbN29m0qRJd3T/l19+mZiYmFsfp06dSuOEklrR0fD0zNpcJhsAhcY8RaYQNWAREcl4FquF99+HTp0gKcm1Nv3BHw645umJ2/MzHSC9XLlyhRdeeIGGDRuyadMmNm3adOtriYmJ2O12Zs2aRaZMmWjTps1tHyMwMJDAwMCMiiz/IT4e2raFX3+vx8mc65g3cCXZn0z9CL+IiEhasVph2jQ4fx7yrf6GIm27E/3iCLKOftl0NPkPXrtZxs6dO6lcufJ/3i537tycP38+RY+pzTLMcTigSxfXahChobBmDVSoYDqViIiIS8zG/WSpXRYrrloV9/1SsrRrYjiVb0ppX/PakeBs2bLRuXPn235t3rx5OJ1O2rVrR9asWTM2mKTeuXPMeHorc75/EH9/mD9fBVhERNxLWK3SRA95m6yjX+ZTnmD2uPv4qRVoawL35bUjwf8mPDyc5ORkbZvsCW7cIKrsvYQf38JLjKLCV0N45FEtAyEiIm7I6eT4xJ+pOKQ5cVctPPEETJ6s1YsymrZNFq9w5JUvCT++BYCXwibySMsYw4lERET+gcVCkf4PMGu2BasVpkwBbVHgvnyyBLdr14727dubjiH/4fhxqD3tSYYxkut+WQhd9SNo+oqIiLi5Fi3g3Xddx28/F8WRLsMgOdlsKPkbn5wOcac0HSLjxMZCnTqwdy9UrQqrv7tI5sKp2+xERETEFKcThnfYR4+5D3IPx7j02LPkmP6B6Vg+QdMhxGPZ7fDww64CnC8f/PADKsAiIuJRLBYY3u8ihSwnAbDPnMWlA5GGU8n/UgkW93L2LEeL3s+BxUfJlMlVgPPnNx1KREQk9fybNOTG+xPZG1CJqvbNPNQnJ4mJplPJTSrB4j4SE4m8ryMlTi5nEzVZMHwr1aqZDiUiInLnsgx6HMumTcSGRrBmDTz3nOlEcpNKsLiNA79eIPqw660iv9DM3N+7kOFEIiIid69MpQC+/tp1PGECzJiBtlZ2AyrB4haio6HVkxFUc27h19ydyLLke8iZ03QsERGRNNGqFQwfDhYcnOj5OpFdBpiO5PO0OkQqaHWI9OFwQJs2sHAhFCoE27ZBjhymU4mIiKQth93JhvwdqHthLgBXx39JyIAeZkN5Ia0OIZ4hPp6RbzpZuBACA2HuXBVgERHxTlabhYrDWgFgx8rXE6JxOAyH8mEqwWKOw8HF+u0o+HpPgrjBxIlQpYrpUCIiIukn5OmenO/5Mi39f6HfgWd54w3TiXyXpkOkgqZDpK0rz71Jtg+GA7C3QFPKnlpiOJGIiEjGmD4dund3HS9a5NplTtKGpkOIW0tKgvELCnGNzNixUvzTF01HEhERyTDdukH//v9/fPpIvNlAPkglWIx49VV47Wg3GoZs48o7nxLwQGPTkURERDLU2LGuaYA1Ly0iqGxR7Os3mY7kU1SCJcMtXQqjR7uOX5paivAhvc0GEhERMSAwEBb0XcQiWhGeeI7Yll1ca4ZKhlAJlgx14WA0jz3mOn7ySWjf3mweERERk/L3bEpk8doArI6uyLq1OlUro6gES4axb95GSNlCtLs4kQrlnbz/vulEIiIihvn7k/OXb5heawJtmUenvtmIijIdyjeoBEvGiI0l5oHOBNtjmUh/fnpkBpkymQ4lIiLiBgoVov2y/pQsaeHsWejRQ7sqZwSVYMkQG3cE8vUV1/ovkUVrkO+5LoYTiYiIuI/gYJg92zVPeNEiGPd+MiQkmI7l1VSCJd1dvQpdewcy0PkRYxvMJ/yXWRAQYDqWiIiIW6lY0bViRAkOUvfFOkT1edl0JK+mzTJSQZtl3Jknn4RJk6BAAdi9G7JmNZ1IRETEPTlj47gaXogsSVcASFqyAv+m9xlO5Vm0WYaY53SydN41Jk1yfTp1qgqwiIjIv7GEZsE57FUADlKCz2ZnMZzIe6kES7q5+sEUSnYsTz3W8Mwz0Fj7YYiIiPyn0FefYUePD6nMDgZMrcbGjaYTeSdNh0gFTYdIOeeRoySUrkhQ8jXsWEnasZegSqVMxxIREfEYjz4KM2dC8eKwY4fr5Dn5b5oOIUYtWGRjS3JlAC637aUCLCIikkoffwz588PhwzBkCJCYaDqSV1EJljR35gz0eL0wDVnF4pYTyDl9rOlIIiIiHidbNvjySwglhqoTenLhvs5aQDgN+ZkOIN7F6YT+/V1bn1evbqPp/P76KRMREblDTe538luuJhS+uAXWQ/yMOQR162Q6llfQSLCkHaeTOd86WbAA/P1dr179VIBFRETunMVCnrFDAIgjhO++ijccyHuoBEuaufrBFLJ3a0kEJxk6FMqWNZ1IRETE8wU92p5Dvd6hDPvotqwbGzaYTuQdtDpEKmh1iH9x5gzXi5Qhc1IsV61Z8D96kMDCeU2nEhER8Ro9esC0aVC6tGu1iMBA04nck1aHkAy18duTXEkKAeD6Ax1UgEVERNLY2LGQKxfs3w9vv206jedTCZa7dvUqdP6wNmXZy8rKz5FrxvumI4mIiHid7Nlh/HiwYidm5Hgu9nnFdCSPptOW5K4NGwYnT0Lhwlmpvvp9CDGdSERExDt17OCkZHgzKkYtxzHFgr3Hg9jq1DQdyyNpJFjuyrZtrlelAJMnQ4gKsIiISLqxWC0U7tMUACtOtr691HAiz6USLHfMsXkrO9q8RqDzBg8/DE2bmk4kIiLi/cJGPMeh6o9Sn9U0X/cqFy+aTuSZVILlzjgcRHXuz+Nn3mCfpSwfPHXEdCIRERHf4OfHPRu+4lrl+kRHw0svmQ7kmVSC5Y7ELN9Ktt93ABCaKxO5axQynEhERMR32GzwySeu4y+/hPXrzebxRCrBckeGfF+DivzGpuD7CPtqgmuLOBEREckwtWpB796u40+7ryN5wxazgTyMNstIBW2W4bJlC9SsCU4n/LrKSYN7LaYjiYiI+KTI41f5pdQAHkmcRlS+CoSf2AZ+vr34lzbLkHRht0P//q4C/NhjqACLiIgYlLNgJprk3gVA+NldxEycaTiR51AJlpRzOPjp5dVs3QqhoTBmjOlAIiIiPs5mI/s3nxBjy0Z/JvDslkdNJ/IYKsGSYje+nEWrd+/lJ5rz0ZP7yZPHdCIRERGx1a3FoV9OMpH+TJ1hY/t204k8g0qwpMyNGyQ851qDpTlLeKT+KcOBRERE5Kbq94XwyCOu4+efd01blH+nEiwpcvJiEANvjOEEBTlfuTn+LbUzhoiIiDsZNQqCgmDVKlg8K9Z0HLenEiwpMnSYhRlJXXi87gFyL/zCdBwRERH5i4IFYfDTCTzH+9TrWoikxb+YjuTWfHsNDUmRLVvg669dx++My4QlXyazgUREROS2Xi7zA5kZDA6Ievx5wk/tcO2sIX+jkWD5V06Hk+eecx136wZVq5rNIyIiIv8sc7cOXCxUDQcWllyqxpUz101HclsqwfKvztZuxwNrh5IrKJa33jKdRkRERP6V1Ur2byfTseh2uiZ+wchxWUwnclvaMS4VfG3HOPvyVdjuvw+AU/lqEHF6I1i0OYaIiIi7+/lneOAB8PeHw4ehUCHTiTKOdoyTu7bl810k4g9A+GsDVIBFREQ8RPPmcN99kJQEb7xhOo17UgmW20pIgM7rBlKCQ2y67yUy9X7EdCQRERFJhZvTGI9NXc35t7Wy01+pBMttTZkCJ09CUr7CVFg0SmeWioiIeJjatZxsyN2GlY57yTr8abhwwXQkt6ISLH9z7RqMHOk6fvVVyKQV0URERDyPxcI997kmAwfZr3P+jU8NB3IvKsHyZ8nJLBy4lAsXnBQpAr16mQ4kIiIidyrnB0M5FVqWHnzJkydfNh3HrWizDPmT65Nn0PmLXkRQm+ju4wgIqG46koiIiNypPHm4tnE3M8pZcCyEjRuhVi3TodyDRoLl/yUlkThsBAB12ECze+MNBxIREZG7Vaq0he7dXcevvGI2iztRCZZbYuKsDEoawy7Kc75iM2wN65uOJCIiImlg+HDXmsHLl8PG1Ymm47gFlWC55ZPJNqZe78TDpXaSa/k3puOIiIhIGilcGJ7sdIkRDKdkkwg4dcp0JONUggVwrQgxdqzreOgrVqw5spkNJCIiImnqtWwfMZw3yZZ4kYsvvms6jnEqwQK41gWOioKiRaFzZ9NpREREJK1lf+1p4m2ZScKP7VvspuMYpxIsJM5fTMArL5Kb87z8MvhpzRARERHvEx7OxfdmUIwjtDg2gf37TQcySyXY1zmdRD/7Gv2vvcvvFKF7tb2mE4mIiEg6KfhsO6q0LYTTCaNGmU5jlkqwj0vac5CwE7sBiMtfEv+KZQwnEhERkfQ0bJjrcuZMOHbMbBaTVIJ93KydpSjMcT7J9DyhY0eAxWI6koiIiKSjatWgWTOw2518/9w6uHrVdCQjVIJ9mNPpWhHiPHmJfuU9Aju1MR1JREREMsCoNhvZRE1e+KEeVydMMx3HCJVgH7ZqFezcCZkzQ79+ptOIiIhIRqlUJzM12AJA4pgPweEwG8gAlWBflZjI2PedAPToAdmzm40jIiIiGcdSsQLnyzZmB5V4LflVEm74XgnWYlg+6tLTrzN80S9k4Tme6d8B8DcdSURERDJQjpXfUa1SGGfOWqj2HXTvbjpRxtJIsC+6do2gaZOozlamWbpTIluk6UQiIiKSwfxzZuXpga4T4seOdZ0r5EtUgn3Q5T1nOJxYGIBLTR+BfPnMBhIREREj+vRxnRu0axesWGE6TcZSCfZBE5eVoLJzG48XW0XucUNNxxERERFDsmWDnj0hN+e52G84nD5tOlKGUQn2MUlJMGECgIWGr92LpWQJ05FERETEoKGl5nKCQjx85E0uvfmJ6TgZRiXYxyxYAOfOQa5c0KmT6TQiIiJiWr72tbFZXKtD+H09Fex2s4EyiEqwL7l0iZnjXCfB9e4NAQGG84iIiIh5efNyukVf3mIoDWzruZFoM50oQ6gE+5BLr3zAzDUFmEFX+rf43XQcERERcRMRP3zMlEJvsSu2MHPmmE6TMVSCfUViIv7TPyOQRLpYZlPgnkDTiURERMRN2GyulSIAJk40myWjqAT7iBtX4plmf4wocnCxTlvIm9d0JBEREXEjvXuDnx9s3Ag7d5pOk/5Ugn3EnCWhDEx4l7oFT5N71kem44iIiIibyZ0bOrVJoAvfYGvTCm7cMB0pXakE+4ibb2107xuErYBGgUVEROTvxiQN4hseofzJRdyYNtt0nHSlEuwDdu92vbXh5we9eplOIyIiIu4q30vdbh2f+3ShwSTpTyXY2yUns+n1n7CRzIMPQp48pgOJiIiIu7LUqsmG+1/hXlbRNeBb03HSlVeX4CtXrjBlyhSaNWtGREQEAQEBFCxYkJ49e3L48GHT8TKEffESHp/bgtMU4JWC003HEREREXdmsVBkxpuss93Lhk1WDh40HSj9eHUJHjRoEH369GHp0qWcPn2apKQkTp06xdSpU6lSpQpbt241HTHdXXh3GgB5uECFe7MZTiMiIiLuLk8eaN7cdTxtmtks6cmrS3D27NlvleCTJ09y7do11q9fT7Vq1bh69SrPP/+86Yjp7ktnT76jPVeyRODXqrnpOCIiIuIBund3Xc6Y4b27KFucTqfTdIiMduHCBfLly0dAQAA3UrH8R2xsLGFhYcTExBAaGpqOCdPG5cuu5YATE2Hn5kQqVtc+ySIiIvLf4m84aZ9rDe2uTqNpq0AifvzEdKQUS2lf88vATG4jd+7cZMuWDT8/7/72Z81yFeCKFVEBFhERkRQLsibybVJbgrlCwk+ZIPYd8IABwNTw6ukQ/2TVqlVcunSJzp07m46SrqZOdV3efEtDRHyb0+kkMTGR69evExMTw6VLl7hw4QJRUVFER0dz7do1EhIScDgcpqOKiGmBgVxt2QWABLsf19b/ZjhQ2vPuodDbuHjxIr169aJQoUK89tpr/3rbhIQEEhISbn0eGxub3vHSzKkvl3FjS278bOV49FGL6Tgiks4uX77MyZMnOXfu3K2Ps2fPcu7cOU6dOsXZs2eJjIwkKSnpPx/LarWSPXt28ubNS0REBPnz5ydv3rx/+ihQoAB58+bFYtHvFxFvlWvEU7ywoS4Tzj3Ex2cz421bDfhUCY6KiqJZs2bExcWxcuVKsmfP/q+3HzVqFCNGjMigdGnI6SR4cD92c5TfM5UjV9hWINB0KhFJI5GRkWzfvp1t27axdetWNm3axNmzZ/90Gz8/PywWC8nJyaT21A+Hw0FUVBRRUVHs3r0bf39/gL8V6KxZs1KtWjWqV69O1apVqVq1KoUKFVIxFvESlnJlyTagLDeGwezZ3rfhls+cGHf69GmaNGnCpUuXWL58OeXLl//P+9xuJDgiIsL9T4zbvh2qVgXgXOlG5N233HAgEblTCQkJrF69mk2bNt0qvOfPnwfAZrMBYDd46rbFYsFms5GcnAxAaGgo1atXv1WOGzVqRLZsWp5RxFMdOQLFi4PNBufPQ3i46UT/TSfG/Y+DBw/StGlTEhISWLlyJWXLlk3R/QIDAwkM9LwR1AM3CvEx4+limU2Vvl1MxxGRVIqMjGTRokUsWLCAn376ifj4eGw2G06n80/zdU2W35ucTuetAgyuPz4rVqxg9erVJCUlYbVaqVu3Lg899BAPPvggxYoVM5hWRFKrWDGoUsU1vjZ3LvTpYzpR2vH6keBt27bxwAMP4Ofnx/LlyylduvQdP5anLJE2fDi8+SY8+CAs+MEJemtSxK05nU727dvHjz/+yLx589iyZQtOpxObzeYWRfduWa2uc7AdDgfFihWjXbt2PPjgg9SuXfvWaLaIuK8pgw9y7v2v6Z55DoUO/gIFCpiO9K9S2te8enWI1atX06hRIwICAli1atVdFWBP4XTCt39s9d25MyrAIm7s6NGjDBkyhEKFClGuXDmGDRt2qwCDe4z0pgWHw3FrBPvIkSOMHTuW+vXrkyNHDnr27MmGDRtSPW9ZRDJO++RZDOdNCl0/QNyX35mOk2a8ugS//fbbxMbGcubMGUqWLInFYvnbR1RUlOmYaWrXLjh4EAIDXSPBIuJekpOT+eGHH2jSpAnFihXj/fff59SpU4CrLPpCGbw5fSImJoavvvqKOnXqUL58eT799FOuXr1qOJ2I/FX2vp0AcGDh9+VHDadJO15dgn2Ow8HRl6eQm/O0aOF1a1qLeLTz588zcuRIChYsSNu2bVm5ciXgPaO9d+pmId63bx/9+vUjd+7cDBgwgL179xpOJiK3lC7N4s7TyM8ZBlrGm06TZrx+TnBacvc5wc4NG7HUqY0dKwfbDKHM/LdNRxLxaU6nk9WrVzNhwgTmzp37txPb5Pb8/PxITk6mTp06PP3007Rv3/7WMm0iYsaJE1C4sGuW5dmzkCeP6UT/THOCfdClLxcAYMNB0aY6A1vEpHXr1lGvXj0aNmzIvHnzsNvtKsApdHN0eNOmTTz88MMUL16cmTNn6t9PxKBChaBGDde5Rz/+aDpN2lAJ9iILMnXmLYZyLEsFgjq0Mh1HxCft2rWLli1bUq9ePTZt2gTwpyXEJOVuThU5deoUjz76KBUqVGDx4sU+MW9axB21bu26XLDAbI60ohLsRT7fWpFXeIulY36DXLlMxxHxKceOHaNr165UqlSJpUuXAprvm1ZujgAfOHDg1guMdevWGU4l4nva1b3A03zEs4ubkDBtluk4d00l2EtcvAgbNriOW2kQWCTDnD9/ngEDBlCyZElmzZr1t80jJO3cfFGxadMm6tWrR8uWLdm1a5fhVCK+o5T1EB/xDI0dy4iaMtd0nLumEuwlFi1yzdOpWtXt17AW8Qp2u53333+fIkWKMGnSJJKTkzXym0Fu/jsvXbqUSpUq0adPH2JjYw2nEvF+ljq1uRqUAwDH/oOu4uHBVIK9QXQ0Zyf/SCau35qvIyLp58CBA9SuXZvBgwcTHx+v8mtIcnIyTqeTL774glKlSrFkyRLTkUS8m58fR17+gnLsprrfThxOz96QSyXYCyTOX8ywTa2JIpweyZ+ZjiPitZKTkxkzZgwVKlRgx44dpuPIH+x2OxcuXKB58+b06tWL6Oho05FEvFaZl1pzMks5Lly0sGWL6TR3RyXYC0T9sTRaZm4QcZ+WRhNJD3v37qVmzZq89NJLJCUlad6vm7l58tz06dMpVaoUixcvNpxIxDsFBEDz5q5jT18qTSXYC/wY2pUpPM7pHBWw1KtrOo6IV0lOTmbUqFFUqlSJXbt2aXkuN2e324mMjKRly5Z0796dK1eumI4k4nUefNB16emvNVWCvcAHh1vRhylsmfIbaFclkTRz8uRJatasybBhw0hOTtbor4e4OSr89ddfU7JkSdasWWM4kYh3aVYzmq7M4Nkd3YgdP9V0nDumEuzhTp2CgwfBaoX77jOdRsR7rFmzhsqVK2v014PZ7XYuXbrEfffdx+TJk03HEfEauRJOMYNudGMGcV9+ZzrOHVMJ9nC//OK6rFEDsmY1GkXEa3z66ac0atSImJgYjf56OIfDgd1up1+/fjz55JMkJiaajiTi+cqVIzY4DwAhB7aCh25prhLsyW7c4NDsHVhw0LSp6TAini8xMZEnn3ySvn37at1fLzR58mQaNWpEZGSk6Sgins1i4dgzH9GQlVTNcQKnxTPrpGemFgAcK1bxztIqnCcPXRM+Nx1HxKNFRkbSqFEjvW3uxZxOJxs3bqRSpUrs3LnTdBwRj1ZiWEc2BDTk6OlADh0ynebOqAR7sMhvXHMhchFJkcpZzYYR8WA7d+6kUqVKbNy4UfN/vdzNNYVr167NnDlzTMcR8ViZM0O9eq7jm1MzPY1KsAfb6KzFfNoQ558Nv6aNTMcR8UhLly6ldu3aXLhwQdMffITdbichIYFOnToxevRo03FEPFaTJq5LlWDJcBMiO/EQ85n2biRky2Y6jojHWbBgAS1btiQxMVEF2MfcHPF/6aWXeO211/QOgMgdaFr/Bk1ZQuOfBmNfvsp0nFTzMx1A7kxyMmzY4DpucJ/NbBgRD/Ttt9/yyCOP4HA4VIB83BtvvMH169cZM2YMFovFdBwRj1ExchlLaA1JcOFLJ7kbNzQdKVU0Euyhdu6Eq1chLAzKlTOdRsSzfP3113Tp0kUFWG557733ePrpp/XzIJIKtvp1bh07PXBTGpVgD7V5aTQAdeu6NsoQkZSZPXs2jz32GE6nU4VH/mTChAk8//zz+rkQSakcOfi1yUg6MIdXKywwnSbVNB3CE8XF0feVcO6jOBeTuwLDTCcS8Qjz5s3jkUceMR1D3NgHH3xAYGAgb7/9tqZGiKSA9dVhfP8L5N4CTid40v82GkP0QM71G7A57ZTmACUynzEdR8QjLFq0iE6dOmkEWP7TO++8w4gRI0zHEPEI1atDQABcuABHj5pOkzoqwR7o9CknG6lJEn6EP1TfdBwRt7dz5046dOiA3W5XAZYUGTFiBNOmTTMdQ8TtBQW5ijCAp00LVgn2QEtoRm020rJONP4d25qOI+LWLly4QIsWLUhKSlIBllR5/PHHWb9+vekYIm7v3rrJVGEbQZPHQVKS6TgpphLsgdaudV1WbxgMmTKZDSPixhISEmjTpg2RkZFaB1hSzel00rp1a06dOmU6iohb67ftcbZRjYc3PQs7dpiOk2IqwR5o82bXZe3aZnOIuDOn00m/fv3YsmULycnJpuOIB7Lb7cTExNCyZUuuXbtmOo6I28rRrPqt42urthhMkjoqwR4mLg4OHHAdV6tmNouIO/vwww+ZOnUqDofDdBTxYMnJyezdu5fu3btrOo3IP8jcrD7zQh5jIOPYmbOJ6TgpphLsYaIfH8wvzsZ8nOUl8vhfMh1HxC0tWbKE559/3nQM8RIOh4Pvv/+eN99803QUEfdUoQLftprOeAby67kSptOkmEqwhwlYt4LGrKBf3LsQGGg6jojbOXjwIB06dNAar5LmXnvtNb7//nvTMUTc0s13p7duNZsjNVSCPYndjvXKZQAu5yoNISGGA4m4l8TERNq1a8eNGzc0DULSnMVioVu3bpw8edJ0FBG3oxIs6ctmo3be38nHGY6M+Mp0GhG389Zbb7F//36tBCHpwul0kpiYSI8ePTQ/WOQvqlRx7RaXdOockZuPm46TIirBHuTyZdduLOfIR6nOlUzHEXEr27dvZ+TIkSonkq6Sk5NZuXIln376qekoIm4lS9Rxztnyc458JL74iuk4KaIS7EG2bXNd3nMPZMtmNouIO0lISODRRx/VPGDJMIMGDeL4cc8Y7RLJEAUKkM3hOmE/aLdnLJOmEuxBbq4/XbWq2Rwi7uaNN97g4MGDmgYhGSYpKYkePXpo7rnITf7+nCndhGU0Zm1YS/CA/zdUgj1FcjLVPuvL03xE82ybTKcRcRtbtmzhnXfe0TQIyVDJycmsXr2aSZMmmY4i4jZOfPwjTVjGID4Aq/tXTItTfzlSLDY2lrCwMGJiYggNDc3YJ9+/H8qUAeBM3U7kXzs7Y59fxA3Fx8dTsWJFjh49qlFgMSIoKIi9e/dStGhR01FEjIuMhFy5XMdxceYWsUppX3P/mi4A2H/bc+s4U/VyBpOIuI833niDI0eOqACLMcnJydpNTuQPOXNC7tyu4337zGZJCZVgD3G0VEtqspH+AVPI2q216Tgixp04cYJ3331XczLFqOTkZNauXcsPP/xgOoqIWyj3xzjdnt3u/8JQJdhD7DqSmc3UZGvFx7FWrmg6johxw4cPNx1BBACr1coLL7xAcnKy6Sgixg27/DybqEHr54uZjvKfVII9xJ4/ZkOU00wIEXbv3s2MGTNUOsQtOBwOjhw5wtSpU01HETGu5PUd1GAL4THHXJOE3ZhKsIe4WYLLlzebQ8QdvPTSS9hsNtMxRG6xWCy88sorXL9+3XQUEaMsFcvjwMJx2z1w4YLpOP9KJdgTnD9P9o2LKcZhypVMMp1GxKg1a9awePFijQKLW3E6nVy8eJHx48ebjiJiVMjo4WQhjqL2I1zK695vX6sEewD7il/59ExLDlOC6qvfNx1HxBin08ngwYM1Cixuyel0MnLkSC5fvmw6iogxWQrnIHuBYAAOHzYc5j+oBHuAmG1Hbh2HVi1uMImIWQsWLGDz5s1aEk3c1vXr13nnnXdMxxAxqvgfVeXIkX+/nWkqwR7gSL4GvMVQfgrthLWCe7+1IJJe7HY7L7zwAlYP2IVIfJfD4eDDDz/k9OnTpqOIGFPsj4UhVILlrm3NVJ9XeItJDWdDyZKm44gY8fPPP3P48GGtCyxuz+Fw8Mknn5iOIWJMk+SfGM4IGk/vBm58/oZKsAe4+UqqmPsvuSeSbsaPH6+5wOIR7HY7kyZNIiEhwXQUESPq7P+cEbxO/eMz4ORJ03H+kUqwB1AJFl939OhRli5dqrnA4jGuXLnCnDlzTMcQMcK/9P8UFjeeE6ES7O4SEzl2yPVWgkqw+KqJEydqLrB4FKvVykcffWQ6hogRIX0foQ3zKcserlS413Scf6S/Km7OPn8BOw5m4iAlqLh/luk4Ihnuxo0bTJkyRaPA4lEcDgdbtmxhx44dpqOIZLjMNSuwOU8b9lGWo6cDTcf5RyrBbi7ut2P4k0wJDpMjh+k0Ihlv1qxZxMbGmo4hkmp+fn5MmDDBdAwRI26+e330qNkc/0Yl2M1dScrCLsoTY8mKrUhB03FEMpTT6WTcuHGaCiEeKTk5ma+++oorV66YjiKS4Qr+UVncebVAv5TcKDExkX379qXZkwYEBFCmTJk0ezxvtq3Gk3TkSerUhnW1nabjiGSozZs389tvv5mOIXLHEhMTmTp1KoMGDTIdRSRDlcx6gWqcJHT1BXi+lek4t5WiEnz27FkqV66cZk9aqFAhfv/99zR7PG928xVUgQKAxWI0i0hG++KLL/Dz8yPZjdeZFPkvn376qUqw+JwnFrdlOBthAZAQD4HuNzc4RSX4poCAAMqXL3/HT5aYmMju3bvv+P6+6E8lWMSHOBwO5s6dqwIsHs3pdHLgwAGOHTtG0aJFTccRyTD2vAXg9z8+OXcOChc2mOb2UlWC8+bNy9atW+/4yX7//XeKFClyx/f3RTdLcESE2RwiGW3r1q1ERUWZjiFy16xWKwsWLODZZ581HUUkwyTd14yJG8KJCy3Ai5kzm45zWzrbxJ2dO8eTv7RjHAOpfu4H02lEMtSCBQu0Q5x4jblz55qOIJKhAp96nP5MZOi1Ydhz5DId57ZSNBKcM2dOZsyYQUhIyF09WVo9js84fpx7L8/jXuDc7xagjelEIhnm+++/19rA4hUcDgfr1q3j8uXLZM+e3XQckQyROzfYbGC3w4ULkC+f6UR/l6ISHBwcTNeuXe/6ydLqcXyF4+TpW0P1mYppUrD4jmPHjnHgwAHTMUTSjMPhYPHixfobKD7DZnMV31OnXFM73bEEazqEG7tSvzVFOUoNNhH8xCOm44hkmB9//FFrA4tXsdls/PCDprWJb8mb13V57pzZHP9Ef2XcWNTVII5TlENhNfAvnN90HJEMM3fuXJxOrYst3sNut7N48WISEhJMRxHJGNevM3tfOc6Rh6qvPGA6zW2lanWIf3Lu3Dk2btzIxYsXSUpKuu1tQkND6datW1o8nc+4eWJ8eLjZHCIZ6cqVK6xdu1YlWLzO9evXWbVqFc2aNTMdRST9ZcpEgeuH8COJC5fPm05zW3dVghMSEujfvz/Tpk37zxNYChUqpBKcSirB4ovWrl2Lw+EwHUMkzfn7+7NixQqVYPENFgvRWQpyPSaR64F5yG06z23cVQkeOHAgX3zxBQD16tWjTJkyZMqU6ba31Rmxqee/Zjn346C0fzg4K2nHOPEJW7du1S5x4pWSkpLYsmWL6RgiGWbKkCMMHQo9G8IXpsPcxh2X4ISEBKZPnw7A559/Tq9evdIslLhU+/o5fmEXSesDgHjTcUQyxJYtW7Q0mnitrVu34nQ6sWhQQ3zAzXey3XXfozs+Me7y5cvEx8cTHBxMz5490zKT/CHwquun5nrmcI0Ci09wOp1s3rxZ84HFa8XFxXH8+HHTMUQyRI4crkt3LcF3PBIcHh6OzWbDarXqFW06WVB2KCc2nePeWgHUNx1GJAOcPXuWS5cumY4hkq62bt1K0aJFTccQSXdeOxLs7+9P8+bNiYuLY+fOnWkYSW6ak+spXmUkBzoNNx1FJENs3brVdASRdOXv78+2bdtMxxDJEBGn1vMWQxl6oi+4YVe8q3WCP/jgA7JmzUrv3r05ffp0WmWSP8TFuS5DQ83mEMko27Ztw88vTVZuFHFLSUlJbNq0yXQMkQyRbf96hjKKHomf4jxy1HScv7mrvzbFixdn3bp11KlTh5IlS1K9enUKFLj99r45c+bkgw8+uJun8zlXr7ouQ0LM5hDJKJs3b9ZJceL1tm3bppPjxCcEZAu+dZx8JQ5/g1lu565K8Llz5+jUqRMxMTEA/Prrr/9420KFCqkEp5JKsPgSnRQnvuLq1ascPXqUYsWKmY4ikq782z3IvYPLcJUQfrmvCO62WO5dleA+ffqwd+9eQkND6d27N6VLl/7HdYJD1ORSZ906Nh98gDhCcHw3EO59yXQikXQVHR3NlStXTMcQyRAHDhxQCRav51+kAJsCC5CQAHH+eE8JTkhIYPHixQAsXryYunXrplkoAWJjyeKMIwtxRDm1RrB4v7Nnz5qOIJJhzp07ZzqCSIYICYGEBLh2zXSSv7vjE+OuXLmCw+Ega9asKsDpwOnnzz5Kc5II/PJq32TxfioF4iv8/f31ok98xs2JADeneLqTu1onOHPmzMTHx+NwOLBa72qhCfmLhPr3U5Z9AMQ8bTiMSAZQKRBfohd94hMSE6ngd4RcXMWxLxRqlDKd6E/uuLn6+fnRuXNn4uPjWbJkSVpmEv78iik4+J9vJ+Itzp49q+XRxCckJSVx5swZ0zFE0t+ZMyw4WpbN1KTAFyNMp/mbuxq+fe+996hcuTKPP/44S5cu1VndaSj+j2nA/v5gs5nNIpIRzp07pyWjxGecOnXKdASR9Jc58/8fX79uLsc/uONhlzNnztC4cWPi4+M5e/YszZo1IyQkhPz589/29vnz52f58uV3HNTXJCW5Lv3dbVE9kXRy9uxZrREsPkPTf8QnBAfzY4F+/H7aj+oVynP7nSTMueMSnJSUxMGDB/903dWrV/923U3x8eZWOLhy5QrTpk1j586d+Pn5UatWLbp27UpQUJCxTP8lYPlPfMjPWJP94MATUMq95tGIpLVTp07hcDhMxxDJEJcuXcJut2PTW33izUJCmFh+Ij+dhi8bQC3Tef7ijktwgQIFOHz4cIpv729oSHP//v00btz4TychfP7554wbN46VK1cSHu6eKy8EbN/AM3wEicCp5irB4vU0R1J8icPhIDIykjx58piOIpKubp7qkZxsNsft3HEJ9vPzc/uFvh0OB506deLcuXPUrl2b3r17k5iYyMcff8yePXvo06cPc+fONR3ztpyJSf//iU4WEh9w+fJl0xFEMpRKsPgCryzB169fx8/Pj4CAgBTdfsuWLVSvXv1On+6O/PTTT+zZs4eKFSvy66+/3hqN7tKlC2XKlGHevHkcOXLELcv8xQ5P8eDnbckbnsz8ymVNxxFJd8nu+BtSJB3pZ158gTuX4DteHeLixYv06NEjRStC/PLLLzz22GN3+lR37OeffwZg4MCBf5qOkS1bNnr16gVwa9c7d3MjRwE2U5OdwXUha1bTcUTSnU6KE1+jEiy+4N1fKrKP0jSb0sF0lL+5qyXSvvnmG1566aV/vc2aNWto27YtCQkJd/NUd2Tv3r0A1K5d+29fq1Onzp9u425urg6hmRDiC5xOp0qw+ByVYPEF+WP3U5oDZLl03HSUv7njipUvXz5q167NmDFjKFSoEP379//bbTZt2kTLli1JSEhgzJgxdxX0TkRGRgKQN2/ev30tX758AERFRf3j/RMSEv5U3mNjY9M44T+7+btRJVh8gQqw+KKkpKT/vpGIh7vhH4Y9IYkkf/fb+euOR4IDAgJYsGABxYsXZ+DAgSxYsOBPX9+xYwfNmzfn6tWrfPHFF3Ts2PGuw6ZWYmLirax/dfO6fxuhHjVqFGFhYbc+IiIi0ifobQRdOkNltlPu+mb33HBbJA1pkwzxRfq5F18wuHskufyj+brvatNR/uaupkOEh4fz008/kT17dh5++GE2bdoEuKYYNG3alOjoaCZOnEi3bt3SJGxqBf+x3/DV25TIm9cF/8uexC+//DIxMTG3PjJyh5+IWWPYTlW+O1UT9u3LsOcVMcFqvatfRSIeSduEiy+YPBkSE+E/Zs8acdd/ee655x5+/PFHnE4nDz74ID/99BP3338/UVFRjB07lr59+6ZFzjtSoIBrb5KjR4/+7Ws3r/u30d3AwEBCQ0P/9JFRLH7/s4C65o2Jl7NYLNo0QHyOSrCIWWky/FKzZk2++eYbLl26RIsWLTh//jwjR45k0KBBafHwd6xatWoALFmy5G9fu7kqxM3buJvrlesygf58FjwQtI6k+AAVAvE1pjaREhGXNHsPsk2bNnz00UeAaxrBsGHD0uqh71j79u2xWCyMGzeOff8zpWDFihXMnj2bkJAQHnjgAYMJ/9m1Zu0ZwAReCRkHRYuajiOS7lSCxdeoBIuYlaK/OidOnKBkyZIpftCxY8cyduzYP11XqFAhDh48mLp0d6ls2bJ0796dqVOnUqVKFerXr09iYiLr1q3DbrfzyiuvEBYWlqGZUsqdF5cWSQ85cuTg2rVrpmOIZJhcuXKZjiDi01JUgp1OZ6rW+b3dbU2sEwwwceJEnE4nM2bMYNmyZYDr1feQIUMYMmSIkUwpcXN6pEqw+IqIiAhOnjxpOoZIhvDz8yNHjhymY4j4tBSV4MKFC9/1eoamloIJCgpi6tSpvPXWW+zatQubzUa1atXInj27kTwppZFg8TUFChTAarXicDhMRxFJd+Hh4VoVRcSwFE/C8/T5evnz5yd//vymY6RY2Pyp7OFd/K8lw8/joHlz05FE0lW+fPmw2WwqweITPOnvkYi38uxm68UyOa5TFtfJfMlnL+o/lHi9vHnzqgCLT7BYLBm6+ZKI3J66lZsKyB7CdTJxlRCCEy36DyVeL1++fNo+WXyCn58f+fLlMx1DxOelaELS+fPnqVev3l1vfZxWj+ML/Hp1I6v/dXJzkSutHjMdRyTdqRSIr3A6nfp5F3EDKRpgjI+PZ926dRQqVOiuniytHsdXhITAlStwm12fRbxO3rx5TUcQyRDJycn6eRdxA6l6lz0xMZGdO3fe8ZOdPXv2ju/ri4KDVYLFd2hkTHyJft5FzEtVCT537hyVK1dOryzyFyEhrkuVYPEFoaGh5M6dmwsXLpiOIpLuypYtazqCiM9LUQkODAykatWqafakegWcArGx9L06hStcJWxRcWj4iOlEIumuZs2aLFy4UKtEiFfLli0bBQoUMB1DxOelqATnzZuXrVu3pncW+V/Xr/Ps6cEAnF7VBlAJFu9XvXp1Fi1aZDqGSLqxWCxUr17d2AZSIvL/tF2NuwoLu3VojYk2l0MkA1WtWlXLpIlXs9ls1KhRw3QMEUEl2H1lysSnDb+mOT+xsPnHptOIZIi0nHYl4o6Sk5P1cy7iJlSC3djRGo+whObst5UzHUUkQ+TKlYs8efKYjiGSrqpVq2Y6goigEuzWwsNdl1FRZnOIZKRatWphtepXk3in7Nmzkz9/ftMxRASVYLd2swRfumQ2h0hG0klD4q0sFgs1atTQz7eIm7jjEnz69Gk6duzIL7/8gtPpTMtM8odcIdcpyAnCT2yDa9dMxxHJENWqVdPJceKVbDYb1atXNx1DRP5wxyU4OTmZ7777jqZNm1K8eHHeeecdLXKfxqrMHsIJCjN9XzXYu9d0HJEMUbduXfz9/U3HEElzycnJNGnSxHQMEfnDHZfgfPnyMX78eMqXL8/Ro0d5+eWXiYiIoGPHjixbtkyjw2kgIG+O//9EE4PFRwQHB3P//fdjs9lMRxFJU2FhYdSuXdt0DBH5wx2X4ICAAAYMGMCuXbtYv349PXr0wN/fn++++44mTZpQvHhxRo8ezcWLF9Myr08JqFqOuTzEeAaQmDvCdByRDNO2bVvtGidexc/PjzZt2uDnl6I9qkQkA1icaThkGxMTw1dffcWnn37Krl27APD396dt27b07duXRo0aefQJAbGxsYSFhRETE0NoaGi6P5/TCZkyQUICHDsGRYqk+1OKuIVz585pe3XxOt9//z3t2rUzHUPE66W0r6Xp6hBhYWE89dRT/Pbbb2zYsIGePXtisViYM2cO999//625wxodThmLBW5uL3/mjNksIhkpb968VKlSxXQMkTTj7+9P06ZNTccQkf+RLkuk3bhxg0OHDnHw4EESExMB11mxN+cOFy5cmNdee01vd6bAzeUkT582m0Mko7Vr107zgsUr2Gw2GjduTEhIiOkoIvI/0rQE7927l2eeeYb8+fPTvXt31q9fT7ly5fj444+5fPky69at45FHHiE+Pp433niDcePGpeXTe6WbI8EqweJrWrduraXSxCs4HA7atm1rOoaI/MVdzwmOj49nzpw5fPrpp6xduxZwnTTXvn17nnzySerXr/+3+4wfP56BAwdSq1YtNmzYcDdPn6Eyek4wwNbyPQjYs53c2ZLIfXl/hjyniDtwOp0ULFiQ03oFKF7gzJkzmucukkFS2tfu+DTV69evM2zYMKZNm8aVK1cAKFy4MH379qV3797kzJnzH+/bvHlzAM6fP3+nT+8z8l89SF52wxUgMRECAkxHEskQFouFjh07Mn78eJKTk03HEbkjVquVSpUqqQCLuKE7ng5x8eJFPvzwQ2JiYmjZsiULFy7k6NGjvPTSS/9agMF1Al3Lli1p1KjRnT69z7DnLUACAZwJLAKXL5uOI5KhevfurQIsHs3hcPDkk0+ajiEit3HH0yHOnz/PRx99RN++fSlUqFBa53JLRqZD/HqNGg0zkb+AlVOnMuQpRdxKgwYNWL9+veYHi0cKCQnhwoULZM6c2XQUEZ+R7kuk5cmTh7fffttnCrApBUoG48TK2bOu2RAivmbgwIEqwOKRbDYbjz/+uAqwiJtKlyXSJO3kzg3BweBwwO+/m04jkvHatGlDrly5TMcQSTW73U7//v1NxxAxYtcuiIszneLfqQS7OYsF7rnHdXz0qNksIib4+/vTv39/rRksHsVms93aJErE19y4ARUrQmgoXLpkOs0/Uwn2AH38vuBtXibnB0NNRxExok+fPqYjiKSK3W5nwIABpmOIGHFz0C5rVsie3WiUf6US7AEePTScl3mHUmummI4iYkTevHlp164dfn53vKqjSIbKly8frVq1Mh1DxIhDh1yXJUq43tF2VyrBHuB6vmIAhMRHQXS02TAihgwYMEDLpYlHsFqtDBgwQFN4xGcVfH8g7/E8j1s+Nx3lX931jnG+xMQSaQA73l/BS4OTcBQpxi9HioBVr13E9zidTmrUqMGOHTu0WoS4teDgYE6cOEGOHDlMRxHJeA4HCQEhBNpvEJWjBOFRBzM8QrovkSYZJ3uHRiylGb+evge7U//JxDdZLBbee+89FWBxaxaLhaFDh6oAi+86e5ZA+w0AEgqWMBzm36lReYACBVy7JSclwYkTptOImHPvvffSvHlzzQ0Wt2SxWAgPD+eZZ54xHUXEnHz5qJrzJI1YTuwA9z6hXyXYA9hsULKk63jfPrNZREwbPXq0RoPFLTmdTkaOHElwcLDpKCLGRMda2R4ZwUoaka99bdNx/pVKsIeoWfwy9ViD/atvTEcRMapChQo8+uijGg0Wt2K1WilatCi9evUyHUXEqL17XZcRERAWZjbLf1EJ9hBvr23AGhrQck5317wIER/25ptvmo4g8icOh4MxY8boxZn4vN27XZfly5vNkRIqwR4isUQ5APwcSXD4sOE0ImYVLlxYS1CJ27DZbFSpUoV27dqZjiJiltNJtpkTuI8V1CwaaTrNf1IJ9hD+D7XiY57iab9PSM4abjqOiHHDhg0jMDDQdAwR7HY777//PhZ33hVAJCOcOUPnNQNYQWN6r+5mOs1/Ugn2EOHPdmVI5o/5OPlJjsblMh1HxLjw8HBNixDjbDYbbdu2pWHDhqajiBjn3L3n1rF/pXIGk6SMSrCHsFqhTBnX8Z49/35bEV/xzDPPULNmTc3DFCMsFgtZsmRh8uTJpqOIuIULucrzOFP4yDKQrJ2amo7zn1SCPUi5P15U3Zx0LuLrbDYb06dP19xgMcLpdPLpp5+SK5fenRMB+C0qP5/zOBNLjiOgZRPTcf6TSrAHqVABwMn5dUchPt50HBG3UKJECd555x3TMcTH2Gw22rdvT8eOHU1HEXEbNwfpyrn/TAhAJdijtD72IZfJzqRlxWDzZtNxRNzGwIEDqVOnjqZFSIawWCyEhoYyceJE01FE3Mq2ba7LSpWMxkgxlWAPkr9kCNmIBiB2xVazYUTciNVqZdq0aSrBkiGcTidTpkwhZ86cpqOIuI/ISKLWHcSCg+rVTYdJGZVgDxJUrxqRfnlYwIMcSixsOo6IWylWrBjvvvuu6Rji5Ww2G126dKF9+/amo4i4lWtffssvp0pxhWzUPjfXdJwUUQn2JBUrMviRc7RhAQsDtCi7yF/179+f+vXra0RY0oXVaiVr1qx8/PHHpqOIuJ2YZVsACCOWLKXyG06TMirBnsRioVo11+FWzYYQ+Rur1crs2bPJnj27VoyQNGexWJg7dy45cuQwHUXE7ewIqc882nIxuDBUrGg6ToqoBHuY/y3BTqfZLCLuKG/evCxatAibzaYdvCRNTZw4kQYNGpiOIeKWPnP2ph3zmDHiOAQFmY6TIirBHqZiRbDZ4MIFOHPKYTqOiFuqVq0aU6dOxalXipIGLBYLTz/9NE888YTpKCJu6+Y71J5yUhyoBHuczInRfB/agz2UhS5dTMcRcVsPP/wwQ4cONR1DPJzNZuO+++5j7NixpqOIuK3z5+H0adfutlWqmE6TcirBniZLFppem0dZ9hH622rNiRD5F2+++SYPPvggVqt+1Unq+fn5UbBgQb777judbCnyLzavuApA6dIQEmI4TCroL4Onsdm4XLouSfhxwlIEoqNNJxJxW1arlZkzZ1KyZEmVGEkVq9VKUFAQixcvJlu2bKbjiLgvp5OGTxTjACX50Pac6TSpohLsgZwTJxNGDJXjN3AtQL+cRf5NSEgIixcvJkuWLFoxQlLM6XQyZ84cSpUqZTqKiHs7fJjQ6xcoySHKWA6YTpMqKsEeqEDtCMIjMmO3w6ZNptOIuL/ChQvz888/ExgYqKkRkiJTpkyhefPmpmOIuL3r52LYQG0S8Sdzs/qm46SK/hp4qHr1XJdr1pjNIeIpatSowdKlSwkICFARln81YcIEevfubTqGiEdYl1idOqynQkQ0WYf2Nx0nVfSXwEPdLMFr15rNIeJJ6taty88//4y/v7+KsNzWBx98QP/+nvWHXMSkm4Nx1e/NDGFhZsOkkv4KeKjGBQ7yBq8yfEVD7PMWmI4j4jHuvfdeFi1apCIsf/Puu+/y7LPPmo4h4lFuluD6njUTAlAJ9ljFOcyrjKS+41civ11hOo6IR2ncuDFLly4lMDBQJ8sJAOPHj2fw4MGmY4h4lMRE2LjRdawSLBnGWr8uDlxbwsbu+t1sGBEP1KBBA1auXEnmzJlVhH2YxWLhs88+Y8CAAaajiHicqG7PsSS+AWMzDaNUeJTpOKmmEuypsmVj8ePziOAk/fPON51GxCPVrFmT1atXExYWpnWEfYzNZsPPz4+vvvpKJ8GJ3KHAX36kAWt4Ov5dLJmCTMdJNZVgD3bPc204TQRr18KNG6bTiHimSpUqsXPnTsqWLas5wj7Cz8+PsLAwVq1axSOPPGI6johniokh9oY/AJHFanvWVnF/0G98D1aqFOTPDwkJWiVC5G5ERESwfv16OnXqZDqKpDOr1UrZsmXZuXMndevWNR1HxGPFEEbxxH1EcBLHhx+ZjnNHVII9mMUCTZq4jn/5xWwWEU+XOXNmZs6cyejRo7FYLBoV9lKdOnVi/fr1REREmI4i4tFWrgS7HTIVjyB/i4qm49wR/Zb3cG0qnaA/E2j+6UNwwLO2KxRxNxaLhRdffJGFCxeSOXNmzRP2ElarFYvFwujRo5k5cyaZM2c2HUnE490cfGva1GyOu6ES7OEax8xlAgNoFDOfuO+WmI4j4hVatGjB1q1bKViwoFaO8HA2m41MmTKxcOFCXnzxRSwWi+lIIl5h6VLX5c13pD2RSrCHy/LQ/beOLy/aYDCJiHcpWbIk27Zto1WrVgAqTx7IYrFQokQJtm3bRosWLUzHEfEal4d/yLtH2vKU5RPuq3jZdJw7phLs6cqV44f7PqAy23mt+EzTaUS8StasWZk3bx4zZ84kNDRUo8IewmazYbPZGD58ODt37qRkyZKmI4l4lYRv5tKWH/jY+RShxJqOc8dUgj2dxULW155lJ5VZuNiK3W46kIh3sVgsPPzwwxw8eJAHH3zw1nXiniwWC6VLl2br1q28/vrrBAQEmI4k4l0SErCdOQHApRwloHBhs3nugkqwF6hbF7Jlg0uXYINmRIiki9y5czN37lxmz55N1qxZNSrsZm5ufjFixAi2b99OpUqVTEcS8UrX7YEUdhynGluIedMzl0a7SSXYC/j5wQMPuI5//NFsFhFvZrFY6NSpEwcPHqRdu3a3rhOzLBYL5cqVY/v27bz66qv4+/ubjiTitZYvhxsJVqIKVaNIv2am49wVlWAv0a5JHO34ntqTu8POnabjiHi1nDlz8u233zJnzhxy5cqlImyIzWYjKCiIkSNHsmXLFsqXL286kojXW7DAddm6tWu/Ak+mEuwlWkTP5Hs60DZmOpc/n2c6johP6NChA8ePH2fMmDGEhYVpg40M4ufnh5+fH08++SS///47Q4cO1eivSAZwOP7/HefWrc1mSQv6je0lMnVsdev4xqLlBpOI+JZMmTIxePBgTpw4wdChQwkKCtJ84XTi5+eHxWKhS5cuHDp0iPHjx5M7d27TsUR8xtknhvP2hV48nGk+DWommI5z11SCvUX+/Kxr+TbN+JleBVWCRTJaWFgYb775JsePH+fJJ5+8NVopd+/mi4pmzZrx22+/MWPGDIoUKWI4lYiPcToJ+W4qvfiSaQmdCSDRdKK7phLsRfJ+9DJLacbytYFcumQ6jYhvypMnD+PHj+fQoUN06dIFi8WikeE7dPNFRM2aNVm3bh0LFy7UvF8RU37/nYA4V7mILNcIsmQxHOjuqQR7kaJFoVIlsNth/nzTaUR8W5EiRZgxYwZHjhzh+eefJ2vWrACaN5wCVqsVf39/unTpwoYNG1i7di116tQxHUvEpx1MLEIOZxQPWheR5a2XTMdJEz7x2zgqKop169Yxd+5c1q9fT3x8vOlI6aZTJ9fl7Nlmc4iIS9GiRRk9ejTnzp1jxowZVK9eHUBTJf7i5r9HoUKFGD16NGfPnmXGjBnUqlVLq2+IuIHZsyGeTNibtSBLq3tNx0kTXl2Cf/75Zxo3bkzu3LmpV68e7du3p27duuTNm5cPP/zQdLx00bG9g1psoOWyQcROmGE6joj8ISgoiK5du7Jx40Z27txJ7969CQoKwmKx+HTJs9lsWCwWHnjgAX7++WeOHTvG4MGDCQ8PNx1NRP7HzcG1zp3N5khLFqfT6TQdIr306NGDadOmER4eTokSJciRIwd79uzh+PHjAIwfP54BAwak+PFiY2MJCwsjJiaG0NDQ9Ip9d/btg7JlAThXrB55D68xHEhE/klsbCxff/01s2fPZu3atdjtdvz8/EhOTjYdLd1YrVacTidOp5Ny5crRoUMHevXqRUREhOloIvIP9uyB8uUhIAAuXoSwMNOJ/l1K+5pXl+DZs2eTK1cuGjZseGukxW6388orr/DOO+9wzz33cOTIkRQ/nkeUYCAqd1nCL+4j0RpIQORZyJ7ddCQR+Q/R0dH8/PPPLFiwgIULFxIXF+c1hfjm9+Hn50ejRo1o06YNrVq1omDBgqajich/cTo5XuQ+5pyowbkGXfjg1yqmE/0nleB/YbfbCQsLIykpiYSElK9z5ykl+OK4b3jh2UR+tLThwPms5MplOpGIpEZSUhLr1q3jxx9/ZO7cufz+++8A+Pv7k5SUZDZcCvj5+WG323E6nWTLlo02bdrQunVrmjRpQkhIiOl4IpIKzo2bsNSuBcDZ8k3Jt2uJ4UT/LaV9zSfPzEhOTsbpdFKiRAnTUdJFrmceZv/XcGULzJ0L/fqZTiQiqeHv70/Dhg1p2LAh7733HkeOHGHTpk1s27aNLVu2sH37dm7cuHHrtiaLsb+//63fqX5+fpQpU4aaNWtStWpVqlatSuXKlbVEnIgHO/3jDvJiww872fp50YRgfLQEjx8/nuvXrzNo0KB/vV1CQsKfRopjY2PTO1qa6dQJtmyBWbNUgkU8mcVioXjx4hQvXpyuXbsC4HA4OHz4MNu2bWP79u1s3ryZ7du3c+3atT/d12azYbVacTgc2O32O3puPz8/nE7n36Zl+Pv7U7Zs2VuFt0qVKpQrV47AwMA7/2ZFxO184ujHFDowsvJc+j38kOk4acqtp0PEx8czPxUL3mbPnp2mTZv+622WLl1Kq1ataN26Nd99992/3vb1119nxIgRf7ve3adDAJw6BYUKgdMJx49D4cKmE4lIenI6ncTGxnLu3DnOnTvH2bNn/3R8+vRpTp8+zdWrV7Hb7SQlJWG327Hb7Vit1ls73NlsNoKCgsiXLx8RERHky5ePvHnzkjdv3j8d58iRQ2sei3g5h8O1B8GJE65BNU9ZGcIr5gSfP3+evHnzpvj2VatWZevWrf/49Z9++on27dvToEED5s+fT1BQ0L8+3u1GgiMiIjyiBAN0qXeawuu+onfxNRQ/uBB8eBkmERERSZ2VK6FRIwgNhfPnIVMm04lSxivmBGfKlInOqXjZcc899/zj12bNmkW3bt24//77mTt37n8WYIDAwECPfmvvg+ie5GUZHAbnuvVY6tU1HUlEREQ8gdPJ1KmuwbMuXTynAKeGW5fgsLAwZs2addePM3HiRAYMGECLFi347rvvPLrYpkb2Z7vBE8sAODFzHYVVgkVERCQFbkyezqAZ4wijB4+16wp433KrXj+ha+TIkfTv35/WrVvz/fff+0wBBgh8uB1zK79BEY7xdvKLpuOIiIiIh4geN5VKzh18xDNUCz1kOk66cOs5wXfrzTffZPjw4eTPn59Ro0bh7+//t9u0a9eOgICAFD2ep6wT/L9WrYL77vO8+TwiIiJiSHw8+3LdS5m4zVwKL0GOiwc86rwir5gTfLfWrVsHwJkzZ+jWrdttbxMZGenVe9Q3aOBaJeLECZg/Hx5+2HQiERERcWfHzgZRNm4TZdjHyo8veFQBTg2vLsENGzYka9as/3obb58eYbVCt27w5pswdSo83NnhulJERETkNqZPd13mb1KGXJ3LmA2Tjrx6OkRa88TpEADH9icwtMw8+jGJigMbkm3c66YjiYiIiBuy211rA588CV99BY8+ajpR6qW0r2lI0AcUzRLJ1zxKQ37F+sUU+MvOTyIiIiIAy769zMmTkD07tG9vOk36Ugn2BQUKcLHGgwBcuBFKwpFThgOJiIiI2zl7lvu65uMHWjOy8UpSsKWCR1MJ9hE5xw+nQ46VlLTvY97OIqbjiIiIiJu58u5nBDgSaM2PdMq5wnScdKcS7CP8alSh3ICGgIVJk0ynEREREXezel84pyiAHSs5hjxhOk66Uwn2IY8/DjYb/Por7NtnOo2IiIi4i4QEeGJHf4pwnDVvr4WCBU1HSncqwT6kQAF40DU1mOkfRUNiotE8IiIi4h7mzYPISMidz496L9Q2HSdDqAT7mOdaHuRzejF8cj7iZ8wxHUdERETcwMSJrssnngA/r95F4v+pBPuYusUv0osvycwNYkd8AFomWkRExKcdmbqWrauvYbO5pk76CpVgH2NtUI+LEVWIIZQFcQ1xxGtKhIiIiM+Kjib/Ew9wighmlnqDAgVMB8o4PjLgLbdYLATP+5pSjfNzOjoLuX6B1q1NhxIRERETYsZ+RljyVTIBDUucNR0nQ2kk2AcFVy3FI32zADB2rOEwIiIiYszUc835kh4kWALJNWqQ6TgZyuJ0alJoSqV0L2pPcPo0FCni2kF52zaoUsV0IhEREclI1665VkK7fBkWTI/mwceymo6UJlLa1zQS7KMKFICOHcFGMuuenQNXrpiOJCIiIhlo+nRXAS5aFFo8ktV0nAynEuzDXqu3nCMU4+k1nYh5/zPTcURERCSDOBzwwQeu42efdW2m5WtUgn1YyfsjKMwJ1yfjxmnzDBERER9xoMc75D+8krBQJz17mk5jhkqwLytRgnM127CEpnS2z+TKtQDTiURERCSdOffspcyMl1lJIzaEP0hIiOlEZqgE+7jcK2bxQvklLLnRgPHjTacRERGR9HZ6xOe3jvM/1shgErNUgn2cNXMQQ4e6jseNg6tXzeYRERGR9NXr0rt04RsO5W9I6At9TccxRiVY6NgRihd3nSE6aZLpNCIiIpJe1q+HZSttfO/XhaD1KyE42HQkY1SCBZsNXnoJSrOPwq8+RsK3P5iOJCIiIungrbdcl927u9YI9mXaLCMVvGmzjL9K2rgNW+3qWHFyMaIquU5sAYvFdCwRERFJIzs2JlCldiBWKxw8CMWKmU6UPrRZhqSKf80qXCpQyXV85jhJh383mkdERETS0PnzFG0YwSheou9DF722AKeGSrC4WCyEfjSS10PeI8Jxgqm/FjGdSERERNJI1EvvEpYQyUuM5o3Q90zHcQsqwXJL4EMtyPrm81wjhDfegPh404lEREQkLfy6MYh4AkmwBhH+9nOm47gFlWD5k379oEABOH0aJk82nUZERETu1tat0OHgWxTnCBffmwF58piO5BZUguVPgoLg1Vddx2PfusH17QfMBhIREZG78sorrsv7HitAxKAOZsO4EZVg+Zue3R28Ej6J9ZHFuNGsDSQnm44kIiIid+DXX2HJEvDzg9dfN53GvagEy9/4B1gYED6L/JwlR9Qhrk+abjqSiIiIpJJz336W9JsLOHniCSha1HQi96ISLH9nsRA+aSQA82nD57trGA4kIiIiqXXhiWG8faA9Wy01eK3rUdNx3I5KsNyW7d56LB23n4eYz8tfl+PiRdOJREREJKUcv+0mz/p5ABTPfIbclfIaTuR+VILlHzV5uhTVqsG1azBihOk0IiIiklJz9pejJQvZZa2I9bVXIXNm05Hcjkqw/COLBd5913U8eTLs3282j4iIiPy3+HgY8pKFxbTkh+HbCRn0hOlIbkklWP5Vw4bQujWUt+/g+r3N4fhx05FERETkX4wbBydOQP788PwLVtfSEPI3KsHynz5pNIdtVKVq5BIu9HzJdBwRERH5B5HH4njrLdfxqFGaBfFvVILlP+Xv2YxrmcIBSNiwDfuVWMOJRERE5G+uXCGo7D2MjnuS+ytc5NFHTQdybyrB8t9CQ2HMuwwNGkvxxL1Mnx9qOpGIiIj8RdSgkWSJj+RJJjE9YhhWtbx/pX8eSZEsA7qT/c1BJBLIsGGuFSNERETEfczfmJc4QkiwZSLvxOGm47g9lWBJsaefhiJF4Nw5eOcd02lERETkpp9/hicODqaM32EuTZgFERGmI7k9lWBJscDA/18y7bPRlzg36QezgURERISEBHjmGddxx6fzkK9va7OBPIRKsKRKu3bwXtkv2ZNUgvD+nXAePGQ6koiIiE97/304dAhy54bXXjOdxnOoBEuqWCzQo95hcnAZf2ci53q8bDqSiIiIz7o8/EMuvv4JVuy89x6EhZlO5DksTqfTaTqEp4iNjSUsLIyYmBhCQ314hYRr14jOV5pFsfX5MN+7/Ho4n9YhFBERyWjHj5NQrAyBjnj2ZalJ6UtrsfhrY4yU9jWNBEvqBQfjv/c3hhb8mq1n8zFqlOlAIiIivmfvmEUEOuIByPVQHRXgVFIJljsSXCAbH3zgOh4zBg4fNptHRETEl9y4Aa2XDqA+qzlYuCnh4183HcnjqATLHXvoIWjWDBIT4Zn+STjPXzAdSURExCeMGQPHjsHx/PXJt2uJa2MrSRWVYLljFguMHw+1/Lby9rLqRN7bARwO07FERES82sGD3JqKOHYsZMliNo+nUgmWu1K8qJ0fs3alEr+R69Ba4t6bbDqSiIiI13Ls+I2JHVeQkOB6N7ZjR9OJPJdKsNwdm42sMycC8BsVGL2iuuFAIiIiXiopiUute/Dh7sZ86fc4n469isViOpTnUgmWu+bX5D4OjV1IDctW3lpSjcWLTScSERHxPpfGzyTn6Z0AtMq1mYLFAswG8nAqwZImSgxqyYBB/gD06wdxcYYDiYiIeBGnE3osf4x+TCTWlpVs338OASrBd0MlWNLMG29AkSJw6hQMHQokJ5uOJCIi4hW++QYWLrbyZUA/zm44ia2Wph/eLZVgSTPBwfDpp5CTi9T5+GEutuplOpKIiIjHi4yEgQNdx6+8AqWqazmItKCtRSRN3d8wmd1Z6pA77igsgfhZ7Qjq0tZ0LBEREc904gRv9U/k0qXilC8PQ4aYDuQ9NBIsacvPj5B3XgUgihxMm6EfMRERkTtit3Ox+WO8tbgS/S0T+fwzp6YBpyE1FElzwU9243DvdyjHHvotbs3SpaYTiYiIeJ4r704h14E1BHOdkaGjqV7mmulIXkUlWNKexULxz4bQ4ak8APTqBVeuGM4kIiLiQZxO6P7LY3zCk9ixkmXudAgJMR3Lq6gES7oZMwZKlIAzZ+Dpp9GWyiIiIin0ySfw44pgng/6hN8X78evUQPTkbyOSrCkm8yZYfp08LckU/jrkZyv3FxFWERE5D8cPAgvvOA6Hj0a7nmghNlAXkolWNJVzZqwvdQjjORV8uz6hdg3x5mOJCIi4raSd++n36Nx3LgBjRvDgAGmE3kvlWBJdyXH9QcgGRuzvkrWYLCIiMjtxMQQ16Aln2+rSPPgNXz5JVjV1NKN/mkl3fk3aciF50fTLPBX+h55gTFjTCcSERFxP+ceeZ5s0ccpynGmRrxCRAGn6UheTSVYMkTu917k0U/qAq7dbjZsMBxIRETEjURGQuutw1lLXa4FZCX3z9PBYjEdy6upBEuG6dkTHn4Y7Hbo0gWuXEwyHUlERMQ4hwN69ICtFwvSr+QqLGvWQKFCpmN5PZVgyTAWC0yaBEWLQqWTP5BcuBjOAwdNxxIRETHqgw9g8WIIDISZ3/qRuUY505F8gkqwZKjQUFjaexY/0JacN04S1aQLxMebjiUiImLEofd/ZOgQOwAffggVKpjN40tUgiXD3fPMg1zKXRqANWfvYfe2RMOJREREMt71KV9TYnBrFtmb8Xir8/TtazqRb1EJlowXHEz2pbOZUO4T2jvm0K5HKDExpkOJiIhkHGdMLI6nXIsA389yPmy9QufBZTCVYDHCUqE8nVc+ScGCFo4cgW7dtJmciIj4jvc+DaVF0g+cJzeRLboT/MQjpiP5HJVgMSY8HL7/HgICYMECeH/EVTQkLCIi3m7FCnjpJVhDA5a8vZ2ccz4xHcknqQSLUdWqwYQJUIKDtHijJhebP6YhYRER8VonT0Lnzq4/dd27Q7eX8kHmzKZj+SSVYDHu8e5JrAtpTln2kWvjj1x5WVvKiYiI90lcvIyPG88jKgoqV4aJE7UfhkkqwWKevz+h30zGgYU9lKX3onZaNU1ERLzL77+T2K4zY460Y2zQUObOsZMpk+lQvk0lWNxCQKumRE2eS8vsG5m3twRPPQVObZkuIiJeYvvTXxKScBmARyvtoXARDQGbphIsbiNXn7Z8NisEqxW++ALGjTOdSERE5O6tXw+1l7zOs3xAVM5S5PppOlhVwUzTfwFxK02awHvvuY6ff87Jby9+DXa72VAiIiJ36PffoW1bSEyycPKhZ8l+YidkzWo2lAAqweKGnn0W+nW/wVfOR6j4blcu9XrBdCQREZFUi4tx8OCDEBkJlSrBjBlgzRRoOpb8QSVY3I7FAh/1/o2Olu8AyDb9Q678ustwKhERkZSzb9pKVERlEvccJE8e+PFHCA42nUr+l0qwuCX/+rW48f4nXLME05b5tB1egcRE06lERERS4PRprjZuTZG4XWykFr+8v4sCBUyHkr/yuRJ8//33Y7FYyKr5OG4vy6AnOLPiEL+Gtmb1aujfXytGiIiI+/vmawe/X8sJQHKpcpRrX9JwIrkdnyrB06ZNY/Xq1QQEBJiOIilUomE+Zs1ynUT7+efw7ruoCYuIiNtauRK6vVKQeqxla7V+5Fw9FwI1D9gd+UwJjoqK4vnnn+fFF18kS5YspuNIKjzwAIwd6zpeNmQpZys9ADdumA0lIiLyF3v2wEMPQXIytOqShaqbJ0LOnKZjyT/wmRI8aNAgwsPDefXVV01HkTvwzDMwrcVsFtGSfLuWENnkES2dJiIibuPizGW0aGYnJgbq1nWtd68tkd2bT5TgZcuWMXPmTD777DMC9ZaEx+r6ZkmS/Fx7TG7cBLu3JxlOJCIiAtcmzSDXo00Ye7YzFUvGs2AB2hLZA3h9Cb5x4wb9+vWjT58+1KtXz3QcuQvWKpXwm/893+UbSNvk72jeNoiTJ02nEhERXxZ/4gK2p/oC0IHvWd73W7JnNxxKUsTrS/CIESOIj49n9OjRqb5vQkICsbGxf/oQswJaNqHxnnGULmvj7FnXfOErV0ynEhERX2S3w2ODc/OQ43uuEkxU5/7kePYx07Ekhdy6BJ8/fx6LxZLij2rVqv3p/rt27eL999/nk08+ITQ0NNXPP2rUKMLCwm59REREpNW3JnchWzb46SfIlw/27YNHWsaQsGKd6VgiIuJDnE4YNAi++w6W+z/AnqnbCP/6I00E9iBuXYLvhsPhoE+fPjz00EO0bt36jh7j5ZdfJiYm5tbHqVOn0jil3KmICFcRLhISyVsb7oMmTUheucZ0LBER8QUOB2+/DePHuz6dPh1qdS8JNpvZXJIqfqYD/Js8efLgvMM1Yffs2cOmTZvYtGkTln94VWaxWMidOzfnz5+/7dcDAwN1Ip0bq1ABfm01hohZO8AB51v3JGfUAWyBbv1jLSIiniwqiouVm7Hp9GtAa8aOhS5dTIeSO+HVI8Hi/SKmjiSyanNOk5/7rv5IvwF+2ktDRETSR2wsUdWak+v0dubSjm86zWPQINOh5E55bQmuVKkSTqfzth85cuQgLCwMp9P5j6PA4iECA8m5+nt2frKeQ9bSfPYZDB6sTeVERCTtzVmYiaUnXFsgXwvOTee3KxpOJHfDa0uw+JDMmWn1ZEE++8z16dix8OabaFc5ERFJMwsXwiPd/enKDJZUeIHQTb9guaeo6VhyF1SCxWv07Akffug6vvbaaC4VqQoXLxrNJCIinm/FCujQwbUd8sOPWLl/+xgsZcuYjiV3ySdLcFRUFNHR0aZjSDp45hlY8sAHjOYlclzYz6VKjTQiLCIidyY5mVPdhtLzwSgSEqBNG5g6VYtAeAufLMHi3Zp83JboLAUAeOdcd8Z/pr0rRUQklZKSiGz6CBEzRvHD9ftpd+8lZs8Gf3/TwSStqASL17EULULY9lX80HQC7/ECAwfCuHGmU4mIiCfZ9MN5kla5NmIqY9nP10N2oVVTvYtKsHglS7F7aP1zf15+2fX5s8/+/3xhERGRf7NqFTTqHsF9zhWcy1QE57wfCHrgPtOxJI2pBIvXsljgrbdg6FDX55MH7edUsftAy+KJiMg/WLECWrSA69ehaPOSZDt/gMA2zU3HknSgEixezWKBkSPhw34HWMl9RBxdxaWKKsIiIvIXCQnsevVbWrZ0nU/dogXMmwdBoQGmk0k6UQkWr2exwMAXgwgMCwLg6MUsvPtJsDbUEBERl+vXiaz/EBVGdubZ+FG0bAlz50JQkOlgkp5UgsUnWIoUJuuOVRwq3pIH+IkX38zCiy9qZzkREYF1w5eQc8tPALxmG8n3H5zUSXA+QCVYfIalSGFKHFrIK2NzAPDee9CnD9jthoOJiIgxU6ZA/bEP8RKjuO6XBduSnwgsXtB0LMkAKsHicwYNgs8/B6sVvvwsmbX3dCfpl1WmY4mISAYbM8Y1GOJ0QnTflwg8uh//xg1Mx5IMohIsPqlXL5j9jYMvLI9z74npOJs3J/77haZjiYhIBnDuP8BHPXcwZIjr85degokTwVYwv9lgkqFUgsVndWiTxAPVIgFwOpy8/Fog2k1bRMS72Tdv42qV+nSZ2owSHGT0aBg1ynUStfgWlWDxXYGB5Fw7j4tNH+XxzN/w4d4mNGgAp0+bDiYiIukhPh62tnuLLPFR5CKSJRVf5MUXTacSU1SCxbcFBJBryVc8v64defLA7t1Qqxbs2oWWjhAR8SKXL0OTJtD0zJfsoRyRpepR+NfppmOJQSrBIkClSrBxI5QuDWfOwPu15nCxQXvXlkEiIuLRjh2DOnVg7VqwhIUR/e1Scm5bAmFhpqOJQSrBIn8oVAjWrYMBFdcw+cZj5Fo7j6jy90FcnOloIiJyJ5KSOPX0GBrWiufgQYiIcBXheh3zQubMptOJYSrBIv8jWzZ4f1QiDj/XNpkLjpXl7Y9CNDNCRMTTxMYSWaMlER8PYXRkTypXdLBxI5QrZzqYuAuVYJG/CHigMUGb17C1bHf6Mplhr1jo1w+SkkwnExGRlPr2nWNk3rkOgPbWeaz5ZDf58hkOJW5FJVjkNqyVK1Jtz1Q+GO+PxQKffgrNmsHl4zGmo4mIyL9ITnZtitR5VCW6MIvYoFxYVywnuE5F09HEzagEi/yLAQPghx8gJAR2rYziaokqXOo2yPVbVkRE3MqVK9CiBXz4oevzmm8+SJaLR/G7t67RXOKe/EwHEHF3Dz4IG9baiavVkYLxx2DGhxyPs1Bk3ljT0UREBMBu51Kfl/jox6L8EvkkmTPD9OnQvj1AiOl04qY0EiySAuUq2ig38mGSLX6cJzf3zh/Ee+9pKWEREeOSk7lYuzU5vniPVyIH0jnXStatu1mARf6ZSrBICmV5vs//tXff4VFVeRjHv5MKSSCBhJpEWoCEEqoUA4QO0kMJriCygoioiLIICLqiK00XBBtFcFFQiqCI0qQXQXqVlhBqKKGE9H73j5FoBBQ0yU0y7+d55tHcc2fuO8fZ5LdnzjkXY+06Pu3yDefxZcQI6N/fegciERHJfYYBU6Y78PnuAAAsGMwaHUHt2ubmkvxBRbDIA3BsHcyobxoxbRrY2Vm/bmvRLJ1r0xZARobZ8UREbEZiIvzznzB8OLzCJHZW6oOx5geKDnvK7GiST6gIFnlAFgsMHQqrV1v3Fe60+3W8hvXlapMQiI42O56ISMGWns7ZDeE88gjMm2cdkJg6zZ6Gp+bj2LaF2ekkH1ERLPIXtWkDB5eGMYqJABTf8T3zXjmqAWERkZwSFUVUvfYUah3E5QOXKFEC1q61DkxYLGaHk/xGRbDI3+Dbwo/05d8T71SMV5hM/9lBdOtm3aZHRESyT1oa7Gv5L0ocXEcp4wrfFX2cfXsNWrUyO5nkVyqCRf4mpy7tcT37M9Vnv4SzM6xYAfXrw4EDaJ6wiEg2uHIF2raFdkfe5Tw+xLiUInDpG/j4avhX/joVwSLZoXRpBgy08OOPUL48nD4N4xt8Q1TFhhinwsxOJyKSb23fDnXrwsaNkOhagp8nrqBo2H4cWwebHU3yORXBItmobl3YuxcGBp9iduqTlDi7h6RqdYg5cNrsaCIi+Ur6jl0cq96TNs2SiYyEgADYvRvajawNZcqYHU8KABXBItmseHGYOS2JdK/SAHyT1onArhXYscPkYCIi+cSNd+diBAUR8PNS3sp4lT59YNcuayEskl1UBIvkALtaNSkesZdLfUcwsfwMzp6z0LQpjB8P6elmpxMRybu++Qa6vVWPdMNaojzht4P5c1Nw092PJZupCBbJKW5ulPl8MlsPuvOPf1iL3zFjYFDjw9x6/b9aNCci8huJiTBkCISEwNaYWkz3fZcbz46h5M+bwcnJ7HhSAFkMwzDMDpFfxMTE4O7uzq1btyhatKjZcSQfMQzrpu7/GpLA5sSHqc7PXK3VmpLrF4Knp9nxRETMk5jIhfe+ov2CJzh61HrolVfgrbdU+8pfc7/1mkaCRXKBxQL9+8PBd9ZSnZ8BuHDwOk+9WEQ3mRMRm5W+YxfXy9XB59V+1D46n9KlrTe/mDRJBbDkPBXBIrnI+7lupHy3luseFXmSz/h0gRM1a8K6dWYnExHJXadOwdj+F/CMOgHA+07DObgjgTZtTA4mNkNFsEguc+rYBs+oE8zYVoNKleDCBestmEc9dZWkhd+YHU9EJEdlZMD770OtWjDxZHcWOfQhqkIDPA5spmR5F7PjiQ1RESxiBgcHgoLg4EHrQhAwaPDpYAr9I4Sr7Z9AcyREpMBJTeXSoi20bg1Dh1oXwrVqBY8cmkGJk9uxBPibnVBsjIpgERO5usKHH8LuSRvpztcAWNas5o3RySQkmBxORCSbGAcPEVX5Ebwea8WtjXtxcbH+7lu7FnwD3MDBweyIYoNUBIvkAfVHtCD+43nEO7ozmBmMm1GKGjXghx/MTiYi8vecPAmzQ1ZS4uweHEljoesADu7PYMgQsFMVIibSx08kL7BYcB3cD9dL4fxzRQ98fCAiAtq2hQF9k4n9eL72FRaRfCUlBd5+GwID4bmI4Ry2BHK9ZAAVf5iFXxWVH2I+fQpF8hJPTzp1gp9/hhdesG6t9tCC8RQZ8gRXqzXHOHnK7IQiIn/s+nWOzthKvXowdiwkJ0Ordo54bPkWz3P7sW/cwOyEIoCKYJE8qUgRmD4ddn97idGWiQAUO7GDIU8lcuaMudlERO4l4ZMFxPr4U/rZblw5cpUSJWDBAli1CnyblANnZ7MjimRSESySh9XrVAbLypXcKF6J/9q9woztgVSrZv2KMTnZ7HQiIlaGAUuWwNqh31Ek6Rqe3GBJ5TEcOwaPP279Vkskr1ERLJLHObZvRfHzh+i+/zWCg63bCo0dCzWqG5zsOw5OnzY7oojYsKNHrVudhYbCM4nvEW1XjCvNehK8+U3dFV7yNBXBIvmBiwtVAguxcaP1q8UyZSAwfBlVFrxBSuVqXBv3odkJRcSWZGSQMOMzpvfeTq1asHEjFCoEQ94ohfOJw5TavMT6i0okD9PGfCL5iMVi/WqxU0eDmwHvwiVwykhmyH/KEpABo0ZB4cJmpxSRgizj8lWuNQ2hZNiPBBMI7CUkxIEpU6B8eQBvcwOK3CeNBIvkQ0XdLZQ7vobr/Yez07MjS9K68eabUK0aLF4MRoZhdkQRKYD27IFmIZ6cD0sCoBaH2DVuNcuW3S6ARfIPFcEi+VXRonh++i4Nr3zL4sUWfHzgzBno3RsWe7/ElU5PwcWLZqcUkfwuI4MzZ6BPH3j4Ydi+055XCr3PtZIBpH6/lrqvdzI7ochfYjEMQ0NG9ykmJgZ3d3du3bpF0aJFzY4jkkV8PLz7LiybeJI9SdVxJI0YJy+u7TtHxeqaIyEiDyg9nYSP55Hw2ngaJWwkPMUXgL59YeJE8C6ToVu+SZ50v/WaPr0iBYSrK/z737Dh4xOkOLkBMCXlOfzrFOall+D6dZMDiki+kZICW7v9F5cXBuAVHc5rKWNp2RL27oXPPwdvb1QAS76nT7BIAePZvzOukWFcHfgqB1v/i9RUeO898PODdyemkfzVCt2CWUTuyjCs6woCAqDLd09zg2IAtGscy7rVadSta3JAkWykIlikIPL0pOTst/n6BzfWrIHAQIiOhqOjP8e5VxeiytUnZdOPZqcUkTzCOHiIHeM3Uq+edV3B6dNQqHQx9j/9MWkbNlP6x2VYHLWhlBQsmhP8ADQnWPKr9HSYPzeFls9WwTf9LAA9Sm6lw/gm9OsHjo4mBxQRUxiJSVx59ElKb15MOBXx5ziF3BwZPhz+9S9wczM7ociD05xgEclkbw9PDnSk9NKPuOpblw3O7Vl2tQkDB1q3VVuwANITU8yOKSK5aPNmCG5XiEObbwJQidMsaD+fiAh44w0VwFLwqQgWsRUWC45dO1Dy7B4an/6CKVOgRAkIC7Ou9l5V+p9cDmxL+obN1omBIlLwXL7Mjh3Qpg00bw5bt8Jbjm8RW7gkMW9MIXTZY3h5mR1SJHdoOsQD0HQIKWji4uD992HFhCNsiw3EDoMo+1J8/+FZHv+nM05OZicUkexgHD7ClaFvU2LzEmob+zlCTRwdYeBAePVV8CmRDM7OZscUyRaaDiEif8rNDUaPhrXzr3LLozwAk9OH88/BzlSqBNOnQ0KCuRlF5K/LyIBly2BKx/WU3rQQeyOdVy0TGTAATp6Ejz4CHx9UAItNUhEsIrh1aUmxqJMkzp6P738GU7o0XLgAL74I/uUSOVehGQkfzLVuHioieZthkJqYxrx5UL069OgBr58fyDU8iS1cgvajavHJJ7rNsYiKYBGxcnCg8MA+DB1ThIgImDEDKlSALtfm8NCZrbi8MIDtjYZz7pzZQUXkrpKTSZwxjyjfukwo+z79+8Px4+DuDi+NccV+zSqKXDtDsfGvmJ1UJE/QnOAHoDnBYmvS0uBc835U3P45ALU4wFH7WvToAS+9BI0amRxQRACIiIAv/xPOqLmVscPgImVpXPI0zw93ZvBg0J8ssSWaEywif5uDA1Tc9hkZW7dz4rHX8WpZi/R06x2lGjeGIdU2cal2e9K+W60dJURymRGfwJYt0L279Y6QY+ZWYgWdAXAo583JTZG88ooKYJF70UjwA9BIsAgcPGi9DfMXX8DilK505VsAlvX7muAp3fD0NDefSIGWkUHqV8u5/u/pnL8ADeI2Zja1bQuvdTnII7XisQtqDBaLiUFFzHO/9ZqK4AegIljkV1ciEnCsF0jxm+Gcx4eKnMbe2ZGePeGZZ6BJgxQsztpjTSS7hIXB7FkG/afUJCD9KAANnfZTu39thg61LoITEU2HEJEcVqqCC8WjTpCycBnhz7xDjdqOJCdb7z7XrBl8VnYUV33rkfDeLO2zJvJXpKWR+tM+Fi+G1q2hcmWY/I6Fj9KfAeC6V1XWLrzBzJkqgEX+Co0EPwCNBIvcm2HA7t0wcyZ882UiJxN98OQGSTjzr39E0ndocRo21De0Ivfj2uj/4vjBFJzjruPNRW7gicUC7dvDc32iae+xE/tH24KdxrJEfk8jwSKSqywWaNAA5syBMz9GkvZQRQAWE8qHXxancWOoUQMmTYKr3+60bkQsIpliY2HePGjVCj6beBH3uEgKkczzRT5j7FjrDhArV0LHPh7Yd2yvAljkb9JI8APQSLDIgzH27mP/SVemra7K4sWQlARg8DPVqMoJLtdsg/vmFbgW09xhsUFpaaSvWcelD75irMeHLPnWOXPmkD/HOWKpwdWHO1Fi4ss4tGhmblaRfEQL43KAimCRv+7WLViyBHZ9uJtZBxoAsJlmdHLbTM+e0K8fBAdrcEtsw+HDEPfEYBofnAlACMv4hhCqVIEnnoC+faG82zXw8jI5qUj+oyI4B6gIFskGkZHcmDQbPv+MSfZjmHztqcymsmVhVaEQStYoScnnemHXppUmEUv+Zxhw6BAnnWuyZKkdixfDoUPQljWsoT0Auys/TsbnC2jQQB95kb9LRXAOUBEsko0MAyMtne0/OfDZZ7BoEbjFXOQiPgCccKjGx88dJTTUemc6jRBLfhQ5eT7Ok8bheSOMILbxI0EAODpC107p/Dd6AGWe7Ypjlw7g7GxyWpGC4X7rNYdczCQi8iuLBYujA02aQJMm8P77cGD8QZLfdsE5PYEv03oxbRpMmwbe3tCrFzxvvE/5FhWwb9MSXFzMfgciWaWnY1jsOHbcwldfWaf/1D2SzjzCAOhtWULRdkH06gXdukHx4vbA/8xMLGLTNBL8ADQSLJIL4uNJXb6SzfH1mbelAsuXW1fNuxDPdTwpRDKXilRm6ycnadcO3N3NDiy2LnX9FqImz6XI1pX09NrM2vMBmW1eDtFcyCjDjSqNKPLS07gNetzEpCK2QSPBIpI/ubri+HgvWgOtn7buKLF2LZye8gOFNicD8E1sK4b0BgcH62K6zp2ht/M3lH7YF+rU0dwJyXFRUbBqFaxYAX4r9jIheR4Agee/Y5NTAK1bW7+96NrVA2f7K5TRwIlInqOR4AegkWAREyUkkL5mHVfnrGC56+O8d7AFJ05Ym+xI5zKlKcE1rnhUYde84zRvYaFIEXMjS8GRtm4TUR8uwmnLOp4qv5EV+324/dezEmGEUZkkBzciur+Mzyfj9NkTMZFGgkWkYHFxwT6kC2VCujAYGAycOmUdiYtY8BMl9l0DYGt0TXp1teDgYF1Q17o1PGZZSCV/JxzatIBixUx9G5IPpKZiODhy9CisXw/r1kHQmg2MSp0BgMeN9Rg8SZ060KkTdOrkR0bcRgoFNSZAi9tE8g2NBD8AjQSL5FHXrhH/2VdEL1rDWtcQ3j7Xj/DwX5tPUIUqnCLJ3oUZE6J5JNiROnWsK/RFwLqL2ZU3Z5KxcBHFwvZQo3gkp6+6ZbY3ZQtbCCbNzpH9nV6jzIev4eNjYmARuSdtkZYDVASL5B8REdYRvP3fnOWjleUB2EJTgtkCQOHC1pHiV+z/SyCHKN6lCYX6hWqlnS04f56Uo6fY696Sbdtg+3br49/Xnud5PgTgUVayufCjNG1q/TahTXAKgVfXYde8Gbi5/ckFRMRMKoJzgIpgkXwoKYmMdRuIWriOw/GVeD/jObZtgxs3rM3bCCKIHwFoU/MyVZuVon59qF8f/ItfxSE1ER56SHcwyMcMA86dg717oc6LTalwYRsxFKEYN8nAPvO8vg4L+TztH9x0L0fki5PxezVUW/eK5EMqgn8nPDycxYsXc+LECYoWLUrTpk3p0aMHdg+wilxFsEjBkJEBx4/D9s2p9HnRC5fUGE5SmaqczHLeGMfJ/Cd1JHGFvdj5/AK8/9mWKlXA3v4eLyymM+LiiX5zGglb9nI23Yc3Paexdy9cs04ZZzG96MVXADT3OECx5rUICoKgIKhbORbnxGjw9TXvDYjI36Yi+DfeeecdxowZQ2pqapbjQUFBbNu27b5fR0WwSAGUnAz79nEtLJr1To+yaxfs2QP79sGcuFBCWQJAdY7wM9Vxc4Pq1aFd2cM8u3cAGQE1KNQvlGL/aK/B4twSHw/79pG4+whhLoHstA/i6FE4ehQO70/j9PWiuJBIOBXxwzo53MEBataEEc7TCY5ejnPLIIqPGIClfDmT34yIZDcVwb+YNWsWzzzzDHZ2djz++OO0bNkSi8XCunXr2LlzJ2FhYff9WiqCRWxHejpce20aad+vpnDEMbrVDGfPfnsSE63tfZjPfJ4A4BUmMdfzFWrUgBo1ICAAuqwcjGsVb9yDa2PfrbOJ7ySfSkiA8HCMsHCuVG7CqZtenDplLXTtt25i8u4WAExjKMOYluWpP/IIjdlBkqMb8ydfolaQGzVrQqFCuf82RCT3qQgGYmNj8fHxISYmhq+++ooePXpkaY+MjKRs2bL3/XoqgkVsW1oanDhhLcQ8Zk6i9YbR2GHQke9ZSYfM84pxgxt4ArDB0pJBFdfj50fmo+WJjymdfBbXKt64DB2IxaWwWW/JXMePQ0QESZHXOde0LxEREBYG4eHQ+Lsx9Do1HrAuUlvNo5lP8yKKKEoCsNW5Ff8JXkf16tYR+po1oVbKbpw93dDcFRHbpH2CgSVLlhATE0OnTp3uKICBByqARUQcHMgstggdCfHPw7FjfOVbheORcOQIHD4MGT+F88smFIQZlQgPtxZ2a9ZYj23nc0qyA4CibzxDKV/w8QFvb2gfv5RWB6eQXros1554GefmjfHyguLFwd5Ig4MHwcsLSpQAFxdzOuJuEhOtd+r7ZSWZYcCt8zGkvzedlEvXuepVnR3VB3LxIly8CBcuwH+39qNm0m6csFCd3qTx65510VSi1y//7mcJp0J5qFQJqlWD6tVLcHHTvyhWtwJNm9RlTaPfh3k4N96xiORzBboI3r59OwChoaGEh4czc+ZMLl68SPny5QkNDaVWrVomJxSRfM3VFerXpzBQp5T1js0AJAfCiYNknAqnm1NZqhb9dYQzLAz8vw6DNLhMKWKTnYgNsx4HKM8x+vIjRMCzO/rx3S8vabFAgPtVjkbXB2Bn6W583PZr3NzIfHRdNRivGyfJcHXjwGvLcCnqgJOTtXj3OLqN4tu/w+LoQFzH3hg1auLgYK1bM6JjcJ07HSMtnaSHqnKz3WOkpVlHvtPToeScCTifOoKRkMCaQcuIT7AQF2edmlvjpzl03/QCzumJ/Nt/EUssoVy/DtevQ9H0NG7wGgD7eZRnGZil+07hQ012Y4dBJZfLOFTwpVIla7Hb0LkuF3f3o1D1Skx5rBHvN/5d3w96J0f+k4qI7SjQRfC5c+cA67B47dq1iYuLy2ybMGECEydO5JVXXrnn85OTk0lOTs78OSYmJufCikjB4ewMgYHYBQZSEigJBAf/pv34FrhwAc9bCZyuax0Vvf2o/8UNOGA9rbCPF8Xi4eZN68iqffS1zJc4fNmLzz7LetmO7KIs+0nFgc4hWacBvMQupjAJgCGzavEVNTPbShPPpV+K1WWE0GPEY1meu4G1tGATAC/sTiSRX0egn8Sef2CdKH31+HWO/eZ50XiQjh32ZOBX+CJd2lhHu28/Kh/vwbVb1XGt4s3x/m6Q5WZ+tYF59+xiEZG/K08XwdHR0QwcOPDPT/xFpUqVmDRpUubPib+sYBk7diz+/v7069cPV1dXNm7cyIIFCxg5ciSNGzemadOmd329CRMmMG7cuL/3JkREfs/fH/z9cQQqABUq/KZtxBRIGg+XL7O4dGkoZB2RvXEDYg64cemjwaRHXadqxQa8Uwvi4n59lPkiHmIhycGNBnWto7Wpqdbne91Ig1vWSzi7OOBm9+tIr5O9PSRZ21wc0yhV3DqV1sHB+k9LlCv8MoYQ0iae9OIuuLlZB8LrXinF1U01SCtanH6dvAjtDJ6etx922G/8DtzdqVKqFMsr/b4j+uRoN4uI/JE8vTDu8uXLlClT5r7Pr1evHnv27Mn8uV27dqxdu5a6devy008/4eDwa83/xhtvMG7cOAYMGMAnn3xy19e720iwr6+vFsaJSN5kGNa5uQkJ1nnDv3X+vHU+RloaBAZCyZK/tqWkwNq11qq31G/ndfzi7Fnr81xdrc97gP3VRURyW4FYGFesWDGWLFly3+cXL148y8++v2x43r179ywFMMBjjz3GuHHj/nCLNGdnZ5x1uyARyS8sFutiubstmPP1vfdNIJycoFOne79uOe2lKyIFT54ugp2dnenZs+dffn6DBg2YM2cO0dHRd7TdPlZIG0eKiIiI2JwC/Z1Wt27dKFSoEJ988glHjhzJPB4TE8PYsWMBeOSRR8yKJyIiIiImKdBFcMmSJRk1ahTR0dHUqVOHJk2a0K5dOypWrMj69espU6YMQ4YMMTumiIiIiOSyPD0dIju8/vrrpKWlMXny5Mx9gwHq16/P559/jtfvF4+IiIiISIGXp3eHyE7R0dHs2bOHxMRE/Pz8CAgIeODX0G2TRURERPK2ArE7RHby8PCgdevWZscQERERkTygQM8JFhERERG5GxXBIiIiImJzVASLiIiIiM1RESwiIiIiNkdFsIiIiIjYHBXBIiIiImJzVASLiIiIiM1RESwiIiIiNkdFsIiIiIjYHBXBIiIiImJzVASLiIiIiM1RESwiIiIiNkdFsIiIiIjYHBXBIiIiImJzVASLiIiIiM1RESwiIiIiNkdFsIiIiIjYHBXBIiIiImJzHMwOkJ8YhgFATEyMyUlERERE5G5u12m367Z7URH8AGJjYwHw9fU1OYmIiIiI/JHY2Fjc3d3v2W4x/qxMlkwZGRlERkZSpEgRLBZLjl8vJiYGX19fzp8/T9GiRXP8emKlfjeH+t0c6ndzqN9zn/rcHGb0u2EYxMbGUrZsWezs7j3zVyPBD8DOzg4fH59cv27RokX1P1gTqN/NoX43h/rdHOr33Kc+N0du9/sfjQDfpoVxIiIiImJzVASLiIiIiM1REZyHOTs78+9//xtnZ2ezo9gU9bs51O/mUL+bQ/2e+9Tn5sjL/a6FcSIiIiJiczQSLCIiIiI2R0WwiIiIiNgcFcEiIiIiYnO0T3A+dfXqVVasWEF4eDheXl60aNGCOnXqmB3LZnzwwQeEhYVRvnx5hg0bZnacAis6Opq1a9dy6tQpYmNjKV++PB07dtRdG7NBVFQUy5YtIyIigmLFitGhQwdq1qxpdqwC7cqVK6xbt46wsDCSk5Px8/Ojc+fOlChRwuxoNiM1NZXXXnuNpKQkWrZsSZcuXcyOVODt27eP9evXc+PGDSpXrky3bt0oXry42bGsDMl3pk+fbri4uBhAlsfgwYPNjmYTNm7cmNnnDRs2NDtOgTVo0CDDycnpjs+5g4OD8dZbb5kdL19btWqV4e7unqVfLRaLMXbsWLOjFVidO3c27O3t7/g8Fy5c2Jg1a5bZ8WzGW2+9ldn3w4cPNztOgXbr1i2ja9eud3zmixQpYqxevdrseIZhGIZ2h8hnPv74Y4YMGQJAhw4dCAoKIiUlhZUrV2JnZ8fOnTtNTliwJScnExgYSMWKFVm9ejUNGzZUn+cQf39/rl69Sps2bfDz88PZ2ZkNGzawefNmABYtWkRoaKjJKfOfyMhIqlatSlxcHG3atKFly5ZERETw6aefkpqaqn7NIW5ubhQpUoRWrVpRsWJFAFauXMnevXsB2LZtG0FBQWZGLPBOnTpFYGAgzZs3Z/Xq1QwfPpx3333X7FgFUkZGBs2bN2fr1q14eHjQu3dvKlasyIULF1i4cCGvv/46zz//vNkxNRKcn1y9etVwdXU1LBaLsXjx4jvaDxw4YEIq2zJmzBjDy8vLOHbsmEaCc9jmzZuN5OTkO44PHz7cAIyuXbvmfqgCYOTIkQZgPPnkk1mOL1myxACMmjVrmhOsgNuwYYORlpaW5VhGRobx2GOPGYDx3HPPmZTMdrRo0cJo2rSpsWzZMo0E57CZM2cagFGxYkXjwoULWdpiY2ONsLAwk5JlpYVx+cj8+fOJj48nNDSUXr163dFeq1YtE1LZjqNHjzJ58mTee+89vLy8zI5T4DVr1gwnJ6c7jj/99NMA3Lp1K7cjFQgrVqwA4LXXXstyvGfPngQEBHD48GHOnj1rRrQCrUWLFtjb22c5ZrFYGDhwIKDPc06bO3cuP/74I7Nnz8ZisZgdp8CbOXMmANOmTcPb2ztLm5ubG5UqVTIj1h20MC4fuf01cJ8+fbh06RJffPEFUVFR+Pn5ERISgqenp8kJCy7DMBg0aBCtW7emT58+XLt2zexINuvcuXMANGjQwOQk+U9KSgonTpygTJkyd/0jFBwczLFjxzh06BDlypUzIaHt0ec550VFRTFixAhef/11qlatyrFjx8yOVKDFxMRw4MCBzAW3W7duZe3atdjZ2dGwYUPat2+PnV3eGINVEZyPnD59GoDExET8/f2JiYnJbBs+fDhffPEFHTt2NCtegTZjxgwOHTrE0aNHzY5i05KTkxk9ejQeHh689NJLZsfJd65fv056evo9d9e4ffzKlSu5GctmRUdH8+abb1KuXDmeeuops+MUWMOGDcPHx4cRI0aYHcUmnDlzhoyMDOrUqcOoUaN45513srQ//PDDrFixglKlSpmU8FcqgnPRxYsX7/gw/JHq1atnfvULkJCQAMCLL76It7c3zz33HE5OTqxatYpdu3YRGhrK8ePHtX3U74SFhfHBBx/c9/n169enb9++mT9funSJ0aNH8/bbb/PQQw/lRMQC6fDhw8yZM+e+z2/atCk9evS4Z3t6ejr9+vXj4MGDLF++nNKlS2dHTJuSlJQEgLOz813bCxUqlOU8yTmJiYmEhIQQFRXFxo0bcXV1NTtSgbR27VoWLVrEjh07cHR0NDuOTbhdqxw7dowNGzbQtWtXHn74YaKiopg/fz67d++mf//+rFq1yuSkaGFcbtq/f/8dW4X80aNjx45Znl+jRg0DMBo1amSkpKRkHs/IyDB69eplAMabb76Z228rz/vtlmb38+jTp0+W53fv3t1o1KiRkZ6ennksKipKC+P+xNdff/1A/f5HC4OSk5ON7t27Gw4ODnddFCr359KlSwZg1K9f/67tt7ePmjNnTi4nsy23bt0ymjVrZri6uhqbNm0yO06BFR8fb1SoUMF4+eWXsxy//btJC+NyxoEDBzJ/r0+dOjVL24ULFwxPT08DME6fPm1OwN/QSHAu8vb2ZurUqfd9/u1tdH77/CNHjtC/f/8s/4/WYrEwaNAglixZorlOd+Hn5/dA/R4QEJD571u2bGHZsmV07NiRl19+OfP47ZGys2fPMmzYMPz8/PLGdi95SM2aNR+o32vXrn3X43FxcYSEhLB582YWLVpE9+7dsymh7fHy8sLJyYkzZ87ctT0iIgKAsmXL5mIq2xIVFUX79u05efIkq1atomnTpmZHKrDee+89IiIiiI2NzXJTo/DwcAA2btzIsGHDaNu2LR06dDApZcFzeyHcbxd+/ratU6dOzJs3j2PHjlGhQgUzImZSEZyLSpQo8bfuLla3bl3WrFlDSkrKHW23j/1+9bGAj4/PX+73yMhIAL7//vu7tl++fJlp06YRHBysIvh3KlWq9Lfvpnf9+nU6dOjAgQMHWLJkCV27ds2ecDbKwcGBGjVqsG/fPg4dOkRgYGBmW0ZGBhs2bMBisWinmRxy9uxZ2rZty6VLl1i9erX2Bc5ht39/z549+67t+/btY9++fbi5uakIzkZeXl74+vpy/vx5UlNT72jPS/WKiuB8pEePHkyYMIGPPvqIJ554Ag8PD8A6Knl7w+/69eubmLDgqVu37l1HM+Pj4xk7dizlypXLXHQh2SsyMpI2bdoQHh7O0qVL6dSpk9mRCoRu3bqxb98+Ro8ezfLly3FwsP4Z+PDDDzlz5gyNGzemTJkyJqcseI4fP07r1q2Ji4tj7dq1NGrUyOxIBV7Pnj3x8/O74/jhw4eZO3cuLVq0oEuXLjRs2NCEdAVbz549mTp1KhMmTGDy5MmZxw8fPszy5cuxs7OjTp06Jia00h3j8pnQ0FCWLFmCl5cXbdu2xdHRkY0bN3Lu3DnKlCnDsWPHcHd3NztmgXft2jVKlCihO8bloOrVq/Pzzz9Tv379u46YlShRgjFjxpiQLH+7efNm5t34AgICCA4OJiIigjVr1mCxWPjhhx9o1aqV2TELlOTkZHx9fYmKiqJ58+Z3HWnXlKrc88033xASEqI7xuWgy5cvU61aNW7evEnt2rUzF8atXLmSlJQUBg0alLmXsKnMnpQsDyYuLu6u9+L29/c3Dh8+bHY8m6GFcTnP3d39DxfSVapUyeyI+daePXuMcuXKZenPQoUKGbNmzTI7WoEUGxv7pwtDg4ODzY5pM7QwLnfs3Lnzjt8zFovF6Nevn5GYmGh2PMMwDEMjwfnU4cOH2b17NykpKQQEBNCkSZM8Mb/GViQlJTFjxgzKlClD7969zY5TIH388cckJyffs93Dw4P+/fvnXqACJiUlhfXr13PmzBmKFStG69atdSfEHJKamsqHH374h+f4+PjQs2fPXEpk28LDw1mxYgX16tXTwsQclpKSwsaNGzl9+jQuLi4EBQXddYqKWVQEi4iIiIjNyRv3rRMRERERyUUqgkVERETE5qgIFhERERGboyJYRERERGyOimARERERsTkqgkVERETE5qgIFhERERGboyJYRERERGyOg9kBRETkr7tw4QLz58/PciwkJISqVatm/rxr1y42bNhA48aNCQ4Ozu2IWXz66adcuXIl8+fy5cvz2GOPmZhIRGyVimARkXzszJkzjB49OssxPz+/LEXwli1bGD16NCNHjjS9CJ42bRoHDx7M/LlVq1YqgkXEFCqCRUQKAF9fXx5//HEA/P39TU5zb0899RSRkZGcO3eOL7/80uw4ImLDVASLiBQAFStWZOLEiWbH+FNDhw4FYNOmTSqCRcRUWhgnIiIiIjZHI8EiIiZZtmwZJ0+epHr16nTu3PmO9lOnTrF06VIcHR154YUXcHJyyvYMhmEwd+5coqKiqFKlCt27dwdgzZo17N+/nw4dOhAYGMjx48fZtm0biYmJVK9enRYtWmCxWDJf56effuLAgQOkpqbSuHFj6tWrl+1ZRUSyk4pgERGTVKlShb59+5KWlsb27dt5+OGHM9sSExPp3r07R44cYdKkSTlSACcnJ9O/f38WLlxI48aNWbFiRWbb119/zcyZM/Hw8GDq1Kn873//y/LcJk2a8N1335GUlERoaChbtmzJ0v70008za9asbM8sIpJdNB1CRMQkNWrU4P333yc1NZXevXtz69atzLbnn3+eI0eO8OijjzJixIhsv3Z0dDTt2rVj4cKFdO/enfXr1+Pp6XnHeePHj+d///sfwcHBPPfcc/To0QMHBwe2bdvG8OHD6dKlCzt27KBz58688MILtGrVCoDZs2ezYMGCbM8tIpJdVASLiJhowIAB9O3bl4iICAYMGADA/PnzmTt3Lt7e3nz22WdZph1kh7NnzxIUFMTmzZsZOnQoS5YsoXDhwnc998KFCyxdupRNmzbxwQcf8NVXX7Fw4UIA5syZw8mTJ9m7dy/ffvst06dPZ926dYwaNQqw7gksIpJXqQgWETHZjBkz8Pf3Z+nSpQwfPpzBgwdjb2/Pl19+iZeXV7Zea9++fTRq1Ihjx47x7rvvMm3aNOzs7v2noFevXpnzhG/r0aMHxYsXB+DVV1+lZs2aWdqffvppAI4cOZKt2UVEspOKYBERk7m6urJ48WIKFy7MlClTiI+P580336Rp06bZep1t27YRHBzMzZs3WbRoEcOHD//T5zRs2PCux8uWLXvPdm9vbwBiYmL+RloRkZylIlhEJA+oVq0a1apVA8DFxYUhQ4Zk+zUOHz5MXFwc3t7ePPLII/f1nKJFi971uL29/T3bb7dlZGT8xaQiIjlPRbCISB4wbtw49u7di5OTEwkJCTzzzDPZfo1nnnmGQYMGcfr0aZo1a8bZs2ez/RoiIvmFimAREZNt2LCBt99+G3d3d3bt2oWfnx+LFy9mxowZ2XodOzs7Zs6cybBhwzh9+jRNmzYlLCwsW68hIpJfqAgWETHRlStX6NOnDxkZGcyZM4datWqxaNEinJyceOmllzh48GC2X3Pq1Km8+uqrnD9/nmbNmvHzzz9n+zVERPI6FcEiIibJyMigb9++XL58mSFDhtCjRw8A6tatyzvvvENSUhK9e/cmLi4u26/99ttv85///IdLly7RvHlzDhw4kO3XEBHJy1QEi4iYZPz48axbt47atWszZcqULG1Dhw6lW7dunDhxgmeffTZHrj9mzBimTp1KVFQULVu2ZNeuXTlyHRGRvEi3TRYRMUFCQgJJSUmMHDmSAQMG4OzsfMc5c+fOJSAggIyMDC5cuICPj89fulbDhg0ZOXIkzZs3v6Nt2LBhlC1bln379rFu3Trq1KmDo6Mj7du3x8PDg1q1at31NZ966ikiIyMpVarUHW12dnaMHDkSR0fHv5RXRCQ3WAzDMMwOISIif822bdto2rQpFSpUYNCgQQCEhIRQtWpVk5Pd3aeffsqVK1c4ffo0s2fPplWrVqxbt87sWCJigzQSLCJSAERERDB69GgA/Pz88mwRPG3atBxZ7Cci8qBUBIuI5GO+vr6MHDkyyzF/f3+T0vy529MobqtcubKJaUTElmk6hIiIiIjYHO0OISIiIiI2R0WwiIiIiNgcFcEiIiIiYnNUBIuIiIiIzVERLCIiIiI2R0WwiIiIiNgcFcEiIiIiYnNUBIuIiIiIzVERLCIiIiI2R0WwiIiIiNgcFcEiIiIiYnP+D5A2m7e85x/YAAAAAElFTkSuQmCC",
      "text/plain": [
       "<Figure size 800x800 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ax = plt.gca()\n",
    "ax.cla() \n",
    "ax.set_ylabel(r\"$\\rm y \\,[km]$\")\n",
    "ax.set_xlabel(r\"$\\rm x \\,[km]$\")\n",
    "ax.plot(CLoes[:800, 1]*np.cos(CLoes[:800, 2]),CLoes[:800, 1]*np.sin(CLoes[:800, 2]),c=\"blue\", linewidth=1.5, linestyle='-');\n",
    "ax.plot(CLoes_ivp.y[1][:800]*np.cos(CLoes_ivp.y[2][:800]),CLoes_ivp.y[1][:800]*np.sin(CLoes_ivp.y[2][:800]),c=\"red\", linewidth=2, linestyle=':');\n",
    "BH = plt.Circle((0, 0), 2, color='black')\n",
    "ax.add_patch(BH);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Das obere Diagramm zeigt die Trajektorie des Probekörpers auf dem ISCO, wobei die blaue Kurve die Ergebnisse der odeint-Methode und die gepunktete rote Kurve die solve_ivp-Simulation zeigt. Beide Simulationen stimmen gut miteinander überein. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Wir stellen nun alle berechneten Trajektorien in einer Animation dar."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.gridspec as gridspec\n",
    "import matplotlib.animation as animation\n",
    "from IPython.display import HTML"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<video width=\"1100\" height=\"500\" controls autoplay loop>\n",
       "  <source type=\"video/mp4\" src=\"data:video/mp4;base64,AAAAIGZ0eXBNNFYgAAACAE00ViBpc29taXNvMmF2YzEAAAAIZnJlZQABdIJtZGF0AAACrwYF//+r\n",
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       "\">\n",
       "  Your browser does not support the video tag.\n",
       "</video>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rval = np.linspace(2.2,30, 1001)\n",
    "step = 100\n",
    "anzframes = 100\n",
    "xylim = 15\n",
    "fig = plt.figure(figsize=(11, 5))\n",
    "gs = gridspec.GridSpec(1, 2, width_ratios=[1,1], wspace=0.21)\n",
    "ax1 = plt.subplot(gs[0])\n",
    "ax2 = plt.subplot(gs[1])\n",
    "\n",
    "def init():\n",
    "    ax1.scatter(ALoes[0, 1]*np.cos(ALoes[0, 2]),ALoes[0, 1]*np.sin(ALoes[0, 2]),c=\"green\", s=60, marker='o')\n",
    "    ax1.scatter(BLoes[0, 1]*np.cos(BLoes[0, 2]),BLoes[0, 1]*np.sin(BLoes[0, 2]),c=\"red\", s=60, marker='o')\n",
    "    ax1.scatter(CLoes[0, 1]*np.cos(CLoes[0, 2]),CLoes[0, 1]*np.sin(CLoes[0, 2]),c=\"black\", s=60, marker='o')\n",
    "    BH = plt.Circle((0, 0), 2, color='black')\n",
    "    ax1.add_patch(BH);\n",
    "    ax2.plot(rval,VeffRezl(rval,4),c=\"green\", linewidth=1.5, linestyle='-')\n",
    "    ax2.scatter(Ar0,VeffRezl(Ar0,4),c=\"green\", s=80, marker='o')\n",
    "    ax2.plot(rval,VeffRezl(rval,3.7),c=\"red\", linewidth=1.5, linestyle='-')\n",
    "    ax2.scatter(Br0,VeffRezl(Br0,3.7),c=\"red\", s=80, marker='o')\n",
    "    ax2.plot(rval,VeffRezl(rval,float(2*sqrt(3))),c=\"black\", linewidth=1.5, linestyle='-')\n",
    "    ax2.scatter(Cr0,VeffRezl(Cr0,float(2*sqrt(3))),c=\"black\", s=80, marker='o')\n",
    "    ax1.set_ylabel(r\"$\\rm y \\,[km]$\")\n",
    "    ax1.set_xlabel(r\"$\\rm x \\,[km]$\")\n",
    "    ax2.set_ylabel(r\"$\\rm V(r)$\")\n",
    "    ax2.set_xlabel(r\"$\\rm Radius \\, r \\,[km]$\")\n",
    "    ax1.set_xlim(-xylim, xylim)\n",
    "    ax1.set_ylim(-xylim, xylim)\n",
    "    ax2.set_xlim(2.3, xylim)\n",
    "    ax2.set_ylim(0.9, 1.01)\n",
    "    return fig,\n",
    "\n",
    "def animate(i):\n",
    "    ax1.cla() \n",
    "    BH = plt.Circle((0, 0), 2, color='black')\n",
    "    ax1.add_patch(BH);\n",
    "    ax1.plot(ALoes[:step*i, 1]*np.cos(ALoes[:step*i, 2]),ALoes[:step*i, 1]*np.sin(ALoes[:step*i, 2]),c=\"green\", linewidth=0.5, linestyle='-')\n",
    "    ax1.scatter(ALoes[step*i, 1]*np.cos(ALoes[step*i, 2]),ALoes[step*i, 1]*np.sin(ALoes[step*i, 2]),c=\"green\", s=60, marker='o')\n",
    "    ax1.plot(BLoes[:step*i, 1]*np.cos(BLoes[:step*i, 2]),BLoes[:step*i, 1]*np.sin(BLoes[:step*i, 2]),c=\"red\", linewidth=0.5, linestyle='-')\n",
    "    ax1.scatter(BLoes[step*i, 1]*np.cos(BLoes[step*i, 2]),BLoes[step*i, 1]*np.sin(BLoes[step*i, 2]),c=\"red\", s=60, marker='o')\n",
    "    ax1.plot(CLoes[:step*i, 1]*np.cos(CLoes[:step*i, 2]),CLoes[:step*i, 1]*np.sin(CLoes[:step*i, 2]),c=\"black\", linewidth=0.5, linestyle='-')\n",
    "    ax1.scatter(CLoes[step*i, 1]*np.cos(CLoes[step*i, 2]),CLoes[step*i, 1]*np.sin(CLoes[step*i, 2]),c=\"black\", s=60, marker='o') \n",
    "    ax1.set_ylabel(r\"$\\rm y \\,[km]$\")\n",
    "    ax1.set_xlabel(r\"$\\rm x \\,[km]$\")\n",
    "    ax2.set_ylabel(r\"$\\rm V(r)$\")\n",
    "    ax2.set_xlabel(r\"$\\rm Radius \\, r \\,[km]$\")\n",
    "    ax1.set_xlim(-xylim, xylim)\n",
    "    ax1.set_ylim(-xylim, xylim)\n",
    "    ax2.set_xlim(2.3, xylim)\n",
    "    ax2.set_ylim(0.9, 1.01)\n",
    "    return fig,\n",
    "\n",
    "ani = animation.FuncAnimation(fig,animate,init_func=init,frames=anzframes,interval=100)\n",
    "plt.close(fig)\n",
    "HTML(ani.to_html5_video())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "bzw. ohne Verwendung von ax1.cla() in jedem Animationsschritt und mit einer Trennung von Animationsobjekten und statischen Elementen."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
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       "\">\n",
       "  Your browser does not support the video tag.\n",
       "</video>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rval = np.linspace(2.2, 30, 1001)\n",
    "step = 100\n",
    "anzframes = 100\n",
    "xylim = 15\n",
    "\n",
    "DATA_CONFIG = [\n",
    "    {\"label\": \"A\", \"data\": ALoes, \"r0\": Ar0, \"param\": 4.0, \"color\": \"green\"},\n",
    "    {\"label\": \"B\", \"data\": BLoes, \"r0\": Br0, \"param\": 3.7, \"color\": \"red\"},\n",
    "    {\"label\": \"C\", \"data\": CLoes, \"r0\": Cr0, \"param\": float(2 * np.sqrt(3)), \"color\": \"black\"}\n",
    "]\n",
    "\n",
    "fig = plt.figure(figsize=(11, 5))\n",
    "gs = gridspec.GridSpec(1, 2, width_ratios=[1, 1], wspace=0.21)\n",
    "ax1 = plt.subplot(gs[0])\n",
    "ax2 = plt.subplot(gs[1])\n",
    "\n",
    "for cfg in DATA_CONFIG:\n",
    "    ax2.plot(rval, VeffRezl(rval, cfg[\"param\"]), c=cfg[\"color\"], linewidth=1.5, linestyle='-')\n",
    "    ax2.scatter(cfg[\"r0\"], VeffRezl(cfg[\"r0\"], cfg[\"param\"]), c=cfg[\"color\"], s=80, marker='o')\n",
    "\n",
    "ax1.set_ylabel(r\"$\\rm y \\,[km]$\")\n",
    "ax1.set_xlabel(r\"$\\rm x \\,[km]$\")\n",
    "ax2.set_ylabel(r\"$\\rm V(r)$\")\n",
    "ax2.set_xlabel(r\"$\\rm Radius \\, r \\,[km]$\")\n",
    "\n",
    "ax1.set_xlim(-xylim, xylim)\n",
    "ax1.set_ylim(-xylim, xylim)\n",
    "ax2.set_xlim(2.3, xylim)\n",
    "ax2.set_ylim(0.9, 1.01)\n",
    "\n",
    "bh_patch = plt.Circle((0, 0), 2, color='black')\n",
    "ax1.add_patch(bh_patch)\n",
    "\n",
    "lines = {}\n",
    "scatters = {}\n",
    "for cfg in DATA_CONFIG:\n",
    "    lbl = cfg[\"label\"]\n",
    "    lines[lbl], = ax1.plot([], [], c=cfg[\"color\"], linewidth=0.5, linestyle='-')\n",
    "    scatters[lbl] = ax1.scatter([], [], c=cfg[\"color\"], s=60, marker='o')\n",
    "\n",
    "def init():\n",
    "    for cfg in DATA_CONFIG:\n",
    "        lbl = cfg[\"label\"]\n",
    "        d = cfg[\"data\"]\n",
    "        x0 = d[0, 1] * np.cos(d[0, 2])\n",
    "        y0 = d[0, 1] * np.sin(d[0, 2])\n",
    "        lines[lbl].set_data([], [])\n",
    "        scatters[lbl].set_offsets(np.c_[x0, y0])  \n",
    "    return [lines[lbl] for lbl in lines] + [scatters[lbl] for lbl in scatters]\n",
    "\n",
    "\n",
    "def animate(i):\n",
    "    idx = step * i\n",
    "    \n",
    "    for cfg in DATA_CONFIG:\n",
    "        lbl = cfg[\"label\"]\n",
    "        d = cfg[\"data\"]\n",
    "        x_path = d[:idx, 1] * np.cos(d[:idx, 2])\n",
    "        y_path = d[:idx, 1] * np.sin(d[:idx, 2])\n",
    "        lines[lbl].set_data(x_path, y_path)\n",
    "        x_head = d[idx, 1] * np.cos(d[idx, 2])\n",
    "        y_head = d[idx, 1] * np.sin(d[idx, 2])\n",
    "        scatters[lbl].set_offsets(np.c_[x_head, y_head])    \n",
    "    return [lines[lbl] for lbl in lines] + [scatters[lbl] for lbl in scatters]\n",
    "\n",
    "ani = animation.FuncAnimation(\n",
    "    fig, animate, init_func=init, frames=anzframes, interval=100, blit=True\n",
    ")\n",
    "plt.close(fig)\n",
    "\n",
    "HTML(ani.to_html5_video())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## b) Bewegung von Licht (Photonen) um ein schwarzes Loch in der Ebene (Die Photonensphäre) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Die Photonensphäre ist der Bereich, in dem masselose Teilchen (Photonen) auf einer Kreisbahn wandern können. Sie liegt exakt bei $r_{PhSph}=3 \\, M$. Die Bahn der Photonen ist jedoch instabil. Winzigste Störungen führen dazu, dass das Photon entweder ins schwarze Loch stürzt oder ins Unendliche entweicht. Da Licht hier kreisen kann, markiert die Photonensphäre die theoretische Grenze für optische Phänomene (wie den sogenannten „Photonenring“ bei astronomischen Aufnahmen). "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In der Literatur wird die Bewegung von Licht um ein schwarzes Loch mittels eines definierten, effektiven Potenzials illustriert: \n",
    "\n",
    "1. General relativity : An introduction for physicists by M. P. Hobson, G. P. Efstathiou and A. N. Lasenby  \n",
    "\n",
    "2. Gravity : An introduction to Einstein's general relativity by James B. Hartle \n",
    "\n",
    "3. Allgemeine Relativitätstheorie by Torsten Fließbach  \n",
    "\n",
    "4. Relativistic hydrodynamics by Luciano Rezzolla and Olindo Zanotti  \n",
    "\n",
    "Dieses Potenzial hängt allein von der Masse des schwarzen Lochs ab:\n",
    "\n",
    "$$\n",
    "\\begin{equation}\n",
    "V_{\\rm Photon} = \\frac{1-2M/r}{r^2}\n",
    "\\end{equation}\n",
    "$$\n",
    "\n",
    "Die folgende Abbildung zeigt das effektive Potenzial als Funktion des Radius bei einer Masse von M=1: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [],
   "source": [
    "VeffPhoton = (1-2*M/r)/r**2\n",
    "VeffPhotonl = lambdify((r),VeffPhoton.subs({(M,set_M)}))\n",
    "rval = np.linspace(1.8, 10, 1001)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 800x800 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.cla()\n",
    "plt.xlim(1.8, 10)\n",
    "plt.ylim(0, 0.04)\n",
    "plt.ylabel(r\"$\\rm V_{\\rm Photon}(r)$\")\n",
    "plt.xlabel(r\"$\\rm Radius \\, r \\,[km]$\")\n",
    "plt.plot(rval,VeffPhotonl(rval), linewidth=1, color=\"black\");\n",
    "plt.scatter(3,VeffPhotonl(3),c=\"orange\", s=80, marker='o');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Wir erkennen, dass es keine stabilen gebundenen Photonen-Trajektorien gibt (fehlendes Minimum beim oberen Potenzial). Es existiert jedoch eine quasi-stabile gebundenen Photonen-Trajektorie bei r=3 km, die sogenannte \"Photonensphäre\" (oranger Punkt in der oberen Abbildung, allgemein bei $r_{PhSph}=3 \\, M$). Wir wollen uns im Folgenden diesen Photonenorbit näher betrachten und in einer Animation darstellen. Wir benutzen dazu wieder die Geodätengleichung, berücksichtigen jedoch bei den Anfangsbedingungen, dass das infinitesimale Weglängenelement für masselose Photonen Null ist ($ds^2=0$). Für die Photonensphäre gelten somit die folgenden Anfangswerte: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/latex": [
       "$\\displaystyle \\left[ - \\sqrt{\\frac{\\left(- 2 M \\mathrm{d}t + \\mathrm{d}r r + \\mathrm{d}t r\\right) \\left(2 M \\mathrm{d}t + \\mathrm{d}r r - \\mathrm{d}t r\\right)}{r^{3} \\left(2 M - r\\right)}}, \\  \\sqrt{\\frac{\\left(- 2 M \\mathrm{d}t + \\mathrm{d}r r + \\mathrm{d}t r\\right) \\left(2 M \\mathrm{d}t + \\mathrm{d}r r - \\mathrm{d}t r\\right)}{r^{3} \\left(2 M - r\\right)}}\\right]$"
      ],
      "text/plain": [
       "⎡      _______________________________________________________________________ ↪\n",
       "⎢     ╱ (-2⋅M⋅\\mathrm{d}t + \\mathrm{d}r⋅r + \\mathrm{d}t⋅r)⋅(2⋅M⋅\\mathrm{d}t +  ↪\n",
       "⎢-   ╱  ────────────────────────────────────────────────────────────────────── ↪\n",
       "⎢   ╱                                                3                         ↪\n",
       "⎣ ╲╱                                                r ⋅(2⋅M - r)               ↪\n",
       "\n",
       "↪ _______________________________       ______________________________________ ↪\n",
       "↪ \\mathrm{d}r⋅r - \\mathrm{d}t⋅r)       ╱ (-2⋅M⋅\\mathrm{d}t + \\mathrm{d}r⋅r + \\ ↪\n",
       "↪ ────────────────────────────── ,    ╱  ───────────────────────────────────── ↪\n",
       "↪                                    ╱                                         ↪\n",
       "↪                                  ╲╱                                          ↪\n",
       "\n",
       "↪ ________________________________________________________________⎤\n",
       "↪ mathrm{d}t⋅r)⋅(2⋅M⋅\\mathrm{d}t + \\mathrm{d}r⋅r - \\mathrm{d}t⋅r) ⎥\n",
       "↪ ─────────────────────────────────────────────────────────────── ⎥\n",
       "↪         3                                                       ⎥\n",
       "↪        r ⋅(2⋅M - r)                                             ⎦"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "EqPh_ds = Eq(ds2.subs({(theta,pi/2),(dtheta,0)}),0)\n",
    "LoesEqPh_ds = solve(EqPh_ds,dphi)\n",
    "LoesEqPh_ds"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Wir stellen die Kreisbahn des Photons in einer Animation dar:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [],
   "source": [
    "t0 = 0.0\n",
    "phi0 = 0.0\n",
    "dr0 = 0.0\n",
    "\n",
    "#Photonensphaere\n",
    "Dr0 = 3.0\n",
    "Ddt0 = 1.0\n",
    "Ddphi0 = float(LoesEqPh_ds[1].subs({(r,3),(dr,0),(dt,1),(M,set_M)}))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [],
   "source": [
    "Dinitialval = np.array([t0,Dr0,phi0,Ddt0,dr0,Ddphi0])\n",
    "DLoes = integrate.odeint(DGLsys, Dinitialval, tauval)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "AAGYAAAAygAAAI0AAADEAAABhwAAALIAAAC/AAAAwgAAAUgAAACxAAAAmgAAAKUAAAEqAAAAvQAA\n",
       "AK8AAADBAAABEwAAANUAAACxAAAAvAAAAMIAAADTAAAAogAAAKoAAACbAAAAxwAAALIAAAAUc3Rj\n",
       "bwAAAAAAAAABAAAAMAAAAGJ1ZHRhAAAAWm1ldGEAAAAAAAAAIWhkbHIAAAAAAAAAAG1kaXJhcHBs\n",
       "AAAAAAAAAAAAAAAALWlsc3QAAAAlqXRvbwAAAB1kYXRhAAAAAQAAAABMYXZmNjAuMTYuMTAw\n",
       "\">\n",
       "  Your browser does not support the video tag.\n",
       "</video>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "step = 15\n",
    "anzframes = 100\n",
    "xylim = 5\n",
    "fig = plt.figure(figsize=(11, 5))\n",
    "gs = gridspec.GridSpec(1, 2, width_ratios=[1,1], wspace=0.25)\n",
    "ax1 = plt.subplot(gs[0])\n",
    "ax2 = plt.subplot(gs[1])\n",
    "\n",
    "def init():\n",
    "    ax1.scatter(DLoes[0, 1]*np.cos(DLoes[0, 2]),DLoes[0, 1]*np.sin(DLoes[0, 2]),c=\"red\", s=60, marker='o')\n",
    "    BH = plt.Circle((0, 0), 2, color='black')\n",
    "    ax1.add_patch(BH)\n",
    "    ax2.plot(rval,VeffPhotonl(rval), linewidth=1, color=\"black\")\n",
    "    ax2.scatter(3,VeffPhotonl(3),c=\"orange\", s=80, marker='o')\n",
    "    ax1.set_ylabel(r\"$\\rm y \\,[km]$\")\n",
    "    ax1.set_xlabel(r\"$\\rm x \\,[km]$\")\n",
    "    ax2.set_ylabel(r\"$\\rm V(r)$\")\n",
    "    ax2.set_xlabel(r\"$\\rm Radius \\, r \\,[km]$\")\n",
    "    ax1.set_xlim(-xylim, xylim)\n",
    "    ax1.set_ylim(-xylim, xylim)\n",
    "    ax2.set_xlim(1.8, 10)\n",
    "    ax2.set_ylim(0, 0.04)\n",
    "    return fig,\n",
    "\n",
    "def animate(i):\n",
    "    ax1.cla() \n",
    "    BH = plt.Circle((0, 0), 2, color='black')\n",
    "    ax1.add_patch(BH);\n",
    "    ax1.plot(DLoes[:step*i, 1]*np.cos(DLoes[:step*i, 2]),DLoes[:step*i, 1]*np.sin(DLoes[:step*i, 2]),c=\"orange\", linewidth=0.5, linestyle='-')\n",
    "    ax1.scatter(DLoes[step*i, 1]*np.cos(DLoes[step*i, 2]),DLoes[step*i, 1]*np.sin(DLoes[step*i, 2]),c=\"orange\", s=60, marker='o')\n",
    "    ax1.set_ylabel(r\"$\\rm y \\,[km]$\")\n",
    "    ax1.set_xlabel(r\"$\\rm x \\,[km]$\")\n",
    "    ax2.set_ylabel(r\"$\\rm V(r)$\")\n",
    "    ax2.set_xlabel(r\"$\\rm Radius \\, r \\,[km]$\")\n",
    "    ax1.set_xlim(-xylim, xylim)\n",
    "    ax1.set_ylim(-xylim, xylim)\n",
    "    ax2.set_xlim(1.8, 10)\n",
    "    ax2.set_ylim(0, 0.04)\n",
    "    return fig,\n",
    "\n",
    "ani = animation.FuncAnimation(fig,animate,init_func=init,frames=anzframes,interval=100)\n",
    "plt.close(fig)\n",
    "HTML(ani.to_html5_video())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Wir betrachten uns die drei kreisförmigen Bahnbewegungen der Probekörper und die oben dargestellte Trajektorie des masselosen Lichtteilchens auf der Photonensphäre in einem eingebetteten Diagramm der räumlichen Hypersphäre der Mannigfaltigkeit der Schwarzschildmetrik:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [],
   "source": [
    "from mpl_toolkits.mplot3d import Axes3D\n",
    "from matplotlib import cm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [],
   "source": [
    "x,y = symbols('x,y')\n",
    "zBett = sqrt(8*M*(r-2*M))\n",
    "R = np.linspace(2.00001, 20, 300)\n",
    "PHI = np.linspace(0, 2*np.pi, 300)\n",
    "R, PHI = np.meshgrid(R, PHI)\n",
    "X = R*np.cos(PHI)\n",
    "Y = R*np.sin(PHI)\n",
    "zlam = lambdify((x,y), zBett.subs({(M,1),(r,sqrt(x**2+y**2))}))\n",
    "ZF = np.array(list(np.real(zlam(X,Y))))\n",
    "rval = np.linspace(2.2,30, 1001)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
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      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "step = 20\n",
    "anzframes = 150\n",
    "xylim = 18\n",
    "zlim = 15\n",
    "fig = plt.figure(figsize=(11, 5))\n",
    "gs = gridspec.GridSpec(1, 2, width_ratios=[2,1], wspace=0.21)\n",
    "ax1 = plt.subplot(gs[0],projection='3d')\n",
    "ax2 = plt.subplot(gs[1])\n",
    "ax1.xaxis.pane.fill = False\n",
    "ax1.yaxis.pane.fill = False\n",
    "ax1.zaxis.pane.fill = False\n",
    "ax1.xaxis.pane.set_edgecolor('white')\n",
    "ax1.yaxis.pane.set_edgecolor('white')\n",
    "ax1.zaxis.pane.set_edgecolor('white')\n",
    "\n",
    "def init():\n",
    "    ax1.plot_surface(X, Y, ZF, cmap=cm.nipy_spectral, linewidth=0, alpha=0.5)\n",
    "    ax1.scatter(ALoes[0, 1]*np.cos(ALoes[0, 2]),ALoes[0, 1]*np.sin(ALoes[0, 2]),np.real(zlam(ALoes[0, 1]*np.cos(ALoes[0, 2]),ALoes[0, 1]*np.sin(ALoes[0, 2]))),c=\"green\", s=60, marker='o')\n",
    "    ax1.contourf(X, Y, ZF,100, zdir='z', offset=0, cmap=cm.nipy_spectral,alpha=0.5)\n",
    "    ax1.scatter(ALoes[0, 1]*np.cos(ALoes[0, 2]),ALoes[0, 1]*np.sin(ALoes[0, 2]),0.1,c=\"green\", s=40, marker='o')\n",
    "    ax1.scatter(BLoes[0, 1]*np.cos(BLoes[0, 2]),BLoes[0, 1]*np.sin(BLoes[0, 2]),np.real(zlam(BLoes[0, 1]*np.cos(BLoes[0, 2]),BLoes[0, 1]*np.sin(BLoes[0, 2]))),c=\"red\", s=60, marker='o')\n",
    "    ax1.scatter(BLoes[0, 1]*np.cos(BLoes[0, 2]),BLoes[0, 1]*np.sin(BLoes[0, 2]),0.1,c=\"red\", s=40, marker='o')\n",
    "    ax1.scatter(CLoes[0, 1]*np.cos(CLoes[0, 2]),CLoes[0, 1]*np.sin(CLoes[0, 2]),np.real(zlam(CLoes[0, 1]*np.cos(CLoes[0, 2]),CLoes[0, 1]*np.sin(CLoes[0, 2]))),c=\"black\", s=60, marker='o')\n",
    "    ax1.scatter(CLoes[0, 1]*np.cos(CLoes[0, 2]),CLoes[0, 1]*np.sin(CLoes[0, 2]),0.1,c=\"black\", s=40, marker='o')\n",
    "    ax1.scatter(DLoes[0, 1]*np.cos(DLoes[0, 2]),DLoes[0, 1]*np.sin(DLoes[0, 2]),np.real(zlam(DLoes[0, 1]*np.cos(DLoes[0, 2]),DLoes[0, 1]*np.sin(DLoes[0, 2]))),c=\"yellow\", s=60, marker='o');\n",
    "    ax1.scatter(DLoes[0, 1]*np.cos(DLoes[0, 2]),DLoes[0, 1]*np.sin(DLoes[0, 2]),0.1,c=\"yellow\", s=40, marker='o')\n",
    "    ax2.plot(rval,VeffRezl(rval,4),c=\"green\", linewidth=1.5, linestyle='-')\n",
    "    ax2.scatter(Ar0,VeffRezl(Ar0,4),c=\"green\", s=80, marker='o')\n",
    "    ax2.plot(rval,VeffRezl(rval,3.7),c=\"red\", linewidth=1.5, linestyle='-')\n",
    "    ax2.scatter(Br0,VeffRezl(Br0,3.7),c=\"red\", s=80, marker='o')\n",
    "    ax2.plot(rval,VeffRezl(rval,float(2*sqrt(3))),c=\"black\", linewidth=1.5, linestyle='-')\n",
    "    ax2.scatter(Cr0,VeffRezl(Cr0,float(2*sqrt(3))),c=\"black\", s=80, marker='o')\n",
    "    ax1.set_ylabel(r\"$\\rm y \\,[km]$\")\n",
    "    ax1.set_xlabel(r\"$\\rm x \\,[km]$\")\n",
    "    ax1.set_zlabel(\"Visualisierung z\");\n",
    "    ax2.set_ylabel(r\"$\\rm V(r)$\")\n",
    "    ax2.set_xlabel(r\"$\\rm Radius \\, r \\,[km]$\")\n",
    "    ax1.view_init(azim=35, elev=20)\n",
    "    ax1.set_xlim(-xylim, xylim)\n",
    "    ax1.set_ylim(-xylim, xylim)\n",
    "    ax1.set_zlim(0, 15)\n",
    "    ax2.set_xlim(2.3, xylim)\n",
    "    ax2.set_ylim(0.9, 1.01)\n",
    "    ax1.grid(False)\n",
    "    return fig,\n",
    "\n",
    "def animate(i):\n",
    "    ax1.cla() \n",
    "    ax1.plot_surface(X, Y, ZF, cmap=cm.nipy_spectral, linewidth=0, alpha=0.15)\n",
    "    ax1.plot(ALoes[:step*i, 1]*np.cos(ALoes[:step*i, 2]),ALoes[:step*i, 1]*np.sin(ALoes[:step*i, 2]),np.real(zlam(ALoes[:step*i, 1]*np.cos(ALoes[:step*i, 2]),ALoes[:step*i, 1]*np.sin(ALoes[:step*i, 2]))),c=\"green\", linewidth=1, linestyle='-')\n",
    "    ax1.scatter(ALoes[step*i, 1]*np.cos(ALoes[step*i, 2]),ALoes[step*i, 1]*np.sin(ALoes[step*i, 2]),np.real(zlam(ALoes[step*i, 1]*np.cos(ALoes[step*i, 2]),ALoes[step*i, 1]*np.sin(ALoes[step*i, 2]))),c=\"green\", s=60, marker='o')\n",
    "    ax1.scatter(ALoes[step*i, 1]*np.cos(ALoes[step*i, 2]),ALoes[step*i, 1]*np.sin(ALoes[step*i, 2]),0.1,c=\"green\", s=40, marker='o')\n",
    "    ax1.plot(ALoes[:step*i, 1]*np.cos(ALoes[:step*i, 2]),ALoes[:step*i, 1]*np.sin(ALoes[:step*i, 2]),0.1,c=\"green\", linewidth=2.5, linestyle='-')\n",
    "    ax1.plot(BLoes[:step*i, 1]*np.cos(BLoes[:step*i, 2]),BLoes[:step*i, 1]*np.sin(BLoes[:step*i, 2]),np.real(zlam(BLoes[:step*i, 1]*np.cos(BLoes[:step*i, 2]),BLoes[:step*i, 1]*np.sin(BLoes[:step*i, 2]))),c=\"red\", linewidth=1, linestyle='-')\n",
    "    ax1.scatter(BLoes[step*i, 1]*np.cos(BLoes[step*i, 2]),BLoes[step*i, 1]*np.sin(BLoes[step*i, 2]),np.real(zlam(BLoes[step*i, 1]*np.cos(BLoes[step*i, 2]),BLoes[step*i, 1]*np.sin(BLoes[step*i, 2]))),c=\"red\", s=60, marker='o')\n",
    "    ax1.scatter(BLoes[step*i, 1]*np.cos(BLoes[step*i, 2]),BLoes[step*i, 1]*np.sin(BLoes[step*i, 2]),0.1,c=\"red\", s=40, marker='o')\n",
    "    ax1.plot(BLoes[:step*i, 1]*np.cos(BLoes[:step*i, 2]),BLoes[:step*i, 1]*np.sin(BLoes[:step*i, 2]),0.1,c=\"red\", linewidth=2.5, linestyle='-')\n",
    "    ax1.plot(CLoes[:step*i, 1]*np.cos(CLoes[:step*i, 2]),CLoes[:step*i, 1]*np.sin(CLoes[:step*i, 2]),np.real(zlam(CLoes[:step*i, 1]*np.cos(CLoes[:step*i, 2]),CLoes[:step*i, 1]*np.sin(CLoes[:step*i, 2]))),c=\"black\", linewidth=1, linestyle='-')\n",
    "    ax1.scatter(CLoes[step*i, 1]*np.cos(CLoes[step*i, 2]),CLoes[step*i, 1]*np.sin(CLoes[step*i, 2]),np.real(zlam(CLoes[step*i, 1]*np.cos(CLoes[step*i, 2]),CLoes[step*i, 1]*np.sin(CLoes[step*i, 2]))),c=\"black\", s=60, marker='o')\n",
    "    ax1.scatter(CLoes[step*i, 1]*np.cos(CLoes[step*i, 2]),CLoes[step*i, 1]*np.sin(CLoes[step*i, 2]),0.1,c=\"black\", s=40, marker='o')\n",
    "    ax1.plot(CLoes[:step*i, 1]*np.cos(CLoes[:step*i, 2]),CLoes[:step*i, 1]*np.sin(CLoes[:step*i, 2]),0.1,c=\"black\", linewidth=2.5, linestyle='-')\n",
    "    ax1.plot(DLoes[:step*i, 1]*np.cos(DLoes[:step*i, 2]),DLoes[:step*i, 1]*np.sin(DLoes[:step*i, 2]),np.real(zlam(DLoes[:step*i, 1]*np.cos(DLoes[:step*i, 2]),DLoes[:step*i, 1]*np.sin(DLoes[:step*i, 2]))),c=\"orange\", linewidth=1, linestyle='-')\n",
    "    ax1.scatter(DLoes[step*i, 1]*np.cos(DLoes[step*i, 2]),DLoes[step*i, 1]*np.sin(DLoes[step*i, 2]),np.real(zlam(DLoes[step*i, 1]*np.cos(DLoes[step*i, 2]),DLoes[step*i, 1]*np.sin(DLoes[step*i, 2]))),c=\"orange\", s=60, marker='o')\n",
    "    ax1.scatter(DLoes[step*i, 1]*np.cos(DLoes[step*i, 2]),DLoes[step*i, 1]*np.sin(DLoes[step*i, 2]),0.1,c=\"orange\", s=40, marker='o')\n",
    "    ax1.plot(DLoes[:step*i, 1]*np.cos(DLoes[:step*i, 2]),DLoes[:step*i, 1]*np.sin(DLoes[:step*i, 2]),0.1,c=\"orange\", linewidth=2.5, linestyle='-')\n",
    "    ax1.contourf(X, Y, ZF,100, zdir='z', offset=0, cmap=cm.nipy_spectral,alpha=0.15)\n",
    "    ax1.set_ylabel(r\"$\\rm y \\,[km]$\")\n",
    "    ax1.set_xlabel(r\"$\\rm x \\,[km]$\")\n",
    "    ax1.set_zlabel(\"Einbettung z\");\n",
    "    ax1.view_init(azim=35, elev=20)\n",
    "    ax1.set_xlim(-xylim, xylim)\n",
    "    ax1.set_ylim(-xylim, xylim)\n",
    "    ax1.set_zlim(0, 15)  \n",
    "    ax1.grid(False)\n",
    "    return fig,\n",
    "\n",
    "ani = animation.FuncAnimation(fig,animate,init_func=init,frames=anzframes,interval=100)\n",
    "plt.close(fig)\n",
    "HTML(ani.to_html5_video())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "bzw. ohne Verwendung von ax1.cla() in jedem Animationsschritt und mit einer Trennung von Animationsobjekten und statischen Elementen."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
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       "\">\n",
       "  Your browser does not support the video tag.\n",
       "</video>"
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      "text/plain": [
       "<IPython.core.display.HTML object>"
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     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "step, anzframes, xylim, zlim = 20, 150, 18, 15\n",
    "\n",
    "DATA_CONFIG = [\n",
    "    {\"data\": ALoes, \"r0\": Ar0, \"param\": 4.0, \"color\": \"green\"},\n",
    "    {\"data\": BLoes, \"r0\": Br0, \"param\": 3.7, \"color\": \"red\"},\n",
    "    {\"data\": CLoes, \"r0\": Cr0, \"param\": float(2 * np.sqrt(3)), \"color\": \"black\"},\n",
    "    {\"data\": DLoes, \"r0\": None, \"param\": None, \"color\": \"orange\"}\n",
    "]\n",
    "\n",
    "fig = plt.figure(figsize=(11, 5))\n",
    "gs = gridspec.GridSpec(1, 2, width_ratios=[2, 1], wspace=0.21)\n",
    "ax1 = plt.subplot(gs[0], projection='3d')\n",
    "ax2 = plt.subplot(gs[1])\n",
    "\n",
    "for axis in [ax1.xaxis, ax1.yaxis, ax1.zaxis]:\n",
    "    axis.pane.fill = False\n",
    "    axis.pane.set_edgecolor('white')\n",
    "\n",
    "ax1.plot_surface(X, Y, ZF, cmap=cm.nipy_spectral, linewidth=0, alpha=0.15)\n",
    "ax1.contourf(X, Y, ZF, 100, zdir='z', offset=0, cmap=cm.nipy_spectral, alpha=0.15)\n",
    "\n",
    "ax1.set_xlabel(r\"$\\rm x \\,[km]$\")\n",
    "ax1.set_ylabel(r\"$\\rm y \\,[km]$\")\n",
    "ax1.set_zlabel(\"Einbettung z\")\n",
    "ax1.view_init(azim=35, elev=20)\n",
    "ax1.set_xlim(-xylim, xylim)\n",
    "ax1.set_ylim(-xylim, xylim)\n",
    "ax1.set_zlim(0, zlim)\n",
    "ax1.grid(False)\n",
    "\n",
    "for cfg in DATA_CONFIG[:3]:\n",
    "    ax2.plot(rval, VeffRezl(rval, cfg[\"param\"]), c=cfg[\"color\"], linewidth=1.5)\n",
    "    ax2.scatter(cfg[\"r0\"], VeffRezl(cfg[\"r0\"], cfg[\"param\"]), c=cfg[\"color\"], s=80)\n",
    "ax2.set_ylabel(r\"$\\rm V(r)$\")\n",
    "ax2.set_xlabel(r\"$\\rm Radius \\, r \\,[km]$\")\n",
    "ax2.set_xlim(2.3, xylim)\n",
    "ax2.set_ylim(0.9, 1.01)\n",
    "\n",
    "plots_3d = []\n",
    "for cfg in DATA_CONFIG:\n",
    "    color = cfg[\"color\"]\n",
    "    ln3d, = ax1.plot([], [], [], c=color, linewidth=1)\n",
    "    ln2d, = ax1.plot([], [], [], c=color, linewidth=2.5) # Projektion auf z=0.1\n",
    "    scat3d = ax1.scatter([], [], [], c=color, s=60)\n",
    "    scat2d = ax1.scatter([], [], [], c=color, s=40)     # Projektion auf z=0.1\n",
    "    \n",
    "    plots_3d.append({\n",
    "        \"data\": cfg[\"data\"],\n",
    "        \"ln3d\": ln3d, \"ln2d\": ln2d,\n",
    "        \"scat3d\": scat3d, \"scat2d\": scat2d\n",
    "    })\n",
    "\n",
    "def init():\n",
    "    for p in plots_3d:\n",
    "        d = p[\"data\"]\n",
    "        x0 = d[0, 1] * np.cos(d[0, 2])\n",
    "        y0 = d[0, 1] * np.sin(d[0, 2])\n",
    "        z0 = np.real(zlam(x0, y0))\n",
    "        p[\"ln3d\"].set_data([], [])\n",
    "        p[\"ln3d\"].set_3d_properties([])\n",
    "        p[\"ln2d\"].set_data([], [])\n",
    "        p[\"ln2d\"].set_3d_properties([])\n",
    "        p[\"scat3d\"].set_offsets(np.array([[x0, y0]]))\n",
    "        p[\"scat3d\"].set_3d_properties([z0], 'z')\n",
    "        p[\"scat2d\"].set_offsets(np.array([[x0, y0]]))\n",
    "        p[\"scat2d\"].set_3d_properties([0.1], 'z')\n",
    "    return fig,\n",
    "\n",
    "def animate(i):\n",
    "    idx = step * i\n",
    "    if idx == 0:\n",
    "        return init()\n",
    "\n",
    "    for p in plots_3d:\n",
    "        d = p[\"data\"]\n",
    "        x = d[:idx, 1] * np.cos(d[:idx, 2])\n",
    "        y = d[:idx, 1] * np.sin(d[:idx, 2])\n",
    "        z = np.real(zlam(x, y))\n",
    "        p[\"ln3d\"].set_data(x, y)\n",
    "        p[\"ln3d\"].set_3d_properties(z)\n",
    "        p[\"ln2d\"].set_data(x, y)\n",
    "        p[\"ln2d\"].set_3d_properties(np.full_like(x, 0.1))\n",
    "        p[\"scat3d\"].set_offsets(np.array([[x[-1], y[-1]]]))\n",
    "        p[\"scat3d\"].set_3d_properties([z[-1]], 'z')\n",
    "        p[\"scat2d\"].set_offsets(np.array([[x[-1], y[-1]]]))\n",
    "        p[\"scat2d\"].set_3d_properties([0.1], 'z')\n",
    "    return fig,\n",
    "\n",
    "ani = animation.FuncAnimation(fig, animate, init_func=init, frames=anzframes, interval=100, blit=True)\n",
    "plt.close(fig)\n",
    "\n",
    "HTML(ani.to_html5_video())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Die Visualisierung kann man auch wieder mittels der \"Plotly Python Open Source Graphing Library\" erstellen ( siehe <A href=\"https://plotly.com/python/\" target=\"_blank\">https://plotly.com/python/</A> ). "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [],
   "source": [
    "import plotly.graph_objects as go"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<iframe\n",
       "    scrolling=\"no\"\n",
       "    width=\"720px\"\n",
       "    height=\"720\"\n",
       "    src=\"iframe_figures/figure_57.html\"\n",
       "    frameborder=\"0\"\n",
       "    allowfullscreen\n",
       "></iframe>\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "trichter = go.Surface(x=X, y=Y, z=ZF, colorscale=\"spectral_r\", showscale=True, name=\"Embedded Surface\", hoverinfo='none')\n",
    "\n",
    "Loes=ALoes\n",
    "circular1_orbit =go.Scatter3d(\n",
    "    x=Loes[:, 1]*np.cos(Loes[:, 2]),\n",
    "    y=Loes[:, 1]*np.sin(Loes[:, 2]),\n",
    "    z=np.real(zlam(Loes[:, 1]*np.cos(Loes[:, 2]),Loes[:, 1]*np.sin(Loes[:, 2]))),\n",
    "    name=\"Circular orbit I\",\n",
    "    mode='lines',\n",
    "    line=dict(color='green', width=8),\n",
    "    hoverinfo='none'\n",
    "    )\n",
    "\n",
    "Loes=BLoes\n",
    "circular2_orbit =go.Scatter3d(\n",
    "    x=Loes[:, 1]*np.cos(Loes[:, 2]),\n",
    "    y=Loes[:, 1]*np.sin(Loes[:, 2]),\n",
    "    z=np.real(zlam(Loes[:, 1]*np.cos(Loes[:, 2]),Loes[:, 1]*np.sin(Loes[:, 2]))),\n",
    "    name=\"Circular orbit II\",\n",
    "    mode='lines',\n",
    "    line=dict(color='red', width=8),\n",
    "    hoverinfo='none'\n",
    "    )\n",
    "\n",
    "Loes=CLoes\n",
    "ISCO_orbit =go.Scatter3d(\n",
    "    x=Loes[:, 1]*np.cos(Loes[:, 2]),\n",
    "    y=Loes[:, 1]*np.sin(Loes[:, 2]),\n",
    "    z=np.real(zlam(Loes[:, 1]*np.cos(Loes[:, 2]),Loes[:, 1]*np.sin(Loes[:, 2]))),\n",
    "    name=\"ISCO\",\n",
    "    mode='lines',\n",
    "    line=dict(color='black', width=8),\n",
    "    hoverinfo='none'\n",
    "    )\n",
    "\n",
    "Loes=DLoes\n",
    "Photon_orbit =go.Scatter3d(\n",
    "    x=Loes[:, 1]*np.cos(Loes[:, 2]),\n",
    "    y=Loes[:, 1]*np.sin(Loes[:, 2]),\n",
    "    z=np.real(zlam(Loes[:, 1]*np.cos(Loes[:, 2]),Loes[:, 1]*np.sin(Loes[:, 2]))),\n",
    "    name=\"Photon orbit\",\n",
    "    mode='lines',\n",
    "    line=dict(color='orange', width=8),\n",
    "    hoverinfo='none'\n",
    "    )\n",
    "\n",
    "layout = go.Layout(autosize=True,\n",
    "                  width=700, height=700,\n",
    "                  margin=dict(l=15, r=50, b=65, t=90),\n",
    "                  scene_aspectmode='cube',\n",
    "                  scene = dict(\n",
    "                    xaxis_showbackground=False,\n",
    "                    yaxis_showbackground=False,\n",
    "                    zaxis_showbackground=False,\n",
    "                    xaxis_gridcolor=\"lightgrey\",\n",
    "                    yaxis_gridcolor=\"lightgrey\",\n",
    "                    zaxis_gridcolor=\"lightgrey\",\n",
    "                    xaxis_range=[-22,22],\n",
    "                    yaxis_range=[-22,22],\n",
    "                    zaxis_range=[0,15],\n",
    "                    xaxis_title='x',\n",
    "                    yaxis_title='y',\n",
    "                    zaxis_title='f'),\n",
    "                  legend=dict(\n",
    "                    yanchor=\"top\",\n",
    "                    y=0.99,\n",
    "                    xanchor=\"left\",\n",
    "                    x=0.01)\n",
    "            )\n",
    "\n",
    "data=[trichter,circular1_orbit,circular2_orbit,ISCO_orbit,Photon_orbit]\n",
    "fig=go.Figure(data=data, layout=layout)\n",
    "\n",
    "fig.show(renderer='iframe')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Zusammenfassung der Zonen um ein nicht-rotierendes, ungeladenes schwarzes Loch\n",
    "\n",
    "<ul>\n",
    "    <li>$r > 6 \\, M$: Stabile Kreisbahnen für massereiche Teilchen sind möglich</li>\n",
    "    <li>$r = r_{\\rm ISCO} = 6 \\, M$: Der ISCO. Die letzte Grenze für stabile Bahnen.</li>\n",
    "    <li>$3 \\, M < r < 6 \\, M$: Nur noch instabile Kreisbahnen für Teilchen mit Masse möglich. Teilchen stürzen unaufhaltsam in das schwarze Loch, wenn sie nicht stark beschleunigen.</li>\n",
    "    <li>$r = r_{PhSph} = 3 \\, M$: Die Photonensphäre. Instabile Kreisbahn für Licht.</li>\n",
    "    <li>$2 \\, M < r < 3 \\, M$: Keine Kreisbahnen mehr möglich, weder für Teilchen mit Masse noch für Licht.</li>\n",
    "    <li>$r = R_{\\rm s} = 2 \\, M$: Der Ereignishorizont. Ab hier ist kein Entkommen mehr möglich.</li>\n",
    "</ul>"
   ]
  }
 ],
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