OOMA_HM03.ipynb

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   "source": [
    "## How long we have to observe stars orbiting the SMBH at the Galactic Center to see 1/2 an orbit at a distance from the black hole of: 0.1\", 0.5\", 1.0\", 2.0\""
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### [Milo] Juan Camilo Buitrago-Casas"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Distance to the center of the Milky Way: 8kpc"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
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   "source": [
    "from math import pi\n",
    "import numpy as np \n",
    "import pandas as pd\n",
    "\n",
    "M = 4.1e6 #Solar masses\n",
    "d = 8e3 #pc\n",
    "r_arcsec = np.array([0.1, 0.5, 1.0, 2.0])\n",
    "r_pc = d * pi/(180. * 3600.) *r_arcsec\n",
    "r_au = r_pc * 206264.81"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### The Kepler's third law in AU and years is:\n",
    "\n",
    "$T^2 = \\frac{a^3}{M}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To observe half of the orbit we need to observe half the period:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\frac{1}{2}T = \\frac{1}{2M}a^{3/2}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "T = 0.5 * (r_au)**(1.5) / M\n",
    "Td = T * 365.25 # days"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
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     "data": {
      "text/plain": [
       "array([  1.00788589,  11.2685068 ,  31.87215029,  90.14805439])"
      ]
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     "execution_count": 44,
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   "source": [
    "Td"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
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   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([   800.00001456,   4000.00007278,   8000.00014556,  16000.00029111])"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "r_au"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.1,  0.5,  1. ,  2. ])"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r_arcsec"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
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   "outputs": [],
   "source": [
    "How_Long = pd.DataFrame({'Dist. [arcsec]':r_arcsec,'semi-major axis [AU]':r_au,'Time half an orbit [days]':Td})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
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    {
     "data": {
      "text/html": [
       "<div>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Dist. [arcsec]</th>\n",
       "      <th>Time half an orbit [days]</th>\n",
       "      <th>semi-major axis [AU]</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>0.1</td>\n",
       "      <td>1.007886</td>\n",
       "      <td>800.000015</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>0.5</td>\n",
       "      <td>11.268507</td>\n",
       "      <td>4000.000073</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>1.0</td>\n",
       "      <td>31.872150</td>\n",
       "      <td>8000.000146</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>2.0</td>\n",
       "      <td>90.148054</td>\n",
       "      <td>16000.000291</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   Dist. [arcsec]  Time half an orbit [days]  semi-major axis [AU]\n",
       "0             0.1                   1.007886            800.000015\n",
       "1             0.5                  11.268507           4000.000073\n",
       "2             1.0                  31.872150           8000.000146\n",
       "3             2.0                  90.148054          16000.000291"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "How_Long"
   ]
  },
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   "cell_type": "code",
   "execution_count": null,
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    "collapsed": true
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   "source": []
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   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
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