## 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
Calculating the derivatives  
\[\tan\frac{f}{2}=\sqrt{\frac{1-\cos f}{1+\cos f}}=\sqrt{\frac{1+e}{1-e}}\tan\frac {E}{2},\]5  
integrating the resulting equations from \(\tilde{y}_{0}\) to \(\tilde{y}_{1}\), and combining all the three equations, the following two equations can be derived for \(\widetilde{J}^{(4)}_{4}(\tilde{y},q)\) and \(\ \widetilde{J}^{(4)}_{3}(\tilde{y},q)\):  
\[\tan\frac{f}{2}=\sqrt{\frac{1-\cos f}{1+\cos f}}=\sqrt{\frac{1+e}{1-e}}\tan\frac {E}{2},\]6  
\[\tan\frac{f}{2}=\sqrt{\frac{1-\cos f}{1+\cos f}}=\sqrt{\frac{1+e}{1-e}}\tan\frac {E}{2},\]7  
\[\tan\frac{f}{2}=\sqrt{\frac{1-\cos f}{1+\cos f}}=\sqrt{\frac{1+e}{1-e}}\tan\frac {E}{2},\]8  
In expressions (105)-(109), the symbol \(\mid_{\tilde{y}=\tilde{y}_{1}}^{\tilde{y}=\tilde{y}_{1}}\), means that the corresponding expression is taken at the value \(\tilde{y}=\tilde{y}_{1}\) of the upper boundary and, from it, the value of the expression at \(\tilde{y}=\tilde{y}_{0}\) (the value at the lower boundary) is subtracted. We should also add to this system of equations the earlier derived equation for \(\widetilde{J}^{(4)}_{2}\):  
\[\tan\frac{f}{2}=\sqrt{\frac{1-\cos f}{1+\cos f}}=\sqrt{\frac{1+e}{1-e}}\tan\frac {E}{2},\]9  
\[\widetilde{T}_{1}=-2i\frac{na}{c}q^{\frac{3}{2}}\int\frac{y^{2}dy}{\sqrt{(1-y^ {2})(1-q^{2}y^{2})}}=-2i\frac{na}{c}q^{\frac{3}{2}}\widetilde{J}_{2}^{(4)}(y; q).\]0The second Equation (109) for \(\widetilde{j}_{3}^{(4)}\) clearly suggests that \(\widetilde{j}_{3}^{(4)}\) is expressed in elementary functions because the first-order elliptic integral \(\widetilde{j}_{1}^{(4)}=\int\frac{\widetilde{y}\widetilde{q}\widetilde{y}}{ \sqrt{(1-\widetilde{y}^{2})(1-q^{2}\widetilde{y}^{2})}}\) can be analytically calculated after performing the Euler substitution:  
\[\widetilde{T}_{1}=-2i\frac{na}{c}q^{\frac{3}{2}}\int\frac{y^{2}dy}{\sqrt{(1-y^ {2})(1-q^{2}y^{2})}}=-2i\frac{na}{c}q^{\frac{3}{2}}\widetilde{J}_{2}^{(4)}(y; q).\]1  
The result is  
\[\widetilde{j}_{1}^{(4)}=-\frac{1}{2q^{3}}\ln\mid\sqrt{(1-\widetilde{y}^{2})(1- q^{2}\widetilde{y}^{2})}-\widetilde{y}^{2}+\frac{(1+q^{2})}{2q^{2}} \begin{array}{c} \widetilde{y}=\widetilde{y}_{1}\\ \widetilde{y}=\widetilde{y}_{0}\end{array} . \tag{112}\]  
Concerning the fourth-order elliptic integral \(\widetilde{j}_{4}^{(4)}\), taking into account the already proved fact that \(\widetilde{j}_{2}^{(4)}\) can be expressed in elementary functions, it would follow that \(\widetilde{j}_{4}^{(4)}\) can also be expressed by elementary functions. However, it is necessary to prove that the zero-order elliptic integral \(\widetilde{j}_{0}^{(4)}\) can be expressed in elementary functions. Further, it will be proved that the integral can be expressed by an analytical formulae, but also, it can be numerically calculated, since the modulus of the elliptic integral is equal to the eccentricity \(e=0.01323881349526\) of the GPS orbit, which is exactly known [13]. The general mathematical theory for the recurrent relations between \(n\)-th order elliptic integrals of any kind and the lower-order elliptic integrals has been developed in the monographs [34; 38; 45]. The Several Parts of the Entire Expression for the Propagation Time for the Case of Space-Oriented Orbits  
Now, we shall collect all the separate terms, which constitute the expression for the propagation time \(T\).