## 2 A Brief Review of the Shapiro Delay Formulae and the Aims of This Paper  
### Standard Shapiro Delay Formulae, Derived from the Null Cone Equation  
The basic theoretical approach to find the propagation time of signals in the near-Earth space with account also of the gravitational field and in the framework of general relativity theory is based on the s.c. Shapiro delay formulae, proposed in 1964 [11] and frequently used also in VLBI (very large baseline interferometry) (see the review article by Sovers, Fanselow and Jacobs [12]). \[T_{AB}=\frac{R_{AB}}{c}+\frac{2 GM_{E}}{c^{3}}\ln\left(\frac{r_{A}+r_{B}+R_{AB }}{r_{A}+r_{B}-R_{AB}}\right), \tag{1}\]  
where the coordinates of the emitting and of the receiving satellite are correspondingly \(\mid x_{A}(t_{A})\mid=r_{A}\) and \(\mid x_{B}(t_{B})\mid=r_{B}\), and \(R_{AB}=\mid x_{A}(t_{A})-x_{B}(t_{B})\mid\) is the Euclidean distance between the signal-emitting satellite and the signal-receiving satellite. Note that the space points of emission \(x_{A}(t_{A})\) and arrival \(x_{B}(t_{B})\) of the signal are taken at the time \(t_{A}\) of emission and \(t_{B}\) of arrival correspondingly. In (1), \(G_{\oplus}M_{\oplus}\) is the geocentric gravitational constant, \(M_{\oplus}\) is the Earth mass and \(G_{\oplus}\) is the gravitational constant; \(t=\) TCG denotes the s.c. geocentric coordinate time (TCG). Further (and also in the above formulae), we shall denote by \(T_{AB}=T_{B}-T_{A}\) the signal propagation time and, thus, we have replaced the times \(t_{A}\) and \(t_{B}\) by \(T_{A}\) and \(T_{B}\), to underline that these are times of emission and reception. The second term in formulae (1) is the Shapiro time delay term, accounting for the signal delay due to the curved space-time. An understandable derivation of the Shapiro delay formulae can be found in the PhD dissertations [13; 14]  
In order to calculate the propagation time \(T_{AB}\) of the signal, it should be remembered that according to general relativity (see the book [15]), the emitter of the first satellite emits a signal propagating on the gravitational null cone after setting up to zero the infinitesimal metric element \(ds^{2}=0\):  
\[ds^{2}=0=g_{00}c^{2}dT^{2}+2g_{oj}cdTdx^{j}+g_{ij}dx^{i}dx^{j}. \tag{2}\]  
For the null cone Equation (2), the solution of this quadratic algebraic equation with respect to the differential \(dT\) can be given as follows [16]:  
\[dT=\pm\frac{1}{c}\frac{1}{\sqrt{-g_{00}}}\sqrt{\left(g_{ij}+\frac{g_{0i}g_{0j }}{-g_{00}}\right)}dx^{i}dx^{j}+\]  
\[+\frac{1}{c}\left(\frac{g_{0j}}{-g_{00}}\right)dx^{j}, \tag{3}\]where the metric tensor components are determined for an Earth reference system with an origin at the centre of the Earth. Further, the propagation time should be found after (1) choosing the metric tensor components \(g_{ij}\), \(g_{0i}\) and \(g_{00}\) in the given expression for the zero infinitesimal metric \(ds^{2}=0\) (2), and (2) choosing the coordinates \(x^{i}\) (\(i,j=1,2,3\)) in the metric, which, additionally, may depend on one, two or more parameters. Note that we have assumed that \(x^{0}=1\). In the present investigation, the coordinates shall be chosen to be related with the satellite's motion along the elliptic orbit. Usually, geostationary satellites are situated on nearly circular orbits. Satellites from the GPS constellation have a very small eccentricity of the orbit of the order \(e=0.01\); for the Russian constellation GLONASS the eccentricity is \(e\approx 0.02\). Let us remember that if \(a\) and \(b\) are, respectively, the large and the small semi-axis of the ellipse, then by definition, \(e=\frac{\sqrt{a^{2}-b^{2}}}{a}=\frac{c}{a}\) (\(0\leq e<1\)), where \(c\) is the distance between the foci of the ellipse. Thus, the eccentricity measures to what extent the ellipse deviates from the circle.