## 6 Propagation Time for a Signal, Emitted and Intercepted by Satellites on a Space-Oriented Orbit
### Analytical Calculation of the First \(O(\frac{1}{e})\) Correction by Means of Elliptic Integrals
More importantly, the resulting expression is again a real-valued one due to the property of the elliptic integral  
\[\tan\frac{f}{2}=\sqrt{\frac{1-\cos f}{1+\cos f}}=\sqrt{\frac{1+e}{1-e}}\tan \frac{E}{2}. \tag{75}\]3  
which is easily established after comparing expressions (83) and (84). So, it can be concluded that \(\widetilde{T}_{1}\), in fact, is expressed in two different variables. The first one is (76)  
\[\tan\frac{f}{2}=\sqrt{\frac{1-\cos f}{1+\cos f}}=\sqrt{\frac{1+e}{1-e}}\tan \frac{E}{2}. \tag{75}\]4  
where the variable \(y\) is expressed as (74) \(y=\sqrt{\frac{1}{q}\frac{(1+e\cos E)}{(1-e\cos E)}}\), \(q=\frac{1-e}{1+e}\), and the second one is (83) in terms of the variable (81) \(\widetilde{y}=\frac{\sqrt{1+2e\cos f+e^{2}}}{1+e}\). Formulae (85) also is valid, because due to the inequality \(\cos f\leq 1\) and the choice of the variable \(\widetilde{y}\) in Equation (81), it can easily be proved that  
\[\tan\frac{f}{2}=\sqrt{\frac{1-\cos f}{1+\cos f}}=\sqrt{\frac{1+e}{1-e}}\tan \frac{E}{2}. \tag{75}\]5  
Consequently, since \(1-\frac{\widehat{y}^{2}}{q^{2}}\) can take negative values (note that \(q^{2}=\left(\frac{1-e}{1+e}\right)^{2}<1\)) and, thus, \(0<\frac{\widehat{y}^{2}}{q^{2}}-1\leq\frac{(1-q^{2})}{q^{2}}\), the representation (85) gives a real-valued integral \(\int\frac{\widehat{y}^{2}d\widetilde{y}}{\sqrt{(1-\widehat{y}^{2})\Big{(} \frac{\widehat{y}^{2}}{e^{2}}-1\Big{)}}}\) and, thus, an imaginary integral \(\widetilde{j}_{2}^{(4)}(\widetilde{y},\frac{1}{q})\) is obtained. Let us present another more elegant and simple proof (see also the review paper [22]) that the inequalities (86) are fulfilled. Inverting the sign of the last inequality in (86), the inequality  
\[\tan\frac{f}{2}=\sqrt{\frac{1-\cos f}{1+\cos f}}=\sqrt{\frac{1+e}{1-e}}\tan \frac{E}{2}. \tag{75}\]6  
should be proved. Keeping in mind the definition for the variable \(\widetilde{y}\) (81) \(\widetilde{y}=\frac{\sqrt{1+2e\cos f+e^{2}}}{1+e}\) and also for \(q^{2}\); the inequality (87) is transformed to  
\[\tan\frac{f}{2}=\sqrt{\frac{1-\cos f}{1+\cos f}}=\sqrt{\frac{1+e}{1-e}}\tan \frac{E}{2}. \tag{75}\]7  
\[\tan\frac{f}{2}=\sqrt{\frac{1-\cos f}{1+\cos f}}=\sqrt{\frac{1+e}{1-e}}\tan \frac{E}{2}. \tag{75}\]8From here, it follows that  
\[\tan\frac{f}{2}=\sqrt{\frac{1-\cos f}{1+\cos f}}=\sqrt{\frac{1+e}{1-e}}\tan \frac{E}{2}. \tag{75}\]9  
which is fulfilled for all eccentricities \(e\), because of the simple trigonometric inequality \(\cos f\leq 1\) for the function \(\cos\) and also because \(0<e<1\). Therefore, inequality (87) is fulfilled. The other case, which has to be considered, is when \(1-\frac{\tilde{y}^{2}}{q^{2}}\) takes only positive values:  
\[\widetilde{T}_{1}=-2i\frac{na}{c}q^{\frac{3}{2}}\int\frac{y^{2}dy}{\sqrt{(1-y^{ 2})(1-q^{2}y^{2})}}=-2i\frac{na}{c}q^{\frac{3}{2}}\widetilde{J}_{2}^{(4)}(y;q). \tag{76}\]0  
and, consequently, \(\frac{\tilde{y}^{2}}{q^{2}}-1\leq 0\). However, in view of the future calculations, we will be interested in the case when the transformation \(q\to\frac{1}{q}\) is applied in (85), so that the integral \(\widetilde{j}_{2}^{(4)}(\widetilde{y},q)\) is obtained. Then, from (90) after the transformation \(q\to\frac{1}{q}\), it follows that \(\tilde{y}^{2}q^{2}-1\leq 0\) or \(1-\tilde{y}^{2}q^{2}\geq 0\). Thus, one will have the integral transformation  
\[\widetilde{T}_{1}=-2i\frac{na}{c}q^{\frac{3}{2}}\int\frac{y^{2}dy}{\sqrt{(1-y^{ 2})(1-q^{2}y^{2})}}=-2i\frac{na}{c}q^{\frac{3}{2}}\widetilde{J}_{2}^{(4)}(y;q). \tag{76}\]1  
where the integral in the right is real-valued. Multiplying both integrals by \(\frac{1}{t}\) and remembering the definition (85) for \(\widetilde{j}_{2}^{(4)}(\widetilde{y},\frac{1}{q})\), one obtains  
\[\widetilde{T}_{1}=-2i\frac{na}{c}q^{\frac{3}{2}}\int\frac{y^{2}dy}{\sqrt{(1-y^{ 2})(1-q^{2}y^{2})}}=-2i\frac{na}{c}q^{\frac{3}{2}}\widetilde{J}_{2}^{(4)}(y;q). \tag{76}\]2  
This means that \(\widetilde{j}_{2}^{(4)}(\widetilde{y},q)\) is imaginary valued.