I am not sure whether I have understood the relativistic range correction which plays an important role for precise Satellite Laser Ranging (SLR) correctly.

The basic observation equations for this kind of measurement looks like \begin{equation} d = \frac{1}{2}c \Delta t + \Delta d_0 + \cdots + \Delta d_r \end{equation} with $d$ being the measured distance, $c$ the speed of light, $\Delta t$ the measured flight time of the laser impulse and $\Delta d_x$ various correction terms. I haven't found a particular formula for the relativistic range correction $\Delta d_r$ but sources like the IERS conventions [0, p.164] provide formulas for the coordinate time delay of propagation: \begin{equation} t_2 - t_1 = \frac{|\vec{x}_2(t_2) - \vec{x}_1(t_1)|}{c} + \sum_J \frac{2 G M_J}{c^3} \ln \left(\frac{r_{J1} + r_{J2} + \rho}{r_{J1} + r_{J2} - \rho} \right) \end{equation} For the definition of the specific terms I refer to [0]. The problem is this is compared to the other correction not in the space but time domain.

Is the transformation of this error term to the space domain thus that simple as saying \begin{equation} \Delta d_r = (t_2 - t_1) c \enspace \mathrm{?} \end{equation}

[0] https://www.iers.org/IERS/EN/DataProducts/Conventions/conventions.html

And specifically Equation 11.17 in section 11.2: http://iers-conventions.obspm.fr/2010/2010_official/chapter11/tn36_c11.pdf



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