ex-13-076.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "6e963640",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import pandas as pd\n",
    "from oetils import init_plotting"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "af0eba87",
   "metadata": {},
   "outputs": [],
   "source": [
    "W = init_plotting(latex=True, sans=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "e6b73ffd",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>a</th>\n",
       "      <th>T</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>57.9</td>\n",
       "      <td>88.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>108.2</td>\n",
       "      <td>224.7</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>149.6</td>\n",
       "      <td>365.2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>227.9</td>\n",
       "      <td>687.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>778.3</td>\n",
       "      <td>4331.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>1426.7</td>\n",
       "      <td>10747.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>2870.7</td>\n",
       "      <td>30589.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>4498.4</td>\n",
       "      <td>59800.0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "        a        T\n",
       "0    57.9     88.0\n",
       "1   108.2    224.7\n",
       "2   149.6    365.2\n",
       "3   227.9    687.0\n",
       "4   778.3   4331.0\n",
       "5  1426.7  10747.0\n",
       "6  2870.7  30589.0\n",
       "7  4498.4  59800.0"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "G = 6.674e-11  # Gravitational constant [N m^2 kg^-2]\n",
    "M = 1.99e30    # Mass of the sun [kg]\n",
    "data = pd.DataFrame({\n",
    "    'a': [57.9, 108.2, 149.6, 227.9, 778.3, 1426.7, 2870.7, 4498.4],  # Semi-major axis [10^6 km]\n",
    "    'T': [88.0, 224.7, 365.2, 687.0, 4331, 10_747, 30_589, 59_800]    # Orbital period [days]\n",
    "})\n",
    "data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "53cc05be",
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
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XXHNNlJaWxqpVq2Lbtm09DQupVCoint9srquri02bNsWuXbviuuuuiylTpkRNTU3U19fHoUOH4uabb46tW7fGkiVL4pJLLslpE0OmUwsGe5pBf00QI6V3U8RQTl3o1ruBoamp6VRKgowOHDgQ+/btiwMHDuS7FACArOQWACBJZBdgNHl4/9H43bU/iA99ZUs8234i47w/+s3quPuTb4ylrzgnh9UBp0puAQCSRHYBgOKieaFArF69Om688caM49/85jfjkksuierq6li/fn3PyQq9GxamTJnSc8rCHXfcEUuWLOn3WlOmTImVK1fGrl27YuvWrVFZWRm1tbXxtre9bUR+tv6sXbs269cD0d8pC/2dxnCq+jYZDOXUhW5969PAwEh4+umn4/HHH4+nn34636UAAGQltwAASSK7AKNBV1c61jc+GIs/+p340S+fyjjvgtMnxH+svjy+8N7FMaXCaQuQNHILAJAksgsAFJcx+S6A51VWVsbu3btP+t7evXujvr4+1q9fHy0tLSedsJBKpXoaGGpra2PNmjUZmxWyqampiU2bNkVjY2MsX748Zs+eHVu3bo3JkycPy8+VyerVq6Ouri6ampqipqZmSKcZ9G1eqKysPKXGgoHez6noW19zc3PU1NQM2/UBAAAAAAD60/zUs/G+2+6OnzyQ/QNBVy2ZFX+9fH5MGj82R5UBAAAAAMVC80IBaWlpibe97W1RW1sbGzdujMbGxoiIk5oWuhsWKisrY+XKlXH99dfHlClTTvm+a2trY+vWrbFo0aKora2NLVu2nPI1X0xVVdWQmhYinv+3amlpOel7y5cvH4aqXmg4mxf6Onjw4Ihdm+JVXV0dXV1dUVLicB0AoLDJLQBAksguQFJ1dnXFLXc+GH/dcG+0He/MOG/mmRPjS1cujl+fd1YOqwNGgtwCACSJ7AIAxUXzQoHZuHFjbNy4MSKeb1ro75SFVatWxVvf+tZhv++qqqpYv359LF++PG677ba48sorh/0+hsv69etf8L1Vq1aNyH0dOHDgpK+nTZs2bNfue9oGDIdx4xzhDQAkg9wCACSJ7AIk0YNPHolrb70r7n7omYxzUqmIa5bOib9YdnFMKPOrQxgN5BYAIElkFwAoLnYgC8S0adPi8OHDJ52yEBEnnbKwatWqmDlz5ojWUVdXF1VVVXHHHXcUdPPCDTfccNLXtbW1UVNTMyL31feEh1PRt/FhOK8NAAAAAAAQ8fxpC1/83s74xDd+Hu0dmU9bqD5rUty0YnG8cs6ZOawOAAAAAChWmhcKxMGDByMicnbKQjY1NTWxefPmnN7nYDQ2Nr7gQ//dp1XwQvfff3/MmzcvysrKeppiIiJOnDgRJ06ciIiI8vLyk27T0dERnZ2dsXz58vjrv/7rk8Yee+yxePbZZ6OkpCTmzJlz0tjDDz8cra2tMWbMmJg1a9ZJY3v27In29vYoKyuLqqqqk8Z2794dx48fj/Hjx8eMGTNOGnvwwQejs7MzJkyYEBdccMFJYw888ECk0+mYPHlynHvuuT3fT6fT8cADD0RERGVlZZx99tkn/dwPPfRQREScdtppceaZv/qFzPHjx3tOwzjjjDPi9NNP7xlra2uLvXv3RkTE9OnTY+rUqT1jzz33XDzyyCMREXHOOefElClTesaeffbZeOyxxyIi4rzzzotJkyb1jLW0tMSTTz4ZEREXXnhhVFRU9IwdPHgwnnrqqYiImDlz5klr9Mwzz8T+/fsjIuKiiy6KsWPH9ow99dRTPc8ns2fPjtLS0p6xJ598suexM2/evJP+LbOt6yOPPBLPPfdc1nUdN25cVFdXnzSWbV0feuihOHHixLCta2dnZzz44IMRMbh1bW9vjz179kRExFlnnXVSg9FQ1/Xw4cPxxBNPRMTQ17W6uvqkvyww0HWdO3fuSY/z7nVNpVIxd+7c3v/MWdd179690dbWNiLrOmnSpDjvvPN6vj/QdZ02bVqcddavjqsf6rq2trbGww8/HBFDX9cLLrggJkyY0DM21HV9+umne073Gcy6Pv7443HkyJERWdfy8vIXNEpmW9edO3dGV1fXC9Y14vnXn4jBrWtHR0fs2rUrIiJOP/30OOOMM3rGBrquZ599dlRWVvaMDXVdDx06FPv27YuIiBkzZsT48eN7xga6rrNmzYoxY34Vt091XUtLS2P27NknjWVb1+bm5jh27FjWda2oqIgLL7zwpLGBrOuUKVPinHPO6fl+rtf16NGj8eijj0ZExLnnnhuTJ0/uGcv1uu7bty8OHToUEYNb10cffTSOHj2adV3Hjh0bF1100Ulj2dZ1165d0dHRMWzr2tXVFTt37oyIiKlTp8b06dN7xrKt67Fjx6K5uTkiIs4888w47bTTesaGuq5HjhyJxx9/PCIGt64HDhyIp59+OiKeP2murKysZ2yg6zpnzpyTjinuXteIF2aqbOvanZVHYl0nTpwY559//kljQ1nX3ll5MOvaOysPdV3PP//8mDhxYs/YUNd1//798cwzz/9F2cGs6xNPPBGHDx+OiOFf1/7eA2Vb1+73QCOxrn2z8kDXte97oKGua+/3QENd177vgU51XbO9t+1vXbO9tz3Vde37Hmio69o7Kw9mXXu/BxrquvZ9DzTUde39Hmgw6zoSexbd65ptz6K/dU3CnsVQ1zXbe6CBrmvf90C5Xtdc7kUNx57FaNiLGq49i0LcixquPQt7UYW9F9X7PVAS9qKan2mLT21+OrY1H4xMSlKpeP8b5sbH3vryaDmwvyeT5GLPwl5UYe9FDWTPIql7Ub3fA42Gvajh2rOwF1XYe1EjuWcxGvaihmvPohD3oga7Z2EvqrD3ogby3nY07EXle89iOPai8r1nMZr3onpn5XzvRY3k52dGw17UcO1Z2Isq7L2o4dizyPde1Eh+fmY07EUN155FIX4uaqT2or7zne9EfX19z9ixY8cinU5HaWnpSc9FEc+vQ0T0jHXnKAZG80KBSafTsXLlylizZs2In7LQbceOHXHw4MG4/PLLI+L54ND3hIBCsmrVqpO+3rhx40lPssOt+4UxadfuduzYsZ4XwMHqDpB9r9fa2nrSi0nfsf6Oc2tvb4/W1tae00V6a2tri2PHjp30xr732IkTJ17w5B8RPdfr23zRPRYRJ72Q9h3rvakR8fzjr3usu7FjIGNdXV09Y52dJ/8Fq87Ozp6xrq6uAY+dOHGiZ6zvv1lHR0fWscceeyxOnDjxgsBz/Pjxntv11T3W3xp0r2t/a9A91rf+iOfXvL29/aRw0a2trS06OjpO2sDrPdbV1XXSRkK37vp7B9+Ik9end7DqO9bR0XHSWO+1y7aufcd6r13fNe+9dtnG+v6bvdi6ZhobiXXtfryOxnXtvXZDXddsj9fBrGvvtRvMunavXSGsa/f1BrOuvceyretwPQ8P9Tk62/PwQJ+j+8o21r3m/a1B91h/a9C95n1/tojn17Wtra3f+2tvb4/jx4/3+3re1tYWnZ2d/b6ed9ffe4O/71jvjcaI7I/X4XgsZ1vXfDyWe69htnXtXrts69rfYyvbura1tQ3ruvZen8Gs62BeX59++uno6OiIsWPHRllZWaJeXwe7rt3Pw0ld14E+Dxd6Vh6p5+H+1rV7rD/t7e1Deg/U1tY26PdAvdcu23ugYs3Kw/netlDWtff6FGtWLoR1zbRn0Xvthrquw/WYHOq6juRjsj8jta7t7e39rmt7e3tPPunr8ccfj+PHj8fpp59+0i8WI079eXg0v75mG7MXNbCxpL2+FsKexXCva1JeX9vaj8WG/340vnr3gejofOHzX7fZZ0+Km6/6tbj0oud/8f20vSh7Ub0MZM8iqXtRw/FYLqS9qL56r2vvPZczzzzTXlQ/Y0l5fe1en6TuRfUeG4m9qOF6Hi7ErDzY90DdY4W8Z+GxXBh7FiO5F5XvPYuRfH3tTyGta/f1crEXNRyfn+k7tm/fvti7d2+MGTPmBY0ZI5GV8/E8PJS9qO73ttker4W8ZzEc+4/D/Vi2F/Urw7FnUSivr4WwZ1GIe1GF9PqalHXdt29f7Nixo9+6GF6aFwrMpk2bYsmSJTm7v+uvvz7WrVsXERELFy6MxsbGaGxsjGXLluWshsFYt25dTyd+RMTq1aujrq4ujxUVvrKyspg5c+aQTl7o/cH33terqKjo98WhrKwsOjs7+32B7r6P/l6gx48fH6WlpRnHMm2CVVRURDqdzjgWEf2+UGUaS6VSPWN9f4ZsYyUlJT1jfd+8lJaW9oz1/TfLNjZmzJiesb4vmmPHjs06duzYsWhvb48DBw6ctIbjxo3r9xc3vccyrWtFRUW/61pWVpZxDcrLy6OkpKTfN1Hjx4/v+ZBif2PpdLrfse76+95f7/XJNtZ3zXuvXbZ17TvWe+36rnnvtcs21vff+sXWNdPYQNa1v+CVbV3Ly8tHbF2zPV4Hs3ZDXdfeazfUdc32eB3MuvZeu8Gsa/fajcS6Zlq7TOtaUVERXV1dg1rX3mO5eB4e6nN0tufhgT5H95VtLNvjtXusvw2ysrKyjGtQXl4eqVSq38dreXl5jBkzJuOaZ3pTPZB1zfVjOdu6FspjOdvjNdu69pdjsq3r+PHjh3VdR2Lt+o49+eST0draGhUVFXH++ecn6vW1P9nWtby8fMTWNRePyYE+Dxd6Vh7Kur7Y83BnZ2e/69q9Zv2tXXl5eZSWlmZc887Ozoxrnuma+Xws95a0rJztvW2mx2v3nP7GTmVdh+ux3Ht9ijUrD+eeRfe6ZtuzyPSY7C9/F9Jjcqjrmq/X1+Fe10zvgcrLy2Ps2LEZP1zQ1tbW7weyvL6+0EjsWRTiXlS+H8uF+Po6GveikvD6et+jLXHVzdvjF48decHP3HOfqYh3XnZWrH3Pr8eE8l/9/Ke6F5XpPZC9qGTuRQ1kzyKpe1HD8VgupL2ovnqPHThwoGfP5cwzz7QX1c9YUl5fu9cu21gh70X1HhuJrDxcz8OFmJUHu2dhLyoZj+XRvheV7z2LkX597atY96KG4/MzfccOHz4czz33XM97hd5GIiuP5OdnhnMv6lTf2yb1sdx77Yb7sWwv6leGY8+iUF5f7UX9SqG+vg73uo7UXtT06dNj/vz5PWODPXkhU5M4L5RK99fuR86VlJREdXV1z3FcuTJt2rSe41dSqVTU1NREU1NTbNu27aQHYSFobm4+6TiXurq62Lhx44jf77Jly6KhoaHn69ra2ti0adOQrrV06dJobGzs+Xokf4YFCxbEjh07Yv78+bF9+/YRuQ8K0/3339+zId33eDsAgEIitwAASSK7AIWo40RXfOa/7ot137ovOjpf+Jflur3kvClx84rLoqbqtBxWB+SL3AIAJInsAkDS+bzu4Dh5oYBUVVXl/D6XLVsWGzZsiFQqFel0OrZt2xZr1qwpuMaFiOc/+N8tV40LEc83eCTx2hSvWbNmRTqd7rf7EACgkMgtAECSyC5Aofn5w4fi6g0/i1880pJxzpjSVHz4t18a1/3uS6Ns7Av/Kh0wOsktAECSyC4AUFw0LxSQysrKnN9nfX19HDx4MLZv3x5VVVWxatWqeOtb35rzOl7MsmXLorm5OSJy27gw3A4ePJjvEigC/R13BQBQiOQWACBJZBegUBw/0RnrvnVffOa/7osTnZkPWH/5BZVxy1W/Fq+4cGoOqwMKgdwCACSJ7AIAxcUrf4Ho6sp8lO9AHDly5KQPxs+YMWPAty30RoB169ZFQ0NDRESsXLky6uvrc3r/I9lUko+GFQAAAAAAIJmamg/ENbfeFb987HDGOWNLS2LN7700/vS3Xxpjx5TksDoAAAAAgOw0L4wSl19+eTQ1NfV8nUqlorKyMlatWhWf/OQn81jZqWloaIg1a9ZERH4aFyIiTjvttJO+Hs7TE/peGwAAAAAAoK/2451xw7//Iv7+O/dHZ1fm0xYWzJgWN191Wbz0/MrcFQcAAAAAMECaF0aJzZs3n/T1wYMHY9u2bXHjjTdGY2NjbNmyJU+VDV1jY2MsW7YsIiJWr14da9euzUsdVVVVw3atlpaWEbs2dNu3b18cP348xo0bF9OnT893OQAAGcktAECSyC5AvmzZ9Uxce+tdsfOJIxnnjBtTEh9988vjQ2+cF2NKnbYAxU5uAQCSRHYBgOKieWGUmDJlygu+njlzZtTV1cVFF10Un/nMZ+LDH/5wnqobvKampli6dGlERKxduzZWr16dt1oqKytP+rpvA8JgNDc3n/S15gVGwqFDh6K1tTUqKiq8qQMACprcAgAkiewC5Frb8RPx8W/8PL70vZ3Rlc582sKi6tPi5hWXxdxzp2ScAxQXuQUASBLZBQCKiz+9UgQqKyvjlltuyXcZA9bc3BxLliyJiIj6+vq8Ni5ERCxatOikrw8ePDhs166pqRm2awEAAAAAAKPDT3c+Hb/2/74bX/juAxkbF8rHlsYn3rYgGv98qcYFAAAAACARnLwwym3evDmampoilUrlu5QBaWlpiaVLl0ZLS0ts3Lgx6urqBny7NWvWxNq1a19wUsKpqqysjKqqqp5TE4Z68oJTF8iVOXPm5LsEAIABkVsAgCSRXYBceO7YifibjffGzZt2RpbDFuKyWWfETSsWx6yzJ+euOCAx5BYAIElkFwAoLpoXCtSRI0di69at0dTUFBERBw4cGNTtW1paem6fSqUS8xf+lyxZEs3NzYNqXIiI2Lp1a6xfvz7q6+tHpK6ampqTmg+am5sH3XzQt+khKWtC8pSUOFQHAEgGuQUASBLZBRhpP77/qXj/bXdH89NHM84ZP640/mrZxbFq6ewo9bwEZCC3AABJIrsAQHHRvFBgjhw5EldddVU0NDQMy/XS6XSkUqm4/vrrh+V6I2np0qXR1NQ06MaFiKE1EwzGFVdccdKaNDU1Dfr++p68sGrVqmGpDQAAAAAASK6j7R3xF7fviA2bH8o679Vzz4wvXbk4qs6alKPKAAAAAACGl+aFArJnz5543eteF83NzZH+/88CTqVSPeP9fa8/6T7nCK9cuTLe8pa3DHO1w2vVqlXR2Ng4pMaFiIhNmzaNaPNCXV1dVFZW9pyecM899wy6zk2bNvX8d2VlZdTW1g5niQAAAAAAQML88P/2xfu/fHc88sxzGedMKBsTH79iflx5+awoKcn+OyIAAAAAgEKmeaGALF26NPbs2XPS9/o2IqRSqRd8rz+pVCpqa2tj5cqV8da3vnVY6xxua9asifXr1w+5cSEiorGxMVauXDmguU1NTdHY2BiVlZUDvk3E800g69at67m/odTYbfny5YO+PQzU448/HsePH49x48bFueeem+9yAAAyklsAgCSRXYDhdLj1eHzs69vjqz/anXXeb77krPjilYvjwjMm5qgyYDSQWwCAJJFdAKC4aF4oENdff300NzdHxPN/lX/58uWxdOnSntMEKisr45ZbbolPf/rTsW3btqisrDzp9i0tLXHw4MFobm6OTZs2RUNDQzQ3N8e0adNy/aMMyvr162PdunWDalzoPv3g4MGD0dLSErfffnu0tLREdXX1i962oaEhli1b1vN1fX19bNu2bUD3e/311/c0LzQ1NUVzc/OAT3vonh/x/FquXbt2QLeDoThy5Ei0trZGRUWFN3UAQEGTWwCAJJFdgOGy6edPxAe+vCUeP9iacc6k8jHxibfXxHt+s/pFT+QG6EtuAQCSRHYBgOKieaEAHD58ONauXRupVCquuuqqWLt2bUyZMuUF8y699NKIiKiuro7Jkyf3e60lS5bEVVddFYcPH44VK1ZEbW1trFu3Lj784Q+P6M8wFI2NjbFq1aqIiJMaCoZq0aJFLzrnqquuOunrpqamaGhoGFDjRHfTwZo1ayLi+RMjNm7cOKDabrjhhp7/Xrt27QuaT2C4+WUWAJAUcgsAkCSyC3AqDj13PK7/l6b45x83Z5239BVnx+ffc2mcd9qEHFUGjEZyCwCQJLILABQPzQsF4I477oiIiJUrV8bNN9+ccV73B96bm5tj/vz5Wa85ZcqU2LhxY6xatSpWr14dlZWVceWVVw5Xyaesqakpli5dOqzXHMgpCN2nNvTWfSLCQKxevTo2bdoUjY2N0dDQMKDGh+55Ec+v8cqVKwd8fzAU8+bNy3cJAAADIrcAAEkiuwCn4jvbH4sPfeWe2NfSlnHOlIqxceM7FsY7Xj3TB3eAUyK3AABJIrsAQHEpyXcBRGzatCmqq6uzNi5EREybNi3S6XRs3bp1wNeur6+PyZMn95wWUAhaWlpiyZIlw3rNysrKAZ1m0F+jwUBOXeht06ZNPQ0Iy5Yt62lM6E9DQ0PPqRIrV66M+vr6Qd0XAAAAAACQXAePHourbvlpXPHZ/8nauPCG+efElk++Kd7561UaFwAAAACAUUvzQgFoamoaUHNB98kCjY2Ng7r+2rVr4+DBg/GZz3xmSPUNt+bm5n5PQDgVAzl1ISJiw4YNPc0KVVVVsXHjxgHftrf6+vrYuHFjVFZWxrJly2Lp0qXR0NAQTU1N0dTUFA0NDbF06dJYtmxZz/1oXAAAAAAAgOLxH1sfjUuu/3Z8/ad7M86ZOmFcbFj1a3HHn7wmzplWkbviAAAAAADyIJVOp9P5LqLYlZSURFNTU8yfP/9F506bNi1SqVQcOHBgwNc/fPhwTJ06NZYuXRrf//73T6FS+tPY2Bj19fXR1NQUzc3NEfH8SRC1tbWxdOnSnlMacm3BggWxY8eOmD9/fmzfvj0vNQAAAAAAQLHZf6Q9rvunrfGNux/JOu93Fp4Xn333JXFW5fgcVQYAAAAADDef1x2cMfkugOcN9K//L1q0KDZv3hw//OEP47Wvfe2AbjNlypSorKyMrVu3nkqJZFBbWxu1tbX5LgN6PProo3Hs2LEoKyuL888/P9/lAABkJLcAAEkiuwAvJp1Ox79teSQ+/I9b45lnj2Wcd9qksvjMuxbFWxZfEKlUKocVAsVCbgEAkkR2AYDiUpLvAnj+r/R3/8X+F7N06dJIp9Oxdu3aQd1HS0tLtLS0DKE6IGmOHj0ahw8fjqNHj+a7FACArOQWACBJZBcgm6cPt8U7v/CTePeX/jdr48JbLr0g7rnhTfHWyy7UuACMGLkFAEgS2QUAiovmhQKwaNGiAZ+KUFdXFxERmzZtih/+8IcDus2ePXsiYuCnOwDJVlpaGmPGjInS0tJ8lwIAkJXcAgAkiewC9CedTsftP90Tiz7y7fiPrY9mnHfG5PL42gdeHf/w/lfHGZPLc1ghUIzkFgAgSWQXACguY/JdAM83JNTX18eKFSt6vnfkyJHYunVrVFVVxYwZM3q+P3PmzKipqYmmpqaoq6uLpqamuPDCC7Nev/uUhsrKypEoHygws2fPzncJAAADIrcAAEkiuwB9PXmoNT701Xviu9sfzzpv+a9dGOveuShOm1SWo8qAYie3AABJIrsAQHFx8kIBWLlyZRw6dCj+7d/+LSKePylh6tSpsXTp0qiuro7bbrvtBfMjIg4dOhRLly6Ne++9N+O1v/nNb8b69esjlUpFbW3tyP0QAAAAAAAARSCdTsc//7g5Lr3+21kbF6ZXjo+v//FvxG3XvErjAgAAAABAaF4oGDfeeGOsWLEiHn744Vi7dm2k0+me/61evfqkuStXrowFCxZEKpWKXbt2RU1NTVx77bXxgx/8IPbu3RsRETt27Ihrrrkmli1b1nO7VatW5fJHAgAAAAAAGFUeP9gadZ/5UVy94a5oae3IOO8PXj0zttzwpnhTzXk5rA4AAAAAoLCNyXcBPK+uri42bdoUVVVVUVVVFalUKuv8DRs2xKJFiyKVSkU6nY76+vqor69/wbx0Oh2pVCpqampixowZI1Q9AAAAAADA6JVOp+OrP9odH/v69jjSlrlp4Zyp4+ML710cr7v4nBxWBwAAAACQDJoXCkh9fX1UV1fHjTfeGOl0uuf7119//Qvm1tTUxB133BHLly/vaXTofZuIiFQq1TO2du3aEawcKCQPP/xwtLe3R3l5eVx44YX5LgcAICO5BQBIEtkFitcjzzwXH/jy3fGD/9uXdd67X1Mdf/v2BTGlYlyOKgPon9wCACSJ7AIAxUXzQoFZvXp1rF69OjZv3hwtLS1RVVUVCxYs6HduXV1d3HnnnbF8+fJoaWl5wWkN3c0M9fX1cfnll4947UBhaG1tjdbW1ujq6sp3KQAAWcktAECSyC5QfLq60vHlH+6KP799exxtP5Fx3vmnVcQXr1wcl7/s7BxWB5CZ3AIAJInsAgDFRfNCgVqyZMmA5tXW1saePXuivr4+1q9fH83NzRERUVlZGbW1tXH99ddnbH4ARqexY8dGWVlZjB07Nt+lAABkJbcAAEkiu0Bx2fP00XjfbXfFj+9/Ouu8FZfPir+5Yn5MGu+5ASgccgsAkCSyCwAUl1S6+8/zA6PKggULYseOHTF//vzYvn17vssBAAAAAICC19WVjvrGB+Ov7tgRrcc7M86bccaE+NKVl8VvvOSsHFYHAAAAABQan9cdHCcvAAAAAAAAUPR27TsS1956d/zswf1Z513zutnxl8vmx4Qyv2YDAAAAABgMu6oAAAAAAAAUrc6urvjS93fGxxt+Hu0dmU9bqD5rYnxpxWXxqjln5rA6AAAAAIDRQ/NCETh8+HBs2LAhamtrY/78+fkuBwAAAAAAoCA88PjhuPbWu+Ke3QcyzkmlIt7/hrnxsbe8IiqctgAAAAAAMGR2WItAc3NzrFmzJlavXq15AYpAc3NztLe3R3l5eVRVVeW7HACAjOQWACBJZBcYXU50dsXff+f+uOHffxHHOroyzpt99uS4acXiWDzrjBxWB3Bq5BYAIElkFwAoLpoXisDBgwcjnU5HS0tLvksBcuDYsWPR1tYWqVQq36UAAGQltwAASSK7wOjxy8da4poNd0XTnoMZ55SkUvGhN86Lj7755VE+rjSH1QGcOrkFAEgS2QUAiovmhSKwadOmiHi+iQEY/crLy0/6fwCAQiW3AABJIrtA8nWc6IrPfvuXceO//190dGY+bWHeuVPi5qsui4VVp+WwOoDhI7cAAEkiuwBAcdG8MIz27t2b7xJO0tLSEo2NjbFu3bpIpVJOXoAiMXPmzHyXAAAwIHILAJAksgsk2y8eORTXbLgr7n34UMY5pSWp+PBvvyRW/97Lomys0xaA5JJbAIAkkV0AoLhoXhhGVVVVBXt8VTqddvICAAAAAABQVI6f6IxP/cd98en/vC9OdKYzznv5BZVx84rL4uIZ03JYHQAAAABAcdG8MIyqqqqiubk532W8QCqVinQ67eQFAAAAAACgaGzfczCuufWuuO/RloxzxpSmYs3vvSz+9LdfEuPGOG0BAAAAAGAkaV4YRpWVlSedvJBOZ/4LPtl0X2Oot+97je7rOHkBAAAAAAAY7Y51dMYN//6L+Ny374/Orsy/a5k/Y2rcvOKyeNkFU3NYHQAAAABA8dK8MIymTZvW0yhQU1MTtbW1cdpppw3qGps2bYrGxsaoqqqK2traIdVx8ODBaGhoiFQqFcuWLYupU226QzHZtWtXtLW1xfjx4+Oiiy7KdzkAABnJLQBAksgukAz37H4mrtlwV+x84kjGOePGlMT1b355/PEb58WY0pIcVgeQG3ILAJAksgsAFBfNC8OoqqoqIiKWLl0a3//+9wd9+w0bNkRjY2OsW7cu/uzP/uyUamlpaYlly5ZFY2NjbNu2LWbMmHFK1wOSo6OjI44fPx5jxniKBwAKm9wCACSJ7AKFre34ifjEN34RX/zeA9GV5WTrRVWnxc1XXRZzz52Sw+oAcktuAQCSRHYBgOLiFX8YVVZW9px2MFjbt2+Pq6++OhoaGuItb3nLsNSyadOmWLVqVSxdujS2bdsWkydPPuXrAoWvoqIiSktLo6ysLN+lAABkJbcAAEkiu0Dh+tmD++PaW++KXfuezTinbGxJfOwtr4j3v2Gu0xaAUU9uAQCSRHYBgOKieWEYnXbaaRERMW3atEHfdvny5XHjjTcOS+NCb/X19bF06dL4yEc+EjfddNOwXhsoTBdeeGG+SwAAGBC5BQBIEtkFCk/rsRPxNw33xk137owshy3E4lmnx00rLovZZ/sjT0BxkFsAgCSRXQCguGheGEZVVVUxc+bMqKysHNTtvvGNb0Q6nY7rrrtuROrauHFjTJs2LdasWSPsFaGdO3fGggUL+h1btWpVXH311TmuCAAAAAAATs1PHngq3nfr3dH89NGMc8aPK42/rLs4rn7d7CgtcdoCAAAAADA0t9xyS9TX1/c7tnPnzhxXk2yaF4bRW9/61njrW9866NvdfvvtsWzZshGo6HmVlZVRV1cXt9xyS9xwww0jdj8Upra2ttixY0e/Y/v27cttMQAAAAAAcAqOtnfEX96xI9Y3PpR13qvmnBlfWrE4qs+alKPKAAAAAIDRat++fRk/i8vgaF4oAE1NTfG2t71tRO/jiiuuiPXr14/ofVCYxo8fH3PmzOl3bPr06TmuBgAAAAAAhuZH9+2L9992dzz8zHMZ50woGxN/c8X8WHH5rCgpSeWwOgAAAABgtJo+fXrMnz+/37GdO3dGW1tbbgtKMM0LBaC5uTmqqqpG9D6qqqpi69atI3ofFKY5c+bE9u3b810GOdT9QpitcQUAoBDILQBAksgukD9H2jriY1/fHl/54a6s817zkrPii1cujhlnTMxRZQCFSW4BAJJEdgEgCa6++uq4+uqr+x1bsGCBUxkGQfNCAaisrIzm5uaMHTnDdR8tLS0jdn2gcHR1dUVnZ2d0dXXluxQAgKzkFgAgSWQXyI/Gnz8RH/jylnjsYGvGOZPKx8Qn3l4T7/nN6kilnLYAILcAAEkiuwBAcdG8UACqqqrinnvuibe85S0jdh9NTU0jfroDUBgmTpwY48aNi3HjxuW7FACArOQWACBJZBfIrZbnjsf1/9IUX/txc9Z5tS8/Oz7/nkvj/NMn5KgygMIntwAASSK7AEBx0bxQABYuXBjr16+PG264YcTuo76+XvMCFInzzz8/3yUAAAyI3AIAJInsArnz3e2Px4e+uiWePNSWcc6UirHxybfXxLt+o8ppCwB9yC0AQJLILgBQXEryXQARq1atikOHDsW11147ItffvHlzbN68OZYtWzYi1wcAAAAAADhVB48ei6tu+Wks/+x/Z21ceP3F58SWT74p/vA11RoXAAAAAAASxMkLBaCmpiZmzpwZ9fX1sXDhwrjyyiuH7dp79+7taVpYvnz5sF0XAAAA+P/Yu/Pwqsszf/x3NkICkgQEcUEgiIgrYZ/O0lbB+rUzXcZQuy/KotYu01ZQZ9rpMm3FjtPFVkHR7otAa9tpOx1Bu08B2VyqghBRXBAEAkICZDm/P/qDAnJOAyTn5HPO63VdXhWem+TGp+ec93mSOw8AAJ3lv1dsjA9/48HYvGNP2pqaXj1i9jvHxFtfNcTQAgAAAABAArl5oZuYO3dupFKpmD59eqfdwPCjH/0ohg0bFjt27IiZM2dGnz59OuXjAgAAAAAAdIaXXt4T77vtD/H2L/8u4+DCP445LZZ9/vXxtr8danABAAAAACChDC90E5MmTYrLLrssUqlUzJ07N/r16xe33HJL7Ny586g+zs6dO2PevHkxfPjwmDJlSqRSqaitrY3Pf/7zXdQ50N08/vjjsWLFinj88cdz3QoAQEZyCwCQJLILdL57lz0T467/eSxc8nTamr69y+Pr17wqvvfBv4+B1RVZ7A4gueQWACBJZBcAKCyluW6Av1iwYEGMGTMmVq1aFdu3b4+ZM2fGzJkzo7a2NkaPHh21tbUxbNiwiIjo27dvRERs27YtGhsb48EHH4yVK1dGQ0NDRESkUqmIiKipqYlFixbl5i8EAAAAAABwmM07muMj31oeP3lwY8a6N48/PW5599jo36dnljoDAAAAAKArGV7oZh544IG48MILY9WqVRHx5yGE9evXHxhKyGT/wML+65Krq6tj8eLFMWTIkC7rF+h+qqqqoqKiInr06JHrVgAAMpJbAIAkkV3g+KVSqVi45On42LdXxLZde9PWnXhCeXzxPePiTeNPz2J3APlDbgEAkkR2AYDCYnihm6mqqooVK1bErFmz4gtf+EJE/GUYYf9wwpEUFRVFUVFRpFKpSKVSMWnSpFiwYEFUVVVlpW+g+zjllFNy3QIAQIfILQBAksgucHw2NTbHh7/xYPx85bMZ66ZMHBw3v2tMnHiC2xYAjpXcAgAkiewCAIWlONcNcGSzZ8+OdevWxWWXXXZgIGG//YMK+/+JiAM1o0ePjkWLFsV9991ncAEAAAAAAMipVCoV3/1dQ4y7/mcZBxdOquoZ3//QP8Td1/ytwQUAAAAAgDzl5oVurLa2NhYsWBA7duyI+fPnx6JFi6KhoSEaGhqisbExqquro2/fvlFbWxuTJ0+O+vr6GDp0aK7bBgAAAAAAiOe2NcUHv74s7nvo+Yx1b/vboXHTO0ZH397lWeoMAAAAAIBcMLyQAFVVVTFt2rSYNm1arlsBEqC9vT1SqVQUFRVFcbELdgCA7ktuAQCSRHaBjkulUvGt3zbEjd9bGTubW9LWnVJTEV9+3/i4ZNSpWewOIP/JLQBAksguAFBYDC8A5Jk1a9ZEU1NTVFZWxsiRI3PdDgBAWnILAJAksgt0zMaXdscH7l4a9z+6KWPdu189LD73trqoquyRpc4ACofcAgAkiewCAIXF8AIAAAAAAADHpb09FV//9br4tx+sil17WtPWDepXGbdeMSEuOu/kLHYHAAAAAEB3YHgBIM/U1NRE7969o6ysLNetAABkJLcAAEkiu0B6T23eFdfetTR++/iLGeuuvPCM+PTlddGnwuMIoCvJLQBAksguAFBYDC8A5JmBAwfmugUAgA6RWwCAJJFd4JXa21Nxx+K18e/zV0fTvra0dYNP7BVfmzohXn22xxFANsgtAECSyC4AUFgMLwAAAAAAAHBU1m3aGe+/a2n835otGeuumnxm/Pu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      "text/plain": [
       "<Figure size 3000x2086.34 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Least squares fit\n",
    "x = np.stack([np.ones(len(data)), data.a**3]).T\n",
    "y = data['T'] ** 2\n",
    "w = np.linalg.lstsq(x, y, rcond=None)[0]\n",
    "\n",
    "# Plot\n",
    "fig, ax = plt.subplots()\n",
    "ax.set_xlabel(r\"$a^3\\quad/\\quad\\mathrm{10^{18}\\ km^3}$\")\n",
    "ax.set_ylabel(r\"$T^2\\quad/\\quad\\mathrm{days^2}$\")\n",
    "ax.plot(data.a**3, data['T']**2, 'r.', label=\"Data\")\n",
    "x = np.linspace((data.a**3).min(), (data.a**3).max())\n",
    "ax.plot(x, sum(w[i] * x**i for i in range(2)), label=\"Fit\")\n",
    "ax.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "d6d53e73",
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 3000x2086.34 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "a = data.a * 10**9\n",
    "T = data['T'] * 24 * 60 * 60\n",
    "\n",
    "# Least squares fit\n",
    "x = np.stack([np.ones(len(data)), np.log10(a)]).T\n",
    "y = np.log10(T)\n",
    "w = np.linalg.lstsq(x, y, rcond=None)[0]\n",
    "\n",
    "# Plot\n",
    "fig, ax = plt.subplots()\n",
    "ax.set_xlabel(r\"$a\\quad/\\quad\\mathrm{m}$\")\n",
    "ax.set_ylabel(r\"$T\\quad/\\quad\\mathrm{s}$\")\n",
    "ax.loglog(a, T, 'r.', label=\"Data\")\n",
    "x = np.linspace(np.log10(a).min(), np.log10(a).max())\n",
    "ax.loglog(10**x, 10**sum(w[i] * x**i for i in range(2)), label=\"Fit\")\n",
    "ax.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "458f3761",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Log-log slope: 1.499, error: -0.093%\n"
     ]
    }
   ],
   "source": [
    "print(f\"Log-log slope: {w[1]:.3f}, error: {(w[1] - 3/2) / (3/2) * 100:.3f}%\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "1d86eb47",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Estimated mass of sun: 1.85e+30 kg, actual mass: 1.99e+30, error: -7.020%)\n"
     ]
    }
   ],
   "source": [
    "# Compute mass of sun\n",
    "M_fit = 4 * np.pi**2 / (10**(2 * w[0]) * G)\n",
    "print(f\"Estimated mass of sun: {M_fit:.3g} kg, actual mass: {M:.3g}, error: {(M_fit - M) / M * 100:.3f}%)\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "117dd3bf",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Estimated semi-major axis of Vesta (from data): 3.54e+11 m\n",
      "Estimated semi-major axis of Vesta (accurate constants): 3.53e+11 m\n"
     ]
    }
   ],
   "source": [
    "# Vesta\n",
    "T = 1325.4 * 24 * 60 * 60\n",
    "a = 10 ** ((np.log10(T) - w[0]) / w[1])\n",
    "print(f\"Estimated semi-major axis of Vesta (from data): {a:.3g} m\")\n",
    "a = np.cbrt(T**2 * G * M / 4 / np.pi**2)\n",
    "print(f\"Estimated semi-major axis of Vesta (accurate constants): {a:.3g} m\")"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.10.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}