## 2 A Brief Review of the Shapiro Delay Formulae and the Aims of This Paper
### Abelian Integrals Versus Elliptic Integrals
#### 3.2.2 Is the Integral (29) an Abelian One? If the square root, representing an irrational function of the variable \(y\), is denoted as  
\[x=\sqrt{\frac{a_{1}y^{3}+a_{2}y^{2}+a_{3}y}{b_{1}y^{3}+b_{2}y^{2}+b_{3}y+b_{4} }}\ =\sqrt{P(y)}, \tag{36}\]  
then the integral  
\[\int xdy=\int\sqrt{P(y)}dy \tag{37}\]  
is a partial case of a class of integrals, often denoted in mathematical literature as \(\int R(x,y)dy\)). In case \(P(y)\) is an algebraic polynomial, integrals \(\int\sqrt{P(y)}dy\) are known in mathematics as abelian integrals (see the monograph by Prasolov and Solovyev [34]), related to the curve  
\[F(x,y):=x^{2}-P(y)=0. \tag{38}\]  
Since in the case of the expression (36), \(P(y)\) is a complicated irrational function of an rational expression of \(y\), the integral (29) is not an abelian one. Further, the expression Equation (25) will not be used, because a justifiable and reasonable physical approximation shall be applied, which will enable us to express the solution in terms of elliptic integrals. Nevertheless, in view of the constantly improving accuracy for measuring the propagation time, this might represent an interesting problem for future mathematical research. In the next sections, when defining the elliptic integrals for the more general case of arbitrary polynomials of the third and fourth degree in the denominator, it will be explained in detail why integrals of the form (29) cannot belong to the class of abelian integrals. The Useful Approximation for a Small Gravitational Potential, Compared to the Square of the Velocity of Light  
Further, we shall be interested in the case  
\[\beta=\frac{2V}{c^{2}}=\frac{2G_{\oplus}M_{\oplus}}{c^{2}a}\ll 1, \tag{39}\]  
which can be assumed to be fulfilled and is consistent with the current experimental data.It shall be proved in the next section that this physical assumption will greatly facilitate the calculations so that the appearance of any non-abelian integrals will be avoided. For the parameters of the \(GPS\) orbit--\(a=26\),\(561\) [km], in the review paper [23], the constant \(\beta\) can exactly be calculated to be \(0.334\times 10^{-9}\), with the velocity of light taken to be \(c=299\),\(792\),\(458\) [\(\frac{m}{8}\)]. The geocentric gravitational constant \(G_{\oplus}M_{\oplus}\) (obtained from the analysis of laser distance measurements of artificial Earth satellites) can be taken to be equal to \(G_{\oplus}M_{\oplus}=(3\),\(986\),\(004.405\pm 1)\times 10^{8}\) [\(\frac{m^{3}}{g^{3}}\)], but the value of \(G_{\oplus}M_{\oplus}\) can vary also in another range from \(G_{\oplus}M_{\oplus}=3\),\(986\),\(056\),\(75236\times 10^{8}\) [\(\frac{m^{3}}{g^{3}}\)] to the value \(G_{\oplus}M_{\oplus}=3\),\(987\),\(999.07898\times 10^{8}\)\([\frac{m^{3}}{g^{2}}]\) due to the uncertainties in measuring the Newton gravitational constant \(G_{\oplus}\). The mass of the Earth can be taken approximately to be \(M_{E}\approx 5.97\times 10^{24}\) [kg]. One of the latest values for \(G_{\oplus}\) from deep space experiments was reported in the paper [39] to be \((6.674+0.0003).10^{-11}\)\([\frac{m^{3}}{kg\cdot g^{2}}]\). The latest value for \(G\) from Encyclopedia Britanica (updated 23 September 2024) is \(6.67\pm 0.00015\times 10^{-11}\)\([\frac{m^{3}}{kg\cdot g^{2}}]\). Derivation of the Expression for the Propagation Time Under the Approximation \(\beta=\frac{2V}{c^{2}}\ll 1\) and Physical Justification of the Obtained Result  
After decomposing the under-integral expression with the square root in Equation (29) under the assumption (39) and leaving only the first-order term in \(\frac{2V}{c^{2}}\), one can obtain  
\[T=\int\frac{v}{c}\sqrt{\frac{(1-\frac{2V}{c^{2}})}{(1+\frac{3V}{c^{2}})}}\,dt \approx\int\frac{v}{c}(1-\frac{2V}{c^{2}})dt=I_{1}+I_{2} \tag{40}\]  
\[=\frac{a}{c}\int\sqrt{1-e^{2}\cos^{2}E}\,dE-\frac{2G_{\oplus}M_{\oplus}}{c^{3} }\int\sqrt{\frac{1+e\cos E}{1-e\cos E}}dE. \tag{41}\]  
This expression is consistent from a physical point of view due to the following reasons:  
1.