## 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
#### 2.2.1 Case A--Plane Elliptical Coordinates of the Satellite  
This is the most simple case, when the two-dimensional \((x,y)\) coordinates of the plane elliptical orbit of the satellite are parametrized in terms of the large semi-major axis \(a\) of the ellipse, the eccentricity \(e\) and the eccentric anomaly angle \(E\). This angle is a dynamical parameter, related to the motion of the satellite on the orbit. The analytical expression for the parametrization of the \((x,y)\) coordinates is  
\[x=a(\cos E-e)\,\quad y=a\sqrt{1-e^{2}}\sin E. \tag{4}\]  
These formulaes are well-known in all basic courses on celestial mechanics (see, especially, [17; 18]. The time coordinate is chosen to be the celestial time \(t_{cel}\), which is related to the eccentric anomaly through the Kepler equation  
\[E-e\sin E=n(t_{cel}-t_{p})=M. \tag{5}\]  
In this equation, \(e\) is the eccentricity of the orbit, \(n=\sqrt{\frac{GM}{a^{3}}}\) is the mean motion, \(M\) is the mean anomaly and \(t_{p}\) is the time of perigee passage for the satellite. The mean anomaly \(M\) is an angular variable, which increases uniformly with time and changes by \(360^{0}\) during one revolution. The mean motion \(n\) has the following geometrical meaning: This is the motion of the satellite along an elliptical orbit, projected onto an uniform motion along a circle with a radius equal to the large axis of the ellipse. Usually, \(M\) is defined with respect to some reference time--this is the time \(t_{p}\) of perigee passage, where the perigee is the point of minimal distance from the foci of the ellipse (the Earth is presumed to be at the foci). Since the eccentric anomaly angle \(E\) will play an important role in the further calculation, let us also remember how this notion is defined: If, from a point on the ellipse, a perpendicular is drawn towards the large axis of the ellipse, then this perpendicular intersects a circle with a centre \(O\) at a point \(P\). Then, the eccentric anomaly represents the angle between the joining line \(OP\) and the semi-major axis (the line of perigee passage). The first main purpose of this paper will be to solve Equation (3) for \(dT\) as a differential equation with respect to the eccentric anomaly angle \(E\), for the given parametrization (4) of the coordinates \((x,y)\) and for the null cone metric of the near-Earth in the standard form  
\[ds^{2}=-c^{2}\left(1+\frac{2V}{c^{2}}\right)(dT)^{2}+\left(1-\frac{2V}{c^{2}} \right)\left((dx)^{2}+(dy)^{2}+(dz)^{2}\right)=0, \tag{6}\]  
where \(V=\frac{G_{\infty}M_{\odot}}{r}\) is the standard gravitational potential of the Earth. No account is taken of any harmonics due to the spherical form of the Earth since the GPS orbits are situated at a distance more than 20,000 km (considered from the centre of the Earth). In fact, the null cone metric is the same, derived from the standard metric for the near-Earth space in basic monographs of gravity theory such as the one by Fock [19]. 2.2 First Aim of the Paper--Expressing the Propagation Time in Terms of a Sum of Elliptic Integrals of Different Kinds (First, Second and Third) for the Case of a Signal Emission by a Satellite on a Plane Elliptical Orbit  
For the first time in [20,21] and in the review paper [22], the solution for the propagation time \(T\) is found to be expressed as a sum of zero-order elliptic integrals of the first, second and of the third kind. Later, it will become clear what the difference is between elliptic integrals of different orders and kinds. The imaginary unit appears in these elliptic integrals, so it is important to prove that the integrals are real-valued quantities, since the propagation time \(T\) is a real and not an imaginary quantity. The physical approximation that is used for finding the solution is \(\beta=\frac{2V}{c^{2}}\ll 1\), which, physically, is related to the assumption about a weak gravity field.