## 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
Fourth-Order Elliptic Integrals and the Second \(O(\frac{1}{c^{3}})\) Correction of the Propagation Time for the Case of a Space-Oriented Orbit  
The First Part of the Second \(O(\frac{1}{c^{3}})\) Correction, Expressed in Terms of Elliptic Integrals of the Second Order  
Taking into account the general equality (67) \(\overline{T}=\widetilde{T}_{1}+\widetilde{T}_{2}=\frac{1}{c}\int vdt-\frac{2 }{c^{3}}\int vVdt\), the second \(O(\frac{1}{c^{3}})\) correction can be written as  
\[\widetilde{T}_{2}=-\int\frac{v}{c}.\frac{2V}{c^{2}}dt=-\frac{2G_{\oplus}M_{ \oplus}}{c^{3}}\int\frac{v_{f}}{r}df\, \tag{95}\]where the radius-vector \(r\) in the orbital plane, the velocity \(v_{f}\), associated with the true anomaly angle \(f\) and the potential \(V\) of the gravitational field around the Earth are defined in the usual way:  
\[v_{f}=\frac{na}{\sqrt{1-e^{2}}}\sqrt{1+2e\cos f+e^{2}}\;,\;\;r=\frac{a(1-e^{2}) }{1+e\cos f},\]  
\[V=\frac{GM}{r}=\frac{GM}{a(1-e^{2})}(1+e\cos f). \tag{96}\]  
The final integral for the second correction is of the form  
\[y=\sqrt{\frac{(1+e\cos E)}{q(1-e\cos E)}},\qquad(q=\frac{1-e}{1+e})\]0  
where  
\[y=\sqrt{\frac{(1+e\cos E)}{q(1-e\cos E)}},\qquad(q=\frac{1-e}{1+e})\]1  
\[y=\sqrt{\frac{(1+e\cos E)}{q(1-e\cos E)}},\qquad(q=\frac{1-e}{1+e})\]2  
In (98), \(\widetilde{T}_{1}\) is the previously calculated integral \(\;\widetilde{T}_{1}=\int\sqrt{1+2e\cos f+e^{2}}df\;\). The second term \(T_{2}^{(2)}\) in the second correction \(\widetilde{T}_{2}\) in (97) is  
\[y=\sqrt{\frac{(1+e\cos E)}{q(1-e\cos E)}},\qquad(q=\frac{1-e}{1+e})\]3  
\[y=\sqrt{\frac{(1+e\cos E)}{q(1-e\cos E)}},\qquad(q=\frac{1-e}{1+e})\]4  
where \(\widetilde{T}_{2}^{(2)}\) is the notation for the more complicated integral  
\[y=\sqrt{\frac{(1+e\cos E)}{q(1-e\cos E)}},\qquad(q=\frac{1-e}{1+e})\]5  
The Second Part of the Second \(O(\frac{1}{c^{3}})\) Correction, Expressed in Terms of Elliptic Integrals of the Second and of the Fourth Order  
It can be proved that  
\[y=\sqrt{\frac{(1+e\cos E)}{q(1-e\cos E)}},\qquad(q=\frac{1-e}{1+e})\]6  
\[y=\sqrt{\frac{(1+e\cos E)}{q(1-e\cos E)}},\qquad(q=\frac{1-e}{1+e})\]7  
\[y=\sqrt{\frac{(1+e\cos E)}{q(1-e\cos E)}},\qquad(q=\frac{1-e}{1+e})\]8  
where  
\[y=\sqrt{\frac{(1+e\cos E)}{q(1-e\cos E)}},\qquad(q=\frac{1-e}{1+e})\]9  
\[\tan\frac{f}{2}=\sqrt{\frac{1-\cos f}{1+\cos f}}=\sqrt{\frac{1+e}{1-e}}\tan\frac {E}{2},\]0is an elliptic integral of the second order in the Legendre form and \(\tilde{y}\) is the notation for the variable \(\tilde{y}=\frac{\tilde{y}}{\tilde{q}}\). The last formulae can also be compared to (85). Analogously, the formulae for the fourth-order elliptic integral \(\widetilde{J}^{(4)}_{4}(\tilde{y},q)\) can be written as  
\[\tan\frac{f}{2}=\sqrt{\frac{1-\cos f}{1+\cos f}}=\sqrt{\frac{1+e}{1-e}}\tan\frac {E}{2},\]1  
Relation Between the Fourth-Order and the Second-Order Elliptic Integrals in the Expression for \(T_{2}^{(2)}\)  
From the recurrent chain (20), after setting up \(n=0\) and after integration, it can easily be established that  
\[\tan\frac{f}{2}=\sqrt{\frac{1-\cos f}{1+\cos f}}=\sqrt{\frac{1+e}{1-e}}\tan\frac {E}{2},\]2  
\[\tan\frac{f}{2}=\sqrt{\frac{1-\cos f}{1+\cos f}}=\sqrt{\frac{1+e}{1-e}}\tan\frac {E}{2},\]3  
We have already proved that \(\widetilde{J}^{(4)}_{2}(\tilde{y},q)\) can be expressed in elementary functions, yet we do not know whether \(\widetilde{J}^{(4)}_{0}(\tilde{y},q)\) or the second-rank integral  
\[\tan\frac{f}{2}=\sqrt{\frac{1-\cos f}{1+\cos f}}=\sqrt{\frac{1+e}{1-e}}\tan\frac {E}{2},\]4  
can also be expressed through elementary functions. This will be proved further.