## 2 A Brief Review of the Shapiro Delay Formulae and the Aims of This Paper
### Abelian Integrals Versus Elliptic Integrals
#### 3.2.2 Is the Integral (29) an Abelian One?
The coefficient \(\frac{a}{c}\) as a ratio of the large semi-major axis of the orbit and the velocity of light \(c=299\),\(792\),\(458\)\([\frac{m}{s}]\) will have a dimension \([m/\frac{m}{s}]=[s]\), as it should be. The second coefficient \(\frac{2G_{\oplus}M_{\oplus}}{c^{3}}\) has a corresponding dimension \([\frac{m^{3}}{g^{2}}:\frac{m^{3}}{g^{3}}]=[s]\), which clearly proves that formulae Equation (41) has the proper dimensions. 2. The formulae, in fact, gives the propagation time \(T\) of the signal, emitted by the satellite at some initial position (given by the eccentric anomaly angle \(E_{init}\)), and the final point (given by \(E_{fin}\)) on the trajectory of the signal. However, there is no real transmission of signals, because the final point \(E_{fin}\) should be given. Let us clarify this moment--if there is a transmission of signals (emission+reception) the point of reception should be determined, depending on the equations for the two null cones in the framework of the approach of the two intersecting null cones, which was developed in the papers [20; 40] for the simplest case of plane elliptic orbit. The method shall be applied for the case of space-oriented orbits in another paper. The integrals (40) and (41) represent curvilinear integrals (depending on the determined-in-advance initial and final points of integration), derived after the application of the null cone equation (in a general form, written as \(F=g_{\alpha}\beta dx^{\alpha}dx^{\beta}=0\)). This propagation time, in fact, can be found from the expression (3) after expressing the differentials \(dT\) and \(dx_{i}\) by means of the differential \(dE\) of the eccentric anomaly angle \(E\). Thus, from (3), after integration, it is obtained for the propagation time \(T\): \[T=\pm\frac{1}{c}\int\frac{1}{\sqrt{-g_{00}}}\sqrt[]{\left(g_{ij}+\frac{g_{0i}g_ {0j}}{-g_{00}}\right)\frac{dx^{i}}{dE}\frac{dx^{j}}{dE}dE}\] \[+\frac{1}{c}\int\left(\frac{g_{0j}}{-g_{00}}\right)\frac{dx^{j}}{dE}dE.\] (42)  
In the same way, the space variables \(x^{i}\) can be parametrized by some other variable. For example, in the next section, by using the celestial transformations (9)-(11) for the case of space-oriented orbits, the true anomaly angle \(f\) will be used and the formulae for \(T\) will be the same as in (42) but with the eccentric anomaly \(E\) replaced by the true anomaly \(f\), i.e., \(E\to f\). This case will be investigated in the next sections. In order to obtain the propagation time \(T\) in the form of the expression (41), the additional assumption \(\beta=\frac{2V}{c^{2}}\ll 1\) has to be taken into account, when transforming expression (42). 3. The expression for the propagation time \(T\) (41) establishes the relation between the eccentric anomaly angle \(E\) and the propagation time \(T-E\Longrightarrow T\). Since \(T\) will be expressed through elliptic integrals and their analytical solutions will be complicated, it might turn out to be impossible to establish the correspondence in the other direction \(T\Rightarrow E\), as was in the case of the \"dual\" correspondence \(t_{cd}\Leftrightarrow E\). The meaning of the correspondence \(E\Longrightarrow T\) is the following: If some initial and final eccentric anomaly angles \(E_{init}\) and \(E_{fin}\) for the position of the satellite are given, then the propagation time of the signal, corresponding to these two positions of the satellite, can be found from the definite integral \[\Delta T=T_{fin}-T_{init}=\frac{a}{c}\int_{E_{init}}^{E_{fin}}\sqrt{1-e^{2} \cos^{2}E}\;dE\] \[-\frac{2G_{\oplus}M_{\oplus}}{c^{3}}\int_{E_{init}}^{E_{fin}}\sqrt{\frac{1+e \cos E}{1-e\cos E}}dE.\] (43) This expression establishes the correspondence \((E_{init},E_{fin})\Longrightarrow T\), accounting for the curved signal trajectory and also for the fact that there is no real transmission of the signal, in the sense that the signal is not intercepted by another (second) satellite. 4.