One can also mention $\dot{G}/G$ estimations deduced, not directly from planetary ephemerides, but from the adjustment of one single planet orbit during the analysis of radar tracking data analysis of one given space mission. This is the case for example with Mercury and the MESSENGER data operated by Genova et al. (2018), noted MSG in Table 6. In this work, it has to be stressed that only Mercury orbit was considered in the analysis. Nevertheless, in all the cases produced in Table 6, no clear indication of a time variation of G is visible, despite the improvements of the planetary ephemerides and of the estimation of the solar mass loss (Pitjeva et al. 2021). Finally, other technics or methods have been used for measuring possible variations of the gravitational constant, G. One can cite for example, the determination of $\dot{G}/G$ deduced from pulsar timing (Stairs 2003; Zhu et al. 2015, 2018; Kramer et al. 2021). The obtained limit is then one order of magnitude greater (e.g. $|\dot{G}/G|<0.9 \times 10^{-12} \text{ yr}^{-1}$ from Zhu et al. 2018) than the one obtained with planetary ephemerides although the gravitational regimes are quite different (strong field regime for the pulsar measurements and weak field for planetary ephemerides).

## 4.4 Massive gravity

Unlike in electromagnetism—like with the Proca theory (de Rham 2014; Proca 1936)—, there is not a unique definition for what massive gravity might entail (de Rham 2014). In field theory, massive interactions typically result in a Yukawa suppression of these interactions at the scale of the Compton wavelength. However, due to its tensorial rather than vectorial nature, this may not necessarily be the case for a fully consistent theory of massive gravity (de Rham 2014). Nonetheless, from a phenomenological perspective, it is possible to test whether a Yukawa suppression of gravitational potentials occurs within the solar system. Formally, this would lead to the following modification of the Newtonian potential (Will 2018a)

$$w = w_{\text{Newton}} \exp(-r/\lambda_g), \qquad (145)$$

which can be developed as Bernus et al. (2019)

$$w = w_{\text{Newton}} \left( 1 + \frac{1}{2} \frac{r^2}{\lambda_g^2} \right) + \mathcal{O}(\lambda_g^{-3}), \qquad (146)$$

after a convenient change of coordinate system that absorbs the constant term in the gravitational potential—which has no impact on the observables.

As discussed in Sect 4.5, this modification is different from a fifth force, for which the new potential is an affine function of the Yukawa suppression instead of a linear function (Fischbach et al. 1992; Will 2014). Fifth forces usually originate from the existence of an additional gravitational field that is massive—e.g. a massive scalar field (Wagoner 1970; Hees et al. 2018)—rather than considering that the field equation on the metric perturbation itself has a mass term. Indeed, Eq. (145) is solution of a massive gravitational potential equation that reads
One can also mention $\dot{G}/G$ estimations deduced, not directly from planetary ephemerides, but from the adjustment of one single planet orbit during the analysis of radar tracking data analysis of one given space mission. This is the case for example with Mercury and the MESSENGER data operated by Genova et al. (2018), noted MSG in Table 6. In this work, it has to be stressed that only Mercury orbit was considered in the analysis. Nevertheless, in all the cases produced in Table 6, no clear indication of a time variation of G is visible, despite the improvements of the planetary ephemerides and of the estimation of the solar mass loss (Pitjeva et al. 2021). Finally, other technics or methods have been used for measuring possible variations of the gravitational constant, G. One can cite for example, the determination of $\dot{G}/G$ deduced from pulsar timing (Stairs 2003; Zhu et al. 2015, 2018; Kramer et al. 2021). The obtained limit is then one order of magnitude greater (e.g. $|\dot{G}/G|<0.9 \times 10^{-12} \text{ yr}^{-1}$ from Zhu et al. 2018) than the one obtained with planetary ephemerides although the gravitational regimes are quite different (strong field regime for the pulsar measurements and weak field for planetary ephemerides).

## 4.4 Massive gravity

Unlike in electromagnetism—like with the Proca theory (de Rham 2014; Proca 1936)—, there is not a unique definition for what massive gravity might entail (de Rham 2014). In field theory, massive interactions typically result in a Yukawa suppression of these interactions at the scale of the Compton wavelength. However, due to its tensorial rather than vectorial nature, this may not necessarily be the case for a fully consistent theory of massive gravity (de Rham 2014). Nonetheless, from a phenomenological perspective, it is possible to test whether a Yukawa suppression of gravitational potentials occurs within the solar system. Formally, this would lead to the following modification of the Newtonian potential (Will 2018a)

$$w = w_{\text{Newton}} \exp(-r/\lambda_g), \qquad (145)$$

which can be developed as Bernus et al. (2019)

$$w = w_{\text{Newton}} \left( 1 + \frac{1}{2} \frac{r^2}{\lambda_g^2} \right) + \mathcal{O}(\lambda_g^{-3}), \qquad (146)$$

after a convenient change of coordinate system that absorbs the constant term in the gravitational potential—which has no impact on the observables.

As discussed in Sect 4.5, this modification is different from a fifth force, for which the new potential is an affine function of the Yukawa suppression instead of a linear function (Fischbach et al. 1992; Will 2014). Fifth forces usually originate from the existence of an additional gravitational field that is massive—e.g. a massive scalar field (Wagoner 1970; Hees et al. 2018)—rather than considering that the field equation on the metric perturbation itself has a mass term. Indeed, Eq. (145) is solution of a massive gravitational potential equation that reads
$$ \Delta w - \frac{w}{\lambda_g^2} = -4\pi G\sigma. \qquad (147) $$

Obviously, as long as $\lambda_g$ is big enough, the gravitational phenomenology in the Newtonian regime can reduce to the one of general relativity to any given level of accuracy. Also, if $\lambda_g$ is large enough that only the leading order correction in Eq. (146) has a significant contribution to the metric, one can assume that only the Newtonian part of the post-Newtonian expansion is modified with respect to the equations of motion in general relativity Eq. (90).

In that situation, the equation of motion only has one extra term with respect to the usual EIHDL equation Eq. (96) that reads Bernus et al. (2019)

$$ \delta a_A^{\lambda_g} = \frac{1}{2\lambda_g^2} \sum_{A \neq T} \frac{\mu_T}{r_{AT}} \mathbf{r}_{AT} + \mathcal{O}(\lambda_g^{-3}), \qquad (148) $$

and further assuming that light still propagates along null geodesics, the Shapiro delay reads (Bernus et al. 2019)

$$ c(t_r - t_e) = c(t_r - t_e)_{\text{GRT}} + \sum_A \frac{\mu_A}{c^2} \frac{1}{2\lambda_g^2} \ln \left[ b^2 \frac{\mathbf{n} \cdot \mathbf{r}_{rA} + r_{rA}}{\mathbf{n} \cdot \mathbf{r}_{eA} + r_{eA}} + \mathbf{n} \cdot (r_{rA} \mathbf{r}_{rA} - r_{eA} \mathbf{r}_{eA}) \right], \qquad (149) $$

where $c(t_r - t_e)_{\text{GRT}}$ corresponds to the general relativity light time given in Eq. (94), $b$ is the minimal distance between the light path and the central body (here the Sun). This expression is an approximation at $c^{-2}$ level, the additional terms induced by gravitational field mass being negligible relative to the present day accuracy for commonly admitted $\lambda_g$ (with $\lambda_g > 2.8 \times 10^{12}$ km (Will 2018a, 1998)).

Likewise, from Eq. (146), the difference between a clock $A$ and a BCRS time $t$ still is

$$ \frac{d\tau_A}{dt} = 1 - \frac{1}{c^2} \left[ \frac{v_A^2}{2} + U(\mathbf{x}_A) \right], \qquad (150) $$

up to terms of order $\mathcal{O}(\lambda_g^{-2}c^{-2})$. For instance, assuming a conservative bound of $\lambda_g > 2 \times 10^{12}$ km, the correction to Eq. (150) at the surface of the Earth would be less than $10^{-17}(\sim R_{\oplus}^2/\lambda_g^2$, where $R_{\oplus}$ is the radius of the Earth) time less than the contribution of the last term in the bracket of Eq. (150)—that is, far beyond what can be achieved with present clocks stability (Guena et al. 2012).

By analogy with standard quantum physics, the Compton length can also be interpreted in terms of a mass of the graviton $m_g$ following the relation:

$$ \lambda_g = \frac{\hbar}{cm_g}, \qquad (151) $$

with $\hbar$ the Planck constant, and $c$ the speed of light. Will (2018a) proposed to use solar system ephemerides to improve the constraints on $\lambda_g$ in the Newtonian limit. The starting point was that a massive gravitational field should lead to a modification
$$ \Delta w - \frac{w}{\lambda_g^2} = -4\pi G\sigma. \qquad (147) $$

Obviously, as long as $\lambda_g$ is big enough, the gravitational phenomenology in the Newtonian regime can reduce to the one of general relativity to any given level of accuracy. Also, if $\lambda_g$ is large enough that only the leading order correction in Eq. (146) has a significant contribution to the metric, one can assume that only the Newtonian part of the post-Newtonian expansion is modified with respect to the equations of motion in general relativity Eq. (90).

In that situation, the equation of motion only has one extra term with respect to the usual EIHDL equation Eq. (96) that reads Bernus et al. (2019)

$$ \delta a_A^{\lambda_g} = \frac{1}{2\lambda_g^2} \sum_{A \neq T} \frac{\mu_T}{r_{AT}} \mathbf{r}_{AT} + \mathcal{O}(\lambda_g^{-3}), \qquad (148) $$

and further assuming that light still propagates along null geodesics, the Shapiro delay reads (Bernus et al. 2019)

$$ c(t_r - t_e) = c(t_r - t_e)_{\text{GRT}} + \sum_A \frac{\mu_A}{c^2} \frac{1}{2\lambda_g^2} \ln \left[ b^2 \frac{\mathbf{n} \cdot \mathbf{r}_{rA} + r_{rA}}{\mathbf{n} \cdot \mathbf{r}_{eA} + r_{eA}} + \mathbf{n} \cdot (r_{rA} \mathbf{r}_{rA} - r_{eA} \mathbf{r}_{eA}) \right], \qquad (149) $$

where $c(t_r - t_e)_{\text{GRT}}$ corresponds to the general relativity light time given in Eq. (94), $b$ is the minimal distance between the light path and the central body (here the Sun). This expression is an approximation at $c^{-2}$ level, the additional terms induced by gravitational field mass being negligible relative to the present day accuracy for commonly admitted $\lambda_g$ (with $\lambda_g > 2.8 \times 10^{12}$ km (Will 2018a, 1998)).

Likewise, from Eq. (146), the difference between a clock A and a BCRS time t still is

$$ \frac{d\tau_A}{dt} = 1 - \frac{1}{c^2} \left[ \frac{v_A^2}{2} + U(\mathbf{x}_A) \right], \qquad (150) $$

up to terms of order $\mathcal{O}(\lambda_g^{-2}c^{-2})$. For instance, assuming a conservative bound of $\lambda_g > 2 \times 10^{12}$ km, the correction to Eq. (150) at the surface of the Earth would be less than $10^{-17}(\sim R_{\oplus}^2/\lambda_g^2$, where $R_{\oplus}$ is the radius of the Earth) time less than the contribution of the last term in the bracket of Eq. (150)—that is, far beyond what can be achieved with present clocks stability (Guena et al. 2012).

By analogy with standard quantum physics, the Compton length can also be interpreted in terms of a mass of the graviton $m_g$ following the relation:

$$ \lambda_g = \frac{\hbar}{cm_g}, \qquad (151) $$

with $\hbar$ the Planck constant, and $c$ the speed of light. Will (2018a) proposed to use solar system ephemerides to improve the constraints on $\lambda_g$ in the Newtonian limit. The starting point was that a massive gravitational field should lead to a modification
of the perihelion advance of solar system bodies. Hence, based on current constraints on the perihelion advance of Mars derived from Mars Reconnaissance Orbiter (MRO) data, Will estimates that the Compton wavelength should be bigger than $(1.4 - 2.7) \times 10^{14}$km (resp. $m_g < (4 - 8) \times 10^{-24}$ eV/c²), depending on the specific analysis. However, making an estimation from quantities derived from ephemerides that assumed other theorical frameworks—here, the perihela advances per orbit and their uncertainty in either general relativity or PPN frameworks—cannot account for the fact that the mass of the graviton is correlated to the various parameters of the ephemeris (e.g. masses, semi-major axes etc.). While a graviton with a non-zero mass may impact the solar system dynamics, so also does a change of the various other parameters of the ephemerides. Because of the correlation between $m_g$ (or $\lambda_g$) and other parameters, any modification induced by a non-null value of $m_g$ may—at least in part—be reabsorbed by the modification of other parameters of the ephemerides.

In order to overcome this issue, Bernus et al. (2019, 2020) and Mariani et al. (2023) have built planetary ephemerides fully developed in the massive gravity framework of Eq. (145) and fitted over the data sample of INPOP17a, INPOP19a and INPOP21a respectively. The results of these investigations are given on Table 7.

Bernus et al. (2019, 2020) had used a method of random walk exploration that is more conservative that the Monte Carlo Markov Chain (MCMC) algorithm used by Mariani et al. (2023). With the same random walk exploration method, but using the updated INPOP21a ephemerides, Mariani et al. (2023) obtained a constraint that is 3 times smaller than Bernus et al. (2020). This improvement is induced by the use in INPOP21a of the latest Juno and Mars orbiter tracking data up to 2020 as well as a fit of the Moon-Earth system to LLR observations also up to 2020. By improving the procedure with MCMC, Mariani et al. (2023) was able to push the limit of detection of the mass of the graviton at a new level, with a constraint at $1.01 \times 10^{-24}$ eVc⁻² (resp. $\lambda_g \ge 122.48 \times 10^{13}$ km) with a 99.7% confidence level.

It is somewhat interesting to compare these constraints to the ones deduced from the observation of gravitational waves. Indeed, it is assumed that a massive gravitational field that leads to Eq. (147) might also modify the dispersion relation of gravitational waves as follows (Abbott et al. 2021; Will 1998)

$$E^2 = p^2 c^2 + m_g c^2, \qquad (152)$$

where E and p are the energy and momentum of the wave.

Such a modified dispersion relation causes gravitational waves frequency modes to propagate at different speeds, leading to an overall modification of the phase morphology of gravitational waves with respect to the general relativity predictions. Since the morphology of gravitational wave phase has been consistent with general relativity so far, it led to severe constraints on the value of $m_g$ that are reproduced on Table 7.

Even if a massive gravity theory actually leads to both the phenomenological consequences represented in Eqs. (147) and (152), there is absolutely no reason for the constraints from ephemerides on the one hand, and from gravitational waves on the other hand, to be at the same level accuracy.
of the perihelion advance of solar system bodies. Hence, based on current constraints on the perihelion advance of Mars derived from Mars Reconnaissance Orbiter (MRO) data, Will estimates that the Compton wavelength should be bigger than $(1.4 - 2.7) \times 10^{14}$km (resp. $m_g < (4 - 8) \times 10^{-24}$ eV/c²), depending on the specific analysis. However, making an estimation from quantities derived from ephemerides that assumed other theorical frameworks—here, the perihela advances per orbit and their uncertainty in either general relativity or PPN frameworks—cannot account for the fact that the mass of the graviton is correlated to the various parameters of the ephemeris (e.g. masses, semi-major axes etc.). While a graviton with a non-zero mass may impact the solar system dynamics, so also does a change of the various other parameters of the ephemerides. Because of the correlation between $m_g$ (or $\lambda_g$) and other parameters, any modification induced by a non-null value of $m_g$ may—at least in part—be reabsorbed by the modification of other parameters of the ephemerides.

In order to overcome this issue, Bernus et al. (2019, 2020) and Mariani et al. (2023) have built planetary ephemerides fully developed in the massive gravity framework of Eq. (145) and fitted over the data sample of INPOP17a, INPOP19a and INPOP21a respectively. The results of these investigations are given on Table 7.

Bernus et al. (2019, 2020) had used a method of random walk exploration that is more conservative that the Monte Carlo Markov Chain (MCMC) algorithm used by Mariani et al. (2023). With the same random walk exploration method, but using the updated INPOP21a ephemerides, Mariani et al. (2023) obtained a constraint that is 3 times smaller than Bernus et al. (2020). This improvement is induced by the use in INPOP21a of the latest Juno and Mars orbiter tracking data up to 2020 as well as a fit of the Moon-Earth system to LLR observations also up to 2020. By improving the procedure with MCMC, Mariani et al. (2023) was able to push the limit of detection of the mass of the graviton at a new level, with a constraint at $1.01 \times 10^{-24}$ eVc⁻² (resp. $\lambda_g \ge 122.48 \times 10^{13}$ km) with a 99.7% confidence level.

It is somewhat interesting to compare these constraints to the ones deduced from the observation of gravitational waves. Indeed, it is assumed that a massive gravitational field that leads to Eq. (147) might also modify the dispersion relation of gravitational waves as follows (Abbott et al. 2021; Will 1998)

$$E^2 = p^2 c^2 + m_g c^2, \qquad (152)$$

where E and p are the energy and momentum of the wave.

Such a modified dispersion relation causes gravitational waves frequency modes to propagate at different speeds, leading to an overall modification of the phase morphology of gravitational waves with respect to the general relativity predictions. Since the morphology of gravitational wave phase has been consistent with general relativity so far, it led to severe constraints on the value of $m_g$ that are reproduced on Table 7.

Even if a massive gravity theory actually leads to both the phenomenological consequences represented in Eqs. (147) and (152), there is absolutely no reason for the constraints from ephemerides on the one hand, and from gravitational waves on the other hand, to be at the same level accuracy.
<table><caption>**Table 7** Limits obtained for the Compton length $\lambda_g$ in km as defined in Eqs. (148) and (149). Are also given the corresponding values in term graviton mass $m_g$ in eV/c². Are also indicated, for comparisons, the values obtained with INPOP17a (Bernus et al. 2019), INPOP19a (Bernus et al. 2020), and INPOP21a (Mariani et al. 2023) as well as the estimations for the dynamical mode from Virgo-Ligo GWTC-1 and GWTC-3 (Abbott et al. 2019, 2021). For INPOP21a, two values at 90% confidence level (CL) are given: the one indicated in the column *RW* corresponds to value obtained with the same method (random walk exploration) as Bernus et al. (2019) and Bernus et al. (2020) and the one given in Column *MC* corresponds to MCMC results</caption><thead><tr><th rowspan="2">CL</th><th rowspan="2">GWTC-1</th><th rowspan="2">GWTC-3</th><th rowspan="2">INPOP17a</th><th rowspan="2">INPOP19a</th><th colspan="2">INPOP21a</th></tr><tr><th>RW</th><th>MCMC</th></tr></thead><tbody><tr><td></td><td>0.90</td><td>0.90</td><td>0.90</td><td>0.90</td><td>0.9</td><td>0.9</td></tr><tr><td>Graviton mass</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>λ<sub>g</sub> × 10<sup>-13</sup> [km]</td><td>2.6</td><td>9.77</td><td>1.83</td><td>3.93</td><td>12.01</td><td>209.67</td></tr><tr><td>m<sub>g</sub> × 10<sup>23</sup> [eV/c<sup>2</sup>]</td><td>4.7</td><td>1.27</td><td>6.76</td><td>3.16</td><td>1.03</td><td>0.059</td></tr><tr><td>Fifth force</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>-λ/√|α| × 10<sup>-13</sup> [km], α > 0</td><td></td><td></td><td>1.83</td><td>3.93</td><td></td><td></td></tr><tr><td>-λ/√|α| × 10<sup>-13</sup> [km], α < 0</td><td></td><td></td><td></td><td>3.77</td><td></td><td></td></tr></tbody></table>

Each type of constraints is relevant on its own right given that they test different phenomenologies—that is, Eq. (145) versus Eq. (152)—which may (or may not) be related, depending on the underlying massive gravity theory that one is considering. For instance, screening mechanisms that kick-in for high density environments—such as the Vainshtein mechanism (Babichev and Deffayet 2013)—may impact Eq. (145) and not Eq. (152). This notably seems to be the case for ghost-free massive gravity (de Rham 2014).

## 4.5 Yukawa potential and fifth force

Whereas one can imagine that the metric field itself has a mass—see Sect. 4.4—it is also possible to imagine the existence of additional gravitational fields that are massive, while the metric field would remain massless. In such situations, the potential often.<fnref n="32" /> becomes an affine function of the Yukawa suppression as follows (Wagoner 1970; Fischbach et al. 1992; Will 2014)

$$w = w_{\text{Newton}}(1 + \alpha \exp(-r/\lambda)), \qquad (153)$$

where $\alpha$ is the strength (relative to gravity) and $\lambda$ the range of the force. This type of modifications of the Newtonian potential is often referred to as a *fifth force* (Will 2014). Depending on whether the additional massive field couples universally to matter or not, the fifth force can be either composition-dependent or independent (Will 2014).

Solar system tests—such as the ones realised with planetary ephemerides—usually focus on composition independent models. Konopliv et al. (2011), constraints

<footnote n="32">³² But not always (Hees et al. 2018)</footnote>
<table><caption>**Table 7** Limits obtained for the Compton length $\lambda_g$ in km as defined in Eqs. (148) and (149). Are also given the corresponding values in term graviton mass $m_g$ in eV/c$^2$. Are also indicated, for comparisons, the values obtained with INPOP17a (Bernus et al. 2019), INPOP19a (Bernus et al. 2020), and INPOP21a (Mariani et al. 2023) as well as the estimations for the dynamical mode from Virgo-Ligo GWTC-1 and GWTC-3 (Abbott et al. 2019, 2021). For INPOP21a, two values at 90% confidence level (CL) are given: the one indicated in the column *RW* corresponds to value obtained with the same method (random walk exploration) as Bernus et al. (2019) and Bernus et al. (2020) and the one given in Column *MC* corresponds to MCMC results</caption><thead><tr><th rowspan="2">CL</th><th rowspan="2">GWTC-1</th><th rowspan="2">GWTC-3</th><th rowspan="2">INPOP17a</th><th rowspan="2">INPOP19a</th><th colspan="2">INPOP21a</th></tr><tr><th>RW</th><th>MCMC</th></tr></thead><tbody><tr><td></td><td>0.90</td><td>0.90</td><td>0.90</td><td>0.90</td><td>0.9</td><td>0.9</td></tr><tr><td>Graviton mass</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>λ<sub>g</sub> × 10<sup>-13</sup> [km]</td><td>2.6</td><td>9.77</td><td>1.83</td><td>3.93</td><td>12.01</td><td>209.67</td></tr><tr><td>m<sub>g</sub> × 10<sup>23</sup> [eV/c<sup>2</sup>]</td><td>4.7</td><td>1.27</td><td>6.76</td><td>3.16</td><td>1.03</td><td>0.059</td></tr><tr><td>Fifth force</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>-λ/√|α| × 10<sup>-13</sup> [km], α > 0</td><td></td><td></td><td>1.83</td><td>3.93</td><td></td><td></td></tr><tr><td>-λ/√|α| × 10<sup>-13</sup> [km], α < 0</td><td></td><td></td><td></td><td>3.77</td><td></td><td></td></tr></tbody></table>

Each type of constraints is relevant on its own right given that they test different phenomenologies—that is, Eq. (145) versus Eq. (152)—which may (or may not) be related, depending on the underlying massive gravity theory that one is considering. For instance, screening mechanisms that kick-in for high density environments—such as the Vainshtein mechanism (Babichev and Deffayet 2013)—may impact Eq. (145) and not Eq. (152). This notably seems to be the case for ghost-free massive gravity (de Rham 2014).

## 4.5 Yukawa potential and fifth force

Whereas one can imagine that the metric field itself has a mass—see Sect. 4.4—it is also possible to imagine the existence of additional gravitational fields that are massive, while the metric field would remain massless. In such situations, the potential often.<fnref n="32" /> becomes an affine function of the Yukawa suppression as follows (Wagoner 1970; Fischbach et al. 1992; Will 2014)

$$w = w_{\text{Newton}}(1 + \alpha \exp(-r/\lambda)), \qquad (153)$$

where $\alpha$ is the strength (relative to gravity) and $\lambda$ the range of the force. This type of modifications of the Newtonian potential is often referred to as a *fifth force* (Will 2014). Depending on whether the additional massive field couples universally to matter or not, the fifth force can be either composition-dependent or independent (Will 2014).

Solar system tests—such as the ones realised with planetary ephemerides—usually focus on composition independent models. Konopliv et al. (2011), constraints

<footnote n="32">³² But not always (Hees et al. 2018)</footnote>
<figure><img src="image_12.png" /><figcaption>**Fig. 10** Observational constraints obtained for the Yukawa potential extracted from A03, standing for Adelberger et al. (2003b). The green lines give the constraints deduced from the INPOP planetary ephemerides graviton tests presented in Table 7: I17a stands for Bernus et al. (2019), I19a for Bernus et al. (2020), I21a and I21a+MC for Mariani et al. (2023). K11 indicates the limits deduced from Mars tracking data analysis by Konopliv et al. (2011)</figcaption></figure>

on the Yukawa potential were deduced from the analysis of Mars orbiter tracking data and from the construction of a Mars updated ephemeris. Figure 10 shows that already in 2011, the constraints from planetary ephemerides were at the level of the constraints from lunar ephemerides. As explained in Bernus et al. (2019), in the limit $\lambda \gg r$ and with $\alpha > 0$, one can use the constraints given in Sect 4.4 for the test of massive graviton, in order to deduce constraints on the Yukawa potential. These constraints are given in Table 7 and Fig. 10. The green lines on this figure show the improvements of the new generation of planetary ephemerides relative to the one use in Konopliv et al. (2011), as well as the new limits obtained at solar system scale. A new limit (labelled *IBC*) obtained by simulating the introduction of BepiColombo MORE experiments as predicted by De Marchi and Cascioli (2020) and Fienga et al. (2022) is also indicated. Let note that, because fifth force models in Eq. (153) depend on two parameters ($\alpha$ and $\lambda$), whereas the massive graviton models in Eq. (145) depend on only one ($\lambda_g$), the mapping between the two breaks down for small Compton wavelength. This explains why the green lines in Fig. 10 are restricted to the right part of the plot.
<figure><img src="image_14.png" /><figcaption>**Fig. 10** Observational constraints obtained for the Yukawa potential extracted from A03, standing for Adelberger et al. (2003b). The green lines give the constraints deduced from the INPOP planetary ephemerides graviton tests presented in Table 7: I17a stands for Bernus et al. (2019), I19a for Bernus et al. (2020), I21a and I21a+MC for Mariani et al. (2023). K11 indicates the limits deduced from Mars tracking data analysis by Konopliv et al. (2011)</figcaption></figure>

on the Yukawa potential were deduced from the analysis of Mars orbiter tracking data and from the construction of a Mars updated ephemeris. Figure 10 shows that already in 2011, the constraints from planetary ephemerides were at the level of the constraints from lunar ephemerides. As explained in Bernus et al. (2019), in the limit $\lambda \gg r$ and with $\alpha > 0$, one can use the constraints given in Sect 4.4 for the test of massive graviton, in order to deduce constraints on the Yukawa potential. These constraints are given in Table 7 and Fig. 10. The green lines on this figure show the improvements of the new generation of planetary ephemerides relative to the one use in Konopliv et al. (2011), as well as the new limits obtained at solar system scale. A new limit (labelled IBC) obtained by simulating the introduction of BepiColombo MORE experiments as predicted by De Marchi and Cascioli (2020) and Fienga et al. (2022) is also indicated. Let note that, because fifth force models in Eq. (153) depend on two parameters ($\alpha$ and $\lambda$), whereas the massive graviton models in Eq. (145) depend on only one ($\lambda_g$), the mapping between the two breaks down for small Compton wavelength. This explains why the green lines in Fig. 10 are restricted to the right part of the plot.
## 4.6 Einstein-dilaton theories

The case of Einstein-dilaton theories is somewhat very interesting for the phenomenology of alternative theories in the solar system, because it allows one to consistently derive the equations of motion, Shapiro delay and conserved quantities in a framework that leads to a violation of both the WEP and the GWEP.

Einstein-dilaton theories are scalar-tensor theories that violate the WEP at the fundamental level, because they possess at least one massless scalar-field that couples non-minimally to matter fields. They are somewhat expected as being the low energy effective gravitational action of string theories (Damour and Polyakov 1994; Damour et al. 2002)<fnref n="33" />—and other higher than four dimensional theories (Overduin and Wesson 1997)—which generically predict the existence of one (or several) additional scalar-field(s) that mediates gravity: the dilaton field (and the moduli fields that come from the compactification of the extra-dimensions in string theories). Tests of Einstein-dilaton theories with planetary motions have been proposed by Damour and Donoghue (2011) based on the equations of Damour and Donoghue (2010). The first results for the massless dilaton with planetary ephemerides have been published in Bernus et al. (2022).

### 4.6.1 Equations of motion and Shapiro delay

The general action for an Einstein-dilaton theory<fnref n="34" /> may be written as (Minazzoli and Hees 2016; Bernus et al. 2022)

$$ S[g, \psi_i, \varphi] = \frac{1}{2\kappa\kappa} \int \left( f(\varphi)R - \frac{\omega(\varphi)}{\varphi} \varphi^\mu \varphi_{,\mu} \right) \sqrt{-g} \, d^4 x \\ + \frac{1}{c} \int (\mathcal{L}_{\text{SM}}[g, \psi_i] + \mathcal{L}_{\text{int}}[g, \psi_i, \varphi]) \sqrt{-g} \, d^4 x \quad (154) $$

where $\mathcal{L}_{\text{SM}}$ is the Lagrangian density of matter described by the standard model of particle physics and $\mathcal{L}_{\text{int}}$ the Lagrangian density of the interactions between the dilaton field $\varphi$ and matter. Such interactions can be parametrized by arbitrary functions of the scalar-field as follows

$$ \mathcal{L}_{\text{int}} = -\frac{D_e(\varphi)\beta_e(e)}{2e} F_{\mu\nu}F^{\mu\nu} - \frac{D_g(\varphi)\beta_3(g_3)}{2g_3} G_{\mu\nu}^a G_a^{\mu\nu} \\ - \sum_{i=e,u,d} (D_{m_i}(\varphi) + \gamma_{m_i} D_g(\varphi)) m_i \bar{\psi}_i \psi_i \quad (155) $$

where $F_{\mu\nu}$ is the Faraday tensor, $G_{\mu\nu}^a$ is the gluons tensor, $e$ and $g_3$ are respectively the

<footnote n="33">³³ Let us note, however, that the current consensus among string theorists is that all the scalar fields acquire a potential through the moduli stabilization mechanism (Douglas and Kachru 2007), such that they mediate gravity with a very small range instead of being of infinite range—unlike the part of gravity that is mediated by the metric field.</footnote>

<footnote n="34">³⁴ With a massless dilaton field. Note that Damour and Donoghue (2010) assume that the phenomenology they derive is valid for light dilaton fields as well. However, even a minute mass can lead to an entirely different phenomenology in the solar system, as one can check in Hees et al. (2018).</footnote>
## 4.6 Einstein-dilaton theories

The case of Einstein-dilaton theories is somewhat very interesting for the phenomenology of alternative theories in the solar system, because it allows one to consistently derive the equations of motion, Shapiro delay and conserved quantities in a framework that leads to a violation of both the WEP and the GWEP.

Einstein-dilaton theories are scalar-tensor theories that violate the WEP at the fundamental level, because they possess at least one massless scalar-field that couples non-minimally to matter fields. They are somewhat expected as being the low energy effective gravitational action of string theories (Damour and Polyakov 1994; Damour et al. 2002)<fnref n="33" />—and other higher than four dimensional theories (Overduin and Wesson 1997)—which generically predict the existence of one (or several) additional scalar-field(s) that mediates gravity: the dilaton field (and the moduli fields that come from the compactification of the extra-dimensions in string theories). Tests of Einstein-dilaton theories with planetary motions have been proposed by Damour and Donoghue (2011) based on the equations of Damour and Donoghue (2010). The first results for the massless dilaton with planetary ephemerides have been published in Bernus et al. (2022).

### 4.6.1 Equations of motion and Shapiro delay

The general action for an Einstein-dilaton theory<fnref n="34" /> may be written as (Minazzoli and Hees 2016; Bernus et al. 2022)

$$ S[g, \psi_i, \varphi] = \frac{1}{2\kappa\kappa} \int \left( f(\varphi)R - \frac{\omega(\varphi)}{\varphi} \varphi^\mu \varphi_{,\mu} \right) \sqrt{-g} \, d^4 x \\ + \frac{1}{c} \int (\mathcal{L}_{\text{SM}}[g, \psi_i] + \mathcal{L}_{\text{int}}[g, \psi_i, \varphi]) \sqrt{-g} \, d^4 x \qquad (154) $$

where $\mathcal{L}_{\text{SM}}$ is the Lagrangian density of matter described by the standard model of particle physics and $\mathcal{L}_{\text{int}}$ the Lagrangian density of the interactions between the dilaton field $\varphi$ and matter. Such interactions can be parametrized by arbitrary functions of the scalar-field as follows

$$ \mathcal{L}_{\text{int}} = -\frac{D_e(\varphi)\beta_e(e)}{2e} F_{\mu\nu}F^{\mu\nu} - \frac{D_g(\varphi)\beta_3(g_3)}{2g_3} G_{\mu\nu}^a G_a^{\mu\nu} \\ - \sum_{i=e,u,d} (D_{m_i}(\varphi) + \gamma_{m_i} D_g(\varphi)) m_i \bar{\psi}_i \psi_i \qquad (155) $$

where $F_{\mu\nu}$ is the Faraday tensor, $G_{\mu\nu}^a$ is the gluons tensor, $e$ and $g_3$ are respectively the

<footnote n="33">³³ Let us note, however, that the current consensus among string theorists is that all the scalar fields acquire a potential through the moduli stabilization mechanism (Douglas and Kachru 2007), such that they mediate gravity with a very small range instead of being of infinite range—unlike the part of gravity that is mediated by the metric field.</footnote>

<footnote n="34">³⁴ With a massless dilaton field. Note that Damour and Donoghue (2010) assume that the phenomenology they derive is valid for light dilaton fields as well. However, even a minute mass can lead to an entirely different phenomenology in the solar system, as one can check in Hees et al. (2018).</footnote>
photons and the gluons coupling constants, $\beta_e(e) = \lambda\delta\ln e/\partial\lambda$ and $\beta_3(g_3) = \lambda\delta\ln g_3/\partial\lambda$ are their respective beta functions relative to the quantum scale invariance violation, where $\lambda$ is the energy scale of the considered physical processes, $m_i$ is the fermion mass, $\psi_i$ their spinor, and $\gamma_{m_i} = -\lambda\delta\ln m_i/\partial\lambda$ is the beta function relative to the dimensional anomaly of the fermion masses coupled to the gluons. The $D_i(\varphi)$ functions describe the different couplings between the matter fields and the dilaton. This Lagrangian density is a straightforward non-linear generalisation of the action considered by Damour and Donoghue (2010).$^{35}$

Assuming a linear coupling—such as in Damour and Donoghue (2010)—and at leading order in the composition dependent effects, the acceleration reads (Bernus et al. 2022)

$$ 
\begin{aligned}
\mathbf{a}_T = & -\sum_{A \neq T} \frac{\mu_A}{r_{AT}^3} \mathbf{r}_{AT} (1 + \delta_T + \delta_{AT}) - \sum_{A \neq T} \frac{\mu_A}{r_{AT}^3 c^2} \mathbf{r}_{AT} \left\{ \gamma v_T^2 + (\gamma + 1) v_A^2 \right. \\
& \left. - 2(1 + \gamma) \mathbf{v}_A \cdot \mathbf{v}_T - \frac{3}{2} \left( \frac{\mathbf{r}_{AT} \cdot \mathbf{v}_A}{r_{AT}} \right)^2 - \frac{1}{2} \mathbf{r}_{AT} \cdot \mathbf{a}_A - 2\gamma \sum_{B \neq T} \frac{\mu_B}{r_{TB}} + \sum_{B \neq A} \frac{\mu_B}{r_{AB}} \right\} \\
& + \sum_{A \neq T} \frac{\mu_A}{c^2 r_{AT}^3} \left[ 2(1 + \gamma) \mathbf{r}_{AT} \cdot \mathbf{v}_T - (1 + 2\gamma) \mathbf{r}_{AT} \cdot \mathbf{v}_A \right] (\mathbf{v}_T - \mathbf{v}_A) \\
& \left. + \frac{3 + 4\gamma}{2} \sum_{A \neq T} \frac{\mu_A}{c^2 r_{AT}^3} \mathbf{a}_A \right\}
\end{aligned}
\quad (156) $$

Equation (156) depends on $\gamma = (1 - \alpha_0^2)/(1 + \alpha_0^2)$ and $\delta_A = \delta_A^d + \delta_A^N$ with $\delta_A^d = \alpha_0\tilde{\alpha}_A/(1 + \alpha_0^2)$ and $\delta_A^N = (\gamma - 1)\omega_A |/\tilde{m}_A c^2$ where $\omega_A$ is the self-gravitational energy of the body A. The fundamental parameters on top of which they are built are $\alpha_0$, the universal coupling constant, and $\tilde{\alpha}_A = d_{\tilde{m}}Q_{\tilde{m}}^A + d_{\tilde{\delta}m}Q_{\tilde{\delta}m}^A + d_{m_e}Q_{m_e}^A + d_eQ_e^A$, where $Q_{\tilde{m}}^A, Q_{\tilde{\delta}m}^A, Q_{m_e}^A$, and $Q_e^A$ are the dilatonic charges, estimated according to the composition of the considered bodies. In these equations, A stands for the planet to consider, $\mu_A$ being its gravitational parameter. Bernus et al. (2022) simplify the problem by considering two average charges for the telluric and gaseous bodies only, because of the similar dilatonic charges for these two classes of objects. This approximation leads to the reduction of the number of the tested parameters from 10 to 3 for the linear coupling case: $\alpha_0, \tilde{\alpha}_T, \tilde{\alpha}_G$, for which T and G stand for telluric and

<footnote>$^{35}$ Although note that in Damour and Donoghue (2010), the dilaton fields do not couple to all the trace terms—that is, the classical part of the trace in addition to all the relevant quantum trace anomalies. However, it has been shown in Nitti and Piazza (2012) that it is much more convenient to consider the parametrization in Eq. (155) because it recovers the fact that in the limit of metric theories, the dilaton field couples to the total trace, as it should. Indeed, with this parametrization, metric theories corresponds to $D_i = D_j\sqrt{i,j}$. The fact that in metric theories, any gravitational scalar degree of freedom must couple to the total trace is a property of conformal couplings. This is consistent to the fact that the mass of a composite object equals the total trace of the fields that compose the particle due to the constraint that the internal stresses all vanish (this is true even if some of the internal forces do not contribute to the trace, such as classical electromagnetism) (Nitti 2022).</footnote>
photons and the gluons coupling constants, $\beta_e(e) = \lambda\partial\ln e/\partial\lambda$ and $\beta_3(g_3) = \lambda\partial\ln g_3/\partial\lambda$ are their respective beta functions relative to the quantum scale invariance violation, where $\lambda$ is the energy scale of the considered physical processes, $m_i$ is the fermion mass, $\psi_i$ their spinor, and $\gamma_{m_i} = -\lambda\partial\ln m_i/\partial\lambda$ is the beta function relative to the dimensional anomaly of the fermion masses coupled to the gluons. The $D_i(\varphi)$ functions describe the different couplings between the matter fields and the dilaton. This Lagrangian density is a straightforward non-linear generalisation of the action considered by Damour and Donoghue (2010).$^{35}$

Assuming a linear coupling—such as in Damour and Donoghue (2010)—and at leading order in the composition dependent effects, the acceleration reads (Bernus et al. 2022)

$$ 
\begin{aligned}
\mathbf{a}_T = & -\sum_{A \neq T} \frac{\mu_A}{r_{AT}^3} \mathbf{r}_{AT} (1 + \delta_T + \delta_{AT}) - \sum_{A \neq T} \frac{\mu_A}{r_{AT}^3 c^2} \mathbf{r}_{AT} \left\{ \gamma v_T^2 + (\gamma + 1)v_A^2 \right. \\
& \left. - 2(1 + \gamma)\mathbf{v}_A \cdot \mathbf{v}_T - \frac{3}{2} \left( \frac{\mathbf{r}_{AT} \cdot \mathbf{v}_A}{r_{AT}} \right)^2 - \frac{1}{2} \mathbf{r}_{AT} \cdot \mathbf{a}_A - 2\gamma \sum_{B \neq T} \frac{\mu_B}{r_{TB}} + \sum_{B \neq A} \frac{\mu_B}{r_{AB}} \right\} \\
& + \sum_{A \neq T} \frac{\mu_A}{c^2 r_{AT}^3} \left[ 2(1 + \gamma)\mathbf{r}_{AT} \cdot \mathbf{v}_T - (1 + 2\gamma)\mathbf{r}_{AT} \cdot \mathbf{v}_A \right] (\mathbf{v}_T - \mathbf{v}_A) \\
& \left. + \frac{3 + 4\gamma}{2} \sum_{A \neq T} \frac{\mu_A}{c^2 r_{AT}^3} \mathbf{a}_A \right\}
\end{aligned}
\quad (156) $$

Equation (156) depends on $\gamma = (1 - \alpha_0^2)/(1 + \alpha_0^2)$ and $\delta_A = \delta_A^d + \delta_A^N$ with $\delta_A^d = \alpha_0 \tilde{\alpha}_A / (1 + \alpha_0^2)$ and $\delta_A^N = (\gamma - 1)\omega_A / \tilde{m}_A c^2$ where $\omega_A$ is the self-gravitational energy of the body A. The fundamental parameters on top of which they are built are $\alpha_0$, the universal coupling constant, and $\tilde{\alpha}_A = d_{\tilde{m}} Q_{\tilde{m}}^A + d_{\tilde{\delta}m} Q_{\tilde{\delta}m}^A + d_{m_e} Q_{m_e}^A + d_e Q_e^A$, where $Q_{\tilde{m}}^A$, $Q_{\tilde{\delta}m}^A$, $Q_{m_e}^A$, and $Q_e^A$ are the dilatonic charges, estimated according to the composition of the considered bodies. In these equations, A stands for the planet to consider, $\mu_A$ being its gravitational parameter. Bernus et al. (2022) simplify the problem by considering two average charges for the telluric and gaseous bodies only, because of the similar dilatonic charges for these two classes of objects. This approximation leads to the reduction of the number of the tested parameters from 10 to 3 for the linear coupling case: $\alpha_0, \tilde{\alpha}_T, \tilde{\alpha}_G$, for which T and G stand for telluric and

<footnote>$^{35}$ Although note that in Damour and Donoghue (2010), the dilaton fields do not couple to all the trace terms—that is, the classical part of the trace in addition to all the relevant quantum trace anomalies. However, it has been shown in Nitti and Piazza (2012) that it is much more convenient to consider the parametrization in Eq. (155) because it recovers the fact that in the limit of metric theories, the dilaton field couples to the total trace, as it should. Indeed, with this parametrization, metric theories corresponds to $D_i = D_j \sqrt{i,j}$. The fact that in metric theories, any gravitational scalar degree of freedom must couple to the total trace is a property of conformal couplings. This is consistent to the fact that the mass of a composite object equals the total trace of the fields that compose the particle due to the constraint that the internal stresses all vanish (this is true even if some of the internal forces do not contribute to the trace, such as classical electromagnetism) (Nitti 2022).</footnote>
gaseous bodies respectively. Let note that $\mu_A$ is here the product of $G$ with the gravitational mass $m_A^G$.

From there, one has to also account for different effects, such as the Nordvedt effect that (in the case of the linear coupling) reads $\delta_A^N = -(1-\gamma)\frac{3\mu_A}{5R_Ac^2}$ with $R_A$ the planet radius, or the modification of the time travel that, at the required level of accuracy, reads (Bernus et al. 2022)

$$c(t_r - t_e) = R + \sum_A (1 + \gamma - \delta_A) \frac{\mu_A}{c^2} \ln \frac{\mathbf{n} \cdot \mathbf{r}_{rA} + r_{rA} + \frac{4\mu_A}{c^2}}{\mathbf{n} \cdot \mathbf{r}_{eA} + r_{eA} + \frac{4\mu_A}{c^2}}. \quad (157)$$

Let us note that

$$(1 + \gamma - \delta_A)\mu_A = (1 + \gamma)\mu_A^I, \quad (158)$$

at the required level of accuracy, where $\mu_A^I$ is the gravitational parameter constructed with the inertial mass, such that $\mu_A^I = (1 - \delta_A)\mu_A$. This means that the mass involved in the Shapiro delay is the inertial mass and not the gravitational mass—as already discussed in Sect. 4.2.5.

### 4.6.2 Conserved quantities and the definition of the SSB

From the Lagrangian formulation of the equations of motion, Bernus et al. (2022) show that the following barycenter constant vector is a first integral of the equations of motion

$$q = G - Vt, \quad (159)$$

where

$$G = \frac{c^2}{h} \sum_A \mu_A z_A \left( 1 - \delta_A + \frac{v_A^2}{2c^2} - \frac{1}{2c^2} \sum_{B \neq A} \frac{\mu_B}{r_{AB}} \right) \quad (160)$$

are the coordinates of the relativistic barycenter of the system and

$$V = \frac{c^2 \mathbf{P}}{h} \quad (161)$$

is the velocity of the barycenter motion. $h$ is the conserved energy—whose value does not affect what follows but can be found in Bernus et al. (2022)—and $\mathbf{P}$ is the conserved linear momentum that reads
gaseous bodies respectively. Let note that $\mu_A$ is here the product of $G$ with the gravitational mass $m_A^G$.

From there, one has to also account for different effects, such as the Nordvedt effect that (in the case of the linear coupling) reads $\delta_A^N = -(1 - \gamma) \frac{3\mu_A}{5R_A c^2}$ with $R_A$ the planet radius, or the modification of the time travel that, at the required level of accuracy, reads (Bernus et al. 2022)

$$c(t_r - t_e) = R + \sum_A (1 + \gamma - \delta_A) \frac{\mu_A}{c^2} \ln \frac{\mathbf{n} \cdot \mathbf{r}_{rA} + r_{rA} + \frac{4\mu_A}{c^2}}{\mathbf{n} \cdot \mathbf{r}_{eA} + r_{eA} + \frac{4\mu_A}{c^2}}. \quad (157)$$

Let us note that

$$(1 + \gamma - \delta_A)\mu_A = (1 + \gamma)\mu_A^I, \quad (158)$$

at the required level of accuracy, where $\mu_A^I$ is the gravitational parameter constructed with the inertial mass, such that $\mu_A^I = (1 - \delta_A)\mu_A$. This means that the mass involved in the Shapiro delay is the inertial mass and not the gravitational mass—as already discussed in Sect. 4.2.5.

### 4.6.2 Conserved quantities and the definition of the SSB

From the Lagrangian formulation of the equations of motion, Bernus et al. (2022) show that the following barycenter constant vector is a first integral of the equations of motion

$$\mathbf{q} = \mathbf{G} - Vt, \quad (159)$$

where

$$\mathbf{G} = \frac{c^2}{h} \sum_A \mu_A \mathbf{z}_A \left( 1 - \delta_A + \frac{v_A^2}{2c^2} - \frac{1}{2c^2} \sum_{B \neq A} \frac{\mu_B}{r_{AB}} \right) \quad (160)$$

are the coordinates of the relativistic barycenter of the system and

$$V = \frac{c^2 \mathbf{P}}{h} \quad (161)$$

is the velocity of the barycenter motion. $h$ is the conserved energy—whose value does not affect what follows but can be found in Bernus et al. (2022)—and $\mathbf{P}$ is the conserved linear momentum that reads
<table><caption>**Table 8** Intervals of possible values for the 3 dilaton parameters as defined in Bernus et al. (2022): $\alpha_0$, the universal coupling, $\alpha_T$ the telluric planet coupling and $\alpha_G$ the gaseous planet coupling</caption><thead><tr><th rowspan="2">Confidence</th><th colspan="2">INPOP19a (Bernus et al. 2022)</th></tr><tr><th>90%</th><th>99.5%</th></tr></thead><tbody><tr><td>α<sub>0</sub>(×10<sup>5</sup>)</td><td>-0.94 ± 5.35</td><td>1.01 ± 23.7</td></tr><tr><td>α<sub>T</sub>(×10<sup>6</sup>)</td><td>0.24 ± 1.62</td><td>0.00 ± 24.5</td></tr><tr><td>α<sub>G</sub>(×10<sup>5</sup>)</td><td>0.01 ± 4.38</td><td>-1.46 ± 12.0</td></tr><tr><td>(γ - 1) × 10<sup>8</sup></td><td>0.2 ± 6</td><td>0.2 ± 11.2</td></tr></tbody></table>

$$ 
\begin{aligned}
\boldsymbol{P} &= \sum_A \boldsymbol{p}_A \\
&= \sum_A \mu_A \boldsymbol{v}_A \left[ 1 - \delta_A + \frac{1}{2c^2} \left( v_A^2 - \sum_{B \neq A} \frac{\mu_B}{r_{AB}} \right) \right] \\
&\quad - \frac{1}{2c^2} \sum_A \sum_{B \neq A} \frac{\mu_A \mu_B}{r_{AB}} (\boldsymbol{n}_{AB} \cdot \boldsymbol{v}_A) \boldsymbol{n}_{AB}.
\end{aligned} 
\quad (162) 
$$

From Eq. (162), one can see that the sum at leading order is over the gravitational parameter based on the inertial mass $\mu_A^I$, because one has $\mu_A^I = \mu_A(1 - \delta_A)$ at leading order. This can be used in order to check what has been discussed in Sect. 4.2.4. Let us note that it is consistent with Klioner (2016), and with Damour and Vokrouhlický (1995) in the $\delta_A = 0$ limit.

### 4.6.3 Results

In Bernus et al. (2022), the planetary ephemerides used were INPOP19a (Fienga et al. 2019) and, as in Fienga et al. (2020b), after integrating Eq. (156) for all bodies, the orbits are adjusted to planetary observations and statistical criteria (cost functions) are applied for selecting the distribution of the tested parameters for which deduced ephemerides are compatible with instrumental uncertainties. The obtained results for the linear coupling parameters are given in Table 8.

At $3-\sigma$, Bernus et al. (2022) obtained constraints at the level of $10^{-4}$ for $\alpha_0$ and $\alpha_G$ and $10^{-5}$ for $\alpha_T$. These results reflect the better accuracy reached for the telluric planets in planetary ephemerides thanks mainly to Mars orbiters. They should be improved by the future Bepi-Colombo measurements of Mercury orbit.

## 4.7 MOND

The modified Newtonian dynamics (MOND) is a framework that modifies Newtonian dynamics $\boldsymbol{a} = \boldsymbol{g}_N$ (where $\boldsymbol{a}$ is the acceleration of a test particle and $\boldsymbol{g}_N$ is the Newtonian gravitational field) by $\boldsymbol{a} = \boldsymbol{g}$ with (Milgrom 1983, 2014)
<table><caption>**Table 8** Intervals of possible values for the 3 dilaton parameters as defined in Bernus et al. (2022): $\alpha_0$, the universal coupling, $\alpha_T$ the telluric planet coupling and $\alpha_G$ the gaseous planet coupling</caption><thead><tr><th rowspan="2">Confidence</th><th colspan="2">INPOP19a (Bernus et al. 2022)</th></tr><tr><th>90%</th><th>99.5%</th></tr></thead><tbody><tr><td>α<sub>0</sub>(×10<sup>5</sup>)</td><td>-0.94 ± 5.35</td><td>1.01 ± 23.7</td></tr><tr><td>α<sub>T</sub>(×10<sup>6</sup>)</td><td>0.24 ± 1.62</td><td>0.00 ± 24.5</td></tr><tr><td>α<sub>G</sub>(×10<sup>5</sup>)</td><td>0.01 ± 4.38</td><td>-1.46 ± 12.0</td></tr><tr><td>(γ - 1) × 10<sup>8</sup></td><td>0.2 ± 6</td><td>0.2 ± 11.2</td></tr></tbody></table>

$$ 
\begin{aligned}
\boldsymbol{P} &= \sum_A \boldsymbol{p}_A \\
&= \sum_A \mu_A \boldsymbol{v}_A \left[ 1 - \delta_A + \frac{1}{2c^2} \left( v_A^2 - \sum_{B \neq A} \frac{\mu_B}{r_{AB}} \right) \right] \\
&\quad - \frac{1}{2c^2} \sum_A \sum_{B \neq A} \frac{\mu_A \mu_B}{r_{AB}} (\boldsymbol{n}_{AB} \cdot \boldsymbol{v}_A) \boldsymbol{n}_{AB}.
\end{aligned} 
\quad (162) 
$$

From Eq. (162), one can see that the sum at leading order is over the gravitational parameter based on the inertial mass $\mu_A^I$, because one has $\mu_A^I = \mu_A(1 - \delta_A)$ at leading order. This can be used in order to check what has been discussed in Sect. 4.2.4. Let us note that it is consistent with Klioner (2016), and with Damour and Vokrouhlický (1995) in the $\delta_A = 0$ limit.

### 4.6.3 Results

In Bernus et al. (2022), the planetary ephemerides used were INPOP19a (Fienga et al. 2019) and, as in Fienga et al. (2020b), after integrating Eq. (156) for all bodies, the orbits are adjusted to planetary observations and statistical criteria (cost functions) are applied for selecting the distribution of the tested parameters for which deduced ephemerides are compatible with instrumental uncertainties. The obtained results for the linear coupling parameters are given in Table 8.

At $3-\sigma$, Bernus et al. (2022) obtained constraints at the level of $10^{-4}$ for $\alpha_0$ and $\alpha_G$ and $10^{-5}$ for $\alpha_T$. These results reflect the better accuracy reached for the telluric planets in planetary ephemerides thanks mainly to Mars orbiters. They should be improved by the future Bepi-Colombo measurements of Mercury orbit.

## 4.7 MOND

The modified Newtonian dynamics (MOND) is a framework that modifies Newtonian dynamics $\boldsymbol{a} = \boldsymbol{g}_N$ (where $\boldsymbol{a}$ is the acceleration of a test particle and $\boldsymbol{g}_N$ is the Newtonian gravitational field) by $\boldsymbol{a} = \boldsymbol{g}$ with (Milgrom 1983, 2014)
$$ \mu \mathbf{g} = \mathbf{g}_N, \qquad (163) $$

where the interpolating function $\mu$ is a function of the ratio $g/a_0$ between the norm of the gravitational field $g$ and the MOND acceleration scale $a_0$. It was developed as an alternative explanation of the galactic rotation curves and the empirical Tully-Fisher relation without relying on dark matter haloes (Milgrom 1983). Several attempts to verify this framework at scales different from the galactic one have been made in the past years (Milgrom 2009; Blanchet and Novak 2011; Magueijo and Bekenstein 2007; Skordis and Złośnik 2021). In the solar system, three main consequences of the MOND phenomenology have been studied (Milgrom 1983; Hees et al. 2014a). Two of them have been shown to be negligible considering uncertainties of planetary mean motions (Sereno and Jetzer 2006). The last one is not negligible however, and is known as the External Field Effect (EFE) (Milgrom 2009, 2010; Blanchet and Novak 2011). The effect stems from the non-linearity of MOND equations, and—at leading order in the multipole expansion (Blanchet and Novak 2011)—it induces an anomalous quadrupolar correction to the Newtonian potential $\delta\Phi(Q_2)$ that reads (Milgrom 2009)

$$ \delta\Phi(Q_2) = -\frac{Q_2}{2} r^2 \left( \cos^2\theta - \frac{1}{3} \right), \qquad (164) $$

where $\theta$ could be the angle pointing either toward the Galactic center, or toward the Newtonian galactic field—depending on the model considered (Hees et al. 2014b). This factor $Q_2$ changes with the shape of the MOND coupling functions and, in the solar system, Blanchet and Novak (2011) estimated that the values of $Q_2$ can vary from $3.8 \times 10^{-26}$ to $4.1 \times 10^{-26}$ s$^{-2}$, and then evaluated that it would therefore lead to additional advances of perihelia up to $5.81 \text{ mas.cy}^{-1}$ for Saturn and even bigger values for Uranus ($-10.94 \text{ mas.cy}^{-1}$). Milgrom (2009) had also proposed additional Saturn perihelion precession rate of about $1.8 \text{ mas.cy}^{-1}$. Nevertheless—as notably emphasized in Blanchet and Novak (2011)—it is not consistent to take into account only a particular MOND effect, like perihelion precession, and to compare it with constraints obtained in other frameworks, such as the PPN framework. To tackle this issue, Hees et al. (2014b) re-adjusted the parameters of planetary ephemeris when the EFE is taken into account. Hees et al. (2014b), the anomalous quadrupolar correction enters directly as a modification of the space-time metric that reads:

$$ ds^2 = \left( -1 + \frac{2\mu}{c^2r} - 2\delta\Phi(Q_2) + 2\left(\frac{\mu}{c^2r}\right)^2 \right) c^2 dt^2 + \left( 1 + \frac{2\mu}{c^2r} + 2\delta\Phi(Q_2) \right) dl^2, \qquad (165) $$

with $dl^2 \equiv dx^2 + dy^2 + dz^2$, because this is what one ought to expect from a relativistic realisation of MOND dynamics. Using this new definition of the metric, Hees et al. (2014b) introduced modifications into the EIHDL equations of motion and in the Shapiro delay. An alternative planetary ephemeris has been built in this new framework and has been adjusted to observations. A specific focus has been brought
$$ \mu \mathbf{g} = \mathbf{g}_N, \qquad (163) $$

where the interpolating function $\mu$ is a function of the ratio $g/a_0$ between the norm of the gravitational field $g$ and the MOND acceleration scale $a_0$. It was developed as an alternative explanation of the galactic rotation curves and the empirical Tully-Fisher relation without relying on dark matter haloes (Milgrom 1983). Several attempts to verify this framework at scales different from the galactic one have been made in the past years (Milgrom 2009; Blanchet and Novak 2011; Magueijo and Bekenstein 2007; Skordis and Złośnik 2021). In the solar system, three main consequences of the MOND phenomenology have been studied (Milgrom 1983; Hees et al. 2014a). Two of them have been shown to be negligible considering uncertainties of planetary mean motions (Sereno and Jetzer 2006). The last one is not negligible however, and is known as the External Field Effect (EFE) (Milgrom 2009, 2010; Blanchet and Novak 2011). The effect stems from the non-linearity of MOND equations, and—at leading order in the multipole expansion (Blanchet and Novak 2011)—it induces an anomalous quadrupolar correction to the Newtonian potential $\delta\Phi(Q_2)$ that reads (Milgrom 2009)

$$ \delta\Phi(Q_2) = -\frac{Q_2}{2} r^2 \left( \cos^2\theta - \frac{1}{3} \right), \qquad (164) $$

where $\theta$ could be the angle pointing either toward the Galactic center, or toward the Newtonian galactic field—depending on the model considered (Hees et al. 2014b). This factor $Q_2$ changes with the shape of the MOND coupling functions and, in the solar system, Blanchet and Novak (2011) estimated that the values of $Q_2$ can vary from $3.8 \times 10^{-26}$ to $4.1 \times 10^{-26}$ s$^{-2}$, and then evaluated that it would therefore lead to additional advances of perihelia up to $5.81 \text{ mas.cy}^{-1}$ for Saturn and even bigger values for Uranus ($-10.94 \text{ mas.cy}^{-1}$). Milgrom (2009) had also proposed additional Saturn perihelion precession rate of about $1.8 \text{ mas.cy}^{-1}$. Nevertheless—as notably emphasized in Blanchet and Novak (2011)—it is not consistent to take into account only a particular MOND effect, like perihelion precession, and to compare it with constraints obtained in other frameworks, such as the PPN framework. To tackle this issue, Hees et al. (2014b) re-adjusted the parameters of planetary ephemeris when the EFE is taken into account. Hees et al. (2014b), the anomalous quadrupolar correction enters directly as a modification of the space-time metric that reads:

$$ ds^2 = \left( -1 + \frac{2\mu}{c^2r} - 2\delta\Phi(Q_2) + 2\left(\frac{\mu}{c^2r}\right)^2 \right) c^2 dt^2 + \left( 1 + \frac{2\mu}{c^2r} + 2\delta\Phi(Q_2) \right) dl^2, \qquad (165) $$

with $dl^2 \equiv dx^2 + dy^2 + dz^2$, because this is what one ought to expect from a relativistic realisation of MOND dynamics. Using this new definition of the metric, Hees et al. (2014b) introduced modifications into the EIHDL equations of motion and in the Shapiro delay. An alternative planetary ephemeris has been built in this new framework and has been adjusted to observations. A specific focus has been brought
on the Saturn orbit as it is supposed to be the most affected by the MOND modification (as also estimated by Hees et al. (2014b),) but also the most constrained by accurate observations (in this case Cassini tracking data). By fitting directly $Q_2$ with the rest of the planetary parameters, Hees et al. (2014b) obtained a constraint on $Q_2$ of about $(3 \pm 3) \times 10^{-27} \text{ s}^{-2}$, which excludes a important interval of $Q_2$ values computed theoretically in Blanchet and Novak (2011) for various MOND interpolating functions $\mu$. Considering other coupling functions, it is still possible to maintain a MOND formalism in the solar system but at the cost of very limited possibilities of couplings. This result is consistent with Milgrom (2009) for which the predicted perihelion precession for Saturn, estimated for coupling functions $\mu_\alpha$ with $\alpha > 1/3$, was significantly smaller than in Blanchet and Novak (2011).

# 5 Inconsistent tests with ephemeris outputs

In the literature, it is possible to find a wide range of *constraints* supposedly deduced from planetary ephemerides. There are of two types:

The first type of tests are those considering residuals obtained after the modifications of the equations of motion of the planetary systems—by, for example, introducing additional terms produced by general relativity-like linearization of an alternative theory—but without re-adjusting the newly modified dynamical system to observations. This approach is equivalent to not considering any attempt to improve the ephemerides as described in Fig. 4 in Sect. 2.2. It is clearly an issue because any modification of the dynamical model must be re-adjusted to observations before any interpretation of the obtained residuals. In essence, any comparison of a simulation, which presupposed the parameters of the solar system bodies inferred in a different theoretical framework, with the observations, will likely overestimate the deviations in the new theoretical framework. This would primarily be because the simulation did not use the most appropriate set of parameters for the solar system bodies-which are, those that minimize the residuals, and which are obtained through readjustment in a given framework. We can refer the reader to the extensive discussion of this very basic concept applied to the massive gravity problem and notably presented in the supplementary materials of Bernus et al. (2019).

The second category of indirect tests involves what we will term as *derived quantities*, as explained in Sect. 5.1. We will discuss these indirect tests in detail in the upcoming section.

## 5.1 Definitions

Several quantities can be provided as outputs of a given planetary ephemeris, notably in order to give an idea of the accuracy of that ephemeris with respect to specific aspects of the solar system phenomenology—such as, for instance, the values of the perihelion and node advances per orbit for a given astronomical body, and their uncertainties; or the secular variation of the gravitational constant $\hat{G}/G$ and its uncertainty. We shall call those quantities *derived quantities* in what follows.
on the Saturn orbit as it is supposed to be the most affected by the MOND modification (as also estimated by Hees et al. (2014b),) but also the most constrained by accurate observations (in this case Cassini tracking data). By fitting directly $Q_2$ with the rest of the planetary parameters, Hees et al. (2014b) obtained a constraint on $Q_2$ of about $(3 \pm 3) \times 10^{-27} \text{ s}^{-2}$, which excludes an important interval of $Q_2$ values computed theoretically in Blanchet and Novak (2011) for various MOND interpolating functions $\mu$. Considering other coupling functions, it is still possible to maintain a MOND formalism in the solar system but at the cost of very limited possibilities of couplings. This result is consistent with Milgrom (2009) for which the predicted perihelion precession for Saturn, estimated for coupling functions $\mu_\alpha$ with $\alpha > 1/3$, was significantly smaller than in Blanchet and Novak (2011).

# 5 Inconsistent tests with ephemeris outputs

In the literature, it is possible to find a wide range of *constraints* supposedly deduced from planetary ephemerides. There are of two types:

The first type of tests are those considering residuals obtained after the modifications of the equations of motion of the planetary systems—by, for example, introducing additional terms produced by general relativity-like linearization of an alternative theory—but without re-adjusting the newly modified dynamical system to observations. This approach is equivalent to not considering any attempt to improve the ephemerides as described in Fig. 4 in Sect. 2.2. It is clearly an issue because any modification of the dynamical model must be re-adjusted to observations before any interpretation of the obtained residuals. In essence, any comparison of a simulation, which presupposed the parameters of the solar system bodies inferred in a different theoretical framework, with the observations, will likely overestimate the deviations in the new theoretical framework. This would primarily be because the simulation did not use the most appropriate set of parameters for the solar system bodies-which are, those that minimize the residuals, and which are obtained through readjustment in a given framework. We can refer the reader to the extensive discussion of this very basic concept applied to the massive gravity problem and notably presented in the supplementary materials of Bernus et al. (2019).

The second category of indirect tests involves what we will term as *derived quantities*, as explained in Sect. 5.1. We will discuss these indirect tests in detail in the upcoming section.

## 5.1 Definitions

Several quantities can be provided as outputs of a given planetary ephemeris, notably in order to give an idea of the accuracy of that ephemeris with respect to specific aspects of the solar system phenomenology—such as, for instance, the values of the perihelion and node advances per orbit for a given astronomical body, and their uncertainties; or the secular variation of the gravitational constant $\hat{G}/G$ and its uncertainty. We shall call those quantities *derived quantities* in what follows.
Derived quantities could also be the values and uncertainties of parameters such as the post-Newtonian parameters $\gamma$ and $\beta$, or of the Compton wavelength in massive gravity, or in the context of a *fifth force*. Table 9 gives a non-exhaustive series of examples of possible interpretations of planetary ephemeris derived quantities for testing alternative theories. It falls outside the scope of this review to detail theories that have not been confronted and re-adjusted against planetary observations but have only considered the derived quantities defined above. The reason for limiting this review to fully tested theories, as presented in Sect. 4, is our belief that the published constraints listed in Table 9 yield to unrealistic constraints, as we will explain in Sects. 5.3 and 5.4. In particular, considering only derived quantities or un-fitted residuals is equivalent to neglecting that these latest were obtained in a given framework (usually general relativity), with instrumental uncertainties and correlations between parameters.

## 5.2 The case of the advance of the perihelia and nodes

It is traditional when one investigates possible laws of gravitation in the solar system to follow the Einstein's steps and to consider supplementary advances in the planetary orbital angles (mainly perihelion and node). Several methods have been developed in the past fifty years in order to estimate possible remaining advances in planetary perihelia and nodes that can be fully explained by general relativity.

The most direct method (e.g. presented in Pitjeva and Pitjev 2013) is the adjustment of a quantity $\Delta\omega$ or $\Delta\Omega$ (respectively supplementary advances of perihelia and node) for all of the planetary orbits or only for some of them, together with the rest of the planetary and relativistic parameters (masses, Sun oblateness, initial conditions, PPN parameters...). One can get an intuition from Eq. (100) that the fit of such derived quantities is affected by strong correlations between various parameters of the ephemeris, such that the results are plagued with biases and underestimated uncertainties.

The second approach is to introduce possible rotations of the planetary planes while considering $\Delta\omega$ or $\Delta\Omega$ fixed, and to build new planetary ephemerides integrated with these fixed additional rotations and fitted to observations. The result is then the limit of possible rotations that one can add without degrading the planetary residuals (see Fienga et al. 2018). The advantage with the method is the uncorrelated estimation of maximum value for $\Delta\omega$ or $\Delta\Omega$, the drawback is that what is obtained is only an upper bound.

Finally, a third method consists in averaging planetary orbits (Park et al. 2017) for obtaining residual precession of the perihelia. In this case, as for the two former methods, the deduced residual precession could be induced by a violation of general relativity, but also—and more likely—by some other sources of uncertainties (i.e unmodeled asteroid perturbations).

Table 10 gathers some of the recently obtained values for $\Delta\omega$, following the three methods described above. However, based on the arguments presented in this Sect. 5, the direct interpretation of these quantities, $\Delta\omega$ or $\Delta\Omega$, in terms of possible violation of general relativity is strongly discouraged.
Derived quantities could also be the values and uncertainties of parameters such as the post-Newtonian parameters $\gamma$ and $\beta$, or of the Compton wavelength in massive gravity, or in the context of a *fifth force*. Table 9 gives a non-exhaustive series of examples of possible interpretations of planetary ephemeris derived quantities for testing alternative theories. It falls outside the scope of this review to detail theories that have not been confronted and re-adjusted against planetary observations but have only considered the derived quantities defined above. The reason for limiting this review to fully tested theories, as presented in Sect. 4, is our belief that the published constraints listed in Table 9 yield to unrealistic constraints, as we will explain in Sects. 5.3 and 5.4. In particular, considering only derived quantities or un-fitted residuals is equivalent to neglecting that these latest were obtained in a given framework (usually general relativity), with instrumental uncertainties and correlations between parameters.

## 5.2 The case of the advance of the perihelia and nodes

It is traditional when one investigates possible laws of gravitation in the solar system to follow the Einstein's steps and to consider supplementary advances in the planetary orbital angles (mainly perihelion and node). Several methods have been developed in the past fifty years in order to estimate possible remaining advances in planetary perihelia and nodes that can be fully explained by general relativity.

The most direct method (e.g. presented in Pitjeva and Pitjev 2013) is the adjustment of a quantity $\Delta\omega$ or $\Delta\Omega$ (respectively supplementary advances of perihelia and node) for all of the planetary orbits or only for some of them, together with the rest of the planetary and relativistic parameters (masses, Sun oblateness, initial conditions, PPN parameters...). One can get an intuition from Eq. (100) that the fit of such derived quantities is affected by strong correlations between various parameters of the ephemeris, such that the results are plagued with biases and underestimated uncertainties.

The second approach is to introduce possible rotations of the planetary planes while considering $\Delta\omega$ or $\Delta\Omega$ fixed, and to build new planetary ephemerides integrated with these fixed additional rotations and fitted to observations. The result is then the limit of possible rotations that one can add without degrading the planetary residuals (see Fienga et al. 2018). The advantage with the method is the uncorrelated estimation of maximum value for $\Delta\omega$ or $\Delta\Omega$, the drawback is that what is obtained is only an upper bound.

Finally, a third method consists in averaging planetary orbits (Park et al. 2017) for obtaining residual precession of the perihelia. In this case, as for the two former methods, the deduced residual precession could be induced by a violation of general relativity, but also—and more likely—by some other sources of uncertainties (i.e unmodeled asteroid perturbations).

Table 10 gathers some of the recently obtained values for $\Delta\omega$, following the three methods described above. However, based on the arguments presented in this Sect. 5, the direct interpretation of these quantities, $\Delta\omega$ or $\Delta\Omega$, in terms of possible violation of general relativity is strongly discouraged.
<table><caption>**Table 9** Examples of interpretation of ephemeris derived parameters. $d\dot{\tau}_{supp}$ and $d\dot{\Omega}_{supp}$ indicate the supplementary advances in perihelia and nodes respectively</caption><thead><tr><th>Theories</th><th>Sections</th><th>Impact on orbits</th><th>References</th></tr></thead><tbody><tr><td>MOND</td><td>4.7</td><td>$d\dot{\tau}_{supp}$, $d\dot{\Omega}_{supp}$</td><td>Blanchet and Novak (2011)</td></tr><tr><td>AWE/chameleons</td><td></td><td>variation of PPN parameters</td><td>Füzfa and Alimi (2007)</td></tr><tr><td></td><td></td><td>EP</td><td>Burrage and Sakstein (2018)</td></tr><tr><td>Scalar field theories</td><td></td><td>$\dot{G}/G$</td><td>Uzan (2003)</td></tr><tr><td></td><td></td><td>Variation of of PPN parameters</td><td></td></tr><tr><td>Dark energy</td><td></td><td>$\dot{G}/G$</td><td>Steinhardt and Wesley (2009)</td></tr><tr><td>Dark matter</td><td>5.2</td><td>linear drift of AU</td><td>Arakida (2010)</td></tr><tr><td></td><td></td><td>$a_{supp}$</td><td>Nordtvedt (1994)</td></tr><tr><td></td><td></td><td>$d\dot{\tau}_{supp}$, $d\dot{\Omega}_{supp}$</td><td>Frère et al. (2008)</td></tr><tr><td>Yukawa, fifth force</td><td>4.5</td><td>$d\dot{\tau}_{supp}$</td><td>Merkowitz (2010)</td></tr><tr><td>f(r)</td><td></td><td>$a_{supp}$</td><td>de Felice and Tsujikawa (2010)</td></tr><tr><td></td><td></td><td>Variation of of PPN parameters</td><td></td></tr><tr><td>Massive graviton</td><td>4.4</td><td>$d\dot{\tau}_{supp}$, $d\dot{\Omega}_{supp}$</td><td>Will (2018a)</td></tr></tbody></table>

### Estimation of the dark matter density in the solar system

An example of misleading conclusion can be taken from the dark matter density estimation. Pitjeva and Pitjev (2013), direct estimations of the density of the dark matter inside the orbit of Saturn have been tested using two different implementations. The first one consists in adding an additional acceleration to the equations of motion of the EPM planetary ephemerides such as:

$$ \mathbf{a}_A^{DM} = \frac{\mu_{M(R)}}{r_{AT}} \mathbf{r}_{AT}, \qquad (166) $$

where $\mu_{M(R)}$ is the gravitational parameter of an additional matter in a sphere of radius $R$ around the Sun. It turned out that such a direct modeling is highly affected by the uncertainties induced by the asteroid masses and no conclusive measurement of the mass of dark matter inside the solar system has been obtained in using Eq. (166). The second attempt in Pitjeva and Pitjev (2013) was from the secular advance of perihelia $d\varpi_A^{DM}$ following the equation from Khriplovich and Pitjeva (2006)

$$ \Delta\varpi_A^{DM} = -3\pi \frac{\rho_{DM}}{M_{sun}} \sqrt{1 - e_A^2}, \qquad (167) $$

where $\Delta\varpi_A^{DM}$ is the supplementary advance of perihelia of the planet A induced by dark matter of density $\rho_{DM}$, supposed uniformly distributed at the planetary distances, $e_A$ being the eccentricity of the orbit. The most stringent constraint gives a density for the dark matter up to the Saturn orbit of about $\rho_{DM} < 1.1 \times 10^{-20} \text{ g cm}^{-3}$, leading to a dark matter mass smaller than $7.1 \times 10^{-11} \text{ solar mass}$. It is interesting to note that this estimation is 5 orders of magnitude higher than the density for local dark matter halo proposed by McMillan (2011), Weber and de Boer (2010) and
<table><caption>**Table 9** Examples of interpretation of ephemeris derived parameters. $d\dot{\tau}_{supp}$ and $d\dot{\Omega}_{supp}$ indicate the supplementary advances in perihelia and nodes respectively</caption><thead><tr><th>Theories</th><th>Sections</th><th>Impact on orbits</th><th>References</th></tr></thead><tbody><tr><td>MOND</td><td>4.7</td><td>$d\dot{\tau}_{supp}$, $d\dot{\Omega}_{supp}$</td><td>Blanchet and Novak (2011)</td></tr><tr><td>AWE/chameleons</td><td></td><td>variation of PPN parameters</td><td>Füzfa and Alimi (2007)</td></tr><tr><td></td><td></td><td>EP</td><td>Burrage and Sakstein (2018)</td></tr><tr><td>Scalar field theories</td><td></td><td>$\dot{G}/G$</td><td>Uzan (2003)</td></tr><tr><td></td><td></td><td>Variation of of PPN parameters</td><td></td></tr><tr><td>Dark energy</td><td></td><td>$\dot{G}/G$</td><td>Steinhardt and Wesley (2009)</td></tr><tr><td>Dark matter</td><td>5.2</td><td>linear drift of AU</td><td>Arakida (2010)</td></tr><tr><td></td><td></td><td>$a_{supp}$</td><td>Nordtvedt (1994)</td></tr><tr><td></td><td></td><td>$d\dot{\tau}_{supp}$, $d\dot{\Omega}_{supp}$</td><td>Frère et al. (2008)</td></tr><tr><td>Yukawa, fifth force</td><td>4.5</td><td>$d\dot{\tau}_{supp}$</td><td>Merkowitz (2010)</td></tr><tr><td>f(r)</td><td></td><td>$a_{supp}$</td><td>de Felice and Tsujikawa (2010)</td></tr><tr><td></td><td></td><td>Variation of of PPN parameters</td><td></td></tr><tr><td>Massive graviton</td><td>4.4</td><td>$d\dot{\tau}_{supp}$, $d\dot{\Omega}_{supp}$</td><td>Will (2018a)</td></tr></tbody></table>

### Estimation of the dark matter density in the solar system

An example of misleading conclusion can be taken from the dark matter density estimation. Pitjeva and Pitjev (2013), direct estimations of the density of the dark matter inside the orbit of Saturn have been tested using two different implementations. The first one consists in adding an additional acceleration to the equations of motion of the EPM planetary ephemerides such as:

$$ \mathbf{a}_A^{DM} = \frac{\mu_{M(R)}}{r_{AT}} \mathbf{r}_{AT}, \qquad (166) $$

where $\mu_{M(R)}$ is the gravitational parameter of an additional matter in a sphere of radius $R$ around the Sun. It turned out that such a direct modeling is highly affected by the uncertainties induced by the asteroid masses and no conclusive measurement of the mass of dark matter inside the solar system has been obtained in using Eq. (166). The second attempt in Pitjeva and Pitjev (2013) was from the secular advance of perihelia $d\varpi_A^{DM}$ following the equation from Khriplovich and Pitjeva (2006)

$$ \Delta\varpi_A^{DM} = -3\pi \frac{\rho_{DM}}{M_{sun}} \sqrt{1 - e_A^2}, \qquad (167) $$

where $\Delta\varpi_A^{DM}$ is the supplementary advance of perihelia of the planet A induced by dark matter of density $\rho_{DM}$, supposed uniformly distributed at the planetary distances, $e_A$ being the eccentricity of the orbit. The most stringent constraint gives a density for the dark matter up to the Saturn orbit of about $\rho_{DM} < 1.1 \times 10^{-20} \text{ g cm}^{-3}$, leading to a dark matter mass smaller than $7.1 \times 10^{-11} \text{ solar mass}$. It is interesting to note that this estimation is 5 orders of magnitude higher than the density for local dark matter halo proposed by McMillan (2011), Weber and de Boer (2010) and
<table><caption>**Table 10** 1-σ uncertainties (mas/yr) on the perihelion advance per orbit. These values were extracted from Pitjeva and Pitjev (2013)—obtained with a full fit of the ephemeris including the perihelion advances— from Fienga et al. (2018) and Fiengaat et al. (2011b)—obtained with fixed values of the maximum advances—and from Park et al. (2017)—obtained by studying averaged Mercury orbit</caption><thead><tr><th>Planet</th><th>Pitjeva and Pitjev (2013)</th><th>Fienga et al. (2011b)</th></tr></thead><tbody><tr><td>Mercury</td><td>0.03</td><td>0.006</td></tr><tr><td>Venus</td><td>0.016</td><td>0.015</td></tr><tr><td>Earth</td><td>0.0019</td><td>0.009</td></tr><tr><td>Mars</td><td>0.00037</td><td>0.0015</td></tr><tr><td>Jupiter</td><td>0.28</td><td>0.42</td></tr><tr><td>Saturn</td><td>0.0047</td><td>0.0065</td></tr><tr><td></td><td>Fienga et al. (2018)</td><td>Park et al. (2017)</td></tr><tr><td>Mercury</td><td>0.02</td><td>0.015</td></tr></tbody></table>

Wardana et al. (2020) and estimated either by galactic simulations or by fit to observations (including Gaia DR2 in Wardana et al. 2020). The third approach is in the interpretation of the secular variation of the Sun gravitational mass as a combination of several phenomena including the fall of dark matter towards the Sun (Press and Spergel 1985; Lundberg and Edsjö 2004; Blennow et al. 2018). In the scenario of dark matter falling into the Sun, the mass of the Sun should decrease less rapidly. However one can discuss the difficulty of disentangling the different contributions from dust and comets falling into the Sun and plasma ejecta that have to be accounted for in the Sun mass equation (see Sect. 4.3 for discussion). An attempt has been proposed by Kardashev et al. (2005) but leading to a constraint of few percents of the sun mass that can be assimilated to falling dark matter. This value is even bigger than the one proposed by Pitjeva and Pitjev (2013) and is therefore not in agreement with the expected estimations for local dark matter density.

Recently, Belbruno and Green (2022) simulated the impact of dark matter located in the galactic halo on the motion of objects in the solar system. Their conclusions are that only objects located in the outer solar system (after 80 AU), or objects situated in saddle points, can allow a detection. This is consistent with what has been already discussed, for example, by Klioner and Soffel (1993). Finally, in Sect. 4.7, the determination of $Q_2$ obtained by Hees et al. (2014b) can also be seen as a measurement of the Galactic potential acting on the solar system, either induced by the stellar population or induced by dark matter. This value presented in Sect. 4.7, shows a clear lack of sensitivity of planetary ephemeris for the detection of the tidal interactions coming from our galaxy.

All these results favour the fact that planetary ephemerides are not yet accurate enough to measure local dark matter influences in the solar system.

## 5.3 What can often be found in the literature

Most of the bounds in the literature do not come from planetary ephemerides developed in a given theoretical or phenomenological framework, but instead they
<table><caption>**Table 10** 1-σ uncertainties (mas/yr) on the perihelion advance per orbit. These values were extracted from Pitjeva and Pitjev (2013)—obtained with a full fit of the ephemeris including the perihelion advances— from Fienga et al. (2018) and Fiengaat et al. (2011b)—obtained with fixed values of the maximum advances—and from Park et al. (2017)—obtained by studying averaged Mercury orbit</caption><thead><tr><th>Planet</th><th>Pitjeva and Pitjev (2013)</th><th>Fienga et al. (2011b)</th></tr></thead><tbody><tr><td>Mercury</td><td>0.03</td><td>0.006</td></tr><tr><td>Venus</td><td>0.016</td><td>0.015</td></tr><tr><td>Earth</td><td>0.0019</td><td>0.009</td></tr><tr><td>Mars</td><td>0.00037</td><td>0.0015</td></tr><tr><td>Jupiter</td><td>0.28</td><td>0.42</td></tr><tr><td>Saturn</td><td>0.0047</td><td>0.0065</td></tr><tr><td></td><td>Fienga et al. (2018)</td><td>Park et al. (2017)</td></tr><tr><td>Mercury</td><td>0.02</td><td>0.015</td></tr></tbody></table>

Wardana et al. (2020) and estimated either by galactic simulations or by fit to observations (including Gaia DR2 in Wardana et al. 2020). The third approach is in the interpretation of the secular variation of the Sun gravitational mass as a combination of several phenomena including the fall of dark matter towards the Sun (Press and Spergel 1985; Lundberg and Edsjö 2004; Blennow et al. 2018). In the scenario of dark matter falling into the Sun, the mass of the Sun should decrease less rapidly. However one can discuss the difficulty of disentangling the different contributions from dust and comets falling into the Sun and plasma ejecta that have to be accounted for in the Sun mass equation (see Sect. 4.3 for discussion). An attempt has been proposed by Kardashev et al. (2005) but leading to a constraint of few percents of the sun mass that can be assimilated to falling dark matter. This value is even bigger than the one proposed by Pitjeva and Pitjev (2013) and is therefore not in agreement with the expected estimations for local dark matter density.

Recently, Belbruno and Green (2022) simulated the impact of dark matter located in the galactic halo on the motion of objects in the solar system. Their conclusions are that only objects located in the outer solar system (after 80 AU), or objects situated in saddle points, can allow a detection. This is consistent with what has been already discussed, for example, by Klioner and Soffel (1993). Finally, in Sect. 4.7, the determination of $Q_2$ obtained by Hees et al. (2014b) can also be seen as a measurement of the Galactic potential acting on the solar system, either induced by the stellar population or induced by dark matter. This value presented in Sect. 4.7, shows a clear lack of sensitivity of planetary ephemeris for the detection of the tidal interactions coming from our galaxy.

All these results favour the fact that planetary ephemerides are not yet accurate enough to measure local dark matter influences in the solar system.

## 5.3 What can often be found in the literature

Most of the bounds in the literature do not come from planetary ephemerides developed in a given theoretical or phenomenological framework, but instead they
use the uncertainty on derived quantities as an input of the maximum tolerated departure from general relativity that would be compatible with observations. For instance, if the uncertainty on the perihelion advance per orbit for a given astronomical body and a given ephemeris is less than a specific value, it is often assumed that any modification of gravity has to induce an effect that is less than this former limit. Therefore, if one computes that a specific theory should induce an effect on, e.g., the perihelion advance per orbit for an astronomical body that is bigger than the uncertainty on this derived quantity, then it is often claimed that this constraint rules out this specific theory.

At first sight, it seems like a reasonable thing to do. Unfortunately, doing this is problematic for the simple reason that derived quantities are obtained assuming a specific theoretical framework, and that there is no guarantee that the adjusted parameters of the ephemeris (masses, initial conditions etc.) would be the same in another theoretical framework. Even more, it is not impossible, a priori, that another theoretical framework actually leads to smaller residuals than general relativity after adjusting the parameters of the ephemeris in this framework—which would mean that this alternative to general relativity is favoured by the data for the considered model of the solar system.

In other words, while one would claim to have derived a constraint on an alternative theory from a derived quantity, actually one could very well have missed a signal in favour of the alternative theory instead—which is precisely the opposite of giving a constraint. This shows that using derived quantities as an input to constrain alternatives to general relativity is not trustworthy in general.

The only way to compare the merit of two different theoretical frameworks is to compare statistically the accuracy of the ephemerides adjusted to the data in each theoretical framework—that is, the statistical amplitude of their residuals. If the ephemeris in a given theoretical framework has significantly better (i.e. smaller) residuals than in another theoretical framework, then it means that the former is favoured by the data within the planetary model considered. Different planetary models—see e.g. Sections 3.3.2 and 3.3.4—might also lead to different answers with this respect.

## 5.4 Why consistency matters

The reason why the adjusted parameters of an ephemeris (masses, initial conditions etc.) are in general different when one considers an alternative theory to general relativity is because all the parameters of the ephemeris, the ones describing gravity (e.g. PPN parameters, Compton wavelength of the Yukawa suppression etc.) and the ones describing the bodies themselves and their orbits (masses, initial conditions, shapes etc.), are more or less correlated to one another. The high degree of correlation is also somewhat accentuated by the specific symmetry of planetary orbits, because most of the motions are close to the ecliptic plane (and with relatively low eccentricities), which limits the disentanglement of the effects of different parameters during the fit of the ephemeris parameters—as one can get an intuition from the approximated analytical expression, for instance, of the advance of the node Eq. (100).
use the uncertainty on derived quantities as an input of the maximum tolerated departure from general relativity that would be compatible with observations. For instance, if the uncertainty on the perihelion advance per orbit for a given astronomical body and a given ephemeris is less than a specific value, it is often assumed that any modification of gravity has to induce an effect that is less than this former limit. Therefore, if one computes that a specific theory should induce an effect on, e.g., the perihelion advance per orbit for an astronomical body that is bigger than the uncertainty on this derived quantity, then it is often claimed that this constraint rules out this specific theory.

At first sight, it seems like a reasonable thing to do. Unfortunately, doing this is problematic for the simple reason that derived quantities are obtained assuming a specific theoretical framework, and that there is no guarantee that the adjusted parameters of the ephemeris (masses, initial conditions etc.) would be the same in another theoretical framework. Even more, it is not impossible, a priori, that another theoretical framework actually leads to smaller residuals than general relativity after adjusting the parameters of the ephemeris in this framework—which would mean that this alternative to general relativity is favoured by the data for the considered model of the solar system.

In other words, while one would claim to have derived a constraint on an alternative theory from a derived quantity, actually one could very well have missed a signal in favour of the alternative theory instead—which is precisely the opposite of giving a constraint. This shows that using derived quantities as an input to constrain alternatives to general relativity is not trustworthy in general.

The only way to compare the merit of two different theoretical frameworks is to compare statistically the accuracy of the ephemerides adjusted to the data in each theoretical framework—that is, the statistical amplitude of their residuals. If the ephemeris in a given theoretical framework has significantly better (i.e. smaller) residuals than in another theoretical framework, then it means that the former is favoured by the data within the planetary model considered. Different planetary models—see e.g. Sections 3.3.2 and 3.3.4—might also lead to different answers with this respect.

## 5.4 Why consistency matters

The reason why the adjusted parameters of an ephemeris (masses, initial conditions etc.) are in general different when one considers an alternative theory to general relativity is because all the parameters of the ephemeris, the ones describing gravity (e.g. PPN parameters, Compton wavelength of the Yukawa suppression etc.) and the ones describing the bodies themselves and their orbits (masses, initial conditions, shapes etc.), are more or less correlated to one another. The high degree of correlation is also somewhat accentuated by the specific symmetry of planetary orbits, because most of the motions are close to the ecliptic plane (and with relatively low eccentricities), which limits the disentanglement of the effects of different parameters during the fit of the ephemeris parameters—as one can get an intuition from the approximated analytical expression, for instance, of the advance of the node Eq. (100).
This is exemplified notably with the oblateness of the Sun $J_2^\odot$ in Figs. 6 and 9.36
Indeed, due to the high level of correlations between the oblateness of the Sun and
the PPN parameters $\gamma$ and $\beta$, both the adjusted value of the $J_2^\odot$ and its uncertainty are
affected when $\gamma$ and $\beta$ are not set to 1 a priori (see Sect. 4.1 for a full discussion).

From a formal perspective, Eq. (100) gives a good illustration of the problem in
terms of perihelion advance per planetary orbit. Because many different parameters
contribute to the advance, if an advance is, say, induced by a parameter related to the
description of gravity beyond what is acceptable in terms of deviation with respect to
the data, the fit will often lead to a modification of other parameters in order to
compensate for this unacceptable contribution—such that, in the end, the final
solution remains as close as possible to the data (i.e. the residuals are minimised).

As long as the statistical properties of the residuals of the ephemerides in distinct
theoretical frameworks are not significantly different, one cannot say which of the
two theoretical frameworks better explains the observations. This tells why one
cannot simply estimate the modification of the perihelion advance per orbit in an
alternative theory and compare it with the output value obtained while assuming
general relativity.

Another example is given by the Compton wavelength of a Yukawa suppression
of the Newtonian potential in a massive gravity framework—see Sect. 4.4. An
illustration of the high degree of correlations between parameters is given by Bernus
et al. (2019) in a table—reproduced here in Table 11—that gathers the correlations
between the Compton wavelength and some of the solar system parameters for the
INPOP17a planetary model.

Furthermore, statistics of the residuals for several planets were also displayed by
Bernus et al. (2019) in order to show their evolution if one assumes that the solar
system parameters are given by fits obtained when assuming general relativity
instead of re-adjusting them in the massive gravity framework. The conclusion of this
investigation is that not adjusting planetary ephemeris parameters within the
framework of massive gravity would have led to an overestimation of the constraint
on the value of the Compton wavelength by about one order of magnitude.

## *The Pioneer anomaly*

Finally, one can discuss the case of the Pioneer anomaly. For some years, the
unexplained supplementary acceleration detected during the navigation of Pioneer 10
and 11 escaping the solar system, keeps the community active in looking for some
possible violations of GR that could produce such a phenomena. The reader can see
Turyshev and Toth (2010) for a complete review. Among the alternative theories that
were proposed to explain the Pioneer anomaly, some also impact the orbits of outer
planets. Most of the authors just consider the effect of the induced modification of the
planetary equations of motion without considering the new adjustment that one
should do for adapting the initial conditions to this new model (Lecian and Montani
2009; Iorio 2009, 2010). Page et al. (2009); Standish (2010) and Fienga et al. (2010)
show that the modification required for explaining the Pioneer acceleration anomaly

<footnote n="36">36 See also Fig. 6 in Milani et al. (2002) for an illustration of the correlation between $J_2^\odot$ and $\beta - 1$ in simulations.</footnote>
This is exemplified notably with the oblateness of the Sun $J_2^\odot$ in Figs. 6 and 9.<fnref n="36" /> Indeed, due to the high level of correlations between the oblateness of the Sun and the PPN parameters $\gamma$ and $\beta$, both the adjusted value of the $J_2^\odot$ and its uncertainty are affected when $\gamma$ and $\beta$ are not set to 1 a priori (see Sect. 4.1 for a full discussion).

From a formal perspective, Eq. (100) gives a good illustration of the problem in terms of perihelion advance per planetary orbit. Because many different parameters contribute to the advance, if an advance is, say, induced by a parameter related to the description of gravity beyond what is acceptable in terms of deviation with respect to the data, the fit will often lead to a modification of other parameters in order to compensate for this unacceptable contribution—such that, in the end, the final solution remains as close as possible to the data (i.e. the residuals are minimised).

As long as the statistical properties of the residuals of the ephemerides in distinct theoretical frameworks are not significantly different, one cannot say which of the two theoretical frameworks better explains the observations. This tells why one cannot simply estimate the modification of the perihelion advance per orbit in an alternative theory and compare it with the output value obtained while assuming general relativity.

Another example is given by the Compton wavelength of a Yukawa suppression of the Newtonian potential in a massive gravity framework—see Sect. 4.4. An illustration of the high degree of correlations between parameters is given by Bernus et al. (2019) in a table—reproduced here in Table 11—that gathers the correlations between the Compton wavelength and some of the solar system parameters for the INPOP17a planetary model.

Furthermore, statistics of the residuals for several planets were also displayed by Bernus et al. (2019) in order to show their evolution if one assumes that the solar system parameters are given by fits obtained when assuming general relativity instead of re-adjusting them in the massive gravity framework. The conclusion of this investigation is that not adjusting planetary ephemeris parameters within the framework of massive gravity would have led to an overestimation of the constraint on the value of the Compton wavelength by about one order of magnitude.

## *The Pioneer anomaly*

Finally, one can discuss the case of the Pioneer anomaly. For some years, the unexplained supplementary acceleration detected during the navigation of Pioneer 10 and 11 escaping the solar system, keeps the community active in looking for some possible violations of GR that could produce such a phenomena. The reader can see Turyshev and Toth (2010) for a complete review. Among the alternative theories that were proposed to explain the Pioneer anomaly, some also impact the orbits of outer planets. Most of the authors just consider the effect of the induced modification of the planetary equations of motion without considering the new adjustment that one should do for adapting the initial conditions to this new model (Lecian and Montani 2009; Iorio 2009, 2010). Page et al. (2009); Standish (2010) and Fienga et al. (2010) show that the modification required for explaining the Pioneer acceleration anomaly

<footnote n="36">³⁶ See also Fig. 6 in Milani et al. (2002) for an illustration of the correlation between $J_2^\odot$ and $\beta - 1$ in simulations.</footnote>
<table><caption>**Table 11** Examples of correlations from Bernus et al. (2019) between various INPOP17b parameters and the Compton wavelength $\lambda_g$. a, EMB and $M_\odot$ state for semi-major axes, the Earth–Moon barycenter and the mass of the Sun respectively</caption><thead><tr><th></th><th>λ<sub>g</sub></th><th>a Mercury</th><th>a Mars</th><th>a Saturn</th><th>a Venus</th><th>a EMB</th><th>GM<sub>☉</sub></th></tr></thead><tbody><tr><td>λ<sub>g</sub></td><td>1</td><td>0.50</td><td>0.49</td><td>0.04</td><td>0.39</td><td>0.05</td><td>0.66</td></tr><tr><td>a Mercury</td><td>...</td><td>1</td><td>0.21</td><td>0.001</td><td>0.97</td><td>0.82</td><td>0.96</td></tr><tr><td>a Mars</td><td>...</td><td>...</td><td>1</td><td>0.03</td><td>0.29</td><td>0.53</td><td>0.06</td></tr><tr><td>a Saturn</td><td>...</td><td>...</td><td>...</td><td>1</td><td>0.003</td><td>0.02</td><td>0.01</td></tr><tr><td>a Venus</td><td>...</td><td>...</td><td>...</td><td>...</td><td>1</td><td>0.86</td><td>0.94</td></tr><tr><td>a EMB</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>1</td><td>0.73</td></tr><tr><td>GM<sub>☉</sub></td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>1</td></tr></tbody></table>

induced, after fit, residuals marginally compatible with the observational accuracies reached at this epoch (before the inputs of the Juno mission). These results restricted severely the possibility of such modifications impacting also planetary orbits. After that, only remained the alternative theories that were affecting the s/c orbit but not the planetary bodies. Another attempt to explain the Pioneer anomaly within the boundaries of conventional physics that does not affect the motion of the planetary bodies was put forward by Kopeikin (2012). In their proposal, the Pioneer effect was perceived as the cosmological consequence of a quadratic divergence between the time scales of electromagnetic wave propagation within the Doppler tracking system and the atomic clocks on Earth. However, Bertolami et al. (2010) and Turyshev et al. (2012) conclusively demonstrated that the Pioneer acceleration can be explained by considering the distinct thermal properties of each spacecraft face. Either way, the Pioneer anomaly thus serves as a compelling illustration of the importance of developing a fully consistent model when testing alternative theories to General Relativity, and ensuring that this model is fitted to observations-a step that is undeniably crucial.

# 6 Future directions

## 6.1 Theory

On the theory side, many aspects of alternative theories remain to be studied and implemented in planetary ephemerides. Theorists do not lack of new ideas, and therefore one should not fall short of new theories to investigate with planetary ephemerides. However, not all existing theories that can lead to significant variations in the solar system have been constrained with planetary ephemerides yet. As an example, not even the full PPN framework Eqs. (114)–(122) have been completely investigated so far with planetary ephemerides. This is not a surprise given the many parameters involved, and given that the more parameters to test, the more difficult, and computationally demanding, the study is—and also the worse the constraints on each parameter are.
<table><caption>**Table 11** Examples of correlations from Bernus et al. (2019) between various INPOP17b parameters and the Compton wavelength $\lambda_g$. a, EMB and $M_\odot$ state for semi-major axes, the Earth–Moon barycenter and the mass of the Sun respectively</caption><thead><tr><th></th><th>λ<sub>g</sub></th><th>a Mercury</th><th>a Mars</th><th>a Saturn</th><th>a Venus</th><th>a EMB</th><th>GM<sub>☉</sub></th></tr></thead><tbody><tr><td>λ<sub>g</sub></td><td>1</td><td>0.50</td><td>0.49</td><td>0.04</td><td>0.39</td><td>0.05</td><td>0.66</td></tr><tr><td>a Mercury</td><td>...</td><td>1</td><td>0.21</td><td>0.001</td><td>0.97</td><td>0.82</td><td>0.96</td></tr><tr><td>a Mars</td><td>...</td><td>...</td><td>1</td><td>0.03</td><td>0.29</td><td>0.53</td><td>0.06</td></tr><tr><td>a Saturn</td><td>...</td><td>...</td><td>...</td><td>1</td><td>0.003</td><td>0.02</td><td>0.01</td></tr><tr><td>a Venus</td><td>...</td><td>...</td><td>...</td><td>...</td><td>1</td><td>0.86</td><td>0.94</td></tr><tr><td>a EMB</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>1</td><td>0.73</td></tr><tr><td>GM<sub>☉</sub></td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>1</td></tr></tbody></table>

induced, after fit, residuals marginally compatible with the observational accuracies reached at this epoch (before the inputs of the Juno mission). These results restricted severely the possibility of such modifications impacting also planetary orbits. After that, only remained the alternative theories that were affecting the s/c orbit but not the planetary bodies. Another attempt to explain the Pioneer anomaly within the boundaries of conventional physics that does not affect the motion of the planetary bodies was put forward by Kopeikin (2012). In their proposal, the Pioneer effect was perceived as the cosmological consequence of a quadratic divergence between the time scales of electromagnetic wave propagation within the Doppler tracking system and the atomic clocks on Earth. However, Bertolami et al. (2010) and Turyshev et al. (2012) conclusively demonstrated that the Pioneer acceleration can be explained by considering the distinct thermal properties of each spacecraft face. Either way, the Pioneer anomaly thus serves as a compelling illustration of the importance of developing a fully consistent model when testing alternative theories to General Relativity, and ensuring that this model is fitted to observations-a step that is undeniably crucial.

# 6 Future directions

## 6.1 Theory

On the theory side, many aspects of alternative theories remain to be studied and implemented in planetary ephemerides. Theorists do not lack of new ideas, and therefore one should not fall short of new theories to investigate with planetary ephemerides. However, not all existing theories that can lead to significant variations in the solar system have been constrained with planetary ephemerides yet. As an example, not even the full PPN framework Eqs. (114)–(122) have been completely investigated so far with planetary ephemerides. This is not a surprise given the many parameters involved, and given that the more parameters to test, the more difficult, and computationally demanding, the study is—and also the worse the constraints on each parameter are.
Another example can also be given with the case of Brans-Dicke-like scalar-tensor theories. A considerable portion of Brans-Dicke-like scalar-tensor theories—that is, theories defined by $L_{int} = 0$ in Eq. (154)<fnref n="37" />—have been exquisitely constrained by observations of binary pulsars, owing to a strong field effect that cannot occur within the solar system. Specifically, with certain selections for the function $\omega(\phi)$, the scalar-field within compact objects like neutron stars can be amplified through a nonlinear effect known as scalarization (Damour and Esposito-Farese 1993; Damour and Esposito-Farèse 1996). This effect should lead to a significant violation of the Strong Equivalence Principle<fnref n="38" />—see Sect. 4.2.1—that is not seen in binary pulsars. For a deeper dive into this captivating subject, we direct the reader to Kramer et al. (2021), Freire et al. (2012) and Voisin et al. (2020).

The PPN framework in Sect. 4.1 encompasses this class of theories with $\gamma = (1 + \omega(\varphi_0))/(2 + \omega(\varphi_0))$, where $\varphi_0$ is the asymptotic value of $\varphi_0$ at the edge of the solar system,<fnref n="39" /> and $\beta - 1 = \omega'(\varphi_0)(3 + 2\omega(\varphi_0))^{-2}(4 + 2\omega(\varphi_0))^{-1}$. The constraints on $\gamma$ and $\beta$ obtained with planetary ephemerides in the PPN framework are given in Fig. 8.

However, those constraints assume that $\gamma$ and $\beta$ are independent parameters in the field equations—which lead to the definition of the time and space coordinate system, the equation of motion [see Eq. (90)] and of the Shapiro delay [see Eq. (97)]—while they are not independent in those scalar-tensor theories whenever $\beta \neq 1$—that is $\omega'(\varphi_0) \neq 0$—given that $\omega(\varphi_0)$ appears in both $\gamma$ and $\beta$.

As a consequence, one cannot directly convert the constraints in Fig. 8 in terms of constraints on $\omega(\varphi_0)$ and $\omega'(\varphi_0)$. Therefore, it would be interesting to test those specific theories with planetary ephemerides in order to compare with the constraints with binary pulsars, in the regions of the theory space where one does not have the non-linear strong-field scalarization effect. For instance, assuming $\omega(\varphi) = \omega_{BD}$—that is, $\omega'(\varphi_0) = 0$ and $\beta = 1$—tests involving pulsars lead to $\omega_{BD} > 130 \times 10^3$ (Voisin et al. 2020)—although note that this value depends on the unknown equations of state of neutron stars.

Considering the constraint on $\gamma$ obtained with the Cassini experiment (Bertotti et al. 2003b), the limit on $\omega(\varphi_0)$ instead is $\omega(\varphi_0) > 40 \times 10^3$. This threshold should also be better than what is currently possible with planetary ephemerides—but perhaps not too far from what will be possible with the additional data from Bepi-Colombo.

A general reminder of Sect. 5 is that, while the PPN framework serves as a highly convenient phenomenological apparatus for testing alternative theories, its limitation lies in its inability to accommodate potential dependencies between PPN parameters (such as the discussed above or the one of Sect. 4.6) that could emerge within a given theory. For example, in Brans-Dicke-like scalar-tensor theories, there is a certain degree of interdependence at the level of the field equations between $\gamma$ and $\beta$ since they both depend on the parameter $\omega(\varphi_0)$ of the theory.

<footnote n="37">³⁷ Indeed, strictly speaking, Brans-Dicke theories are an even more restricted group, which is such that $L_{int} = 0$ and $\omega(\varphi) = \omega_{BD}$ in Eq. (154).</footnote>

<footnote n="38">³⁸ Even, occasionally, with large values of $\omega(\varphi_0)$.</footnote>

<footnote n="39">³⁹ Which could vary over cosmological times.</footnote>
Another example can also be given with the case of Brans-Dicke-like scalar-tensor theories. A considerable portion of Brans-Dicke-like scalar-tensor theories—that is, theories defined by $\mathcal{L}_{\text{int}} = 0$ in Eq. (154)<fnref n="37" />—have been exquisitely constrained by observations of binary pulsars, owing to a strong field effect that cannot occur within the solar system. Specifically, with certain selections for the function $\omega(\phi)$, the scalar-field within compact objects like neutron stars can be amplified through a nonlinear effect known as scalarization (Damour and Esposito-Farese 1993; Damour and Esposito-Farèse 1996). This effect should lead to a significant violation of the Strong Equivalence Principle<fnref n="38" />—see Sect. 4.2.1—that is not seen in binary pulsars. For a deeper dive into this captivating subject, we direct the reader to Kramer et al. (2021), Freire et al. (2012) and Voisin et al. (2020).

The PPN framework in Sect. 4.1 encompasses this class of theories with $\gamma = (1 + \omega(\varphi_0))/(2 + \omega(\varphi_0))$, where $\varphi_0$ is the asymptotic value of $\varphi_0$ at the edge of the solar system,<fnref n="39" /> and $\beta - 1 = \omega'(\varphi_0)(3 + 2\omega(\varphi_0))^{-2}(4 + 2\omega(\varphi_0))^{-1}$. The constraints on $\gamma$ and $\beta$ obtained with planetary ephemerides in the PPN framework are given in Fig. 8.

However, those constraints assume that $\gamma$ and $\beta$ are independent parameters in the field equations—which lead to the definition of the time and space coordinate system, the equation of motion [see Eq. (90)] and of the Shapiro delay [see Eq. (97)]—while they are not independent in those scalar-tensor theories whenever $\beta \neq 1$—that is $\omega'(\varphi_0) \neq 0$—given that $\omega(\varphi_0)$ appears in both $\gamma$ and $\beta$.

As a consequence, one cannot directly convert the constraints in Fig. 8 in terms of constraints on $\omega(\varphi_0)$ and $\omega'(\varphi_0)$. Therefore, it would be interesting to test those specific theories with planetary ephemerides in order to compare with the constraints with binary pulsars, in the regions of the theory space where one does not have the non-linear strong-field scalarization effect. For instance, assuming $\omega(\varphi) = \omega_{BD}$—that is, $\omega'(\varphi_0) = 0$ and $\beta = 1$—tests involving pulsars lead to $\omega_{BD} > 130 \times 10^3$ (Voisin et al. 2020)—although note that this value depends on the unknown equations of state of neutron stars.

Considering the constraint on $\gamma$ obtained with the Cassini experiment (Bertotti et al. 2003b), the limit on $\omega(\varphi_0)$ instead is $\omega(\varphi_0) > 40 \times 10^3$. This threshold should also be better than what is currently possible with planetary ephemerides—but perhaps not too far from what will be possible with the additional data from Bepi-Colombo.

A general reminder of Sect. 5 is that, while the PPN framework serves as a highly convenient phenomenological apparatus for testing alternative theories, its limitation lies in its inability to accommodate potential dependencies between PPN parameters (such as the discussed above or the one of Sect. 4.6) that could emerge within a given theory. For example, in Brans-Dicke-like scalar-tensor theories, there is a certain degree of interdependence at the level of the field equations between $\gamma$ and $\beta$ since they both depend on the parameter $\omega(\varphi_0)$ of the theory.

<footnote n="37">³⁷ Indeed, strictly speaking, Brans-Dicke theories are an even more restricted group, which is such that $\mathcal{L}_{\text{int}} = 0$ and $\omega(\varphi) = \omega_{BD}$ in Eq. (154).</footnote>

<footnote n="38">³⁸ Even, occasionally, with large values of $\omega(\varphi_0)$.</footnote>

<footnote n="39">³⁹ Which could vary over cosmological times.</footnote>
Consequently, one has to keep in mind that it is not generally possible to directly translate constraints obtained on PPN parameters to potential underlying theories—unless the theory predicts the parameters to be independent at the level of the field equations of the considered theory (e.g. Scalar-Tensor, massive gravity, etc.). Therefore, to establish constraints on the parameters of a specific theory—like $\omega(\varphi_0)$ and $\omega'(\varphi_0)$— one should directly test that particular theory.

## 6.2 Observation

In terms of observations, much progress is anticipated, particularly in light of active or future planetary missions such as BepiColombo. This mission will orbit Mercury for more than a year and will provide unprecedented accurate measures of the Mercury-Earth distance, and consequently, will produce stringent new limits on deviations from general relativity. A lot of publications propose to include Bepi-Colombo range simulations in order to provide possible new constraints on classic general relativity tests, such as advance of the Mercury perihelion, PPN parameter estimations, SEP or alternative tests (Milani et al. 2002; Ashby et al. 2007; De Marchi et al. 2016; Imperi et al. 2018; De Marchi and Cascioli 2020; van der Zwaard and Dirkx 2022; Fienga et al. 2021a, 2022). As previously explained, a specific care should be taken on the consistency between the definition of the considered framework (e.g. harmonics versus non-harmonics gauges, definition of the solar system barycenter...), the tests performed and the claimed accuracy. The question of consistencies and correlations between astronomical constraints and general relativity tests in the full PPN context will then become even more urgent to address.

On the other side of the solar system, missions towards gas giants and outer solar system will also be interesting for testing another types of general relativity violation such as dark sector violations or dark matter clumps (Bergé et al. 2021, 2018).

Finally, among more exploratory projects, LISA-like configurations for interplanetary laser distance measurements between telluric planets (Earth, Mars and Venus) have been proposed as a way to gain accuracy in planetary ephemerides and sensitivity to general relativity violations such as the secular variations of the gravitational constant (Smith et al. 2018). Despite the technical challenges of such project (Bills and Skillman 2022; Bills and Gorski 2022), the outcome of these measurements would indeed impact the global accuracy of the ephemerides, improve significantly the Bepi-Colombo results but also allow for better constraints on the distribution of mass in the solar system.

## 7 Summary

This paper describes how the planetary ephemerides are built in the framework of General Relativity and how they can be used to test alternative theories. It focuses specifically on the dependencies that exist behind the definition of the reference frame (space and time) in which the planetary ephemeris is described, the equations of motion that govern the orbits of solar system bodies and electromagnetic waves. This paper then summarizes the results obtained considering consistent modifications
Consequently, one has to keep in mind that it is not generally possible to directly translate constraints obtained on PPN parameters to potential underlying theories—unless the theory predicts the parameters to be independent at the level of the field equations of the considered theory (e.g. Scalar-Tensor, massive gravity, etc.). Therefore, to establish constraints on the parameters of a specific theory—like $\omega(\varphi_0)$ and $\omega'(\varphi_0)$— one should directly test that particular theory.

## 6.2 Observation

In terms of observations, much progress is anticipated, particularly in light of active or future planetary missions such as BepiColombo. This mission will orbit Mercury for more than a year and will provide unprecedented accurate measures of the Mercury-Earth distance, and consequently, will produce stringent new limits on deviations from general relativity. A lot of publications propose to include Bepi-Colombo range simulations in order to provide possible new constraints on classic general relativity tests, such as advance of the Mercury perihelion, PPN parameter estimations, SEP or alternative tests (Milani et al. 2002; Ashby et al. 2007; De Marchi et al. 2016; Imperi et al. 2018; De Marchi and Cascioli 2020; van der Zwaard and Dirkx 2022; Fienga et al. 2021a, 2022). As previously explained, a specific care should be taken on the consistency between the definition of the considered framework (e.g. harmonics versus non-harmonics gauges, definition of the solar system barycenter...), the tests performed and the claimed accuracy. The question of consistencies and correlations between astronomical constraints and general relativity tests in the full PPN context will then become even more urgent to address.

On the other side of the solar system, missions towards gas giants and outer solar system will also be interesting for testing another types of general relativity violation such as dark sector violations or dark matter clumps (Bergé et al. 2021, 2018).

Finally, among more exploratory projects, LISA-like configurations for interplanetary laser distance measurements between telluric planets (Earth, Mars and Venus) have been proposed as a way to gain accuracy in planetary ephemerides and sensitivity to general relativity violations such as the secular variations of the gravitational constant (Smith et al. 2018). Despite the technical challenges of such project (Bills and Skillman 2022; Bills and Gorski 2022), the outcome of these measurements would indeed impact the global accuracy of the ephemerides, improve significantly the Bepi-Colombo results but also allow for better constraints on the distribution of mass in the solar system.

## 7 Summary

This paper describes how the planetary ephemerides are built in the framework of General Relativity and how they can be used to test alternative theories. It focuses specifically on the dependencies that exist behind the definition of the reference frame (space and time) in which the planetary ephemeris is described, the equations of motion that govern the orbits of solar system bodies and electromagnetic waves. This paper then summarizes the results obtained considering consistent modifications
of the ephemeris framework with direct comparisons with the observations of planetary systems. The PPN formalism is the one that has been the most heavily tested, and the results of its confrontation with planetary astrometry constitutes the most developed part. The paper then moves on to specific alternatives to general relativity such as Einstein-dilaton theories, a massive graviton phenomenology and MOND. The paper finally concludes on some comments and recommendations regarding misinterpreted estimations of the advance of perihelia, giving examples such as the Pioneer anomaly interpretation or some attempts to measure dark matter in the solar system.

As we hope this paper demonstrates, the consistency of the planetary ephemeris framework is a crucial aspect in the field of testing alternative theories. Misinterpretations of results obtained in the general relativity framework can lead to significant errors (e.g. Pioneer anomaly) or over optimistic constraints (e.g. mass of the graviton).

# Appendix A: Derivation of the Shapiro delay at the $c^{-4}$ level, and related issues

In the following lines, we indicate some arguments stressing the complexity of keeping a consistent framework for the derivation of the Shapiro delay at the $c^{-4}$ level in the general data analysis framework of planetary ephemerides.

Indeed, although there are several $c^{-4}$ formulae given in the literature already, e.g. Ashby and Bertotti (2010), Deng and Xie (2012), Linet and Teyssandier (2013), Hees et al. (2014a), Linet and Teyssandier (2016), Cappuccio et al. (2021) and Zschocke (2022), most of them are not consistent with the coordinate system recommended by the IAU—the harmonic gauge—that is used in planetary ephemerides in order to describe the motion of celestial bodies. Hence, one still has to convert those propagation time formulae in the harmonic coordinate system. For instance, many $c^{-4}$ Shapiro equations are derived from an isotropic metric (Ashby and Bertotti 2010; Linet and Teyssandier 2013; Hees et al. 2014a; Linet and Teyssandier 2016; Cappuccio et al. 2021), whereas the metric in harmonic coordinates is not isotropic at the full $c^{-4}$ level, even for a spherical object at the center of the coordinate system—see, e.g., Minazzoli (2012) for the difference between harmonic and isotropic coordinate metrics at the $c^{-4}$ level in the framework of general relativity and scalar-tensor theories. A second order ($c^{-4}$) propagation time formula with harmonic coordinates has recently been derived in Zschocke (2022), but it assumes general relativity and a single body at rest. In general, most of the propagation time formulae in the literature indeed simplify the problem by assuming staticity of the celestial bodies during the propagation of light, whereas one may have to go beyond this
of the ephemeris framework with direct comparisons with the observations of planetary systems. The PPN formalism is the one that has been the most heavily tested, and the results of its confrontation with planetary astrometry constitutes the most developed part. The paper then moves on to specific alternatives to general relativity such as Einstein-dilaton theories, a massive graviton phenomenology and MOND. The paper finally concludes on some comments and recommendations regarding misinterpreted estimations of the advance of perihelia, giving examples such as the Pioneer anomaly interpretation or some attempts to measure dark matter in the solar system.

As we hope this paper demonstrates, the consistency of the planetary ephemeris framework is a crucial aspect in the field of testing alternative theories. Misinterpretations of results obtained in the general relativity framework can lead to significant errors (e.g. Pioneer anomaly) or over optimistic constraints (e.g. mass of the graviton).

# Appendix A: Derivation of the Shapiro delay at the $c^{-4}$ level, and related issues

In the following lines, we indicate some arguments stressing the complexity of keeping a consistent framework for the derivation of the Shapiro delay at the $c^{-4}$ level in the general data analysis framework of planetary ephemerides.

Indeed, although there are several $c^{-4}$ formulae given in the literature already, e.g. Ashby and Bertotti (2010), Deng and Xie (2012), Linet and Teyssandier (2013), Hees et al. (2014a), Linet and Teyssandier (2016), Cappuccio et al. (2021) and Zschocke (2022), most of them are not consistent with the coordinate system recommended by the IAU—the harmonic gauge—that is used in planetary ephemerides in order to describe the motion of celestial bodies. Hence, one still has to convert those propagation time formulae in the harmonic coordinate system. For instance, many $c^{-4}$ Shapiro equations are derived from an isotropic metric (Ashby and Bertotti 2010; Linet and Teyssandier 2013; Hees et al. 2014a; Linet and Teyssandier 2016; Cappuccio et al. 2021), whereas the metric in harmonic coordinates is not isotropic at the full $c^{-4}$ level, even for a spherical object at the center of the coordinate system—see, e.g., Minazzoli (2012) for the difference between harmonic and isotropic coordinate metrics at the $c^{-4}$ level in the framework of general relativity and scalar-tensor theories. A second order ($c^{-4}$) propagation time formula with harmonic coordinates has recently been derived in Zschocke (2022), but it assumes general relativity and a single body at rest. In general, most of the propagation time formulae in the literature indeed simplify the problem by assuming staticity of the celestial bodies during the propagation of light, whereas one may have to go beyond this
approximation at the required level of accuracy (Bertone et al. 2014; Zschocke 2016). A $c^{-4}$ propagation formula in harmonic coordinates$^{40}$ for light in the solar system—that is, with many moving bodies—has been derived in Minazzoli and Chauvineau (2011) and Deng and Xie (2012) in the framework of general relativity and scalar-tensor theories, but the actual propagation time that results from it remains to be derived. Otherwise, a $c^{-4}$ propagation time formula for light rays in harmonic coordinates, but restricted to one arbitrarily moving pointlike body, has been derived in Zschocke (2016).

**Acknowledgements** The authors thank Aurélien Hees and Vincenzo Mariani for their valuable inputs on the initial versions of the paper. OM expresses also gratitude to Francesco Nitti, Federico Piazza, and Aurélien Hees for stimulating discussions on the relationship between composite particle mass and quantum trace anomalies. AF thanks L. Blanchet, A. Hees, E. Gourgoulhon, S. Reynaud, P. Wolf and C. Will for their supports, comments, discussions all these years. AF is also debtfull to J. Laskar and M. Gastineau for the development of the INPOP planetary ephemerides. Finally, AF is grateful to the French Space Agency and Observatoire de la Côte d'azur for the financial supports.

**Open Access** This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

## References

Abbott BP et al [LIGO Scientific Collaboration and Virgo Collaboration] (2019) GWTC-1: a gravitational-wave transient catalog of compact binary mergers observed by LIGO and Virgo during the first and second observing runs. Phys Rev X 9(3):031040. https://doi.org/10.1103/PhysRevX.9.031040. arXiv:1811.12907

Abbott R et al [LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration] (2023) GWTC-3: compact binary coalescences observed by LIGO and Virgo during the second part of the third observing run. Phys Rev X 13(4):041039. https://doi.org/10.1103/PhysRevX.13.041039

Adelberger EG (2001) New tests of Einstein's equivalence principle and Newton's inverse-square law. Class Quantum Grav 18(13):2397-2405. https://doi.org/10.1088/0264-9381/18/13/302

Adelberger EG, Fischbach E, Krause DE, Newman RD (2003) Constraining the couplings of massive pseudoscalars using gravity and optical experiments. Phys Rev D 68(6):062002. https://doi.org/10.1103/PhysRevD.68.062002

Adelberger EG, Heckel BR, Nelson AE (2003) Tests of the gravitational inverse-square law. Annu Rev Nucl Part Sci 53:77-121. https://doi.org/10.1146/annurev.nucl.53.041002.110503. arXiv:hep-ph/0307284

Anderson JD, Keesey MSW, Lau EL, Standish JEM, Newhall XX (1978) Tests of general relativity using astrometric and radio metric observations of the planets. Acta Astronaut 5:43-61

<footnote>$^{40}$ Note that there can be two distinct versions of harmonic coordinates in scalar-tensor theories, based on whether the harmonic gauge condition $g^{\alpha\beta}\Gamma_{\alpha\beta}^{\sigma} = 0$ is imposed on the metric in the Jordan frame or in the Einstein frame (Minazzoli and Chauvineau 2011; Deng and Xie 2012). This further complicates the discussion on the coordinate systems to be used in alternative theories.</footnote>
approximation at the required level of accuracy (Bertone et al. 2014; Zschocke 2016). A $c^{-4}$ propagation formula in harmonic coordinates$^{40}$ for light in the solar system—that is, with many moving bodies—has been derived in Minazzoli and Chauvineau (2011) and Deng and Xie (2012) in the framework of general relativity and scalar-tensor theories, but the actual propagation time that results from it remains to be derived. Otherwise, a $c^{-4}$ propagation time formula for light rays in harmonic coordinates, but restricted to one arbitrarely moving pointlike body, has been derived in Zschocke (2016).

**Acknowledgements** The authors thank Aurélien Hees and Vincenzo Mariani for their valuable inputs on the initial versions of the paper. OM expresses also gratitude to Francesco Nitti, Federico Piazza, and Aurélien Hees for stimulating discussions on the relationship between composite particle mass and quantum trace anomalies. AF thanks L. Blanchet, A. Hees, E. Gourgoulhon, S. Reynaud, P. Wolf and C. Will for their supports, comments, discussions all these years. AF is also debtfull to J. Laskar and M. Gastineau for the development of the INPOP planetary ephemerides. Finally, AF is grateful to the French Space Agency and Observatoire de la Côte d'azur for the financial supports.

**Open Access** This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

## References

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Abbott R et al [LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration] (2023) GWTC-3: compact binary coalescences observed by LIGO and Virgo during the second part of the third observing run. Phys Rev X 13(4):041039. https://doi.org/10.1103/PhysRevX.13.041039

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Adelberger EG, Fischbach E, Krause DE, Newman RD (2003) Constraining the couplings of massive pseudoscalars using gravity and optical experiments. Phys Rev D 68(6):062002. https://doi.org/10.1103/PhysRevD.68.062002

Adelberger EG, Heckel BR, Nelson AE (2003) Tests of the gravitational inverse-square law. Annu Rev Nucl Part Sci 53:77-121. https://doi.org/10.1146/annurev.nucl.53.041002.110503. arXiv:hep-ph/0307284

Anderson JD, Keesey MSW, Lau EL, Standish JEM, Newhall XX (1978) Tests of general relativity using astrometric and radio metric observations of the planets. Acta Astronaut 5:43-61

<footnote>$^{40}$ Note that there can be two distinct versions of harmonic coordinates in scalar-tensor theories, based on whether the harmonic gauge condition $g^{\alpha\beta}\Gamma_{\alpha\beta}^{\sigma} = 0$ is imposed on the metric in the Jordan frame or in the Einstein frame (Minazzoli and Chauvineau 2011; Deng and Xie 2012). This further complicates the discussion on the coordinate systems to be used in alternative theories.</footnote>
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Archinal BA, Acton CH, A'Hearn MF, Conrad A, Consolmagno GJ, Duxbury T, Hestroffer D, Hilton JL, Kirk RL, Klioner SA, McCarthy D, Meech K, Oberst J, Ping J, Seidelmann PK, Tholen DJ, Thomas PC, Williams IP (2018) Report of the IAU working group on cartographic coordinates and rotational elements: 2015. Celest Mech Dyn Astron 130(3):22. https://doi.org/10.1007/s10569-017-9805-5

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Ash ME, Shapiro II, Smith WB (1967) Astronomical constants and planetary ephemerides deduced from radar and optical observations. *Astron J* 72:338. https://doi.org/10.1086/110230

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Ashby N, Bender PL, Wahr JM (2007) Future gravitational physics tests from ranging to the BepiColombo Mercury planetary orbiter. *Phys Rev D* 75(2):022001. https://doi.org/10.1103/PhysRevD.75.022001

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Bernus L, Minazzoli O, Fienga A, Hees A, Gastineau M, Laskar J, Deram P, Di Ruscio A (2022) Constraining massless Dilaton theory at Solar system scales with the planetary ephemeris INPOP. *Phys Rev D* 105(4):044057. https://doi.org/10.1103/PhysRevD.105.044057

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Damour T, Polyakov AM (1994) String theory and gravity. Gen Relat Grav 26:1171-1176. https://doi.org/10.1007/BF02106709. arXiv:gr-qc/9411069

Damour T, Vokrouhlický D (1995) Conservation laws for systems of extended bodies in the first post-Newtonian approximation. Phys Rev D 52(8):4455-4461. https://doi.org/10.1103/PhysRevD.52.4455. arXiv:gr-qc/9503041

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Damour T, Donoghue JF (2010) Equivalence principle violations and couplings of a light Dilaton. Phys. Rev. D 82(8):084033. https://doi.org/10.1103/PhysRevD.82.084033. arXiv:1007.2792

Damour T, Donoghue JF (2011) Spatial variation of fundamental couplings and lunar laser ranging. Class Quantum Grav 28(16):162001. https://doi.org/10.1088/0264-9381/28/16/162001

Damour T, Esposito-Farese G (1992) Tensor-multi-scalar theories of gravitation. Class Quantum Grav 9 (9):2093-2176. https://doi.org/10.1088/0264-9381/9/9/015

Damour T, Esposito-Farese G (1993) Nonperturbative strong-field effects in tensor-scalar theories of gravitation. Phys Rev Lett 70(15):2220-2223. https://doi.org/10.1103/PhysRevLett.70.2220

Damour T, Esposito-Farèse G (1996) Tensor-scalar gravity and binary-pulsar experiments. Phys Rev D 54 (2):1474-1491. https://doi.org/10.1103/PhysRevD.54.1474

Damour T, Nordtvedt K (1993) Tensor-scalar cosmological models and their relaxation toward general relativity. Phys Rev D 48:3436-3450. https://doi.org/10.1103/PhysRevD.48.3436

Damour T, Polyakov AM (1994) String theory and gravity. Gen Relat Grav 26:1171-1176. https://doi.org/ 10.1007/BF02106709. arXiv:gr-qc/9411069

Damour T, Vokrouhlický D (1995) Conservation laws for systems of extended bodies in the first post-Newtonian approximation. Phys Rev D 52(8):4455-4461. https://doi.org/10.1103/PhysRevD.52. 4455. arXiv:gr-qc/9503041

Damour T, Soffel M, Xu C (1991) General-relativistic celestial mechanics. I. Method and definition of reference systems. Phys Rev D 43(10):3273-3307. https://doi.org/10.1103/PhysRevD.43.3273
Damour T, Piazza F, Veneziano G (2002) Violations of the equivalence principle in a dilaton-runaway scenario. Phys Rev D 66(4):046007. https://doi.org/10.1103/PhysRevD.66.046007. arXiv:hep-th/0205111

Damour T, Polyakov AM (1994) The string dilation and a least coupling principle. Nucl Phys B 423 (2):532–558. https://doi.org/10.1016/0550-3213(94)90143-0

de Felice A, Tsujikawa S (2010) f(R) theories. Living Rev Relativ 13:3. https://doi.org/10.12942/lrr-2010-3

De Marchi F, Cascioli G (2020) Testing general relativity in the solar system: present and future perspectives. Class Quantum Grav 37(9):095007. https://doi.org/10.1088/1361-6382/ab6ae0. arXiv:1911.05561

De Marchi F, Tommei G, Milani A, Schettino G (2016) Constraining the Nordtvedt parameter with the BepiColombo Radioscience experiment. Phys. Rev. D. 93(12):123014. https://doi.org/10.1103/PhysRevD.93.123014. arXiv:1605.07822

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Deram P, Fienga A, Verma AK, Gastineau M, Laskar J (2022) Gaia-DR2 asteroid observations and INPOP planetary ephemerides. Celest Mech Dyn Astron 134(3):32. https://doi.org/10.1007/s10569-022-10084-6. arXiv:2203.01586

Devine CJ, Dunham DW (1966) The ephemerides of the earth-moon barycenter, Venus, Mars and Mercury considering the earth and the moon as separate bodies. Tech. rep., JPL Technical Memorandum 33-232

Di Ruscio A, Fienga A, Durante D, Iess L, Laskar J, Gastineau M (2020) Analysis of Cassini radio tracking data for the construction of INPOP19a: a new estimate of the Kuiper belt mass. Astron Astrophys 640:A7. https://doi.org/10.1051/0004-6361/202037920

Di Ruscio A, Durante D, Iess L (2020a) Mass of the saturnian system from Cassini tracking data, private communication

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Durante D, Parisi M, Serra D, Zannoni M, Notaro V, Racioppa P, Buccino DR, Lari G, Gomez Casajus L, Iess L, Folkner WM, Tommei G, Tortora P, Bolton SJ (2020) Jupiter's gravity field halfway through the Juno Mission. Geophys Res Lett 47(4):e86572. https://doi.org/10.1029/2019GL086572

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Einstein A, Infeld L, B H (1938) The gravitational equations and the problem of motion. Ann Math 39(1):65–100. https://doi.org/10.2307/1968714

Fey AL, Gordon D, Jacobs CS, Ma C, Gaume RA, Arias EF, Bianco G, Boboltz DA, Böckmann S, Bolotin S, Charlot P, Collioud A, Engelhardt G, Gipson J, Gontier AM, Heinkelmann R, Kurdubov S, Lambert S, Lytvyn S, MacMillan DS, Malkin Z, Nothnagel A, Ojha R, Skurikhina E, Sokolova J, Souchay J, Sovers OJ, Tesmer V, Titov O, Wang G, Zharov V (2015) The second realization of the international celestial reference frame by very long baseline interferometry. Astron J 150(2):58. https://doi.org/10.1088/0004-6256/150/2/58

Fienga A, Simon JL (2005) Analytical and numerical studies of asteroid perturbations on solar system planet dynamics. Astron Astrophys 429:361–367. https://doi.org/10.1051/0004-6361:20048159

Fienga A, Manche H, Laskar J, Gastineau M (2008) INPOP06: a new numerical planetary ephemeris. Astron Astrophys 477:315–327. https://doi.org/10.1051/0004-6361:20066607

Fienga A, Laskar J, Morley T, Manche H, Kuchynka P, Le Poncin-Lafitte C, Budnik F, Gastineau M, Somenzi L (2009) INPOP08, a 4-D planetary ephemeris: from asteroid and time-scale computations to ESA Mars Express and Venus Express contributions. Astron Astrophys 507:1675–1686. https://doi.org/10.1051/0004-6361/200911755. arXiv:0906.2860

Fienga A, Laskar J, Kuchynka P, Manche H, Desvignes G, Gastineau M, Cognard I, Theureau G (2011) The INPOP10a planetary ephemeris and its applications in fundamental physics. Celest Mech Dyn Astron 111:363–385. https://doi.org/10.1007/s10569-011-9377-8. arXiv:1108.5546
Damour T, Piazza F, Veneziano G (2002) Violations of the equivalence principle in a dilaton-runaway scenario. Phys Rev D 66(4):046007. https://doi.org/10.1103/PhysRevD.66.046007. arXiv:hep-th/0205111

Damour T, Polyakov AM (1994) The string dilation and a least coupling principle. Nucl Phys B 423 (2):532–558. https://doi.org/10.1016/0550-3213(94)90143-0

de Felice A, Tsujikawa S (2010) f(R) theories. Living Rev Relativ 13:3. https://doi.org/10.12942/lrr-2010-3

De Marchi F, Cascioli G (2020) Testing general relativity in the solar system: present and future perspectives. Class Quantum Grav 37(9):095007. https://doi.org/10.1088/1361-6382/ab6ae0. arXiv:1911.05561

De Marchi F, Tommei G, Milani A, Schettino G (2016) Constraining the Nordtvedt parameter with the BepiColombo Radioscience experiment. Phys. Rev. D. 93(12):123014. https://doi.org/10.1103/PhysRevD.93.123014. arXiv:1605.07822

de Rham C (2014) Massive gravity. Living Rev Relat 17:7. https://doi.org/10.12942/lrr-2014-7. arXiv:1401.4173

Deng XM, Xie Y (2012) Two-post-Newtonian light propagation in the scalar-tensor theory: an *n*-point mass case. Phys Rev D 86:044007. https://doi.org/10.1103/PhysRevD.86.044007

Deram P, Fienga A, Verma AK, Gastineau M, Laskar J (2022) Gaia-DR2 asteroid observations and INPOP planetary ephemerides. Celest Mech Dyn Astron 134(3):32. https://doi.org/10.1007/s10569-022-10084-6. arXiv:2203.01586

Devine CJ, Dunham DW (1966) The ephemerides of the earth-moon barycenter, Venus, Mars and Mercury considering the earth and the moon as separate bodies. Tech. rep., JPL Technical Memorandum 33-232

Di Ruscio A, Fienga A, Durante D, Iess L, Laskar J, Gastineau M (2020) Analysis of Cassini radio tracking data for the construction of INPOP19a: a new estimate of the Kuiper belt mass. Astron Astrophys 640:A7. https://doi.org/10.1051/0004-6361/202037920

Di Ruscio A, Durante D, Iess L (2020a) Mass of the saturnian system from Cassini tracking data, private communication

Donivan FF, Newhall XX (1981) Delta VLBI observations of Mars Viking Lander I. Bull Am Astron Soc 13:555

Douglas MR, Kachru S (2007) Flux compactification. Rev Mod Phys 79:733–796. https://doi.org/10.1103/RevModPhys.79.733

Durante D, Parisi M, Serra D, Zannoni M, Notaro V, Racioppa P, Buccino DR, Lari G, Gomez Casajus L, Iess L, Folkner WM, Tommei G, Tortora P, Bolton SJ (2020) Jupiter's gravity field halfway through the Juno Mission. Geophys Res Lett 47(4):e86572. https://doi.org/10.1029/2019GL086572

Einstein A (1915) Zur allgemeinen Relativitätstheorie. Sitzungsber Königl Preuss Akad Wiss Berlin 1915:778–786

Einstein A (1918) Prinzipielles zur allgemeinen relativitätstheorie. Ann Phys 360(4):241–244. https://doi.org/10.1002/andp.19183600402

Einstein A, Infeld L, B H (1938) The gravitational equations and the problem of motion. Ann Math 39(1):65–100. https://doi.org/10.2307/1968714

Fey AL, Gordon D, Jacobs CS, Ma C, Gaume RA, Arias EF, Bianco G, Boboltz DA, Böckmann S, Bolotin S, Charlot P, Collioud A, Engelhardt G, Gipson J, Gontier AM, Heinkelmann R, Kurdubov S, Lambert S, Lytvyn S, MacMillan DS, Malkin Z, Nothnagel A, Ojha R, Skurikhina E, Sokolova J, Souchay J, Sovers OJ, Tesmer V, Titov O, Wang G, Zharov V (2015) The second realization of the international celestial reference frame by very long baseline interferometry. Astron J 150(2):58. https://doi.org/10.1088/0004-6256/150/2/58

Fienga A, Simon JL (2005) Analytical and numerical studies of asteroid perturbations on solar system planet dynamics. Astron Astrophys 429:361–367. https://doi.org/10.1051/0004-6361:20048159

Fienga A, Manche H, Laskar J, Gastineau M (2008) INPOP06: a new numerical planetary ephemeris. Astron Astrophys 477:315–327. https://doi.org/10.1051/0004-6361:20066607

Fienga A, Laskar J, Morley T, Manche H, Kuchynka P, Le Poncin-Lafitte C, Budnik F, Gastineau M, Somenzi L (2009) INPOP08, a 4-D planetary ephemeris: from asteroid and time-scale computations to ESA Mars Express and Venus Express contributions. Astron Astrophys 507:1675–1686. https://doi.org/10.1051/0004-6361/200911755. arXiv:0906.2860

Fienga A, Laskar J, Kuchynka P, Manche H, Desvignes G, Gastineau M, Cognard I, Theureau G (2011) The INPOP10a planetary ephemeris and its applications in fundamental physics. Celest Mech Dyn Astron 111:363–385. https://doi.org/10.1007/s10569-011-9377-8. arXiv:1108.5546
Fienga A, Laskar J, Exertier P, Manche H, Gastineau M (2015) Numerical estimation of the sensitivity of INPOP planetary ephemerides to general relativity parameters. Celest Mech Dyn Astron 123:325–349. https://doi.org/10.1007/s10569-015-9639-y

Fienga A, Avdellidou C, Hanuš J (2020) Asteroid masses obtained with INPOP planetary ephemerides. Mon Not R Astron Soc 492(1):589–602. https://doi.org/10.1093/mnras/stz307

Fienga A, Di Ruscio A, Bernus L, Deram P, Durante D, Laskar J, Iess L (2020) New constraints on the location of P9 obtained with the INPOP19a planetary ephemeris. Astron Astrophys 640:A6. https://doi.org/10.1051/0004-6361/202037919

Fienga A, Bernus L, Minazzoli O, Hees A, Bigot L, Herrera C, Herrera C, Mariani V, Di Ruscio A, Mary D (2022) INPOP Planetary ephemerides and applications in the frame of the BepiColombo mission including new constraints on the graviton mass and dilaton parameters. In: Auge E, Dumarchez J, Tran Thanh Van J (eds) Proceedings, 56nd Rencontres de Moriond on Gravitation: La Thuile, Italy, January 30–February 6, 2022, ARISF. arxiv:2211.04881

Fienga A, Bigot L, Mary D, Deram P, Di Ruscio A, Bernus L, Gastineau M, Laskar J (2021a) Evolution of INPOP planetary ephemerides and Bepi-Colombo simulations. In: Celletti A et al (eds) Multi-scale (time and mass) dynamics of space objects. IAU Symposium, vol 364. Cambridge University Press, pp 31–51

Fienga A, Deram P, Di Ruscio A, Viswanathan V, Camargo JIB, Bernus L, Gastineau M, Laskar J (2021b) INPOP21a planetary ephemerides. Notes Sci Tech Inst Mec Celest 110

Fienga A, Deram P, Viswanathan V, Di Ruscio A, Bernus L, Durante D, Gastineau M, Laskar J (2019) INPOP19a planetary ephemerides. Notes Sci Tech Inst Mec Celest 109

Fienga A, Laskar J, Kuchynka P, Le Poncin-Lafitte C, Manche H, Gastineau M (2010) Gravity tests with INPOP planetary ephemerides. In: Klioner SA, Seidelmann PK, Soffel MH (eds) Relativity in fundamental astronomy: dynamics, reference frames, and data analysis, IAU Symposium, vol 261. Cambridge University Press, pp 159–169. https://doi.org/10.1017/S1743921309990330. arXiv:0906.3962

Fienga A, Laskar J, Manche H, Gastineau M (2018) Tests of GR with INPOP15a planetary ephemerides: estimations of possible supplementary advances of perihelia for Mercury and Saturn. In: Bianchi M, Jansen RT, Ruffini R (eds) Proceedings of the 14th Marcel Grossmann meeting. World Scientific, pp 3694–3695. https://doi.org/10.1142/9789813226609_0482

Fienga A, Laskar J, Manche H, Gastineau M, Verma A (2014) New INPOP release: INPOP13c. https://www.imcce.fr/recherche/equipes/asd/inpop/download13c

Fienga A, Laskar J, Manche H, Kuchynka P, Desvignes G, Gastineau M, Cognard I, Thereau G (2011b) The planetary ephemerides INPOP10a and its applications in fundamental physics. Celest Mech Dyn Astron 111:363. https://doi.org/10.1007/s10569-011-9377-8

Fischbach E, Gillies GT, Krause DE, Schwan JG, Talmadge C (1992) Non-Newtonian gravity and new weak forces: an index of measurements and theory. Metrologia 29(3):213–260. https://doi.org/10.1088/0026-1394/29/3/001

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Fienga A, Laskar J, Exertier P, Manche H, Gastineau M (2015) Numerical estimation of the sensitivity of INPOP planetary ephemerides to general relativity parameters. Celest Mech Dyn Astron 123:325–349. https://doi.org/10.1007/s10569-015-9639-y

Fienga A, Avdellidou C, Hanuš J (2020) Asteroid masses obtained with INPOP planetary ephemerides. Mon Not R Astron Soc 492(1):589–602. https://doi.org/10.1093/mnras/stz3407

Fienga A, Di Ruscio A, Bernus L, Deram P, Durante D, Laskar J, Iess L (2020) New constraints on the location of P9 obtained with the INPOP19a planetary ephemeris. Astron Astrophys 640:A6. https://doi.org/10.1051/0004-6361/202037919

Fienga A, Bernus L, Minazzoli O, Hees A, Bigot L, Herrera C, Herrera C, Mariani V, Di Ruscio A, Mary D (2022) INPOP Planetary ephemerides and applications in the frame of the BepiColombo mission including new constraints on the graviton mass and dilaton parameters. In: Auge E, Dumarchez J, Tran Thanh Van J (eds) Proceedings, 56nd Rencontres de Moriond on Gravitation: La Thuile, Italy, January 30–February 6, 2022, ARISF. arxiv:2211.04881

Fienga A, Bigot L, Mary D, Deram P, Di Ruscio A, Bernus L, Gastineau M, Laskar J (2021a) Evolution of INPOP planetary ephemerides and Bepi-Colombo simulations. In: Celletti A et al (eds) Multi-scale (time and mass) dynamics of space objects. IAU Symposium, vol 364. Cambridge University Press, pp 31–51

Fienga A, Deram P, Di Ruscio A, Viswanathan V, Camargo JIB, Bernus L, Gastineau M, Laskar J (2021b) INPOP21a planetary ephemerides. Notes Sci Tech Inst Mec Celest 110

Fienga A, Deram P, Viswanathan V, Di Ruscio A, Bernus L, Durante D, Gastineau M, Laskar J (2019) INPOP19a planetary ephemerides. Notes Sci Tech Inst Mec Celest 109

Fienga A, Laskar J, Kuchynka P, Le Poncin-Lafitte C, Manche H, Gastineau M (2010) Gravity tests with INPOP planetary ephemerides. In: Klioner SA, Seidelmann PK, Soffel MH (eds) Relativity in fundamental astronomy: dynamics, reference frames, and data analysis, IAU Symposium, vol 261. Cambridge University Press, pp 159–169. https://doi.org/10.1017/S1743921309990330. arXiv:0906.3962

Fienga A, Laskar J, Manche H, Gastineau M (2018) Tests of GR with INPOP15a planetary ephemerides: estimations of possible supplementary advances of perihelia for Mercury and Saturn. In: Bianchi M, Jansen RT, Ruffini R (eds) Proceedings of the 14th Marcel Grossmann meeting. World Scientific, pp 3694–3695. https://doi.org/10.1142/9789813226609_0482

Fienga A, Laskar J, Manche H, Gastineau M, Verma A (2014) New INPOP release: INPOP13c. https://www.imcce.fr/recherche/equipes/asd/inpop/download13c

Fienga A, Laskar J, Manche H, Kuchynka P, Desvignes G, Gastineau M, Cognard I, Thereau G (2011b) The planetary ephemerides INPOP10a and its applications in fundamental physics. Celest Mech Dyn Astron 111:363. https://doi.org/10.1007/s10569-011-9377-8

Fischbach E, Gillies GT, Krause DE, Schwan JG, Talmadge C (1992) Non-Newtonian gravity and new weak forces: an index of measurements and theory. Metrologia 29(3):213–260. https://doi.org/10.1088/0026-1394/29/3/001

Folkner WM, Williams JG, Boggs DH (2008) JPL planetary and lunar ephemerides DE421. JPL Interoffice Memorandum IOM 343R-08-003

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Folkner WM, Williams JG, Boggs DH, Park RS, Kuchynka P (2014) The planetary and lunar ephemerides 2950 DE430 and DE431. 1 IPN Progress Report 42–196

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Genova A, Mazarico E, Goossens S, Lemoine FG, Neumann GA, Smith DE, Zuber MT (2018) Solar system expansion and strong equivalence principle as seen by the NASA MESSENGER mission. Nat Commun 9:289. https://doi.org/10.1038/s41467-017-02558-1

Godun RM, Nisbet-Jones PBR, Jones JM, King SA, Johnson LAM, Margolis HS, Szymaniec K, Lea SN, Bongs K, Gill P (2014) Frequency ratio of two optical clock transitions in $^{171}$Yb$^+$ and constraints on the time variation of fundamental constants. Phys Rev Lett 113(21):210801. https://doi.org/10.1103/PhysRevLett.113.210801. arXiv:1407.0164

Guena J, Abgrall M, Rovera D, Laurent P, Chupin B, Lours M, Santarelli G, Rosenbusch P, Tobar ME, Li R, Gibble K, Clairon A, Bize S (2012) Progress in atomic fountains at LNE-SYRTE. IEEE Trans Ultrason Ferroelectr Freq Control 59(3):391-409. https://doi.org/10.1109/TUFFC.2012.2208

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Hees A, Folkner WM, Jacobson RA, Park RS (2014b) Constraints on modified Newtonian dynamics theories from radio tracking data of the Cassini spacecraft. Phys Rev D 89(10):102002. https://doi.org/10.1103/PhysRevD.89.102002. arXiv:1402.6950

Hees A, Minazzoli O, Savalle E, Stadnik YV, Wolf P (2018) Violation of the equivalence principle from light scalar dark matter. Phys Rev D 98(6):064051. https://doi.org/10.1103/PhysRevD.98.064051. arXiv:1807.04512

Hellings RW, Adams PJ, Anderson JD, Keesey MS, Lau EL, Standish JEM, Canuto VM, Goldman I (1983) Experimental test of the variability of G using Viking Lander ranging data. Phys Rev Lett 51 (18):1609-1612. https://doi.org/10.1103/PhysRevLett.51.1609

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Iorio L (2010) The perihelion precession of Saturn, planet X/Nemesis and MOND. Open Astron J 3:1-6. https://doi.org/10.2174/1874381101003010001. arXiv:0907.4514

Iorio L, Lichtenegger HIM, Ruggiero ML, Corda C (2011) Phenomenology of the Lense-Thirring effect in the solar system. Astrophys Space Sci 331(2):351-395. https://doi.org/10.1007/s10509-010-0489-5. arXiv:1009.3225

Ivashkin VV (2021) The influence of the Earth's oblateness on the energy integral and some characteristics of a spacecraft's orbit. Cosm Res 59(5):324-327. https://doi.org/10.1134/S001095252105004X

Jackson JD (1998) Classical electrodynamics, 3rd edn. Wiley

Jacobson R (2021a) The orbits of the major saturnian satellites, the trajectories of spacecraft at Saturn, the gravity field of the saturnian system, and the orientation of Saturn's pole, private communication to Horizon/NAIF/Dragonfly

Jacobson R (2021b) The orbits of the regular Jovian satellites and the orientation of the pole of Jupiter, private communication to Horizon/NAIF

Jacobson RA (2009) The orbits of the Neptunian satellites and the orientation of the pole of Neptune. Astron J 137(5):4322-4329. https://doi.org/10.1088/0004-6256/137/5/4322

Jacobson RA (2014) The orbits of the Uranian satellites and rings, the gravity field of the Uranian system, and the orientation of the pole of Uranus. Astron J 148(5):76. https://doi.org/10.1088/0004-6256/148/5/76
Gaillot AM, Le Verrier UJ (1913) Addition à la théorie du mouvement de Jupiter de Le Verrier: tables rectifiées du mouvement de Jupiter. Ann l'Observ Paris 31:1

Gao S, Wald RM (2000) Theorems on gravitational time delay and related issues. Class Quantum Grav 17 (24):4999-5008. https://doi.org/10.1088/0264-9381/17/24/305. arXiv:gr-qc/0007021

Genova A, Mazarico E, Goossens S, Lemoine FG, Neumann GA, Smith DE, Zuber MT (2018) Solar system expansion and strong equivalence principle as seen by the NASA MESSENGER mission. Nat Commun 9:289. https://doi.org/10.1038/s41467-017-02558-1

Godun RM, Nisbet-Jones PBR, Jones JM, King SA, Johnson LAM, Margolis HS, Szymaniec K, Lea SN, Bongs K, Gill P (2014) Frequency ratio of two optical clock transitions in $^{171}$Yb$^+$ and constraints on the time variation of fundamental constants. Phys Rev Lett 113(21):210801. https://doi.org/10.1103/PhysRevLett.113.210801. arXiv:1407.0164

Guena J, Abgrall M, Rovera D, Laurent P, Chupin B, Lours M, Santarelli G, Rosenbusch P, Tobar ME, Li R, Gibble K, Clairon A, Bize S (2012) Progress in atomic fountains at LNE-SYRTE. IEEE Trans Ultrason Ferroelectr Freq Control 59(3):391-409. https://doi.org/10.1109/TUFFC.2012.2208

Guéna J, Abgrall M, Rovera D, Rosenbusch P, Tobar ME, Laurent P, Clairon A, Bize S (2012) Improved tests of local position invariance using Rb87 and Cs133 fountains. Phys Rev Lett 109(8):080801. https://doi.org/10.1103/PhysRevLett.109.080801. arXiv:1205.4235

Hees A, Bertone S, Le Poncin-Lafitte C (2014a) Relativistic formulation of coordinate light time, Doppler, and astrometric observables up to the second post-Minkowskian order. Phys Rev D 89(6):064045. https://doi.org/10.1103/PhysRevD.89.064045. arXiv:1401.7622

Hees A, Folkner WM, Jacobson RA, Park RS (2014b) Constraints on modified Newtonian dynamics theories from radio tracking data of the Cassini spacecraft. Phys Rev D 89(10):102002. https://doi.org/10.1103/PhysRevD.89.102002. arXiv:1402.6950

Hees A, Minazzoli O, Savalle E, Stadnik YV, Wolf P (2018) Violation of the equivalence principle from light scalar dark matter. Phys Rev D 98(6):064051. https://doi.org/10.1103/PhysRevD.98.064051. arXiv:1807.04512

Hellings RW, Adams PJ, Anderson JD, Keesey MS, Lau EL, Standish JEM, Canuto VM, Goldman I (1983) Experimental test of the variability of G using Viking Lander ranging data. Phys Rev Lett 51 (18):1609-1612. https://doi.org/10.1103/PhysRevLett.51.1609

Hinterbichler K, Khoury J (2010) Screening long-range forces through local symmetry restoration. Phys Rev Lett 104(23):231301. https://doi.org/10.1103/PhysRevLett.104.231301. arXiv:1001.4525

Huntemann N, Lipphardt B, Tamm C, Gerginov V, Weyers S, Peik E (2014) Improved limit on a temporal variation of $m_p/m_e$ from comparisons of Yb$^+$ and Cs atomic clocks. Phys Rev Lett 113(21):210802. https://doi.org/10.1103/PhysRevLett.113.210802. arXiv:1407.4408

Imperi L, Iess L, Mariani MJ (2018) An analysis of the geodesy and relativity experiments of BepiColombo. Icarus 301:9025. https://doi.org/10.1016/j.icarus.2017.09.008

Iorio L (2009) The recently determined anomalous perihelion precession of Saturn. Astron J 137(3):3615-3618. https://doi.org/10.1088/0004-6256/137/3/3615. arXiv:0811.0756

Iorio L (2010) The perihelion precession of Saturn, planet X/Nemesis and MOND. Open Astron J 3:1-6. https://doi.org/10.2174/1874381101003010001. arXiv:0907.4514

Iorio L, Lichtenegger HIM, Ruggiero ML, Corda C (2011) Phenomenology of the Lense-Thirring effect in the solar system. Astrophys Space Sci 331(2):351-395. https://doi.org/10.1007/s10509-010-0489-5. arXiv:1009.3225

Ivashkin VV (2021) The influence of the Earth's oblateness on the energy integral and some characteristics of a spacecraft's orbit. Cosm Res 59(5):324-327. https://doi.org/10.1134/S001095252105004X

Jackson JD (1998) Classical electrodynamics, 3rd edn. Wiley

Jacobson R (2021a) The orbits of the major saturnian satellites, the trajectories of spacecraft at Saturn, the gravity field of the saturnian system, and the orientation of Saturn's pole, private communication to Horizon/NAIF/Dragonfly

Jacobson R (2021b) The orbits of the regular Jovian satellites and the orientation of the pole of Jupiter, private communication to Horizon/NAIF

Jacobson RA (2009) The orbits of the Neptunian satellites and the orientation of the pole of Neptune. Astron J 137(5):4322-4329. https://doi.org/10.1088/0004-6256/137/5/4322

Jacobson RA (2014) The orbits of the Uranian satellites and rings, the gravity field of the Uranian system, and the orientation of the pole of Uranus. Astron J 148(5):76. https://doi.org/10.1088/0004-6256/148/5/76
Jones DL, Fomalont E, Dhawan V, Romney J, Folkner WM, Lanyi G, Border J, Jacobson RA (2011) Very long baseline array astrometric observations of the Cassini spacecraft at Saturn. Astron J 141:29. https://doi.org/10.1088/0004-6256/141/2/29. arXiv:1012.0264

Jones D, Folkner W, Park R, Jacobs C, Romney J, Dhawan V (2019) Constraining Jupiter's orbit with VLBA astrometry of Juno. In: American Astronomical Society Meeting Abstracts, vol 233, p 302.01

Kardashev NS, Tutukov AV, Fedorova AV (2005) Limits on the mass of dark matter in the Sun from a model for the modern Sun and its previous evolution. Astron Rep 49(2):134-143. https://doi.org/10.1134/1.1862360

Kaula WM (1966) Theory of satellite geodesy: applications of satellites to geodesy. Blaisdell Publishing, Waltham, MA

Khoury J, Weltman A (2004) Chameleon cosmology. Phys Rev D 69(4):044026. https://doi.org/10.1103/PhysRevD.69.044026. arXiv:astro-ph/0309411

Khriplovich IB, Pitjeva EV (2006) Upper limits on density of dark matter in Solar system. Int J Mod Phys D 15(4):615-618. https://doi.org/10.1142/S0218271806008462. arXiv:astro-ph/0604122

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Konopliv AS, Asmar SW, Folkner WM, Karatekin Ö, Nunes DC, Smrekar SE, Yoder CF, Zuber MT (2011) Mars high resolution gravity fields from MRO, Mars seasonal gravity, and other dynamical parameters. Icarus 211:401-428. https://doi.org/10.1016/j.icarus.2010.10.004

Konopliv AS, Park RS, Folkner WM (2016) An improved JPL Mars gravity field and orientation from Mars orbiter and lander tracking data. Icarus 274:253-260. https://doi.org/10.1016/j.icarus.2016.02.052

Konopliv AS, Park RS, Ermakov AI (2020) The Mercury gravity field, orientation, love number, and ephemeris from the MESSENGER radiometric tracking data. Icarus 335:113386. https://doi.org/10.1016/j.icarus.2019.07.020

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Kopeikin SM (2019) Covariant equations of motion of extended bodies with arbitrary mass and spin multipoles. Phys Rev D 99(8):084008. https://doi.org/10.1103/PhysRevD.99.084008

Kopeikin S, Vlasov I (2004) Parametrized post-Newtonian theory of reference frames, multipolar expansions and equations of motion in the N-body problem. Phys Rep 400(4-6):209-318. https://doi.org/10.1016/j.physrep.2004.08.004. arXiv:gr-qc/0403068

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Jones D, Folkner W, Park R, Jacobs C, Romney J, Dhawan V (2019) Constraining Jupiter's orbit with VLBA astrometry of Juno. In: American Astronomical Society Meeting Abstracts, vol 233, p 302.01

Kardashev NS, Tutukov AV, Fedorova AV (2005) Limits on the mass of dark matter in the Sun from a model for the modern Sun and its previous evolution. Astron Rep 49(2):134-143. https://doi.org/10.1134/1.1862360

Kaula WM (1966) Theory of satellite geodesy: applications of satellites to geodesy. Blaisdell Publishing, Waltham, MA

Khoury J, Weltman A (2004) Chameleon cosmology. Phys Rev D 69(4):044026. https://doi.org/10.1103/PhysRevD.69.044026. arXiv:astro-ph/0309411

Khriplovich IB, Pitjeva EV (2006) Upper limits on density of dark matter in Solar system. Int J Mod Phys D 15(4):615-618. https://doi.org/10.1142/S0218271806008462. arXiv:astro-ph/0604122

Klioner SA (2016) Parametrized post-Newtonian equations of motion of N mass monopoles with the SEP violation. arXiv e-prints arXiv:1607.00183

Klioner SA (2008) Relativistic scaling of astronomical quantities and the system of astronomical units. Astron Astrophys 478(3):951-958. https://doi.org/10.1051/0004-6361:20077786

Klioner SA, Kopeikin SM (1992) Microarcsecond astrometry in space: relativistic effects and reduction of observations. Astron J 104:897. https://doi.org/10.1086/116284

Klioner S, Soffel M (1993) Note on the influence of dark matter on the motion of the solar system. Phys Lett A 184(1):41-44. https://doi.org/10.1016/0375-9601(93)90343-X

Klioner SA, Soffel MH (2000) Relativistic celestial mechanics with PPN parameters. Phys Rev D 62 (2):024019. https://doi.org/10.1103/PhysRevD.62.024019. arXiv:gr-qc/9906123

Konopliv AS, Banerdt WB, Sjogren WL (1999) Venus gravity: 180th degree and order model. Icarus 139 (1):3-18. https://doi.org/10.1006/icar.1999.6086

Konopliv AS, Yoder CF, Standish EM, Yuan DN, Sjogren WL (2006) A global solution for the Mars static and seasonal gravity, Mars orientation, Photos and Deimos masses, and Mars ephemeris. Icarus 182:23-50. https://doi.org/10.1016/j.icarus.2005.12.025

Konopliv AS, Asmar SW, Folkner WM, Karatekin Ö, Nunes DC, Smrekar SE, Yoder CF, Zuber MT (2011) Mars high resolution gravity fields from MRO, Mars seasonal gravity, and other dynamical parameters. Icarus 211:401-428. https://doi.org/10.1016/j.icarus.2010.10.004

Konopliv AS, Park RS, Folkner WM (2016) An improved JPL Mars gravity field and orientation from Mars orbiter and lander tracking data. Icarus 274:253-260. https://doi.org/10.1016/j.icarus.2016.02.052

Konopliv AS, Park RS, Ermakov AI (2020) The Mercury gravity field, orientation, love number, and ephemeris from the MESSENGER radiometric tracking data. Icarus 335:113386. https://doi.org/10.1016/j.icarus.2019.07.020

Kopeikin SM (2012) Celestial ephemerides in an expanding universe. Phys Rev D 86(6):064004. https://doi.org/10.1103/PhysRevD.86.064004

Kopeikin SM (2019) Covariant equations of motion of extended bodies with arbitrary mass and spin multipoles. Phys Rev D 99(8):084008. https://doi.org/10.1103/PhysRevD.99.084008

Kopeikin S, Vlasov I (2004) Parametrized post-Newtonian theory of reference frames, multipolar expansions and equations of motion in the N-body problem. Phys Rep 400(4-6):209-318. https://doi.org/10.1016/j.physrep.2004.08.004. arXiv:gr-qc/0403068

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