## 6 Propagation Time for a Signal, Emitted and Intercepted by Satellites on a Space-Oriented Orbit
### An Earlier Example for an Elliptic Integral, Expressed Also Analytically
These terms are (83), earlier proved to be a real-valued integral  
\[\widetilde{T}_{1}=\frac{2 na(1+e)}{cq\sqrt{1-e^{2}}}\int\frac{\widetilde{y}^{2} d\widetilde{y}}{\sqrt{(1-\widetilde{y}^{2})\left(\frac{\widetilde{y}^{2}}{q^{2 }}-1\right)}},\]  
the expression (98) for \(T_{2}^{(1)}\)  
\[T_{2}^{(1)}=-\frac{2G_{\oplus}M_{\oplus}}{c^{3}}\cdot\frac{n}{(1-e^{2})^{\frac {3}{2}}}\widetilde{T}_{1},\]  
expression (102) for \(T_{2}^{(2)}\)  
\[T_{2}^{(2)}=-inq^{\frac{5}{2}}\frac{2G_{\oplus}M_{\oplus}}{c^{3}}(1+e^{2}) \widetilde{j}_{2}^{(4)}(\widetilde{y},q)\]  
\[+inq^{\frac{3}{2}}\frac{2G_{\oplus}M_{\oplus}}{c^{3}}\frac{(1+e^{2})}{(1-e^{2} )}\widetilde{j}_{4}^{(4)}(\widetilde{y},q),\]  
expression (103) for \(\widetilde{j}_{2}^{(4)}(\widetilde{y},q)\)  
\[\widetilde{j}_{2}^{(4)}(\widetilde{y},q)=\int\frac{\widetilde{y}^{2}d\widetilde {y}}{\sqrt{(1-\widetilde{y}^{2})(1-q^{2}\widetilde{y}^{2})}}\]  
\[=\frac{q^{3}}{i}\int\frac{\widetilde{y}^{2}d\widetilde{y}}{\sqrt{(\widetilde{ y}^{2}-1)(1-q^{2}\widetilde{y}^{2})}}=\frac{q^{3}}{i}\,\widetilde{j}_{2}^{(4)}( \widetilde{y},q),\]expression (104) for \(\,\,\hat{J}_{4}^{(4)}(\widetilde{y},q)\)  
\[\widetilde{J}_{4}^{(4)}(\widetilde{y},q)=\frac{q^{5}}{i}\int\frac{\widehat{y}^{4 }d\widehat{y}}{\sqrt{(\widehat{y}^{2}-1)(1-q^{2}\widehat{y}^{2})}}.\]  
Note that this expression for \(\widetilde{J}_{4}^{(4)}(\widetilde{y},q)\) resembles the previous expression \(\widetilde{J}_{2}^{(4)}(\widetilde{y},q)\), since both of them have one and the same denominators \(\sqrt{(\widehat{y}^{2}-1)(1-q^{2}\widehat{y}^{2})}\), which determine whether both formulaes are imaginary- or real- valued. Earlier, by formulaes (82)-(86) and (90)-(92), it was proved that \(\widetilde{J}_{2}^{(4)}(\widetilde{y},q)\) is an imaginary-valued expression. Consequently, it can be concluded that expression (104) for \(\widetilde{J}_{4}^{(4)}(\widetilde{y},q)\) is also imaginary-valued. This conclusion can be made also from the chain of elliptic integrals, establishing the relation between elliptic integrals of zero, second and fourth order (108):  
\[\widetilde{J}_{4}^{(4)}(\widetilde{y},q)=\frac{1}{3q^{2}}\bigg{[}\widetilde{y }\sqrt{  
In the following numerical calculations, we shall use only the first three parameters \(a,e\) and \(M\), but in other publications, the approach will be extended and it will be required to find the propagation time dependent on the full set of six Kepler parameters \((a,e,M,I,\Omega,\omega)\). Instead of \(M\), either the eccentric anomaly angle \(E\) or the true anomaly angle \(f\) will be used. In fact, all the Kepler parameters might be used for finding the propagation time of the signal, and this is a problem to be solved in future.  
For finding the propagation time for the case of a signal, emitted by a satellite, moving along a plane elliptical orbit with an eccentricity \(e\), most important is to find the eccentric anomaly angle \(E\) from the Kepler equation \(M=E-e\sin E\). The calculation will be based on formulaes (2.54) and (2.55) in Ch. 2 of the monograph [48].  
The iterative sequence of the formulaes is  
\[E_{(i+1)}=M+e\sin E_{(i)}\quad,\quad i=0,1,2, \tag{114}\]  
where, for the first approximation, it is assumed \(E_{0}=M=-0.3134513508155\) [rad].