## 2 A Brief Review of the Shapiro Delay Formulae and the Aims of This Paper
### Standard Shapiro Delay Formulae, Derived from the Null Cone Equation
#### 2.2.1 Case A--Plane Elliptical Coordinates of the Satellite
This assumption is consistent with the experimental data and the calculation of the numerical quantity \(\beta=\frac{2G_{\infty}M_{\odot}}{c^{2}}=0.334\times 10^{-9}\ll 1\) by some authors [23] and also with the relativistic effects of the order of \(c^{-2}\). Further in the text, this accuracy will be denoted as \(O(c^{-2})\). This is a consequence of the experimental fact that a GPS constellation has to keep time to an accuracy of about four nanoseconds per day (\(4\) ns\(/\)d = \(4\times 10^{-9}\) s\(/\)d) [23], which is compatible with the fractional time stability of \(5\times 10^{-14}\), maintained by atomic clocks. Naturally, the fractional stability has been constantly improving in the past and present years with the perspective of becoming smaller and smaller. 2.3 Accuracy of the Relativistic Effects and the Possibility of Finding the Solution of the Null Cone Equation  
The last fact is mentioned due to the following interesting consistency between the experimentally measured accuracy of relativistic effects and the found solution of the null cone equation (6) for the propagation time \(T\) in terms of the eccentric anomaly angle \(E\). It will be proved in Section 3.4 that it is possible for this solution to be expressed analytically only under the assumption \(\beta=\frac{2V}{c^{2}}\ll 1\). The necessity to implement such an assumption, physically expressing the fact of the weak gravitational field, is related also to the fulfillment of the equivalence principle only for weak gravitational fields and slow motions--an important topic, discussed in the monograph by Fock [19]. 2.4 Second Aim of the Paper--Theoretical Justification of the Approach from the Viewpoint of General Relativity Theory and Differential Geometry  
The second main purpose of the paper is to justify the theoretical approach and to provide the reasoning for choosing as parametrization variables in the null cone Equation (6) the coordinates (4), parametrizing the satellites trajectory along the orbit in terms only of three variables, \(a\), \(e\) and \(E\), among which only the eccentric anomaly angle \(E\) is the dynamical parameter. The need to justify the theoretical approach will refer not only to the simplified case of the Kepler coordinates (4), but also to the next (second) case of a space-oriented orbit, when the disposition of the orbit in space will be described by six Kepler parameters. They will be determined in the next section. From a conceptual point of view, this is a very non-trivial problem, since the presence of the Kepler coordinates (4) in the null cone Equation (6) may raise the question of whether it is correct to use coordinates, not related to the trajectory of the signal, but to the motion of the satellite. Following such wrong logic, then, it might turn out that the calculated time \(T\) is not a propagation time of a signal, but the time of motion for the satellite. Fortunately, this line of reasoning is not correct, because the essential fact is the null cone Equation (6) and not the parametrization of the space coordinates \(x,y\) or \(x,y,z\). This will be confirmed by (1) a theorem in the monograph by Fock [19], which, further, will be formulated in detail [22], and (2) a theorem from differential geometry [24] with a meaning that is very close to the meaning in the monograph by Fock (cited also in the papers [22; 25]). In fact, here, the problem can be stated in a different way, since the propagation time \(T\) will be found as a solution from the null cone equation (remember also that the Shapiro delay formulae (1) is also derived from the null cone equation), but it should be derived from the null geodesics. So, the question is as follows: Should the null geodesic equation (which, as known, is difficult to solve exactly) be used or one may use instead only the null cone equation?