Improvement of the planetary ephemeris and test of general relativity with MESSENGER

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Mars Global Surveyor: Radioscience data analysis

Introduction

Mars is the most explored terrestrial planet in the solar system. Many space missions have been attempted to Mars than to any other place in the solar system except the Moon. These missions include flyby missions (Mariner, Rosetta), orbiter missions (Mars Global Surveyor, Mars Odyssey, Mars Express, Mars Reconnaissance Orbiter ), and lander missions (Viking, Mars Exploration Rovers, Mars Science Laboratory). Obtaining scientific information is the primary reason for launching and operating such deep-space missions. Usually, science ob-jectives of such missions involve: high resolution imaging of the planet surface, studies of the gravity and topography, studies of the atmosphere and the interior of the Mars, studies of bio-logical, geological, and geochemical processes, etc. To achieve the mission objectives, some of these investigations are performed by using a dedicate science instrument aboard the spacecraft, which measure a particular physical phenomenon, for example:
• The thermal emission spectrometer: is used to measure the infrared spectrum of energy emitted by the planet. This information is used to study the composition of rock, soil, atmospheric dust, clouds, etc.
• The orbital laser altimeter: is used to measure the time, takes for a transmitted laser beam to reach the surface, reflect, and return. This information provides the topographic maps of Mars.
• The magnetometer: is used to determine a magnetic field, and its strength and orientation.
On the other hand, some objectives are undertaken as opportunities arise to take advantage of a spacecraft special capabilities or unique location or other circumstance. For example:
• With the radioscience experiment, which measures the Doppler shift of radio signals, it is then possible to determine the gravity field by computing the change in the speed of the spacecraft, associated with the high and low concentration of the mass below and at the surface of the Mars. Moreover, unique location such as spacecraft occultation allows the radio signals pass through the Martian atmosphere on their way to Earth. Hence, perturbations in the signals induced by the atmosphere allows to derive the atmospheric and ionospheric characteristics of the Mars. Brief description of such investigations are discussed in Chapter 2.
This chapter deals with the radioscience data analyses to precisely construct the MGS or-bit. Such analyses has been already performed by several authors, such as Yuan et al. (2001); Lemoine et al. (2001); Konopliv et al. (2006); Marty et al. (2009). We have therefore cho-sen MGS as an academic case to test our understanding of the raw radiometric data (ODF, see Chapter 2) and their analysis with GINS by comparing our results with the one found in the above literature. Moreover, these analyses also allowed us to derive solar corona model and to perform corona physic studies. Derivation of such models and their application to planetary ephemerides are discussed in Chapter 4 and in Verma et al. (2013a).
The outline of this chapter is as follows: in the Section 3.2 an overview of the MGS mission is discussed. Data processing and dynamic modeling used for the orbit construction is described in the Section 3.3. Results obtained during the orbit computation and their comparison with the estimations found in the literature are discussed in the Section 3.4. The supplementary tests, which include: (a) comparison between the GINS solution and the JPL light time solution, and (b) the impact of the macro-model on the orbit reconstruction, are discussed in Section 3.5. Conclusion and prospectives of these results are reported in Section 3.6.

Mission overview

Mission design

MGS was a NASA’s space mission to Mars. It was launched from the Cape Canaveral Air Station in Florida on 7th November 1996 aboard a Delta II rocket. The MGS began its Mars orbit insertion on 12th September 1997. Figure 3.1 illustrates the summary of the MGS mission from launch to the Mars orbit insertion (MOI) in an elliptical orbit, the initial areobraking period, the science-phase period, the aerobraking resumption, and mapping in the circular orbit (Albee et al., 2001).
The MOI represents an extremely crucial maneuver, because any failure would result in a fly-by mission. The MOI slows down the spacecraft and allows Mars to capture the probe into an elliptical orbit. Near the point of closest approach on spacecraft in-bound trajectory, the main engines fired for approximately 20 to 25 minutes to slow down the spacecraft. This In order to attain the mapping orbit, MGS spacecraft was designed to facilitate the use of aerobraking. Aerobraking is the utilization of atmospheric drag on the spacecraft to reduce the energy of the orbit. The friction caused by the passage of the spacecraft through the atmosphere provides a velocity change at periapsis, which results in the lowering of the apoapsis altitude. After almost sixteen months of orbit insertion, the aerobraking events were utilized to convert the elliptical orbit into an almost circular 2-hour low altitude sun synchronous polar orbit with an average altitude of 378 km. Thus, MGS started its low altitude mapping orbital phase in March 1999 and lost communication with the ground station on 2nd November 2006.

Spacecraft geometry

MGS was designed to carry science payloads to Mars, to maintain proper pointing and orbit as a three-axis stabilized platform for acquiring mapping data and return to the Earth.
In order to meet the strength mass requirements, the spacecraft structure was constructed of lightweight composite materials. It was divided into four sub-assemblies (Figure 3.2, extracted from Albee et al. (2001)):
• Equipment module: it houses the avionics system and the science instruments. MGS carried six on-board instruments, that are Magnetometer/Electron Reflectometer, Mars orbiter camera, Mars orbiter laser altimeter, Mars relay, Thermal Emissions Spectrometer, and ultra-stable oscillator. All science instruments are bolted to the nadir equipment deck, mounted above the equipment module on the +Z panel.
• Propulsion module: it serves as the adapter between the launch vehicle and contains the propellant tanks, main engines, and attitude control thrusters. This module bolts beneath the equipment module on the -Z panel.
• Solar arrays: two solar arrays provide power for the MGS spacecraft. Each array mounts close to the top of the propulsion module on the +Y and -Y panels, near the interface between the propulsion and equipment modules. Rectangular shaped, metal drag f la ps mounted onto the ends of both arrays. These f la ps were only used to increase the total surface area of the array structure to increase the spacecraft’s ballistic coefficient during aerobraking.
• High Gain Antenna: a parabolic high gain antenna (HGA) structure was also bolt to the outside of the propulsion module. It was used to make high rate communication with the Earth. When fully deployed, the 1.5 meter diameter HGA sits at the end of a 2.0 meter long boom, mounted to the +X panel of the propulsion module.
As described in Chapter 2 (Section 2.5.1), the precise computation of the spacecraft tra-jectory relies on the non-gravitational accelerations that are acting in the spacecraft. These non-gravitational forces depend on the shape, size, surface properties, and orientations of the spacecraft. Thus, an accurate spacecraft geometry is an essential information for modeling such forces precisely. The characteristics of the MGS macro-model are given in Table 3.1 (Lemoine et al., 1999).

Radioscience data

MGS is the first operational planetary mission to employ exclusively X-band technology for radioscience observations, tracking, and spacecraft command and communication (Tyler et al., 2001). The radioscience instrument used for this purpose is the spacecraft telecommunications subsystem, augmented by an ultra-stable oscillator, and the normal MGS transmitter and re-ceiver. The ultra-stable oscillator typically have frequency stabilities on the order of 1×10−13 for time intervals of 1 to 100 seconds (Cash et al., 2008). The oscillator provides the refer-ence frequency for the radioscience experiments and operates on the X-band 7164.624 MHz uplink and 8416.368 MHz downlink frequency. The radioscience data are then collected by the DSN and consist of one-way Doppler, two- and three-way ramped Doppler, and two-way range observations (see Chapter 2 for full details of radioscience data).
Figure 5.2.3 illustrates the MGS tracking strategy during the mapping period for sharing the orbit between: i) the occultation studies when MGS was near the limb of the planet, and ii) the gravity studies during the MGS Earth side pass (Tyler et al., 2001). Pole-to-Pole two- and three-way Doppler observations provide the primary information of Mars gravity field. Whereas, one-way Doppler observations obtained during the occultation period provide an information about the Martian atmosphere and ionosphere. Table 3.2 gives the summary of the data coverage obtained during the mapping period and used to construct the MGS orbit.

Data processing and dynamic modeling

The radioscience observations, used for computing the MGS orbit, are available on the NASA Planetary Data System (PDS) Geoscience website1. These observations are analyzed with the help of the GINS software. As described in Section 2.5 of Chapter 2, GINS numerically inte-grates the equations of motion (Eqs. 2.62) and the associated variational equations (Eqs. 2.83). It simultaneously retrieves the physical parameters of the force model using an iterative least-squares technique (see Section 2.5.3). The modeling of the MGS orbit includes gravitational and non-gravitational forces that are acting on the spacecraft (see Section 2.5.1). In addition to these forces, third body perturbations due to Phobos and Deimos are also included.
The data processing and the dynamic modeling are done as follows:
• In order to have access to the planet positions and velocities, planetary ephemeris has been used for both measurements and force models (e.g, DE405, INPOP10b).
• The Mars geopotential is modeled in terms of spherical harmonic. This model is given by Eq. 2.65 as described in section 2.5.1. Fully normalized spherical harmonic coefficients of MGS95J solution2 has been used. MGS95J is a 95×95 spherical harmonics model which was derived from 6 years of MGS tracking data plus 3 years of measurements on the Mars Odyssey (ODY) spacecraft (Konopliv et al., 2006).
• The rotation model which defines the orientation of the Mars is taken from the Konopliv et al. (2006).
• Earth kinematics and polar motion effects are taken according to the IERS standards (McCarthy and Petit, 2003).
• The Phobos and Deimos ephemerides are taken as developed by Lainey et al. (2007).
• The complex geometry of the spacecraft is treated as a combination of flat plates arrange in the shape of box, with attached solar arrays and drag flaps, and attached HGA. This Box-Wing model includes six plates for the spacecraft bus, four plates to represent the front and back side of the +Y and -Y solar arrays, and a parabolic HGA. The surface area and reflectivity of this macro-model are given in Table 3.1. Moreover, in addition to this macro-model, a Spherical macro-model has been also used to reconstruct the MGS orbit and to understand the impact of macro-model over the orbit reconstruction (see Section 3.5).
• Using such configurations of the MGS macro-models, solar radiation pressure (Eq. 2.74), atmospheric drag (Eq. 2.75), and Mars radiation pressure (Eqs. 2.78 and 2.79) forces are computed separately for each plate and HGA. Vectorial sum of all these components are then compute to calculate the total force acting on the spacecraft.
• In addition to the macro-model characteristics, orientations of the spacecraft are also taken in account. The attitude of spacecraft, and of its articulated panels and antenna in inertial frame are defined in terms of quaternions. These quaternions are extracted from the SPICE Navigation and Ancillary Information Facility (NAIF) C-Kernels3.
• MGS periodically fires its thruster to desaturate the reactions wheels, which absorb an-gular momentum from disturbance torque acting on the spacecraft. Thus, empirical ac-celerations are modeled over the duration of each angular momentum wheel desaturation (AMD) event (Marty, 2011). Constant radial, along-track, and cross-track accelerations are applied over the duration of each AMD event, and are estimated as part of each orbit determination solution.
• The relativistic effects on the measurements and on the spacecraft dynamics are modeled based on the PPN formulation as described in Sections 2.4 and 2.5.1 of Chapter 2.
• The tropospheric delay corrections to the measurements are also included. Computation of this delay uses meteorological data (pressure, temperature and humidity) collected every half-hour at the DSN sites.

Solve-for parameters

For the orbit computation and for the parameter estimation, a multi-arc approach is used to get independent estimates of the MGS accelerations. In this method, orbital fits are obtained from short data-arcs of two days with two hours (approx. one orbital period of MGS) of overlapping period. The short day data-arcs are used to accounting the model imperfections (see Section 3.5) and overlapping period are used to estimate the quality of the spacecraft orbit determina-tion by taking orbit overlap differences between two successive data-arcs (see Section 3.4). In order to account the effect of shortest wavelengths of Mars gravity field on the MGS motion, we integrate the equations of motion using the time-step of 20s. An iterative least-squares fit is then performed on the complete set of Doppler- and range-tracking data-arcs.
Solve-for parameters which have been estimated during the MGS orbit determination are:
• The initial state vector components at the epoch for each arc. Prior values of these vectors are taken from the SPICE NAIF kernels.
• Scale factor, FD , for the drag force. One FD per arc is computed for accounting mis-modeling of the drag force.
• Scale factor, FS , for the solar radiation pressure force. One FS per arc is also computed for accounting mis-modeling in the solar radiation pressure.
• Empirical delta accelerations, radial, along-track, and cross-track, are computed at the AMD epochs. The information of these epochs are given by Marty (2011).
• For each arc, one offset per DSN station for two or three-way Doppler measurements.
• One offset per arc for one-way Doppler measurements accounting for the ultra stable oscillator stability uncertainty.
• One bias per DSN station for accounting the uncertainties on the DSN antenna center position or the instrumental delays (such as the range group delay of the transponder)
• One range bias per arc for ranging measurements to account the systematic geometric positions error (ephemerides error) between the Earth and the Mars.

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Orbit computation results

Acceleration budget

Accurate orbit determination of planetary spacecraft requires good knowledge of gravitational and non-gravitational forces which act on the spacecraft. These forces are precisely modeled in the GINS software as described in Chapter 2. Figure 3.4 illustrates an average of various accelerations experienced by the MGS spacecraft, that are:
• Accelerations due to gravitational potential: the first two columns of the Figure 3.4 represent the accelerations owing to the gravitational attraction of Mars. These are the most dominating forces that are acting on the MGS spacecraft. G M/r + J2 in Figure 3.4 represents the mean and zonal coefficients contribution in the accelerations, which are related to Clm by the relation Jl = -Cl,0. However, gravity in the same figure represents the tesseral (l , m) and sectoral (l = m) coefficients contribution in the accelerations (see Eq. 2.65). An average acceleration for potentials G M/r + J2 and gravity is estimated as 3.0 and 8.9 × 10−4 m/s2 respectively.
• Accelerations due to third body attractions: the third and fourth columns of the Fig-ure 3.4 represent the accelerations owing to the Sun and the Moon, and planets-satellite attractions (see Eq. 2.71), respectively. The Sun-Moon attraction is the third most dom-inating gravitational acceleration, whereas planets-satellite causes a smaller perturbation in the MGS orbit. Average accelerations estimated for Sun-Moon and planets-satellite attractions are 6.0 × 10−8 and 2.7 × 10−12 m/s2, respectively.
• Accelerations due to general relativity: the fifth column of the Figure 3.4 represents the accelerations owing to the contribution of general relativity (see Eq. 2.72). An average value of this acceleration is estimated as 1.1 × 10−9 m/s2.
• Accelerations due to maneuvers: the sixth, seventh, and eighth columns of the Figure 3.4 represent empirical accelerations that are estimated over the duration of each AMD (see Eq. 2.80). Average value of radial (x-stoc), along-track (y-stoc), and cross-track (z-stoc) accelerations are estimated as 1.0×10−4, 1.2×10−4, and 1.8×10−4 m/s2 respectively.
• Accelerations due to atmospheric drag: the ninth column of the Figure 3.4 represents the acceleration owing to the resistance of the Mars atmosphere (see Eq. 2.75). It is one of the largest non-gravitational accelerations acting on the low altitude spacecraft. For MGS, an average value of this acceleration is estimated as 4.8 × 10−9 m/s2.
• Accelerations due to solar radiation pressure: the tenth column of the Figure 3.4 rep-resents the acceleration due to the solar radiation pressure (see Eq. 2.74). It is the largest non-gravitational acceleration acting on the MGS spacecraft with an average value of 4.5 × 10−8 m/s2.
• Accelerations due to Mars radiation: the eleventh and twelfth columns represent the accelerations due to the Infra-Red radiation (see Eq. 2.79) and Albedo (see Eq. 2.78) of the Mars. These are the smallest non-gravitational accelerations acting on the MGS spacecraft with average values of 1.1 × 10−9 and 3.0 × 10−9 m/s2 respectively.
• Accelerations due to solid planetary tides: the thirteen column represents the accelera-tions owing to the contribution of solid planetary tides (see Eq. 2.69). An average value of this acceleration is estimated as 4.18 × 10−9 m/s2.

Doppler and range postfit residuals

In general, the Doppler data are mainly used for the computation of spacecraft orbit. They are sensitive to the modeling of the spacecraft dynamics and provide strong constraints on the orbit construction. However, range data are also used to assist the orbit computation. Unlike Doppler data, range data are more sensitive to the positions of the planet in the solar system and provide strong constraints to the planetary ephemerides.
In Figure 3.5, the peaks and the gaps in the postfit residuals correspond to solar conjunction periods. Excluding these periods, the rms value of the postfit Doppler- and two- way range residuals for each data-arc is varying from1.8 to 5.8 mHz4,5 and 0.4 to 1.2m, respectively. These estimations are comparable with Yuan et al. (2001); Lemoine et al. (2001); Marty et al. (2009), see Table 3.3. The mean value of the estimated Doppler offset for each DSN station tracking pass is of the order of a few tenths of mHz, which is lower than the Doppler postfit residuals for each data-arc. This implies that there is no large offset in the modeling of the Doppler shift measurements at each tracking DSN station.

DSN station position and ephemeris bias

To account for the uncertainties on the DSN antenna center position or in the instrumental de-lays, one bias per station is adjusted on the range measurements for each data-arc. For comput-ing this bias, GINS used station coordinates that are given in the 2008 International Terrestrial Reference Frame (ITRF). These coordinates are then corrected from the continental drift, tides and then projected into an inertial frame through the EOP.
The adjustment of the station bias has been done simultaneously with the orbit fit. It is done independently for each station which are participating in the data-arc. This adjustment then absorbs the error at each station like uncalibrated delay in the wires. The panel a of Figure 3.9 shows the variations with time of the computed station bias for each station. A mean and 1-σ value of the station bias is estimated as 2±1 meter, which is compatible with the Konopliv et al. (2006).
In addition to the station bias, one ephemeris bias (so called range bias) is computed for each data-arc. The range bias represents the systematic error in the geometric positions between the Earth and Mars. Similar to the station bias, the range bias is also estimated from the range measurements of each data-arc. The panel b of Figure 3.9 shows the variations with time of the estimated range bias per station compared to distances estimated with INPOP10b ephemeris. The peaks and gaps shown in Figure 3.9 demonstrate the effect of the solar conjunction on the range bias.
Moreover, such estimation of range bias are very crucial for the construction of planetary ephemeris and also to perform solar corona studies (Verma et al., 2013a). The complete de-scription of solar corona investigations (that are performed with these range bias) and its impact on the planetary ephemeris and the asteroids mass determination are discussed in Chapter 4.
One of the important information brought by the radioscience analysis is the range bias measure-ments between the Earth and the planet. These measurements are important for the estimation of the planet orbit. However, the range measurement accuracy is limited by the calibration of the radio signal delays at the tracking antennas and by the accuracy of the spacecraft orbit re-construction. In order to check the accuracy of our estimations of the range bias, we computed the range bias (separately from the GINS) from the light time data7 provided by the JPL and compared them with the one obtained from the GINS.
The JPL data represent a round-trip light time for each range measurement of MGS space-craft made by the DSN relative to Mars system barycenter. Unlike to the radioscience data, the light-time is a processed data from the JPL Orbit Determination Program (ODP). ODP es-timated the MGS orbit from the Doppler tracking data and then measured the Mars position relative to an Earth station by adjusting the spacecraft range measurements for the position of the MGS relative to the Mars center-of-mass (Konopliv et al., 2006). In the ODP software, a calibration for the tropospheric and ionospheric path delay at the DSN stations has been applied based on calibration data specific to the time of each measurement. Moreover, a calibration for the electronic delay in the spacecraft transponder and a calibration for the DSN tracking station measured for each tracking pass are also applied in the ODP (Konopliv et al., 2006). However, no calibration or model for solar plasma has been applied for this JPL release of the MGS light-time.
In addition to MGS, JPL also provides light time data for Odyssey and MRO missions on an irregular time basis. In order to use these light time in the INPOP construction, we therefore modeled a precise light time solution (based on the algorithms given in Section 2.4.2 of Chapter 2) to compute the round-trip time delay from Earth station to planet barycenter (in this case Mars) using INPOP planetary ephemerides. Except solar corona, all corrections which introduced perturbations in the radio signal have been taken in account. With this configuration of light time solution, we are then able to analyze the effect of the solar corona over the ranging data. The solar corona model derived from this analysis is discussed in Chapter 4.
The range bias obtained from the light time solution using INPOP10b ephemeris are shown in panel a of Figure 3.10. Panel b of the same figure represents the range bias obtained from the GINS software. As one can see in Figure 3.10, both range bias show a similar behavior. The major difference between both solutions is the density of the data. JPL light time data sets are denser than the range bias obtained with GINS as this latest estimated one bias over each two days data-arc when the JPL provides one light time measurement for each range measurement. To plot the approximated differences between both range measurements, we computed an average values of the JPL light time solution over each two days. The differences between GINS range bias and averaged JPL light time are plotted in panel c. In this panel, one can notice meter-level fluctuations in the differences, especially during 2004. Such fluctuations may be explained, by the degradations in the computation of the atmospheric drag forces (see Figure 3.7 and Section 3.4.4.1), and by the different approaches and softwares that have been used for the analysis of the MGS radiometric data. Although, the average differences between both solutions was estimated as -0.08±1.28m, which is less than the current accuracy of the planetary ephemerides.
As mentioned earlier, non-gravitational forces which are acting on the spacecraft are function of spacecraft model characteristics. These forces are however not as important in amplitude as the gravitational forces as shown in Figure 3.4. Although, despite of their smaller contri-butions, these forces are extremely important for the precise computation of spacecraft orbit and the detection of geophysical signatures. However, in practice, the complete information of the spacecraft shape (also called macro-model) is either not precisely known or not pub-licly available. Therefore, the motivation of this study is to understand the impact of in-perfect macro-model over the spacecraft orbit and estimated parameters.

Table of contents :

1. Introduction 
1.1. Introduction to planetary ephemerides
1.2. INPOP
1.2.1. INPOP construction
1.2.2. INPOP evolution
1.3. Importances of the direct analysis of radioscience data for INPOP
2. The radioscience observables and their computation 
2.1. Introduction
2.2. The radioscience experiments
2.2.1. Planetary atmosphere
2.2.2. Planetary gravity
2.2.3. Solar corona
2.2.4. Celestial mechanics
2.3. Radiometric data
2.3.1. ODF contents
2.3.1.1. Group 1
2.3.1.2. Group 2
2.3.1.3. Group 3
2.3.1.3.1. Time-tags
2.3.1.3.2. Format IDs
2.3.1.3.3. Observables
2.3.1.4. Group 4
2.3.1.4.1. Ramp tables
2.3.1.5. Group 5
2.3.1.6. Group 6
2.3.1.7. Group 7
2.4. Observation Model
2.4.1. Time scales
2.4.1.1. Universal Time (UT or UT1)
2.4.1.2. Coordinated Universal Time (UTC)
2.4.1.3. International Atomic Time (TAI)
2.4.1.4. Terrestrial Time (TT)
2.4.1.5. Barycentric Dynamical Time (TDB)
2.4.2. Light time solution
2.4.2.1. Time conversion
2.4.2.2. Down-leg τU computation
2.4.2.3. Up-leg τU computation
2.4.2.4. Light time corrections, δτD and δτU
2.4.2.4.1. Relativistic correction δτRC
2.4.2.4.2. Solar Corona correction δτSC
2.4.2.4.3. Media corrections δτMC
2.4.2.5. Total light time delay
2.4.2.5.1. Round-trip delay
2.4.2.5.2. One-way delay
2.4.3. Doppler and range observables
2.4.3.1. Two-way (F2) and Three-way (F3) Doppler
2.4.3.1.1. Ramped
2.4.3.1.2. Unramped
2.4.3.2. One-way (F1) Doppler
2.4.3.3. Two-way (ρ2,3) Range
2.5. GINS: orbit determination software
2.5.1. Dynamic model
2.5.1.1. Gravitational forces
2.5.1.1.1. Gravitational potential
2.5.1.1.2. Solid planetary tides
2.5.1.1.3. Sun, Moon and planets perturbation
2.5.1.1.4. General relativity
2.5.1.2. Non-Gravitational forces
2.5.1.2.1. Solar radiation pressure
2.5.1.2.2. Atmospheric drag and lift
2.5.1.2.3. Thermal radiation
2.5.1.2.4. Albedo and infrared radiation
2.5.1.2.5. Motor burn
2.5.2. Variational equations
2.5.3. Parameter estimation
3. Mars Global Surveyor: Radioscience data analysis 
3.1. Introduction
3.2. Mission overview
3.2.1. Mission design
3.2.2. Spacecraft geometry
3.2.3. Radioscience data
3.3. Orbit determination
3.3.1. Data processing and dynamic modeling
3.3.2. Solve-for parameters
3.4. Orbit computation results
3.4.1. Acceleration budget
3.4.2. Doppler and range postfit residuals
3.4.3. Orbit overlap
3.4.4. Estimated parameters
3.4.4.1. FS and FD scale factors
3.4.4.2. DSN station position and ephemeris bias
3.5. Supplementary investigations
3.5.1. GINS solution vs JPL Light time solutions
3.5.2. Box-Wing macro-model vs Spherical macro-model
3.6. Conclusion and prospectives
4. Solar corona correction of radio signals and its application to planetary ephemeris 
4.1. Introduction
4.2. The solar cycle
4.2.1. Magnetic field of the Sun
4.2.2. Sunspots
4.2.3. Solar maxima
4.2.4. Solar minima
4.3. The solar wind
4.3.1. Fast solar wind
4.3.2. Slow solar wind
4.4. Radio signal perturbation
4.5. Solar corona correction of radio signals and its application to planetary ephemeris
4.6. Conclusion
4.7. Verma et al. (2013a)
5. Improvement of the planetary ephemeris and test of general relativity with MESSENGER
5.1. Introduction
5.2. MESSENGER data analysis
5.2.1. Mission design
5.2.2. Spacecraft geometry
5.2.3. Radioscience data
5.2.4. Dynamical modeling and orbit determination processes
5.3. Orbit determination
5.3.1. Acceleration budget
5.3.2. Significance of MESSENGER observation for INPOP
5.3.3. Evolution of INPOP with the accuracy of MESSENGER orbit
5.3.3.1. Case I: First guess orbit for Messenger and INPOP12a
5.3.3.1.1. Description
5.3.3.1.2. Results
5.3.3.2. Case II: New Mercury orientation model and INPOP12b
5.3.3.2.1. Description
5.3.3.2.2. Results
5.3.3.3. Case III: Group delay and INPOP12c
5.3.3.3.1. Description
5.3.3.3.2. Results
5.3.3.4. Case IV: New gravity field HgM002 and INPOP12d
5.3.3.4.1. Description
5.3.3.4.2. Results
5.3.3.5. Case V: Extension of the mission and INPOP13a
5.3.3.5.1. Description
5.3.3.5.2. Results
5.3.3.5.3. Comparisons
5.3.3.5.4. INPOP13a ephemeris
5.4. Verma et al. (2013b)
6. General conclusions

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