## 6 Propagation Time for a Signal, Emitted and Intercepted by Satellites on a Space-Oriented Orbit  
### Three-Dimensional Orbit Parametrization and the General Formulae for the Orbit Parametrization  
The formulaes for the three-dimensional parametrization of space-oriented orbits previously were given by the expressions (9)-(11). Again, after using the null cone Equation (40) \(T=\int\frac{v}{c}\sqrt{\frac{(1-\frac{2V}{c^{2}})}{(1+\frac{2V}{c^{2}})}}\ dt\approx\int\frac{v}{c}(1-\frac{2V}{c^{2}})dt\), remembering relation (15) \(vdt_{col}=v_{f}df\) and also expression (16) \(v_{f}=\frac{na}{\sqrt{1-e^{2}}}\sqrt{1+e^{2}+2e\cos f}\) for the velocity \(v_{f}\), associated with the true anomaly angle \(f\), the following expression can be obtained for the propagation time \(\overline{T}\) for the three-dimensional case:  
\[\overline{T}=\int\frac{v}{c}(1-\frac{2V}{c^{2}})dt=\widetilde{T}_{1}+ \widetilde{T}_{2}=\frac{1}{c}\int vdt-\frac{2}{c^{3}}\int vVdt. \tag{67}\]  
We denote the propagation time for the three-dimensional case (9)-(11) by \(\overline{T}\), in order to distinguish it from the propagation time \(T\) for the case of a satellite, moving along a plane elliptical orbit. Analytical Calculation of the First Elliptic Integral (First \(O(\frac{1}{c})\) Correction) Without the Use of Elliptic Integrals  
Let us take the first \(O(\frac{1}{c})\) propagation time correction  
\[\widetilde{T}_{1}=\frac{1}{c}\int v/df=\frac{na}{c\sqrt{1-e^{2}}}\int\sqrt{1+ e^{2}+2e\cos f}df=\frac{na}{c\sqrt{1-e^{2}}}\widetilde{T}_{1}^{(1)} \tag{68}\]and let us perform the series of six subsequent variable transformations  
\[\sqrt{1+e^{2}+2e\cos f}=y\quad,\quad\overline{y}=(e+1)^{2}-y^{2}, \tag{69}\]  
\[\sqrt{1-\frac{\overline{y}q^{2}}{\overline{y}+B_{1}}}=z\;\;,\quad z^{2}=z_{1} \;,\;1-z_{1}=m\;\;,\;\;\overline{m}=m-\frac{q^{2}}{2}\;. \tag{70}\]  
The first transformation \(\sqrt{1+e^{2}+2e\cos f}=y\) in (69) will be used only in this paragraph; further, another transformation \(\widetilde{y}(1+e)=\sqrt{1+e^{2}+2e\cos f}\) will be applied. After applying the sequence of transformations (69) and (70), the following sum of two integrals for \(\widetilde{T}_{1}\) will be obtained:  
\[\widetilde{T}_{1}=-\frac{nai}{c\sqrt{(1-e^{2})q}}\left[\int\frac{d\overline{m }}{\left(\overline{m}-\frac{q^{2}}{2}\right)\sqrt{\overline{m}-\frac{q^{2}}{2 }}}+\int\frac{d\overline{m}}{\left(\overline{m}+\frac{q^{2}}{2}\right)\sqrt{ \overline{m}-\frac{q^{2}}{2}}}\right], \tag{71}\]  
where, again, the notation \(q=\frac{1-e}{1+e}\) has been used and each of the integrals inside the square bracket will be denoted correspondingly as \(\widetilde{T}_{1}^{(1)}\) and \(\widetilde{T}_{1}^{(2)}\). Making use of the analytically calculated integral from the book [46],  
\[\int\frac{dx}{(x\pm p)\sqrt{a+2 bx+cx^{2}}}\]  
\[=-\frac{1}{\sqrt{a\mp 2 bp+cp^{2}}}\ln\Biggl{\{}\frac{\sqrt{a+2 bx+cx^{2}}+ \sqrt{a\mp 2 bp+cp^{2}}}{x\pm p}+\frac{b\mp cp}{\sqrt{a\mp 2 bp+cp^{2}}} \Biggr{\}}, \tag{72}\]  
one can derive for the second integral \(\widetilde{T}_{1}^{(2)}\) in Equation (71) the following expression:  
\[\widetilde{T}_{1}^{(2)}=-\frac{na(1+e)\sqrt{2}}{c\sqrt{1-e^{2}}\sqrt{q}\sqrt{3 e^{2}+2e+3}}\ln\Biggl{[}\left(\frac{\sqrt{2m_{1}(f_{b};r_{b})}+\frac{1}{2}m_{2}(f_{b };r_{b})}{\sqrt{2m_{1}(f_{a};r_{a})}+\frac{1}{2}m_{2}(f_{a};r_{a})}\right) \left(\frac{m_{3}(f_{a};r_{a})}{m3(f_{b};r_{b})}\right)\Biggr{]}. \tag{73}\]  
In (73), \(m_{1}(f;r)\), \(m_{2}(f;r)\), \(m_{3}(f;r)\) are expressions, written in terms either of the initial and final true anomaly angles \(f_{a}\) and \(f_{b}\) or of the initial distance \(r_{a}\) (at which the emission of the signal takes place) and the final point \(r_{b}\) of reception, corresponding to the propagation time \(\widetilde{T}_{1}^{(2)}\). The first integral \(\widetilde{T}_{1}^{(1)}\) can be calculated analogously. It should be noted that both \(\widetilde{T}_{1}^{(1)}\) and \(\widetilde{T}_{1}^{(2)}\) are real-valued expressions and moreover, the logarithmic term is again present, as in the original Shapiro delay formulae (1).