The expression for the acceleration of a near-earth satellite as presented in the IERS Technical note is given by \begin{equation} \label{eq:problemeq} \tag{1} \frac{d^2\mathbf{r}}{dt^2} = \frac{GM_E}{c^2r^3} \left\{\left[2(\beta+\gamma)\frac{GM_E}{r} - \gamma \dot{\mathbf{r}} \cdot \dot{\mathbf{r}} \right] \mathbf{r} + 2(1+\gamma)(\mathbf{r}\cdot\dot{\mathbf{r}})\dot{\mathbf{r}} \right\}. \end{equation}
We are working in the Parametrised Post-Newtonian formalism, hence the dimensionless constants $\beta,\gamma$.
From what I can gather from here and here this expression can be derived from the "Schwarzschild isotropic one-body point mass metric".
Now, I know what the Schwarzschild metric looks like in isotropic coordinates but I can't see where Eq. (\ref{eq:problemeq}) comes from.
Another point to make is that in GR the dimensionless parameters $\beta,\gamma$ are equal to unity and when substituted above gives a well known formula for Schwarzschild precession which is given by \begin{equation} \label{eq:problemeq2} \tag{1} \frac{d^2\mathbf{r}}{dt^2} = \frac{GM_E}{c^2r^3} \left\{\left[4\frac{GM_E}{r} - \dot{\mathbf{r}} \cdot \dot{\mathbf{r}} \right] \mathbf{r} + 4(\mathbf{r}\cdot\dot{\mathbf{r}})\dot{\mathbf{r}} \right\}. \end{equation}
Again, I don't recall this either.
Any suggestions?
