## 6 Propagation Time for a Signal, Emitted and Intercepted by Satellites on a Space-Oriented Orbit
### Numerical Calculation of the First \(O(\frac{1}{c})\) Correction in the Propagation Time  
The corresponding elliptic integral (59) of the second kind for the eccentric anomaly approximation \(E_{(3)}\) is  
\[T_{1}^{(E_{3})}=\int\limits_{0}^{E_{(3)}}\sqrt{1-e^{2}\cos^{2}E}dE=-0.317557268 125933936045. \tag{122}\]  
The second Shapiro delay term contains ten zeroes after the decimal dot, so the overall result of the calculation of both terms in (127) is changed by the Shapiro term at and after the nanosecond level (\(1\) ns = \(10^{-9}\) s). The second Shapiro delay term in (128) is equal to \(2.99618\times 10^{-9}[\mathrm{s}]\approx 2.99618[\mathrm{ns}]\), which is smaller than the timing error for a receiver at the geometric center of a tetrahedron satellite configuration, measured to be 10 [ns] [49]. Here it should be clarified that the calculation in (128) and in (129) is of an approximate value because one should take the numerical values for \(T\) after performing the integration of the elliptic integral. For example, from formulae (125)  
\[\frac{a}{c}T_{1}^{(E_{6})}=-0.0281342485273829[\mathrm{s}],\]  
obtained after integrating the first term in (127), it is seen that the first number \(0.0885884561709019072629880[\mathrm{s}]\) in (128) has been diminished to the number \(0.0281342485273829[\mathrm{s}]\) in (125), which represents a substantial change. In the literature (see [50]) time transfer has been discussed between a geostationary satellite and a station on the rotating Earth. This time transfer amounts to \(80[\mathrm{ps}]=0.008[\mathrm{ns}]\) for the \(c^{-3}\) terms, accounting for the Shapiro time delay (in [50] called a gravitational time delay). However, two-way time transfer between two stations at Braunschweig (Germany) and FTZ (Darmstadt, Germany) can be not so accurate and can amount to \(300[\mathrm{ps}]=0,3[\mathrm{ns}]\), the uncertainties due to the indeterminacies in the position of the emitter (the station) and the receiver (the satellite), or vice versa. From the numerical calculation for the above case of inter-satellite communications, the relativistic time transfer (i.e., the propagation time of the signal) will be of another magnitude, depending also on the given final value for \(E^{(6)}\). For the moment, we do not have a numerical estimation of the second integral term in (41), consequently we take the whole under-integral expression. It should be noted also that in the previous calculations of the iterations \(E_{(3)}\) and \(E_{(6)}\) of the eccentric anomaly angle, it was taken with a negative sign, due to the accepted convention. After obtaining the result of the calculation of \(T_{1}^{(E_{3})}\) and \(T_{1}^{(E_{6})}\) from the integrals (122) and (123), they were with a negative sign. This means that after the integrating of the first term in (127), there will be a negative sign before the first number in (128) and the negative sign of the second term in (129) remains. This is so because after calculating \(\frac{2 GM}{c^{3}}\sqrt{\frac{1+c\cos E_{(3)}}{1-c\cos E_{(3)}}}\), the obtained number is positive. Thus, the two numbers in (128) should be added, giving a total number of \(-0.0885884562008637167248048\) in (129), greater than the first number in (128) after the ninth digit after the decimal dot. Although approximate, this result, based on the simple application of the celestial mechanics approach in the calculation of the propagation time, is interesting, if compared with the calculations, based on the formulae (1) \(T=\frac{R_{AB}}{c}+2\frac{G_{c}-M_{E}}{c^{3}}\ln\left(\frac{r_{A}+r_{B}+R_{AB} }{r_{A}+r_{B}-R_{AB}}\right)\). From the literature, it can be seen that relativistic effects on light propagation from satellite laser ranging (SLR) data are measured with an accuracy of 1 \(\upmu\)m (1 micrometer), which corresponds to 0.01 ps (see also paper [50]).