\begin{equation} \vec{ \textbf{F}}_g = G \frac{m_1 m_2}{r^2} \hat{\textbf{r}} \qquad (\textnormal{Force of gravity}) \tag{13.1} \end{equation} \begin{equation} w = F_g = G \frac{m \, m_E}{R_E^2} \qquad (\textnormal{Weight near earth's surface}) \tag{13.2} \end{equation} \begin{equation} g = \frac{G \, m_E}{R_E^2} \qquad (\textnormal{Acceleration near earth's surface}) \tag{13.3} \end{equation} \begin{equation} U = - \frac{G \, m \, m_E}{r} \qquad (\textnormal{Gravitational potential energy}) \tag{13.4} \end{equation} \begin{equation} v = \sqrt{\frac{G m_E}{r}} \qquad (\textnormal{Circular orbit}) \tag{13.5} \end{equation} \begin{equation} T = \frac{2 \pi r}{v} = 2 \pi r \sqrt{\frac{r}{G m_E}} = \frac{2 \, \pi \, r^{3/2}}{\sqrt{G m_E}} \qquad (\textnormal{Circular orbit}) \tag{13.6} \end{equation} \begin{equation} \textnormal{1. Each planet moves in an elliptical orbit, with the sun at one focus of the ellipse} \end{equation} \begin{equation} \textnormal{2. A line from the sun to a planet sweeps out equal areas in equal times.} \end{equation} \begin{equation} \textnormal{3. The periods of the planets are proportional to the } \frac{3}{2} \textnormal{ powers of the major axis of the orbit.} \end{equation} \begin{equation} R_S = \frac{2 G M}{c^2} \qquad (\textnormal{Schwarzschild radius}) \tag{13.7} \end{equation}
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