## 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
In fact, it is about 108.8 m, which is \(0.000004098763343794755826290507\sim 0.409\times 10^{-5}\) times smaller than the distance, given by the number (132)  
\[\frac{a}{c}T_{1}^{(E_{6})}=-0.0281342485273829[\mathrm{s}],\]1  
travelled by the light or radio signal for the calculated propagation time of \(0.0281342485273829\) s.  
Comparison of the Result in Formulae (134) with a Result in Other Papers. Approximations in the Calculations of the Shapiro Delay Term  
The result of 8.98 [mm] in (134) was obtained by substracting the propagation distance, obtained from the first term in the formula (41), initially, by taking into account the integrated term, and afterwards, by calculating the numerical value only for the under-integral expression (see also (128)). However, in the known review article by Ashby [51; 52], it was proved that at the level of several millimeters, spatial curvature effects have to be considered, which is evident from the simple equality  
\[\frac{a}{c}T_{1}^{(E_{6})}=-0.0281342485273829[\mathrm{s}],\]2  
In this paper, the second term was estimated to be \(4.43\times\ln(4.2)\approx 6.3\) mm. It might be concluded, yet, that in the first term in (134), spatial curvature effects are important and have to be included. In the Shapiro formulae, the first term is the Euclidean distance, divided by the light velocity. From a formal point of view, it might be believed that the first terms \(\frac{a}{c}\sqrt{1-e^{2}\cos^{2}E}\) in (128) and \(\frac{d}{c}\int\limits_{0}^{E_{(0)}}\sqrt{1-e^{2}\cos^{2}E}dE\) in (125) should correspond to the first term \(\frac{R}{c}\) in the Shapiro formulae. Therefore, since it might be suspected that curvature effects may appear yet in these formulas, it is not quite correct to consider any correspondence \(\frac{a}{c}\sqrt{1-e^{2}\cos^{2}E}\Rightarrow\frac{R}{c}\) or \(\frac{a}{c}\int\limits_{0}^{E_{(0)}}\sqrt{1-e^{2}\cos^{2}E}dE\Rightarrow\frac {R}{c}\). In view of this, in a recent paper [25], a new modification of the Shapiro formulae has been proposed, where the first term \(\frac{R}{c}\) is replaced by the term \(\frac{1}{c}\int\limits_{path}\sqrt{\left(\dot{x}\right)^{2}+\left(\dot{y} \right)^{2}+\left(\dot{z}\right)^{2}}ds\). In this term, \(s\) is some parameter, which parametrizes some curve, for example, the space curve, determining the space coordinates on the satellite orbit. Comparison of the Propagation Time with the Atomic Time of the Atomic Clocks of the Satellites in the Near-Earth Space--Consistency of the Results  
Now, let us point out one another numerical fact, which is interesting, because it confirms the consistency of the approach of calculating the propagation time on the base of using the parametrization of the satellite orbit. In the book [53] it was pointed out that at an altitude of 20,184 km, due to the difference in the gravitational potential, the atomic time of the satellite clock runs faster by 45 \(\upmu\)s/d (microseconds per day, 1 \(\upmu\)s = 10\({}^{-6}\) s). Consequently, for one second, the atomic time will run faster by \(0.5208333\times 10^{-9}\) s. In the preceding sections, it was proved that the dominant part in the propagation time \(dT\) (i.e., the part without the Shapiro delay term, which is very small) is \(dT=0.0281341332790419\ [\text{s}]\), given by the number (125) in Section 8.2. Corresponding to this propagation time interval \(dT\), the atomic time is \(dT=0.0146531934\times 10^{-9}\) s and can easily be found from 45 \(\upmu\)s/d(keeping in mind how many seconds are contained in 24 h, i.e., 1 day). The ratio of the two time intervals is equal to  
\[\frac{a}{c}T_{1}^{(E_{6})}=-0.0281342485273829[\mathrm{s}],\]3  
The very small atomic time interval, compared to the propagation interval, means that the atomic time can serve as a standard for measuring the propagation time, because it will be able to detect changes even at the nanosecond level. It should be noted that this value is of the order of \(10^{-9}\), but it is a little greater than the value \(\beta=\frac{2V}{c^{2}}=0.334\times 10^{-9}\).