## 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
Similarly, if a clock on an orbit is compared to a clock on the Earth's surface, due to the net effect of time dilation and gravitational redshift, the satellite clock will appear to run fast by 38 us/d [53], which also is considered to be an enormous difference for an atomic clock with precision of a few nanoseconds. So, for this value, a similar number will be obtained for the ratio \(\frac{d\tau}{dT}\), again confirming that this nanosecond (\(10^{-9}\) s) precision of the atomic clock (no matter whether it is on the Earth's surface or on a satellite in a near-Earth orbit) is sufficient for measuring the propagation time \(T\) of the signal. Justification of the Theoretical Approach in This Paper from the Viewpoint of General Relativity Theory and Differential Geometry  
As mentioned in the introduction, the basic approach of the calculation of the propagation time in this paper is based on finding the solution for the propagation time from the null cone equation and parametrizing the space coordinates for both cases, considered as plane elliptical orbits or space-oriented orbits by the orbital coordinates. So, in fact, the light-like geodesics are not used in this approach. In the previous chapter, it was demonstrated that the solution for the propagation time does not give any time, related to the motion of the satellite, but the calculated propagation time is of the order of microseconds (1 us = \(10^{-6}\) s) or nanoseconds (1 ns = \(10^{-9}\) s). This is the first fact in favor of the correctness of using the approach of the null cone equation. The second fact is related to the proper dimension of seconds of the expression for the propagation time for the case of a signal, emitted by a satellite on a plane elliptical orbit. Now we shall present the third argument, related to a theorem, proved in the book on general relativity theory by Fock [19]. This theorem is related to known ones from basic textbooks on differential geometry [24; 26]. Let us first begin with the theorem in [19]  
**Theorem 1**. : _Let \(F=\frac{1}{2}g_{\alpha}\beta\frac{dx}{ds}\ \frac{dx^{\beta}}{ds}\). \(F=0\) is the null cone equation and the extremum of the integral \(s=\int\limits_{s_{1}}^{s_{2}}Lds\) is searched \((L=\sqrt{2F})\); then, the Lagrange equation of motion will give_  
\[\frac{a}{c}T_{1}^{(E_{6})}=-0.0281342485273829[\mathrm{s}],\]4  
_From the last equation, the geodesic equations \((\alpha,\beta,\
u=0,1,2,3)\)_  
\[\frac{a}{c}T_{1}^{(E_{6})}=-0.0281342485273829[\mathrm{s}],\]5  
_can be obtained. Since the constant in (139) can be set up to zero, this would mean that the geodesic Equation (140) is compatible with the null cone equation \(F=\frac{1}{2}g_{\alpha}\frac{dx^{\alpha}}{ds}\ \frac{dx^{\beta}}{ds}=0\)._  
It is important to acquire the proper understanding of this statement: It means that every solution of the null equation (including in the proper parametrization of the differentials,related to the trajectory of motion of the satellite) is also a solution of the light-like geodesic Equation (140). In the \"backward direction\", however, the statement is not true, meaning that there might be other solutions of the geodesic equation, which are not solutions of the null cone equation. The meaning of the geodesic as the shortest path, joining two given points (but only in the local sense, i.e a sufficiently short part of the geodesics), is confirmed also by the following theorem in the monograph by Prasolov [26]. Let us present the exact formulation of this theorem:  
**Theorem 2**. : _Any point \(p\) on a surface \(S\) has a neighborhood \(U\) such that the length of a geodesic \(\gamma\) going from the point \(p\) to some point \(q\) and lying entirely in the neighborhood \(U\) does not exceed the length of any curve \(\alpha\) on \(S\), joining the points \(p\) and \(q\).