## 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  
Let us calculate the integral (68) \(\widetilde{T}_{1}=\frac{na}{c\sqrt{1-e^{2}}}\int\sqrt{1+e^{2}+2e\cos fdf}\) by making use of the substitution  
\[y=\sqrt{\frac{1}{q}\frac{(1+e\cos E)}{(1-e\cos E)}}\;,\;q=\frac{1-e}{1+e} \tag{74}\]  
and also of the well-known relation from celestial mechanics between the eccentric anomaly angle \(E\) and the true anomaly angle \(f\)[18]:  
\[\tan\frac{f}{2}=\sqrt{\frac{1-\cos f}{1+\cos f}}=\sqrt{\frac{1+e}{1-e}}\tan \frac{E}{2}. \tag{75}\]Then, we can write the integral \(\widetilde{T}_{1}\) in the form of an elliptic integral of the second order and of the first kind in the Legendre form  
\[\widetilde{T}_{1}=-2i\frac{na}{c}q^{\frac{3}{2}}\int\frac{y^{2}dy}{\sqrt{(1-y^{ 2})(1-q^{2}y^{2})}}=-2i\frac{na}{c}q^{\frac{3}{2}}\widetilde{J}_{2}^{(4)}(y;q). \tag{76}\]  
Thus, the integral (68) is very interesting from a mathematical point of view, because it becomes evident that it is an elliptic one after performing the transformations (74) and (75). The integral \(\widetilde{J}_{2}^{(4)}(y;q)\) in (76) can also be represented as  
\[\widetilde{J}_{2}^{(4)}(y;q)=\int\frac{y^{2}dy}{\sqrt{(1-y^{2})(1-q^{2}y^{2})}} \tag{77}\]  
\[=-\frac{1}{q^{2}}\int\frac{(1-q^{2}y^{2})dy}{\sqrt{(1-y^{2})(1-q^{2}y^{2})}}+ \frac{1}{q^{2}}\int\frac{dy}{\sqrt{(1-y^{2})(1-q^{2}y^{2})}} \tag{78}\]  
\[=-\frac{1}{q^{2}}\int\sqrt{1-q^{2}\sin^{2}\varphi}d\varphi+\frac{1}{q^{2}} \int\frac{dy}{\sqrt{(1-y^{2})(1-q^{2}y^{2})}}. \tag{79}\]  
The first integral in Equation (79) is an elliptic integral of the second kind (denoted usually by \(E(\varphi)=\int\sqrt{1-q^{2}\sin^{2}\varphi}d\varphi\), where \(x=\sin\varphi\)). So we obtain a relation between the second-order elliptic integral \(\widetilde{J}_{2}^{(4)}(y;q)\) of the first kind in the Legendre form, the zero-order elliptic integral of the first kind in the Legendre form \(\widetilde{J}_{0}^{(4)}(y;q)\) (the second integral in (79)) and the elliptic integral \(E(\varphi)\) (see the monograph [34]):  
\[\widetilde{J}_{2}^{(4)}(y;q)=\int\frac{y^{2}dy}{\sqrt{(1-y^{2})(1-q^{2}y^{2})} }=-\frac{1}{q^{2}}E(\varphi)+\frac{1}{q^{2}}\widetilde{J}_{0}^{(4)}(y;q). \tag{80}\]  
Since, in the preceding section, an analytical expression was obtained for \(\widetilde{T}_{1}\) without the use of any elliptic functions, the obtained result in Equation (79) means that for the investigated case, elliptic integrals of second order can be expressed through elementary functions. Second Analytical Calculation of the First Time \(O(\frac{1}{c})\) Correction in Terms of Second-Order Elliptic Integrals and Proof of the Real-Valuedness of the \(O(\frac{1}{c})\) Correction  
This second calculation will not make any use of the eccentric anomaly angle variable \(E\). The main purpose of this second analytical calculation will be prove the real-valuedness of the expression (76)  
\[\widetilde{T}_{1}=-2i\frac{na}{c}q^{\frac{3}{2}}\int\frac{y^{2}dy}{\sqrt{(1-y ^{2})(1-q^{2}y^{2})}}=-2i\frac{na}{c}q^{\frac{3}{2}}\widetilde{J}_{2}^{(4)}(y, q),\]  
which means that the second-order elliptic integral in the Legendre form should be imaginary-valued. Let us apply the variable transformation  
\[\widetilde{y}=\frac{\sqrt{1+2e\cos f+e^{2}}}{1+e} \tag{81}\]  
in expression (68) for \(\widetilde{T}_{1}\), after which the integral \(\widetilde{T}_{1}\) acquires the form  
\[\widetilde{T}_{1}=\frac{na}{c\sqrt{1-e^{2}}}\int\sqrt{1+e^{2}+2e\cos f}df \tag{82}\]\[\tan\frac{f}{2}=\sqrt{\frac{1-\cos f}{1+\cos f}}=\sqrt{\frac{1+e}{1-e}}\tan \frac{E}{2}. \tag{75}\]0  
\[\tan\frac{f}{2}=\sqrt{\frac{1-\cos f}{1+\cos f}}=\sqrt{\frac{1+e}{1-e}}\tan \frac{E}{2}. \tag{75}\]1  
\[\tan\frac{f}{2}=\sqrt{\frac{1-\cos f}{1+\cos f}}=\sqrt{\frac{1+e}{1-e}}\tan \frac{E}{2}. \tag{75}\]2  
Note that the coefficient in front of the integral is modified in comparison with the one in Equation (76).