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I have seen two different forms of the vis-viva equation:

\begin{align} v^2 &= GM\left( \frac{2}{r} - \frac{1}{a} \right) \tag{1} \\ v^2 &= G(M+m)\left( \frac{2}{r} - \frac{1}{a} \right). \tag{2} \end{align}

I realize if the mass of $m$ is negligible compared to $M$ often the $m$ is dropped in $M+m$ but I do not care about that.

I am confused about 2 things: First, what is the difference between (1) and (2) [is the difference related to orbital energy vs. specific orbital energy?]. And second, how do you derive (2)?

You can derive (1) from orbital energy:

$$e = \frac{1}{2}mv^2 - \frac{mGM}{r} = -\frac{mGM}{2a}$$

And I know that specific orbital energy is given by: $$-\frac{mGM}{2a} \cdot \frac{M+m}{mM} = -\frac{G(M+m)}{2a} = - \frac{\mu}{2a}$$

I'm having trouble connecting the dots.

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  • $\begingroup$ Go to the URL you referenced, add a $m$ to the $v^{2}/2$ term near the beginning, then carry it through the calculation - the web page will connect the dots for you. $\endgroup$ – Cinaed Simson Nov 13 at 7:14
  • $\begingroup$ WP. Have you defined your terms in (2)? Look at the earth-moon system. $\endgroup$ – Cosmas Zachos Nov 13 at 14:08

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