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Neptune was discovered as the something that was perturbing the angular velocity of Uranus. Here we solve for and visualize that pertubation assuming the planetary orbits remain circular. When Neptune is in front it speeds up the angular velocity of \n", " Uranus, and when Neptune is behind, it slows down the angular velocity. \n", " \n", " \n", "Calling $\\bf{F_T}({\\bf r_{su}},{\\bf r_{nu}})$ the sum of the force of the Sun on Uranus and the force of Neptune on Uranus, then:\n", "\n", "\\begin{eqnarray*}\n", "\\bf{F_T}({\\bf r_{su}},{\\bf r_{nu}})& =& {\\bf F_{su}}({\\bf r_{su}}) + {\\bf F_{nu}}({\\bf r_{nu}})\n", "\\\\ \n", "&=& \\frac{G M_s m_u}{\\vert {\\bf r_{su}\\vert^3}}\\,{\\bf r_{su}}+\n", " \\frac{G m_n m_u}{\\vert {\\bf r_{nu}\\vert^3}}\\,{\\bf r_{nu}}\n", "\\end{eqnarray*}" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "data": { "text/html": [ "
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