1 Introduction

After more than one century, the general relativity developed by Albert Einstein is an extraordinarily successful theory of gravity and has passed a wealth of tests from our Solar System to binary pulsars systems and exoplanets [1,2,3,4,5,6,7,8,9,10]. These great successes, however, have never stopped alternatives being able to stand up for explaining some exotic phenomena such as the dark matter and the dark energy. Among these modified gravity theories, the most eminent case is a scalar–tensor gravity because it is the most natural and the simplest way to modify the general relativity. Some modified gravity theories, such as the noncommutative geometry theories, the string theory, the extra-dimensional theory and the braneworld, which try to unify microscopic physics and gravity and to explain the accelerating expansion of the universe, demand one scalar field besides the metric tensor (see [11] for a review). Especially, Damour and Nordtvedt pointed out that the scalar–tensor gravity are shown to generically contain an attractor mechanism toward the general relativity at the beginning of the matter-dominated era [12].

By means of observations and experiments, probing the scalar–tensor theory on different distance scales is still required for deeper understanding of the nature of spacetime and for testing possible violations of the general relativity. For this reason, we will investigate the (un)testability of one scalar–tensor gravity for the Solar System tests in the present paper. On the other hand, it is worth emphasizing that one framework in view of such a purpose is called by the parametrized post-Newtonian formalism [1, 2]. In the formalism, there exist ten parametrized post-Newtonian parameters. And their values stand for possible deviations from the general relativity. In the standard parametrized post-Newtonian formalism pioneered by Will and Nordtvedt [13, 14], the parametrized post-Newtonian parameters are all numerical values and independent of the radial distance r. Generally, the scalar–tensor theory has two nonzero parametri-zed post-Newtonian parameters such as \(\beta \) and \(\gamma \), which have been derived in a lot of literatures [15,16,17,18,19,20,21,22,23,24,25,26,27].

Recently, by considering a general free scalar–tensor gravity with three free coupling functions \(F(\phi )\), \(W(\phi )\) and \(V(\phi )\) [28, 29], the authors of Ref. [27] constrained one Yukawa correction in the gravity by borrowing the measured values of the parametrized post-Newtonian parameters \(\gamma -1=(2.1\pm 2.3)\times 10^{-5}\) [30] of the Cassini tracking experiment and \(\beta -1=(-4.1\pm 7.8)\times 10^{-5}\) [2] of the lunar laser ranging experiment. Firstly, they defined two parametrized post-Newtonian parameters \(\gamma (r)\) and \(\beta (r)\) in the general free scalar–tensor gravity, which depend on the radial distance r and include two Yukawa-type parameters. Then, by comparing between their \(\gamma (r)\) and \(\beta (r)\) and the above measured values of two parametrized post-Newtonian parameters in the Solar System tests, the authors in Ref. [27] claimed that they gave the tightest bound on the gravity by the Cassini tracking experiment.

In this paper, we will treat this issue as a more rigorous way. Concretely, corresponding astronomical experiments have been physically modelled by considering the lightlike and the timelike geodesics in the general free scalar–tensor gravity. Contrary to the wrong results in Ref. [27], we will show that the Cassini tracking experiment is insensitive to the general free scaler–tensor gravity. Furthermore, we will also find that the light deflection and the time delay are all insensitive to the gravity. However, due to an additional Yukawa-type advance in the periastron shift, we will derive very much improved lower bounds on the Yukawa-type parameters of this gravity by using the secular periastron precession of binary pulsars and the supplementary perihelion precessions provided by EPM2011 and INPOP10a ephemerides.

The paper is organized as follows: In the next Section we devoted to briefly reviewing the general free scalar–tensor gravity and its first post-Newtonian metric for the light propagation for completeness. Section 3 describes the null geode-sic, which is applied to model the Cassini tracking experiment, the time delay experiment and the light deflection experiment. Section 4 exhibits the timelike particle’s periastron shift and obtain very much improved lower bounds on the Yukawa-type parameters of this gravity. Finally, our results and the discussion are outlined in Sect. 5.

2 General free scalar–tensor gravity

Based on the work of Ref. [27], a general free scalar–tensor gravity with three free coupling functions \(F(\phi )\), \(W(\phi )\) and \(V(\phi )\) has the following action

$$\begin{aligned} S= & {} \frac{1}{16\pi G_{N}}\!\!\int \textrm{d}^{4}x\sqrt{-g}\bigg [F(\phi )R\!-\!W(\phi )\partial _{\mu }\phi \partial ^{\mu }\phi -V(\phi )\bigg ]\nonumber \\{} & {} +S_{m}(\psi _{m},g_{\alpha \beta }), \end{aligned}$$
(1)

where \(\phi \) is a scalar field and \(G_{N}\) is the Newtonian gravitational constant. Here we adopt \(c=\hbar =1\). \(\psi _{m}\) is the matter field and only interacts with the metric field \(g_{\alpha \beta }\).

By using the post-Newtonian approximation approach, from the action (1), the metric coefficients had been derived in Ref. [27] with the parameterized post-Newtonian gauge [1, 31] in the Solar System. Specifically, the metric has

$$\begin{aligned} \textrm{d}s^{2}=(-1+\overset{(2)}{h}_{00})\textrm{d}t^{2}+(\delta _{ij}+\overset{(2)}{h}_{ij})\textrm{d}x^{i}\textrm{d}x^{j}, \end{aligned}$$
(2)

and the metric coefficients could be found as

$$\begin{aligned} \overset{(2)}{h}_{00}= & {} \frac{2G_{N}M}{F_{0}r}\bigg [1+\frac{F'^{2}_{0}}{(3F'^2+2FW)_{0}}e^{-m_{s}r}\bigg ], \end{aligned}$$
(3)
$$\begin{aligned} \overset{(2)}{h}_{ij}= & {} \frac{2G_{N}M}{F_{0}r}\delta _{ij}\bigg [1-\frac{F'^{2}_{0}}{(3F'^2+2FW)_{0}}e^{-m_{s}r}\bigg ], \end{aligned}$$
(4)

in Ref. [27] [see its equations (26) and (27) in the page 4]. In the above metric, “0” denotes the value of a quantity at a scalar background \(\phi _{0}\) and a prime indicates differentiation with respect to the scalar field \(\phi \). \(m_{s}\) is the effective mass of the scalar field. It is worth mentioning that the metric coefficients in (2) is adequate for modeling astronomical experiments for the light propagation in the first post-Newtonian approximation and higher-order terms belong to the second post-Newtonian approximation (see Ref. [24] for details).

In the metric (3) and (4), we can see that there exists a Yukawa-type correction in the gravity, such as \(\alpha e^{-m_{s}r}\), namely

$$\begin{aligned} \alpha\equiv & {} \frac{F'^{2}_{0}}{(3F'^2+2FW)_{0}},~~~~ m_{s}\equiv \alpha ^{1/2}m_{0}, \end{aligned}$$
(5)

where \(\alpha \) and \(1/m_{s}\) are the dimensionless coupling strength parameter and the length-scale for the Yukawa correction respectively [32, 33]. The two Yukawa-type parameters in Eq. (5) had been given by Ref. [27] [see its equations (37) and (40) in the page 6]. And \(m_0\) is the mediator mass [27]. For the purpose of constraining two Yukawa-type parameters \(\alpha \) and \(m_0\), the authors defined the parametrized post-Newtonian parameter \(\gamma (r)\) as follows

$$\begin{aligned} \overset{(2)}{h}_{ij}= & {} \gamma (r)\overset{(2)}{h}_{00}\delta _{ij}, \end{aligned}$$
(6)
$$\begin{aligned} \gamma (r)= & {} 1-\frac{2F'^{2}_{0}e^{-m_{s}r}}{(3F'^2+2FW)_{0}+F'^{2}_{0}e^{-m_{s}r}}, \end{aligned}$$
(7)

in Ref. [27] [see its equations (28) and (30) in the page 4]. By borrowing the result of the Cassini tracking experiment \(\gamma -1=(2.1\pm 2.3)\times 10^{-5}\) [30] and comparing it with Eq. (7), the authors in Ref. [27] claimed that they gave the tightest bound on the coupling strength \(\alpha \) of the gravity by the Cassini tracking experiment.

It is worth noting that \(\gamma (r)\) [Eq. (7)] defined in Ref. [27] depends not only on the Yukawa-type parameters but also on the radial distance r. This character is totally different from the standard parameterized post-Newtonian parameter \(\gamma \) [13, 14], which is a number value not a function of the radial distance r. Further analysis, we find the approach in Ref. [27] is not rigorous because they do not model the observation of the light propagation in the general free scalar–tensor gravity. In the next section, we will prove that the Cassini tracking experiment is insensitive to the Yukawa-type parameters of the general free scalar–tensor gravity when we fully consider the null geodesics. Furthermore, we also find that the light deflection and the time delay are all independent of the gravity. For \(r>0\), based on the metric (2) given by Ref. [27], we can easily obtain the following useful relationship

$$\begin{aligned}{}[\gamma (r)+1]\overset{(2)}{h}_{00}=4\frac{\mathscr {G}M}{r}, \end{aligned}$$
(8)

where \(\mathscr {G}\equiv G_{N}/F_0\). The above equation does not include the Yukawa-type correction and plays a very important role for hiding the Yukawa effects in the general free scalar–tensor gravity for the light propagation such as the Cassini tracking experiment.

3 Models of null geodesics

In the section, with the help of Eq. (8), we will show that the light propagation is insensitive to the Yukawa-type parameters of the general free scaler–tensor gravity by modeling the observation of the light propagation, which are Cassini tracking experiment, the time delay experiment and the light deflection experiment. It means that the observations of the light propagation in the general free scaler–tensor gravity are the same form as the prediction by the general relativity.

3.1 Cassini tracking

In the isotropic Cartesian coordinates, the metric (2) yields

$$\begin{aligned} \textrm{d}s^{2}=(-1+\overset{(2)}{h}_{00})\textrm{d}t^{2}+[1+\gamma (r)\overset{(2)}{h}_{00}](\textrm{d}x^2+\textrm{d}y^2+\textrm{d}z^2), \end{aligned}$$
(9)

where \(r=\sqrt{x^{2}+y^{2}+z^{2}}\). In the Born approximation, a light ray is derived by \(y=d\), \(z=0\) and \(r=\sqrt{x^{2}+d^{2}}\), where d is the closest approach as the ray passing the Sun [see Chap. 11.7 in Ref. [34]]. For the null geodesics, we have \(\textrm{d}s^{2}=0\) and obtain

$$\begin{aligned} (1-\overset{(2)}{h}_{00})\textrm{d}t^2=[1+\gamma (r)\overset{(2)}{h}_{00}]\textrm{d}x^{2}. \end{aligned}$$
(10)

Then, we have

$$\begin{aligned} \textrm{d}t=\bigg \{1+\frac{1}{2}[\gamma (r)+1]\overset{(2)}{h}_{00}\bigg \}\textrm{d}x, \end{aligned}$$
(11)

in the first post-Newtonian approximation. By using the relationship Eq. (8), the above equation reads

$$\begin{aligned} \textrm{d}t=\bigg [1+2\frac{\mathscr {G}M}{\sqrt{x^{2}+d^{2}}}\bigg ]\textrm{d}x+(\mathscr {G}^2), \end{aligned}$$
(12)

which does not depend on two Yukawa-type parameters \(\alpha \) and \(m_0\). From Eq. (12), it gives the gravitational time delay as

$$\begin{aligned} t(X,d)= & {} \int ^{X}_{0}\frac{\textrm{d}t}{\textrm{d}x}\textrm{d}x\nonumber \\= & {} X+2\mathscr {G}M\ln \bigg (\frac{X+\sqrt{X^2+d^2}}{d}\bigg )+(\mathscr {G}^2), \end{aligned}$$
(13)

between \(x=0\) and \(x=X\). It suggests that the time delay is independent of the Yukawa-type parameters of the gravity and has the same form as the prediction by the general relativity [see Chap. 11.7 in Ref. [34]].

When we substitute \(X=\sqrt{r^2-d^2}\) into the above equation, it yields

$$\begin{aligned} t(r,d)=\sqrt{r^2-d^2}+2\mathscr {G}M\ln \bigg (\frac{r+\sqrt{r^2-d^2}}{d}\bigg )+(\mathscr {G}^2).\nonumber \\ \end{aligned}$$
(14)

In the superior conjunction such as the Cassini tracking experiment [30], the emitter is on the opposite side of the Sun as seen from the receiver. Then, we have \(r_{R}\gg d\) and \(r_{E}\gg d\), where \(r_{R}\) denotes the distance between the Sun and the reflector and \(r_{E}\) denotes the distance between the Sun and the emitter. Therefore, the round-trip time delay in the general free scaler–tensor gravity is

$$\begin{aligned} \Delta t_{\textrm{SC}}= & {} 2t(r_E,d)+2t(r_R,d)\nonumber \\= & {} 2(r_E+r_R)+4\mathscr {G}M\ln \frac{4r_Er_R}{d^2}. \end{aligned}$$
(15)

It is shown that Eq. (15) is identical with the time delay of the general relativity [35] at the superior conjunction.

The observable in the Cassini tracking experiment is the relative change \(\nu (t)\) in the frequency around the superior conjunction, namely \(\textrm{d}\Delta t_{\textrm{SC}}/\textrm{d}t\). Because two Yukawa-type parameters of the gravity do not appear in Eq. (15), it is not difficult to find that the Cassini tracking experiment does not include the Yukawa-type parameters. This experiment begins 12 days before the superior conjunction and ends 12 days after the superior conjunction. On the middle day during this period of the experiment, the closest approach d is about \(1.5R_{\odot }\) [see Ref. [30] for details]. Then, based on Eq. (15), we obtain

$$\begin{aligned} y_{\textrm{SC}}= & {} \frac{\nu (t)-\nu _{0}}{\nu _0}=\frac{\textrm{d}\Delta t_{\textrm{SC}}}{\textrm{d}t} =-8\frac{\mathscr {G}M}{d}v_{\bigoplus }, \end{aligned}$$
(16)

where \(\textrm{d}d(t)/\textrm{d}t\simeq v_{\bigoplus }\) during the Cassini tracking experiment. Here \(\nu _0\) is the frequency transmitted in the ground station. Contrary to the result in Ref. [27], it is seen that the Cassini tracking experiment is insensitive to the Yukawa-type parameters of the general free scaler–tensor gravity. And the modeling result for this experiment is also exactly the same as the prediction by the general relativity. Since the Cassini tracking experiment is independent of the Yukawa-type parameters, we can not constrain the Yukawa coupling strength \(\alpha \) of the gravity by the Cassini tracking, which is obviously contrary to the statements of Ref. [27].

3.2 Light deflection

In the isotropic spherical coordinates, the metric (2) yields

$$\begin{aligned} \textrm{d}s^{2}= & {} (-1+\overset{(2)}{h}_{00})\textrm{d}t^{2}\nonumber \\{} & {} +[1+\gamma (r)\overset{(2)}{h}_{00}][\textrm{d}r^2+r^2(\textrm{d}\theta ^2+\sin ^2\theta \textrm{d}\varphi ^2)], \end{aligned}$$
(17)

it can be written as

$$\begin{aligned} 0= & {} (-1+\overset{(2)}{h}_{00})\bigg (\frac{\textrm{d}t}{\textrm{d}\lambda }\bigg )^2\nonumber \\{} & {} +[1+\gamma (r)\overset{(2)}{h}_{00}]\bigg [\bigg (\frac{\textrm{d}r}{\textrm{d}\lambda }\bigg )^2+r^2\bigg (\frac{\textrm{d}\varphi }{\textrm{d}\lambda }\bigg )^2\bigg ], \end{aligned}$$
(18)

for the null geodesic in the equatorial plane with \(\theta =\pi /2\) [36] and \(\lambda \) is an affine parameter. Along the path of light, there are two conserved quantities as follows

$$\begin{aligned} E=(1-\overset{(2)}{h}_{00})\frac{\textrm{d}t}{\textrm{d}\lambda },~~~~ L=[1+\gamma (r)\overset{(2)}{h}_{00}]r^2\frac{\textrm{d}\varphi }{\textrm{d}\lambda }. \end{aligned}$$
(19)

From Eqs. (18) and (19), we derive

$$\begin{aligned} \frac{\textrm{d}\varphi }{\textrm{d}r}= & {} \frac{1}{r^{2}}(1-\overset{(2)}{h}_{00})^{1/2} [1+\gamma (r)\overset{(2)}{h}_{00}]^{-1/2}\nonumber \\{} & {} \times \bigg \{\frac{1}{b^2}-\frac{1}{r^2}(1-\overset{(2)}{h}_{00})[1+\gamma (r)\overset{(2)}{h}_{00}]^{-1}\bigg \}^{-1/2}, \end{aligned}$$
(20)

where \(b\equiv L/E\) and we neglect the minus solution just as done by Ref. [36]. For the closest approach, we have \(\textrm{d}r/\textrm{d}\varphi |_{r=d}\) \(=0\) and give the expression for b. Equation (20) can be simplified as

$$\begin{aligned} \frac{\textrm{d}\varphi }{\textrm{d}r}= & {} \frac{d}{r\sqrt{r^2-d^2}} +\frac{rd}{2(r^2-d^2)^{3/2}}\bigg \{[\gamma (d)+1]\overset{(2)}{h}_{00}|_{r=d}\nonumber \\{} & {} -[\gamma (r)+1]\overset{(2)}{h}_{00}\bigg \}. \end{aligned}$$
(21)

Then, we have

$$\begin{aligned} \frac{\textrm{d}\varphi }{\textrm{d}r}=\frac{d}{r\sqrt{r^2-d^2}}+\frac{2\mathscr {G}M(r-d)}{(r^2-d^2)^{3/2}}+(\mathscr {G}^2), \end{aligned}$$
(22)

with the help of the useful relationship Eq. (8). The above equation does not depend on the Yukawa-type parameters either. The corresponding light deflection in the general free scaler–tensor gravity is

$$\begin{aligned} \Delta \varphi =2\int ^{\infty }_{d}\frac{\textrm{d}\varphi }{\textrm{d}r}\textrm{d}r-\pi =4\frac{\mathscr {G}M}{d}+(\mathscr {G}^2), \end{aligned}$$
(23)

in the first post-Newtonian approximation. The above formula is just the light deflection of the general relativity in the isotropic coordinates [see its equation (6) in Ref. [37]]. It is shown that we cannot distinguish the gravity from the general relativity by the light deflection.

In summary, based on the metric (2) given by Ref. [27], the experiments for the light propagation are insensitive to the Yukawa-type parameters of the general free scaler–tensor gravity (see Table 1). These experiments include Cassini tracking, the time delay and the light deflection. In the next section, the timelike geodesic will be calculated and the Yukwawa-type parameters’ bound will be improved.

4 Models of timelike geodesic

Table 1 Improved bounds on the Yukawa-type parameters \(\alpha \) and \(m_0\) of the general free scalar–tensor gravity in the present work
Full size table

Due to the Yukawa correction, an additional periastron shift of a planet in the Solar System or one binary system was given by

$$\begin{aligned} \frac{\delta \theta _{5}}{2\pi }=\frac{\alpha }{2}(m_{s}p)^2e^{-m_{s}p}, \end{aligned}$$
(24)

in its equation (54) of Ref. [27] with treating the Yukawa force as a small perturbation. Here \(p=a(1-e^2)\) is the semilatus rectum, a is the semi-major axis and e is the eccentricity. And \(\alpha \) and \(m_{s}\) are defined in Eq. (5) following Ref. [27] [see its equations (37) and (40) in the page 6]. Furthermore, the secular periastron advance based on Eq. (24) is

$$\begin{aligned} \frac{\textrm{d}\omega }{\textrm{d}t}\bigg |_{\textrm{Yukawa}}=\frac{\delta \theta _{5}}{T}=\frac{n\alpha }{2}(m_{s}p)^2e^{-m_{s}p}, \end{aligned}$$
(25)

where T is the orbital period and \(n^2a^3=\mathscr {G}M\). And \(M =M_1+M_2\) if the system is a point-mass binary. By the aid of the averaging method [44] in celestial mechanics, the authors of Ref. [23] obtained the same form as Eq. (25) [in its equation (74)]. Note that the symbol \(\lambda \) in Ref. [23] equals to \(1/m_s\) in the work.

With the additional Yukawa correction in the gravitational potential, the corresponding exact orbital precessions due to the Yukawa potential are considered and discussed in the binary pulsar and extraterrestrial solar-system [45] and in the OPERA experiment around the Earth [46]. Based on the Solar System planetary motions given by ephemerides EPM2004 (IAA RAS, Russia) and and INPOP (IMCCE, France), the author of Refs. [47, 48] constrained the upper bounds on the Yukawa parameters with considerring the general relativistic gravitomagnetic effect of the Sun.

The periastron advance for binary pulsars can reach several degrees per year, which is \(\sim 10^5\) more than the perihelion advance of Mercury. Given that, the authors in Ref. [23] used \(\dot{\omega }\) of four well-observed binary pulsars data (PSR B1913+16 [38], PSR B1534+12 [39], PSR J0737-3039 [40], and PSR B2127+11C [41]) to constrain two Yukawa parameters: \(\alpha =(2.40\pm 0.02)\times 10^{-8}\) and \(1/m_s=(3.97\pm 0.01)\times 10^8\) m. Based on Eq. (5) and the result of Ref. [23], we obtain

$$\begin{aligned} \alpha= & {} (2.40\pm 0.02)\times 10^{-8}, \end{aligned}$$
(26)
$$\begin{aligned} m^{-1}_{0}= & {} (6.2\pm 0.5)\times 10^4~\textrm{m}, \end{aligned}$$
(27)

in the general free scalar–tensor gravity by using the four double neutron star binaries (see Table 1). The bound on the coupling strength \(\alpha \) is tighter than the previous wrong results constrained by Ref. [27] by about 3 orders of magnitude.

On the other hand, two high-precision ephemerides EPM2011 [42] and INPOP10a [43] were used to probe modified gravities [49,50,51,52] and planetary science [53, 54], and references therein. By employing the supplementary advances \(\dot{\omega }\) given by EPM2011 and INPOP10a ephemerides, the authors of Ref. [55] constrained the upper bounds on the Yukawa parameters \(\alpha \) and \(1/m_{s}\) based on Eq. (25), which are \(\alpha \lesssim 4\times 10^{-11}\) and \(1/m_s\lesssim 0.2\) au, then it gives

$$\begin{aligned} \alpha\lesssim & {} 4\times 10^{-11}, \end{aligned}$$
(28)
$$\begin{aligned} m^{-1}_{0}\lesssim & {} 1.9\times 10^5~\textrm{m}. \end{aligned}$$
(29)

in the general free scalar–tensor gravity according to Eq. (5) and the result of Ref. [55]. The bound on the coupling strength \(\alpha \) is tighter than the previous wrong results constrained by Ref. [27] by about 6 orders of magnitude.

It is worth stressing that real constraints can only be obtained by explicitly modeling the Yukawa (or whatsover modified) potential. And it will estimate the Yukawa parameters in global-solve-for fits to the ephemerides data. Simply comparing residuals made without modeling it to theoretical predictions like Ref. [55] can be yield just insights on the possibilities offered by ephemerides, not, strictly speaking, real constraints (see section 4 of Ref. [56] for more details).

A summary of our results are given in Table 1. In Ref. [27], the authors give the tightest bound on the coupling strength \(\alpha \le 10^{-5}\) mainly based on the Cassini tracking experiment for different values of the mediator mass \(m_0\). However, we find that the Cassini tracking experiment is insensitive to the Yukawa-type parameters of the general free scaler–tensor gravity. Due to the increase of the accuracy and precision of observations, in the present work, the bounds on two Yukawa-tpye parameters \(\alpha \) and \(m_0\) given by binary pulsars and the Solar System ephemerides are tighter than the previous wrong results constrained by Ref. [27] by about 3–6 orders of magnitude (see Table 1 for more details).

5 Conclusions and discussion

We consider the (un)testability of the general free scalar–tensor gravity for the Solar System tests through a more rigorous method. Unlike Zhang’s approach [27], in the present work, corresponding astronomical experiments have been physically modelled by fully considering the lightlike and the timelike geodesics in the gravity. The authors in Ref. [27] claimed that they gave the tightest bound on the gravity by the Cassini tracking experiment through borrowing the measured values of the parametrized post-Newtonian parameters.

Contrary to the statement of Ref. [27], we find that the Cassini tracking experiment is insensitive to the general free scaler–tensor gravity. Furthermore, it is also shown that the time delay and the light deflection are all independent of the gravity. However, using the secular periastron precession of binary pulsars and the supplementary perihelion precessions provided by EPM2011 and INPOP10a ephemerides, we derive very much improved bounds on the Yukawa-type parameters of this gravity. Table 1 lists our main results in this work.

Our work suggests that it should be treated with caution when the one want to use the measured values of the parametrized post-Newtonian parameters to constrain the model parameters for one gravity. In the standard parametrized post-Newtonian formalism pioneered by Will and Nordtvedt [13, 14], the parametrized post-Newtonian parameters, e.g. \(\gamma \), are all numerical values and independent of the radial distance r. In Zhang et al.’s work [27], however, the parametrized post-Newtonian parameter \(\gamma (r)\) they defined depends on r, which involves the light propagation along the ray path. Besides, it should be also emphasized that the parametrized post-Newtonian parameters are not measurable quantities. This is why we model corresponding measurable quantities for the various astronomical experiments and compare them with observations. Actually, it has been already pointed out by the authors in Refs. [57, 58] that a comparison between observations and the solutions of the field equations contains the light propagation’s solution, and it must be fully considered in measurable quantities.