# Testing General Relativity with the Radio Science Experiment of the BepiColombo mission to Mercury

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## Abstract

**:**

## 1. Introduction

- (a)
- spacecraft state vector (position and velocity) at given times (Mercurycentric orbit determination);
- (b)
- (c)
- parameters defining the model of the Mercury’s rotation (rotation experiment);
- (d)
- (e)
- state vector of Mercury and Earth-Moon Barycenter (EMB) orbits at some reference epoch, in order to improve the ephemerides (Mercury and EMB orbit determination);
- (f)
- post-Newtonian (PN) parameters [12,13,21,22], together with some related parameters, like the solar oblateness factor ${J}_{2\odot}$, the solar gravitational factor ${\mu}_{\odot}=G{M}_{\odot}$, where G is the gravitational constant and ${M}_{\odot}$ the Sun’s mass, and its time variation, $\zeta =(1/{\mu}_{\odot})\phantom{\rule{0.166667em}{0ex}}d{\mu}_{\odot}/dt$, in order to test gravitational theories (relativity experiment).

## 2. The ORBIT14 Software

#### 2.1. Global Structure

**data simulator**(short: simulator) and

**differential corrector**(short: corrector). Code is written in

`Fortran90`language. The simulator is needed to predict possible scientific results of the experiment. It generates simulated observables (range and range-rate, accelerometer readings) and the nominal value for orbital elements of the Mercurycentric orbit of the spacecraft, of Mercury and of the EMB orbits. The program structure of the simulator is quite simple if compared with the differential corrector, the most demanding part being the implementation of the dynamical, observational and error models.

#### 2.2. Non-Linear Least Squares Fit

#### 2.3. Pure and Constrained Multi-Arc Strategy

**multi-arc strategy**(see, e.g., [25]). According to this method, every single arc of observations has its own set of initial conditions (position and velocity at the reference central epoch of the considered time interval), as it belongs to a different object. In this way, due to lack of knowledge in the dynamical models, the actual errors in the orbit propagation can be reduced by an over-parameterization of the initial conditions. A different choice has been made in ORBIT14, implementing the so called

**constrained multi-arc strategy**[10,14,27]. The method is based on the idea that each observed arc belongs to the same object (the spacecraft). First of all, an extended arc is defined as the observed arc broadened to half the preceding and to half the following periods without tracking, as shown in Figure 3. The orbits of two consecutive extended arcs should coincide at the connection time in the middle of the non-observed interval. We refer to [10,27] for a complete description of the constrained multi-arc strategy.

**Global Parameters ($\mathbf{g}$)**: parameters that affect the dynamical equations of every observed (and extended) arc. The PN parameters and the spherical harmonic coefficients of Mercury are an example.**Local Parameters (${\mathbf{l}}^{k}$)**: parameters that affect the dynamical equations of a single observed arc k. The state vector of the Mercurycentric orbit associated with the arc and the desaturation manoeuvres applied during the tracking are few examples.**Local External Parameters (${\mathbf{le}}^{k,k+1}$)**: parameters that affect only the dynamical equations in the period without tracking between two subsequent observed arcs k and $k+1$. These are the desaturation maneuvres taking place out of the observed arcs.

**internally constrained multi-arc strategy**. In this case, we consider the confidence ellipsoids associated with ${\mathbf{X}}_{0}^{k}$ and ${\mathbf{X}}_{0}^{k+1}$ at ${t}_{0}^{k}$ and ${t}_{0}^{k+1}$, respectively, and we propagate them to ${t}_{c}^{k}$ through the corresponding state transition matrices. This means that we expect ${\mathbf{d}}^{k,k+1}$ to be normally distributed with mean $\Phi ({t}_{c}^{k};{t}_{0}^{k+1},{\mathbf{X}}_{0}^{k+1})-\Phi ({t}_{c}^{k};{t}_{0}^{k},{\mathbf{X}}_{0}^{k})$ and covariance

**apriori constrained multi-arc strategy**and it takes care of the degeneracy in orbit determination due to the orbit geometry (details can be found in [28]). In particular, we deal with an approximated version of the exact symmetry described in [28], where the small parameter of the perturbation is the angle of displacement of the Earth-Mercury vector in an inertial frame. In this case, the normal matrix has one eigenvalue significantly smaller than the others. As a consequence of this weakness, the confidence ellipsoid associated with the discrepancy and defined by ${C}^{k,k+1}$ could be very elongated. The basic idea of this approach is to constrain the discrepancy ${\mathbf{d}}^{k,k+1}$ inside a sphere of given radius, that can be suitably shrunk by varying μ. This can be interpreted as adding apriori observations. On the contrary, in the internally constrained multi-arc strategy, the discrepancy is constrained inside the intersection of the two ellipsoids propagated from ${t}_{0}^{k}$ and ${t}_{0}^{k+1}$. All the details are extensively explained in [27]. For the results presented in this review, we will always adopt an apriori constrained multi-arc strategy. Finally, it can be noted that in the multi-arc method the residuals

**ξ**depend only on global and local parameters, and this applies to the target function Q defined in Equation (1) as well.

## 3. Mathematical Models

- the Eddington parameters β and γ. β accounts for the modification of the non-linear three-body general relativistic interaction and γ parametrizes the velocity-dependent modification of the two-body interaction and accounts also for the space-time curvature through the Shapiro effect [30]. These are the only non-zero PN parameters in GR (they are both equal to unity);
- the Nordtvedt parameter η. The effect of η in the equations of motion is to produce a polarization of the Mercury and Earth orbits in the gravitational field of the other planets and it is related to possible violations of the Strong Equivalence Principle (see, e.g., the discussion in [31]);
- the preferred frame effects parameters ${\alpha}_{1}$ and ${\alpha}_{2}$. They phenomenologically describe the effects due to the presence of a gravitationally preferred frame; we follow the standard assumption to identify the preferred frame with the rest frame of the cosmic microwave background [32].

#### 3.1. Computation of Observables

- the Mercurycentric position of the spacecraft, ${\mathbf{x}}_{sat}$;
- the SSB positions of Mercury and of the EMB, ${\mathbf{x}}_{M}$ and ${\mathbf{x}}_{EM}$;
- the geocentric position of the ground antenna, ${\mathbf{x}}_{ant}$;
- the position of the Earth barycenter with respect to the EMB, ${\mathbf{x}}_{E}$.

#### 3.2. Dynamical Relativistic Models

**N-body Newtonian Lagrangian:**$${L}_{New}=\frac{1}{2}\sum _{i}\phantom{\rule{0.166667em}{0ex}}{\mu}_{i}\phantom{\rule{0.166667em}{0ex}}{v}_{i}^{2}+\frac{1}{2}\phantom{\rule{0.166667em}{0ex}}\sum _{i}\phantom{\rule{0.166667em}{0ex}}\sum _{j\ne i}\frac{{\mu}_{i}{\mu}_{j}}{{r}_{ij}}\phantom{\rule{0.166667em}{0ex}}.$$**Post-Newtonian General Relativistic Lagrangian:**$$\begin{array}{ccc}\hfill {L}_{GR}& =& \frac{1}{8\phantom{\rule{0.166667em}{0ex}}{c}^{2}}\phantom{\rule{0.166667em}{0ex}}\sum _{i}\phantom{\rule{0.166667em}{0ex}}{\mu}_{i}{v}_{i}^{4}-\frac{1}{2\phantom{\rule{0.166667em}{0ex}}{c}^{2}}\phantom{\rule{0.166667em}{0ex}}\sum _{i}\phantom{\rule{0.166667em}{0ex}}\sum _{j\ne i}\phantom{\rule{0.166667em}{0ex}}\sum _{k\ne i}\frac{{\mu}_{i}\phantom{\rule{0.166667em}{0ex}}{\mu}_{j}\phantom{\rule{0.166667em}{0ex}}{\mu}_{k}}{{r}_{ij}\phantom{\rule{0.166667em}{0ex}}{r}_{ik}}+\hfill \\ & +& \frac{1}{2}\sum _{i}\sum _{j\ne i}\frac{{\mu}_{i}\phantom{\rule{0.166667em}{0ex}}{\mu}_{j}}{{r}_{ij}}\left[\frac{3}{2{c}^{2}}({v}_{i}^{2}+{v}_{j}^{2})-\frac{7}{2\phantom{\rule{0.166667em}{0ex}}{c}^{2}}({\mathbf{v}}_{\mathbf{i}}\xb7{\mathbf{v}}_{\mathbf{j}})-\frac{1}{2\phantom{\rule{0.166667em}{0ex}}{c}^{2}}({\mathbf{n}}_{\mathbf{ij}}\xb7{\mathbf{v}}_{\mathbf{i}})({\mathbf{n}}_{\mathbf{ij}}\xb7{\mathbf{v}}_{\mathbf{j}})\right]\phantom{\rule{0.166667em}{0ex}}.\hfill \end{array}$$**Lagrangian for PN parameter γ:**$${L}_{\gamma}=\frac{1}{2\phantom{\rule{0.166667em}{0ex}}{c}^{2}}\sum _{i}\sum _{j\ne i}\frac{{\mu}_{i}\phantom{\rule{0.166667em}{0ex}}{\mu}_{j}}{{r}_{ij}}{({\mathbf{v}}_{\mathbf{i}}-{\mathbf{v}}_{\mathbf{j}})}^{2}\phantom{\rule{0.166667em}{0ex}}.$$**Lagrangian for PN parameter β:**$${L}_{\beta}=-\frac{1}{{c}^{2}}\sum _{i}\sum _{j\ne i}\sum _{k\ne i}\frac{{\mu}_{i}\phantom{\rule{0.166667em}{0ex}}{\mu}_{j}\phantom{\rule{0.166667em}{0ex}}{\mu}_{k}}{{r}_{ij}\phantom{\rule{0.166667em}{0ex}}{r}_{ik}}\phantom{\rule{0.166667em}{0ex}}.$$**Lagrangian for parameter ζ:**${L}_{\zeta}$ describes the effect of a time variation of the gravitational parameter of the Sun, ${\mu}_{\odot}$:$${\mu}_{\odot}={\mu}_{\odot}({t}_{0})+{\dot{\mu}}_{\odot}({t}_{0})(t-{t}_{0})+\dots \phantom{\rule{0.166667em}{0ex}};$$$$\zeta =\frac{{\dot{\mu}}_{\odot}({t}_{0})}{{\mu}_{\odot}({t}_{0})}=\frac{d}{dt}ln{\mu}_{\odot}({t}_{0})\phantom{\rule{0.166667em}{0ex}},$$$${L}_{\zeta}=(t-{t}_{0})\sum _{i\ne 0}\frac{{\mu}_{\odot}{\mu}_{i}}{{r}_{0i}}\phantom{\rule{0.166667em}{0ex}}.$$**Lagrangian for ${\mathbf{J}}_{\mathbf{2}\odot}$ effect:**$${L}_{{J}_{2\odot}}=-\frac{1}{2}\sum _{i\ne 0}\frac{{\mu}_{0}\phantom{\rule{0.166667em}{0ex}}{\mu}_{i}}{{r}_{0i}}{\left(\frac{{R}_{\odot}}{{r}_{0i}}\right)}^{2}[3{({\mathbf{n}}_{0i}\xb7{\mathbf{e}}_{0})}^{2}-1]\phantom{\rule{0.166667em}{0ex}},$$**Lagrangian for preferred frame effects, PN ${\alpha}_{1}$ and ${\alpha}_{2}$:**$${L}_{{\alpha}_{1}}=-\frac{1}{4\phantom{\rule{0.166667em}{0ex}}{c}^{2}}\sum _{j}\sum _{i\ne j}\frac{{\mu}_{i}\phantom{\rule{0.166667em}{0ex}}{\mu}_{j}}{{r}_{ij}}\left({\mathbf{z}}_{i}\xb7{\mathbf{z}}_{j}\right),$$$${L}_{{\alpha}_{2}}=\frac{1}{4\phantom{\rule{0.166667em}{0ex}}{c}^{2}}\sum _{j}\sum _{i\ne j}\frac{{\mu}_{i}\phantom{\rule{0.166667em}{0ex}}{\mu}_{j}}{{r}_{ij}}\left[\left({\mathbf{z}}_{i}\xb7{\mathbf{z}}_{j}\right)-\left({\mathbf{n}}_{\mathbf{i}\phantom{\rule{0.166667em}{0ex}}\mathbf{j}}\xb7{\mathbf{z}}_{i}\right)\left({\mathbf{n}}_{\mathbf{i}\phantom{\rule{0.166667em}{0ex}}\mathbf{j}}\xb7{\mathbf{z}}_{j}\right)\right],$$$$\begin{array}{ccc}\hfill {L}_{\alpha}={\alpha}_{1}\phantom{\rule{0.166667em}{0ex}}{L}_{{\alpha}_{1}}+{\alpha}_{2}\phantom{\rule{0.166667em}{0ex}}{L}_{{\alpha}_{2}}& =& \frac{{\alpha}_{2}-{\alpha}_{1}}{4\phantom{\rule{0.166667em}{0ex}}{c}^{2}}\sum _{j}\sum _{i\ne j}\frac{{\mu}_{i}\phantom{\rule{0.166667em}{0ex}}{\mu}_{j}}{{r}_{ij}}\phantom{\rule{0.166667em}{0ex}}({\mathbf{v}}_{\mathbf{i}}+\mathbf{w})\xb7({\mathbf{v}}_{\mathbf{j}}+\mathbf{w})+\hfill \\ & -& \frac{{\alpha}_{2}}{4\phantom{\rule{0.166667em}{0ex}}{c}^{2}}\sum _{j}\sum _{i\ne j}({\mathbf{r}}_{\mathbf{j}\phantom{\rule{0.166667em}{0ex}}\mathbf{i}}\xb7({\mathbf{v}}_{\mathbf{j}}+\mathbf{w}))\phantom{\rule{0.166667em}{0ex}}({\mathbf{r}}_{\mathbf{j}\phantom{\rule{0.166667em}{0ex}}\mathbf{i}}\xb7({\mathbf{v}}_{\mathbf{i}}+\mathbf{w}))\frac{{\mu}_{i}\phantom{\rule{0.166667em}{0ex}}{\mu}_{j}}{{r}_{ij}^{3}}\phantom{\rule{0.166667em}{0ex}}.\hfill \end{array}$$**Lagrangian for possible violation of the equivalence principle, PN η:**With the Lagrangian multiplied by G, the Newtonian kinetic energy is:$$T=\frac{1}{2}\sum _{i}\phantom{\rule{0.166667em}{0ex}}{\mu}_{i}\phantom{\rule{0.166667em}{0ex}}{v}_{i}^{2}\phantom{\rule{4pt}{0ex}},$$$${\mu}_{i}=(1+\eta {\Omega}_{i})\phantom{\rule{0.166667em}{0ex}}{\mu}_{i}^{I}\u27fa{\mu}_{i}^{I}=(1-\eta {\Omega}_{i})\phantom{\rule{0.166667em}{0ex}}{\mu}_{i}+\mathcal{O}({\eta}^{2})$$$${L}_{\eta}=-\frac{1}{2}\phantom{\rule{0.277778em}{0ex}}\sum _{i}\phantom{\rule{0.166667em}{0ex}}{\Omega}_{i}\phantom{\rule{0.166667em}{0ex}}{\mu}_{i}\phantom{\rule{0.166667em}{0ex}}{v}_{i}^{2}\phantom{\rule{4pt}{0ex}}.$$

#### 3.3. Mercurycentric Dynamical Model

#### 3.3.1. Mercury Gravity Field (Static Part)

#### 3.3.2. Tidal Perturbations

#### 3.3.3. Sun and Planetary Perturbations

#### 3.3.4. Rotational Dynamics

#### 3.3.5. Non-Gravitational Perturbations

**ε**represents the contribution of all the error sources in the ISA readings. As already highlighted, one of the main goals of the radio science experiment is to perform a very accurate orbit determination of the Mercurycentric motion of the spacecraft. To this aim, what really matters is to remove in the most suitable way any bias introduced in the accelerometer readings by instrumental errors. For this reason, in our analysis we mainly focus on the techniques to handle these error terms instead of accurately modeling the non-gravitational perturbations themselves.

## 4. Simulation Scenario and Assumptions

#### 4.1. Observables Error Models

#### 4.1.1. Range and Range-Rate

#### 4.1.2. Accelerometer Readings and Calibration Strategy

**ε**, we assume the model provided by ISA team (private communications). It consists of a random background with some periodic terms superimposed: the main ones are a thermal term, resulting in a sinusoid at Mercury sidereal period ($7.6\times {10}^{6}$ s) and a resonant term, resulting in a sinusoid at the orbital period of the spacecraft ($8.3\times {10}^{3}$ s). All the details on the adopted model and the effects of the main components on the Mercurycentric orbit determination are described in [10].

**ψ**, to be added in the solve for list, and of time, such that $\mathbf{\epsilon}(t)-c(\mathbf{\psi};t)\simeq 0$. In such a way, the calibration function $c(\mathbf{\psi};t)$ absorbs most of the accelerometer error and the physical parameters of interest for the radio science experiment are, in principle, not anymore biased. In the ORBIT14 software we implemented a novel calibration strategy, in which the calibration function is represented by a ${\mathcal{C}}^{1}$ cubic spline. All the details can be found in [19]. As a consequence, six additional parameters per arc (two per direction) are determined. We point out, as extensively discussed in [10], that this calibration strategy is able to absorb the low frequencies (i.e., longer than one day) error terms and the random component; in fact, the coefficients of the spline polynomials are computed once per arc, hence features with a periodicity lower than one day cannot be accounted for. This means that the resonant term, which shows a periodicity significantly lower than one day (about 2.3 h), is not absorbed by calibration at all. While this term results highly critical for what concerns the gravimetry and rotation experiments, we will see that its amplitude is not significantly detrimental for the relativity experiment.

#### 4.2. Desaturation Maneuvres

#### 4.3. Metric Theories of Gravitation

#### 4.4. Rank Deficiencies in the Mercury and EMB Orbit Determination Problem

## 5. Results

- Global dynamical:
- -
- PN parameters: β, γ, η, ${\alpha}_{1}$, ${\alpha}_{2}$;
- -
- other parameters of interest for the relativity experiment: ${\mu}_{\odot}$, ζ, ${J}_{2\odot}$;
- -
- the state vectors of Mercury and EMB (8 components): (${x}_{M},{y}_{M},{z}_{M};{\dot{x}}_{M},{\dot{y}}_{M},{\dot{z}}_{M}$); (${\dot{x}}_{EM},{\dot{y}}_{EM}$);
- -
- normalized harmonic coefficients of the gravity field of Mercury up to degree and order 25 and the Love number ${k}_{2}$;
- -
- rotational parameters: ${\delta}_{1}$, ${\delta}_{2}$, ${\epsilon}_{1}$, ${\epsilon}_{2}$;
- -
- six accelerometer calibration coefficients for each arc, plus 6+6 boundary conditions;

- Local dynamical:
- -
- state vector of the Mercurycentric orbit of the spacecraft, in the Ecliptic J2000 inertial reference frame, at the central time of each observed arc;
- -
- three dump manoeuvre components, $\Delta \mathbf{v}$, taking place during tracking, for each observed arc;

- External local dynamical:
- -
- three dump manoeuvre components, $\Delta \mathbf{v}$, taking place in the period without tracking between each pair of consecutive observed arcs.

#### 5.1. The Relativity Experiment Results

#### 5.2. Results for Gravimetry and Rotation

- relativity simulations: we removed from the solve-for list the gravimetry and rotational parameters, i.e. the gravity field spherical harmonic coefficients, Love number ${k}_{2}$, the angles $({\delta}_{1},{\delta}_{2})$, the libration amplitudes ${\epsilon}_{1}$, ${\epsilon}_{2}$;
- gravimetry and rotation simulations: we removed from the solve-for list the PN and related parameters.

## 6. Discussion and Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Block diagram of the ORBIT14 code: simulation and differential corrections stages. Green arrows refer to simulator inputs/outputs and orange arrows to corrector inputs/outputs. The input option files for simulator and corrector are similar and include, for example, the state vector of the spacecraft at the initial epoch, the number of considered arcs, the time steps for the orbit propagation of the spacecraft, Mercury and EMB, the time sampling for range, range-rate and accelerometer data.

**Figure 2.**Block diagram of a differential corrector decomposed in three steps: (1) in “cor_par_setup” all the input options are read, data are split for the following parallel computation and the orbits of Mercury and EMB are propagated; (2) “cor_par_arc” contains most of the computationally expensive processing and is parallelized, by executing multiple copies of the same code, without need for interprocess communication; at this stage, the orbit of the spacecraft is propagated at each arc, the light-time computation is performed and residuals and normal matrix are given as output for the next step; (3) in “cor_solve” the covariance matrix and the LS solution are computed.

**Figure 3.**Schematic representation of observed and extended arc. The times ${t}_{i}$ ($i=1,..,4$) are the central epoch of each arc; the black bars correspond to dark intervals, without tracking from Earth. See the text for more explanation.

**Figure 4.**Vectors involved in the multiple dynamics for the tracking of the spacecraft from the Earth.

**Figure 5.**The difference in the observables range and range-rate for one pass of Mercury above the horizon for a ground station, by using an hybrid model in which the position and velocity of the orbiter have not been transformed to TDB-compatible quantities and a correct model in which all quantities are TDB-compatible. Gaps of the signal are due to spacecraft passage behind Mercury as seen for the Earth station. (

**Top**): for a hybrid model with the satellite position and velocity not transformed to TDB-compatible; (

**Bottom**): for a hybrid model with the position and velocity of the antenna not transformed to TDB-compatible.

**Table 1.**Simulation results for the parameters of interest in the MORE relativity experiment (errors on ${\mu}_{\odot}$ are in cm${}^{3}$/s${}^{2}$, on ζ in y${}^{-1}$).

Parameter | Formal Error | True Error | T/F Error Ratio | Current Accuracy |
---|---|---|---|---|

β | $7.3\times {10}^{-7}$ | $2.6\times {10}^{-6}$ | 3.6 | $7\times {10}^{-5}$ [52] |

γ | $9.3\times {10}^{-7}$ | $1.1\times {10}^{-6}$ | 1.2 | $2.3\times {10}^{-5}$ [48] |

η | $2.2\times {10}^{-6}$ | $1.1\times {10}^{-5}$ | 4.9 | $4.5\times {10}^{-4}$ [53] |

${\alpha}_{1}$ | $4.9\times {10}^{-7}$ | $4.9\times {10}^{-7}$ | 1.0 | $6.0\times {10}^{-6}$ [54] |

${\alpha}_{2}$ | $8.3\times {10}^{-8}$ | $1.0\times {10}^{-7}$ | 1.2 | $3.5\times {10}^{-5}$ [54] |

${\mu}_{\odot}$ | $4.2\times {10}^{13}$ | $4.2\times {10}^{13}$ | 1.0 | ${10}^{16}$, $8\times {10}^{15}$ [55,56] |

ζ | $2.3\times {10}^{-14}$ | $3.6\times {10}^{-14}$ | 1.5 | $4.3\times {10}^{-14}$ [57] |

${J}_{2\odot}$ | $4.1\times {10}^{-10}$ | $4.1\times {10}^{-10}$ | 1.0 | $1.2\times {10}^{-8}$ [52] |

**Table 2.**Correlations between PN and related parameters (values higher than 0.7 are highlighted in bold).

β | γ | η | ${\mathit{\alpha}}_{1}$ | ${\mathit{\alpha}}_{2}$ | ${\mathit{\mu}}_{\odot}$ | ζ | ${\mathit{J}}_{2\odot}$ | |
---|---|---|---|---|---|---|---|---|

${J}_{2\odot}$ | 0.15 | 0.21 | 0.11 | 0.90 | 0.29 | 0.89 | 0.10 | – |

ζ | <0.1 | 0.28 | <0.1 | <0.1 | 0.17 | <0.1 | – | |

${\mu}_{\odot}$ | 0.20 | 0.14 | 0.14 | 0.84 | <0.1 | – | ||

${\alpha}_{2}$ | 0.35 | 0.28 | 0.36 | 0.27 | – | |||

${\alpha}_{1}$ | 0.35 | 0.12 | 0.22 | – | ||||

η | 0.96 | 0.60 | – | |||||

γ | 0.77 | – | ||||||

β | – |

**Table 3.**Comparison between the results in Schettino & Tommei (2016) (this paper) and previous results of the relativity experiment (${\mu}_{\odot}$ in cm${}^{3}$/s${}^{2}$).

Parameter | Schettino & Tommei (2016) | Milani et al. (2002) [12] | Iess et al. (2009) [6] |
---|---|---|---|

β | $7.3\times {10}^{-7}$ | $9.2\times {10}^{-7}$ | $2\times {10}^{-6}$ |

γ | $9.3\times {10}^{-7}$ | $2\times {10}^{-6}$ (SCE) | $2\times {10}^{-6}$ |

η | $2.2\times {10}^{-6}$ | $3.3\times {10}^{-6}$ | $8\times {10}^{-6}$ |

${\alpha}_{1}$ | $4.9\times {10}^{-7}$ | $7.1\times {10}^{-7}$ | – |

${\alpha}_{2}$ | $8.3\times {10}^{-8}$ | $1.9\times {10}^{-7}$ | – |

${\mu}_{\odot}$ | $4.2\times {10}^{13}$ | $4.1\times {10}^{13}$ | – |

${J}_{2\odot}$ | $4.1\times {10}^{-10}$ | $6.2\times {10}^{-10}$ | $2\times {10}^{-9}$ |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Schettino, G.; Tommei, G.
Testing General Relativity with the Radio Science Experiment of the BepiColombo mission to Mercury. *Universe* **2016**, *2*, 21.
https://doi.org/10.3390/universe2030021

**AMA Style**

Schettino G, Tommei G.
Testing General Relativity with the Radio Science Experiment of the BepiColombo mission to Mercury. *Universe*. 2016; 2(3):21.
https://doi.org/10.3390/universe2030021

**Chicago/Turabian Style**

Schettino, Giulia, and Giacomo Tommei.
2016. "Testing General Relativity with the Radio Science Experiment of the BepiColombo mission to Mercury" *Universe* 2, no. 3: 21.
https://doi.org/10.3390/universe2030021