## 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?
It should be stressed that \(T\) was found as a solution of the null cone Equation (6) and because of that, it is the time for propagation of the signal and definitely does not represent the time, related to the real motion of the satellite. This statement is also confirmed by the fact that the null cone equation \(F=0\) is proved to be compatible with the geodesic equation [19] for light-like geodesics: \[\int xdy=\int\sqrt{P(y)}dy \tag{37}\]0 (44) where \(\tau\) is the proper time along the light ray and \(\
u,\alpha,\beta=0,1,2,3\). In other words, the zero-length (light-like) geodesics (representing the trajectory of the signal) are determined also by the null cone equation \(g_{\alpha\beta}dx^{\alpha}dx^{\beta}=0\), because the null cone equation is a first integral of the geodesic Equation (44). This point will be clarified in more detail in a separate chapter, since it is confirmed also by a theorem in differential geometry [24; 26]. This was explained also in the introduction of this paper and also in the recent paper [25]. Furthermore, the numerical calculations in the next sections will confirm that the propagation time of the signal is much smaller than the time of motion of the satellite from the initial value \(E_{init}\) of the eccentric anomaly angle to the final value \(E_{fin}\). 5. Now, let \(x,y,z\) be the space coordinates, reached by the signal after it has been emitted. From the null cone Equation (6), after using the approximation \(\beta=\frac{2V}{c^{2}}\ll 1\), it can be obtained that \[\int xdy=\int\sqrt{P(y)}dy \tag{37}\]1 (45) Since \(cdT\) is the infinitesimal distance, travelled by the light signal, the above inequality means that the distance travelled by light is much greater than the Euclidean distance. In other words, the light trajectory signal is a curved one, in accord with the meaning of the Shapiro delay formulae (1) regarding the delay of the signal under the action of the gravitational field. The trigonometric function \(\cos^{2}E\) has the same values for \(E=\alpha\) and \(E=180-\alpha\), so the correspondence eccentric anomaly angle \(E\) and the propagation time \(T\) are not reliable for angles in the second quadrant. The same refers to values of \(E\) in the first and the fourth quadrant since \(\cos E=\cos(360-E)\). 7. The propagation time \(T\) in Equation (41) should be a real-valued expression, since time cannot be complex. This fact is not evident from the beginning of the calculations, but it will be proved in the next sections. Remarkably, this important property will turn out to be valid also for the next case of a signal, emitted and received by satellites on a space-oriented orbit. 8. The approximation \(\beta=\frac{2V}{c^{2}}\ll 1\) with account of the vector \(r\) in the orbital plane is \[\int xdy=\int\sqrt{P(y)}dy \tag{37}\]2 (46) which is fulfilled, because \(\beta=\frac{2G_{\oplus}M_{\oplus}}{c^{2}a}\ll 1\). For the parameters of a \(GPS\) orbit, the constant \(\beta\) can exactly be calculated to be \(0.33\times 10^{-9}\), which justifies the above strong inequality. In the strict mathematical sense and for the given case, the above inequality (46) takes place when \(\cos E\ll\frac{1}{e}-\frac{2G_{\oplus}M_{\oplus}}{c^{2}a}\). This is always fulfilled for the case of the small eccentricity of the GPS orbits (\(e=0.01323881349526\)), because the first term will be a large number of the order \(\frac{1}{13}\times 10^{3}\), while the second term is a very small number of the order \(1.4\times 10^{-8}\). Other values for the parameter \(\beta\) have been obtained in the literature. For example, in the review article by J. Pascual Sanchez [23], it was obtained for the approximate radius of the satellite orbit \(r_{s}=26\),561 [km] that \[\int xdy=\int\sqrt{P(y)}dy \tag{37}\]3 (47) If multiplied by 2 (in order to obtain the constant \(\beta=\frac{2G_{\oplus}M_{\oplus}}{c^{2}a}\)), this will give \(0.334\times 10^{-9}\), which is in full accord (up to the second digit) with the estimate in this paper \(\beta=0.33\times 10^{-9}\).