Equations of Motion for the $N$-body Problem in the Astrocentric Frame

Question

I’ve been studying Tremaine's Dynamics of Planetary Systems and have hit a roadblock.

In Chapter 4, titled 'The $N$-body Problem,' there is a derivation of the $N$-body equations of motion, which can be expressed as follows:

$$ \ddot{\vec{r}}_{j} = \sum_{k = 0 \\ k \neq j}^{N-1} \frac{Gm_{k} \left(\vec{r}_{k} - \vec{r}_{j} \right)}{|\vec{r}_{j} - \vec{r}_{k}|^{3}} \, , j = 0, ..., N-1. $$

To rewrite these equations of motion in the astrocentric frame, we introduce $\vec{r}^{\, \star}_{m} = \vec{r}_{m} - \vec{r}_{0}$. Substituting this into the equation above allows us to express it as follows: $$ \ddot{\vec{r}}^{\, \star}_{j} = \sum_{k = 0 \\ k \neq j}^{N-1} \frac{Gm_{k} \left(\vec{r}^{\, \star}_{k} - \vec{r}^{\, \star}_{j} \right)}{|\vec{r}^{\, \star}_{j} - \vec{r}^{\, \star}_{k}|^{3}} - \sum_{k = 1}^{N-1} \frac{Gm_{k} \vec{r}^{\, \star}_{k}}{|\vec{r}^{\, \star}_{k}|^{3}} \\ \iff \ddot{\vec{r}}^{\, \star}_{j} = - \frac{Gm_{0} \vec{r}^{\, \star}_{j}}{|\vec{r}^{\, \star}_{j}|^{3}} + \sum_{k = 1 \\ k \neq j}^{N-1} Gm_{k}\left[\frac{\left(\vec{r}^{\, \star}_{k} - \vec{r}^{\, \star}_{j} \right)}{|\vec{r}^{\, \star}_{j} - \vec{r}^{\, \star}_{k}|^{3}} - \frac{ \vec{r}^{\, \star}_{k}}{|\vec{r}^{\, \star}_{k}|^{3}} \right] \, , j = 0, ..., N-1. $$

However, Equation (4.5) in Tremaine's Dynamics of Planetary Systems includes an extra factor of $m_{j}$ in the first term. Why is that?

$$ \ddot{\vec{r}}^{\, \star}_{j} = - \frac{G\left(m_{0} + m_{j}\right)\vec{r}^{\, \star}_{j}}{|\vec{r}^{\, \star}_{j}|^{3}} + \sum_{k = 1 \\ k \neq j}^{N-1} Gm_{k}\left[\frac{\left(\vec{r}^{\, \star}_{k} - \vec{r}^{\, \star}_{j} \right)}{|\vec{r}^{\, \star}_{j} - \vec{r}^{\, \star}_{k}|^{3}} - \frac{ \vec{r}^{\, \star}_{k}}{|\vec{r}^{\, \star}_{k}|^{3}} \right] \, , j = 0, ..., N-1. $$

Answer 1

I'll be quoting two of the equations in the question and addressing them in turn.

$$ \ddot{\vec{r}}^{\, \star}_{j} = \sum_{k = 0 \\ k \neq j}^{N-1} \frac{Gm_{k} \left(\vec{r}^{\, \star}_{k} - \vec{r}^{\, \star}_{j} \right)}{|\vec{r}^{\, \star}_{j} - \vec{r}^{\, \star}_{k}|^{3}} - \sum_{k = 1}^{N-1} \frac{Gm_{k} \vec{r}^{\, \star}_{k}}{|\vec{r}^{\, \star}_{k}|^{3}} \, , j = 0, ..., N-1\tag{1} $$

That is correct. Note that $\ddot{\vec r}^{\,\star}_0$ is tautologically zero from equation (1), so we might as well exclude $j=0$ as a special case (which is what I will do later on).

$$\iff \ddot{\vec{r}}^{\, \star}_{j} = - \frac{Gm_{0} \vec{r}^{\, \star}_{j}}{|\vec{r}^{\, \star}_{j}|^{3}} + \sum_{k = 1 \\ k \neq j}^{N-1} Gm_{k}\left[\frac{\left(\vec{r}^{\, \star}_{k} - \vec{r}^{\, \star}_{j} \right)}{|\vec{r}^{\, \star}_{j} - \vec{r}^{\, \star}_{k}|^{3}} - \frac{ \vec{r}^{\, \star}_{k}}{|\vec{r}^{\, \star}_{k}|^{3}} \right] \, , j = 0, ..., N-1 \tag{2} $$

That is not correct, for two reasons. You have included $j=0$ (your equation (2) results in $\frac00$ for $j=0$), and you have combined sums incorrectly.

Going back to equation (1), note that the second sum in this equation includes $k=j$ for non-zero $j$. We need to single out the $k=j$ element of the second sum, resulting in

$$ \ddot{\vec{r}}^{\, \star}_{j} = - \frac{Gm_{j} \vec{r}^{\, \star}_{j}}{|\vec{r}^{\, \star}_{j}|^{3}} + \sum_{k = 0 \\ k \neq j}^{N-1} \frac{Gm_{k} \left(\vec{r}^{\, \star}_{k} - \vec{r}^{\, \star}_{j} \right)}{|\vec{r}^{\, \star}_{j} - \vec{r}^{\, \star}_{k}|^{3}} - \sum_{k = 1 \\ k \neq j}^{N-1} \frac{Gm_{k} \vec{r}^{\, \star}_{k}}{|\vec{r}^{\, \star}_{k}|^{3}} \, , j = 1, ..., N-1 \tag{3} $$

In order to combine the two sums, the $k=0$ element in the first sum in equation (3) also needs to be treated as a special term, resulting in

$$ \ddot{\vec{r}}^{\, \star}_{j} = - \frac{G(m_{0}+m_j)\, \vec{r}^{\, \star}_{j}}{|\vec{r}^{\, \star}_{j}|^{3}} + \sum_{k = 1 \\ k \neq j}^{N-1} \frac{Gm_{k} \left(\vec{r}^{\, \star}_{k} - \vec{r}^{\, \star}_{j} \right)}{|\vec{r}^{\, \star}_{j} - \vec{r}^{\, \star}_{k}|^{3}} - \sum_{k = 1 \\ k \neq j}^{N-1} \frac{Gm_{k} \vec{r}^{\, \star}_{k}}{|\vec{r}^{\, \star}_{k}|^{3}} \, , j = 1, ..., N-1\tag{4} $$ Now the two sums can be combined, resulting in $$ \ddot{\vec{r}}^{\, \star}_{j} = - \frac{G(m_{0}+m_j)\, \vec{r}^{\, \star}_{j}}{|\vec{r}^{\, \star}_{j}|^{3}} + \sum_{k = 1 \\ k \neq j}^{N-1} Gm_{k} \left(\frac{\left(\vec{r}^{\, \star}_{k} - \vec{r}^{\, \star}_{j} \right)}{|\vec{r}^{\, \star}_{j} - \vec{r}^{\, \star}_{k}|^{3}} - \frac{\vec{r}^{\, \star}_{k}}{|\vec{r}^{\, \star}_{k}|^{3}}\right) \, , j = 1, ..., N-1\tag{5} $$ This is equation (4.5) in Tremaine's Dynamics of Planetary Systems. Note that equations (3) to (5) exclude $j=0$. That's okay because we already know that $\ddot{\vec r}^{\,\star}_0$ is tautologically zero from equation (1).