## 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
However, the null cone equation is compatible with the light-like geodesic equation in the sense that each solution of the null cone equation is a solution also of the light-like geodesic equation according to the theorem in the book by Fock [19], so it represents a geodesic. But then if it is a light-like geodesic, according to another theorem in [26], it is the shortest path (length) from one point of the geodesic to another point on the same geodesic. This is consistent with the Fermat principle about the propagation of light, asserting that light travels between two points for a minimal time [27]. Since the light velocity is constant, light will travel along the shortest path between the two points. In the book by Mishtenko and Fomenko [24], the theorem is formulated in the sense that each extremal of the null cone functional  
\[E(\gamma(s))=\int\mid\dot{\gamma}^{2}(s)\mid ds=\int g_{\bar{ij}}(x)\dot{x}^{ i}(s)\dot{x}^{j}(s)ds \tag{7}\]  
is also an extremal of the length functional  
\[L(\gamma(s))=\int\sqrt[]{g_{\bar{ij}}(x)\dot{x}^{i}(s)\dot{x}^{j}(s)}ds, \tag{8}\]  
but the statement in the inverse direction is not always fulfilled. This means that the extremals of the null cone functional constitute a subset of the extremals of the length functional--see also the arguments in [22; 25]. The length functional of the type (8) can be obtained also after the integration of the null cone equation (6) for the case when the space coordinates \(x,y,z\) are parametrized by several variables (see the next section)--in differential geometry, well studied are the s.c. first and second fundamental forms [24; 26]. In subsequent papers, this will also be studied in detail, but this paper will focus only on finding the solutions for the propagation time in terms of elliptic integrals. 2.5 Third Aim of the Paper--Finding the Solution for the Propagation Time \(T\) for the Case of Space-Oriented - Orbits - in Terms of Elliptic Integrals of Higher Order  
Space-oriented) orbits are well known from celestial mechanics (further, we shall use the coordinate representation from the book [18]) and in this case, the position of the orbit in space is characterized by six Kepler parameters \((f,a,e,\Omega,I,\omega)\), which are, in fact, the orbital coordinates. The analogues of the coordinate transformation (4) for this case of space-oriented orbits are (see the cite [28] and also the known monographs [29; 30; 31; 32])  
\[x=\frac{a(1-e^{2})}{1+e\cos f}[\cos\Omega\cos(\omega+f)-\sin\Omega\sin(\omega+f) \cos I], \tag{9}\]  
\[y=\frac{a(1-e^{2})}{1+e\cos f}[\sin\Omega\cos(\omega+f)+\cos\Omega\sin(\omega+ f)\cos I], \tag{10}\]  
\[z=\frac{a(1-e^{2})}{1+e\cos f}\sin(\omega+f)\sin I, \tag{11}\]  
where \(r=\frac{a(1-e^{2})}{1+e\cos f}\) is the radius vector in the orbital plane, the angle \(\Omega\) of the longitude of the right ascension of the ascending node is the angle between the line of nodes and the direction to the vernal equinox, the argument of perigee (periapsis) \(\omega\) is the angle within the orbital plane from the ascending node to the perigee in the direction of the satellite motion \((0\leq\omega\leq 360^{0})\) and the angle \(I\) is the inclination of the orbit with respect to the equatorial plane. From celestial mechanics, it is also known that there is a relation between the eccentric anomaly angle \(E\) and the true anomaly angle \(f\)  
\[\tan\frac{f}{2}=\sqrt{\frac{1+e}{1-e}}\tan\frac{E}{2}\Longrightarrow f=2\arctan \left[\sqrt{\frac{1+e}{1-e}}\tan\frac{E}{2}\right]. \tag{12}\]  
2.6 Elliptic Integrals of the Second and of the Fourth Order with an Integration over the True Anomaly Angle \(f\)  
Finding the solutions for the propagation time \(T\) of the signal in terms of elliptic integrals of the second and the fourth order (as shall be explained, the order of the elliptic integral is related to the order of the polynomial in the nominator) is the third main goal of this paper.