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ISA (Italian Spring Accelerometer): A new configuration for the Radio Science Experiment
Advances in Space Research 38 (2006) 639–646 www.elsevier.com/locate/asr
ISA (Italian Spring Accelerometer): A new configuration for the Radio Science Experiment V. Iafolla, D.M. Lucchesi *, S. Nozzoli Istituto di Fisica dello Spazio Interplanetario (IFSI/INAF), Via Fosso del Cavaliere 100, 00133 Roma, Italy Received 11 October 2004; received in revised form 14 March 2005; accepted 2 March 2006
Abstract We describe the accelerometer to be used in the Radio Science Experiment conceived for the BepiColombo space mission to Mercury. The key roˆle of the ISA is outlined in order to remove from the list of unknowns the strong non-gravitational perturbations acting on the satellite. The main characteristics of the three-axis accelerometer are given, with particular emphasis on its configuration inside the planned spacecraft and to the attenuation of the thermal perturbing effects that represent one of the major difficulties in a gravimetry experiment. 2006 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Planetology; Accelerometer; Space instrumentation, Non-gravitational perturbations; Thermal analysis; General relativity
1. Introduction The Radio Science Experiment (RSE) is one of the main experiments considered for the BepiColombo ESA cornerstone mission to Mercury. The RSE is devoted to the planet study and to test the General Relativity (GR) theory to an unprecedented level of accuracy. For such measurements, it is necessary to allocate a set of dedicated instrumentation on-board the Mercury Planetary Orbiter (MPO). These instruments allow the determination of the MPO orbit around the planet and the accurate determination of Mercury’s center-of-mass around the Sun (Milani et al., 2001, 2002), by which one can estimate several parameters related to the planet structure and verify the theory of GR. In particular, the scientific goals to achieve are: (1) the rotation state of Mercury, in order to constrain the size and physical state of the core of the planet, (2) the global gravity field of Mercury and its temporal variations due to solar tides, in order to constrain the internal structure of the planet, (3) the local gravity anomalies, in order to *
Corresponding author. Tel.: +39 06 49934367. E-mail address: [email protected] (D.M. Lucchesi).
constrain the mantle structure of the planet and the interface between mantle and crust, and (4) the orbit of Mercury’s center-of-mass around the Sun and the propagation of electromagnetic waves between the Earth and the spacecraft, in order to improve the determination of the parameterized post-newtonian (PPN) parameters of GR. These goals can be achieved by: (1) range and range-rate derivations of the MPO position with respect to Earthbound radar station(s) (Iess and Boscagli, 2001), (2) the determination of the non-gravitational signals on the MPO by means of an on-board accelerometer (Iafolla and Nozzoli, 2001), (3) the determination of the MPO absolute attitude by means of a Star-Tracker, and (4) the determination of angular displacements of reference points on the solid surface of the planet by means of a High Resolution Camera. The MPO spacecraft will be nadir pointing and threeaxis stabilised in a 400 · 1500 km polar orbit around Mercury. In Table 1 we give the spacecraft orbital parameters used in our numerical simulations. The measurement of the non-gravitational signals will use a high sensitivity accelerometer. The accelerometer allows to remove the perturbing non-gravitational accelerations from the list of unknowns and to transform, a posteriori, the spacecraft
0273-1177/$30 2006 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2006.03.006
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Table 1 The MPO orbital parameters Orbital parameter
Symbol
Numerical value
Semimajor axis Eccentricity Inclination Orbital period Ascending node longitude Argument of pericenter Nodal rate Pericenter rate
a e I P X x X_ x_
3389 km 0.162 90 8355 s 0 deg 0.7 deg 0 deg/day 0.0915 deg/day
The values used for the spacecraft initial ascending node longitude and argument of pericenter are only indicative.
into a drag-free satellite. The Italian Spring Accelerometer (ISA) is a three-axis instrument with an intrinsic noise level p of 1010 g¯/ Hz (g¯@ 9.8 m/s2) in the frequency band of 3 Æ 105–101 Hz. In order to reach the ambitious goals of the RSE, we need to determine the MPO orbit with an accuracy of about 1 m over the spacecraft orbital period (about 2.3 h). This corresponds to an accuracy of about 108 m/s2 for the component of the acceleration necessary to reconstruct the MPO orbit. Therefore ISA meets the BepiColombo science requirements. The rest of the paper is organised as follows. In Section 2, we describe the accelerometer characteristics and performances. We demonstrate the superiority of the accelerometer readings with respect to the modelling of the nongravitational accelerations. In Section 3, we analyse the problem of the accelerometer positioning inside the MPO, showing a new configuration for the three sensitive elements that minimises the inertial and gravitational perturbing effects. In Section 4, we address the problem of the impact of the thermal signals on the accelerometer and the way to attenuate or remove such effects. Finally, in Section 5, the conclusions are given and the future work is outlined. 2. The Italian Spring Accelerometer The Italian Spring Accelerometer (ISA), developed at IFSI/CNR (Fuligni and Iafolla, 1997; Fuligni et al., 1997), has been considered to fly on-board the MPO. ISA is a three-axis room temperature accelerometerpand its final intrinsic noise will be equal to 5 Æ 1013 g¯/ Hz. In Fig. 1, one of the mechanical sensitive units is shown. An accuracy of 109 g¯ in the frequency band 3 Æ 105– 1 10 Hz has already been obtained for an accelerometer located in the Gran Sasso underground laboratories of the INFN (Istituto Nazionale di Fisica Nucleare), a result which fits very well the requirements imposed by the RSE. In Table 2, we report the accelerometer characteristics necessary to fulfil the RSE. The key roˆle of the accelerometer is to remove, from the list of unknowns, the non-gravitational perturbations, i.e., the strong non-conservative accelerations acting on the MPO produced by the Sun’s visible radiation and by
Fig. 1. One of the three ISA sensitive elements and its control capacitors. The sensitive unit has been built by working a single piece of Aluminum 5056. The proof-mass is connected to an external rigid frame by a crankshaped suspension, i.e., by two torsional springs; the sensitive axis is perpendicular to the proof-mass face.
Table 2 ISA characteristics for the BepiColombo RSE Intrinsic noise Frequency range Resonance frequency Dynamic range Mass Volume mechanical part Volume electronic part Power dissipation Supply voltage A/D converter Bit rate Sensitivity to temperature
p 109 g¯/ Hz 1 (10 –3 · 105) Hz 3.5 Hz 3000 4.6 kg (310 · 170 · 130) mm (170 · 130 · 50) mm 7.4 W 5V 16 bit 9600 bit/s 5 Æ 108 g¯/C
Mercury’s infrared and visible radiation. To fulfil the RSE, the non-gravitational accelerations must be considered to a level of 108 m/s2, which has been taken as the accelerometer accuracy. The largest non-gravitational acceleration is due to direct solar radiation pressure, which has a maximum value of about 106 m/s2 in the strong radiation environment of Mercury (i.e., about two orders-of-magnitude larger than the accuracy requested for the space mission). The acceleration due to solar radiation is defined as: 2 A USun R ^ ~ aSun ¼ C R S ð1Þ m c R where CR represents the satellite radiation coefficient, A/m the area-to-mass ratio of the satellite (about 1.9 Æ 102 m2/
V. Iafolla et al. / Advances in Space Research 38 (2006) 639–646
kg), c is the speed of light, and USun is the solar irradiance at the average distance R of Mercury from the Sun. The last term accounts for the modulation due to Mercury’s eccentric orbit around the Sun (em @ 0.206). In our numerical simulations of the solar radiation we assumed equal to 1 the radiation coefficient. Of course, this is not true, but no matter this value is because we are only interested to the main behaviours of the solar radiation acceleration and to prove the superiority of the accelerometer readings with respect to the modelling approach. In Fig. 2, we plot the transversal component of the solar radiation pressure over a time span of about three orbital revolutions of the MPO spacecraft around Mercury, while
Transversal acceleration (m/s2)
1,5x10-6 1,0x10-6 5,0x10-7 0,0 -5,0x10-7 -1,0x10-6 -1,5x10-6 0
5000
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Fig. 2. Transversal component of the solar radiation pressure over three orbital revolutions of the MPO around Mercury. The numerical integration has been performed by integrating Eq. (1) with a 1 degree step-size over three consecutive orbits of the satellite.
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in Fig. 3, we show the corresponding Fourier analysis. The MPO orbital period around Mercury is about 2.3 h, and three orbital revolutions correspond to the typical arc length that will be used for the spacecraft orbit determination during the RSE. From Fig. 2 we see that the peaks of the transversal acceleration are about 106 m/s2 in magnitude, while in Fig. 3, the maximum value for the disturbing acceleration component corresponds to the spectral line at the MPO orbital period. Therefore, with the ISA we are able to reduce the impact of the solar radiation by two orders-of-magnitude. Assuming a 10% modelling of the solar radiation, we are able to gain at least a factor 10 with respect to the modelling approach. In reality, under these conditions, we are able to gain more than a factor 10 because, with an on-board accelerometer, we are able to detect the variations of the solar radiation pressure during the MPO eclipses by Mercury, i.e., the penumbra effects, which are very difficult to model correctly. However, the 10% modelling assumption will be possible probably only during a small fraction of the BepiColombo 1 year nominal mission. The reason is that the MPO will be a three axes satellite of complex shape (giving rise to intricate shadow effects among the various surfaces) and characterised by a degradation of its optical properties because of the strong radiation environment of Mercury (the solar irradiance will vary between 14,448 W/m2 and 6272 W/m2, respectively when Mercury at perihelion and aphelion) Therefore, the radiation coefficient will degrade very quickly. A forthcoming paper is devoted to the study of the impact of both the solar radiation pressure and Mercury’s albedo on the MPO orbit.
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Fig. 4. Simplified scheme of an ISA single axis element. The mechanical oscillator with the pick-up system is shown together with the amplifier and the transformer. The two sensing capacitors, C1 and C2, and the two fixed external capacitors, Ca and Cb, used in a bridge configuration, provide the extraction of the signal.
The fundamental components of the accelerometer are three mechanical oscillators with a low resonance frequency (f0 = 3.5 Hz). In Fig. 4, we show a simplified scheme of an ISA single-axis mechanical oscillator plus the pick-up system. The displacement of the proof-mass due to a perturbing acceleration is detected by means of capacitive transducer in a bridge configuration, followed by a low noise amplifier. The capacitive bridge is biased at high frequency (fp = 10 kHz), so that accelerations at frequencies fs, acting on the proof-mass, produce the bridge unbalance. At the output of the capacitive bridge, the signal is seen as a modulation of the bias voltage at the two side bands f± = fp ± fs in the frequency domain. The bridge is driven by the transformer with an alternate voltage. The mechanical signal produces the variation of the two detector capacitors (unbalance of the bridge) as well as a residual output voltage that is modulated at the signal frequency. The output of the capacitive bridge is sent to a low-noise amplifier, characterised by very high input impedance Zi, an equivalent generator of voltage noise en, and an equivalent generator of current noise in, see Fig. 4 for a circuit diagram of the ISA. 3. ISA positioning inside the MPO In order to reduce the gravity-gradient forces on the ISA structure, and the effects of the apparent forces on its sensing masses due to the MPO rotation, the ideal location for the accelerometer inside the MPO is with its center-of-mass coincident with that of the MPO spacecraft. The ISA is a three-axis accelerometer, with three sensitive units, of which only one can be positioned on the spacecraft center-of-mass. Therefore, we need to determine the best configuration of the accelerometer structure with respect to the MPO center-of-mass (by which we mean the nominal position for the satellite center-of-mass, not considering its displacements due to the fuel consumption-and-sloshing and to the High Gain Antenna (HGA) movements).
Once this best configuration is determined it is also necessary to determine the maximum tolerable distance between the accelerometer center-of-mass and that of the spacecraft (Iafolla et al., 2006). Such information will also be useful for the location of the fuel tank around the MPO nominal center-of-mass. Eq. (2) gives the resultant acceleration on a point close to the MPO center-of-mass: _ € ~ ~ ~ ~ ð~ RÞ ~ ANGP ; R ¼ ðr gÞ~ Rx x~ RÞ ~ x_ ~ R 2ð~ x~ ð2Þ where ~ R is the displacement vector between the MPO center-of-mass and one of the ISA proof-mass center-of-mass, ~ and x ~_ are, respective~ g represents Mercury gravity field, x ly, the MPO angular rate and angular acceleration, and finally, ~ ANGP represents the linear acceleration due to the non-gravitational perturbations. From Eq. (2), we can recognise the contribution due to the gravity-gradients of Mercury, that due to the centrifugal and tangential accelerations, that due to the Coriolis acceleration, and finally, the component due to the non-conservative accelerations of non-gravitational origin. Eq. (3) gives the resultant perturbing acceleration on the ISA proof-masses due to the gravity-gradient tensor plus the contribution of the apparent forces for every axis: 0
1 0 2 1 n ½ð3 þ 5e2 þ 10ecos MÞDX ð2esin M þ 5e2 sin 2M ÞDY aX 1 2 C B C B 2 2 5 2 ~ a ¼ @ aY A ¼ @ n ð2e sin M þ 5e sin 2M ÞDX þ 2 e þ ecos M þ 2 e cos 2M DY A: n2 1 þ 32 e2 þ 3ecos M þ 92 e2 cos2M DZ aZ ð3Þ
The XYZ reference frame is the MPO Gauss co-moving frame, i.e., X is along the radial direction from Mercury’s center-of-mass (zenith pointing), Z is in the out-ofplane direction (along the MPO rotation axis), Y is the transversal direction perpendicular to the other two (in the orbital plane). The other quantities are, respectively, n the satellite mean motion (7.52 Æ 104 rad/s), e and M the satellite eccentricity and mean anomaly, finally, (DX, DY, DZ) are the components of the vector ~ R previously introduced.
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The apparent accelerations have been computed considering the main contribution arising from the nominal angu~ ¼ ð0; 0; x0 Þ and lar rate and angular acceleration, i.e., x ~_ ¼ ð0; 0; x_ 0 Þ, while there is no contribution from the x Coriolis acceleration because the proof-masses act as onedimensional harmonic oscillators (no movements along the direction orthogonal to the sensitive axis). For the full expressions of the gravity field and apparent accelerations ~ and x ~_ , we refer to Iafolla tensor, as well as for those of x et al. (2006). Neglecting the MPO center-of-mass movements (which are considered when we estimate the angular rate and angular acceleration corrections necessary to guarantee nadir pointing (Iafolla et al., 2006)), in order to compute the accuracy in the accelerometer location we need only to consider the contribution of the periodic terms inside Eq. (3). As we can see from Eq. (3), the amplitude of the acceleration along the MPO rotation axis is unaffected by the displacements along the radial and transversal directions. At the same time, the amplitudes of the accelerations along these two directions do not depend on the shifting along the rotation axis. Therefore, for the best configuration of the accelerometer, this result suggests a configuration of the three sensitive masses aligned along the rotation axis of the MPO, with ISA center-of-mass coincident with the MPO one, see Fig. 5. When comparing the perturbing acceleration of Eq. (3) with the requested accelerometer accuracy (109 g¯), we have been able to obtain the following
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position-and-error matrix for (units are in meters): 0 1 0 ~ D~ R0X þ D~ Rt RtX R0X þ ~ B C B B~ C B ~ ~ ~ @ R0Y þ Rt A þ @ DR0Y þ DRtY ~ R0Z þ ~ Rt 0 0 þ Xt B ¼ @0 þ Xt
D~ R0Z þ D~ RtZ
the ISA inside the MPO 1 C C A 1
0 þ Yt
X 0X þ Z t
0 þ Yt
C Y 0Y þ Z t A
0 þ X t 0 þ Y t Z 0Z þ Z t 0 1 5 103 7:5 103 15 103 B C 3 þB 7:5 103 15 103 C @ 5 10 A; 3 3 3 5 10 7:5 10 15 10
ð4Þ
where we have also included the contribution ~ Rt ¼ ðX t ; Y t ; Z t Þ due to the spacecraft center-of-mass movements. The three vectors ð~ R0X ; ~ R0Y ; ~ R0Z Þ define the position of the center-of-mass of the proof-masses with sensitive axis respectively along the radial direction, the transversal direction and the out-of-plane direction (i.e., along the MPO rotation axis), with X0X = Y0Y = 5 cm and Z0Z = 0 the displacements along the rotation axis. These vectors define a time-invariant contribution to the accelerometer position matrix, while ~ Rt is responsible of the timevariant contribution. In our analyses we estimated for the components of ~ Rt a value of a few cm, but the industrial contractor will be responsible for the final estimate of such numbers. The other numbers represent the uncertainties in
Fig. 5. Configuration of the ISA. The three sensitive units are aligned along the MPO rotation axis in order to reduce the perturbative effects of Mercury gravity-gradients and the apparent forces (nominal case). The center-of-mass of the central unit, the one with sensitive axis along the rotation axis, is coincident with the MPO center-of-mass. The three sensitive axes X, Y and Z of the accelerometer are aligned, respectively, along the radial direction (i.e., along the nadir-zenith direction), along the transversal direction (in the orbital plane) and along the out-of-plane direction (i.e., along the MPO rotation axis).
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consider are: (1) the temperature change over one orbital period (about 2.3 h) of the MPO around Mercury, (2) the temperature change over half of the sidereal period (about 88 days) of Mercury around the Sun (the thermal effects at half of the sidereal period of Mercury around the Sun plays a significant roˆle). The exact impact of the cited thermal disturbances on the accelerometer structure is dependent on the accelerometer thermal stability to the temperature variations. In the case of ISA, this thermal stability is about 5 Æ 108 g¯/C. The simulations of the gravimetry and general relativity experiments have been performed following three different assumptions (Milani et al., 2003): (1) no modelling of the 44 days thermal effect is performed, i.e., 100% of the thermal signal considered equal to 11.8 C peak to peak, (2) the thermal signal at half of the sidereal period is decreased to 10% of its value, (3) the thermal signal at half of the sidereal period is decreased to 1% of its value. In Fig. 6, we show the results of the simulations in the case of the gravimetry experiment (courtesy of Prof. A. Milani). The error is given as an RMS value for all the normalized harmonic coefficients of a given degree ‘. When passing from 100% to 10% calibration, the error decreases by a factor 10 over the entire spectrum up to ‘ = 25. This linearity proves that the 44 days thermal effect is by far the largest source of systematic effect.
the positions of the masses. In reality, the accuracy in the positioning of each mass along the rotation axis is higher, but we refer to Eq. (4) as a conservative way to estimate the combinations of the errors due to the location of the sensitive masses in the accelerometer box, plus the error in positioning this box on the MPO mounting plate, plus the errors in the MPO center-of-mass knowledge. In particular, the vectors D~ R0 (one for each axis of the ISA) represent the time-invariant contribution of the position errors, while the vectors D~ Rt (again one for each axis of the ISA) represent the time-variant contribution of the position errors. We can also choose a configuration with Z0Z 6¼ 0, i.e., with the ISA configuration of Fig. 5 shifted along the rotation axis, useful if the positioning of the accelerometer in the center-of-mass of the MPO must be avoided. The limits of the maximum shift along the Z axis of the accelerometer center-of-mass with respect to the spacecraft center-ofmass are reported in Iafolla et al. (2006). 4. Attenuation of the thermal effects A good attenuation (or elimination) of the thermal effects acting on the accelerometer are an essential prerequisite in order to fulfil both the gravimetry and the general relativity experiments. The main temperature variations to
SIMULATION SUMMARY 22-24-25 –5
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Fig. 6. Degree variance as a function of the degree ‘. The red line gives the simulated field while the lower pink line gives the formal uncertainty. The other lines give the error when we consider the thermal effects, especially the long-term thermal effect at half of the sidereal period of Mercury around the Sun. With no attenuation (blue line), i.e., 100% of the thermal effect, the signal-to-noise ratio becomes 1 at degree ‘ = 20. On the other hand, with a 10% attenuation (black line), we are able to reach the degree ‘ = 25 with a signal-to-noise ratio still above 1 (and about 10 to the degree ‘ = 20). Finally, the 99% attenuation (i.e., only 1% residual thermal effect at half of the sidereal period (upper pink line)) will tune toward the degree ‘ = 30 of the expansion in spherical harmonics of the planet gravity field with a signal-to-noise ratio still greater than 1 (and larger than 10 to the degree ‘ = 20). The higher the degree ‘ the higher the accuracy in constraining the planet internal structure. Moreover, the higher the degree, the more accurate will be the resolution of the gravity field. With the 10% calibration, we will reach a spatial resolution of about 400 km (‘ @ 20) and about 300 km (‘ @ 25) with a 1% calibration. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this paper.)
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To reduce the effects of the temperature below the required precision, it is necessary to reduce the thermal variations on the internal part of the accelerometer. The thermal attenuation can be obtained using an active thermal control. A passive system has been considered, but it has not been able to attenuate the thermal variation at the long periods, such as half of the revolution period of Mercury. Therefore, the thermal variation must be reduced by filtering the data out of the required frequency band, or correcting them by measuring the temperature with a precision of 0.1 C. The active control has the advantage to attenuate the temperature variations at very low frequency, but it requires additional power dissipation. In Fig. 7, we report the main results of the thermal analysis simulation for the active thermal control. Of course, the attenuation factor obtained from the simulation (about 10,000) is not easy to achieve experimentally, hence we use a factor 700 obtained from the experimental activity through an active thermal control of the accelerometer. On the contrary, the power dissipation obtained in the simulation is retained as an acceptable value. These results demonstrate that for the Accelerometer Set (AS) – the more sensitive part of the accelerometer to the temperature – an attenuation factor larger than 700 has been obtained experimentally both at the revolution period of Mercury around the Sun and at the orbital period of the MPO around Mercury. The power dissipation necessary for the thermal stabilization equals 3.69 W.
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5. Conclusions The ISA characteristics and roˆle in the case of the RSE has been described in the present paper. In particular, we have demonstrated that the accelerometer accuracy (about 109 g¯) is enough to remove the perturbing effects due to the solar radiation pressure, the largest non-gravitational perturbation. It has been shown that the best geometrical configuration of the accelerometer is with the three sensitive proof-masses aligned along the spacecraft rotation axis and with the center-of-masses of the accelerometer and the MPO coincident; in this way the perturbing effects due to Mercury’s gravity-gradients and to the MPO apparent accelerations are less than the instrument accuracy. However, we have examined the possibility of shifting the accelerometer along the spacecraft rotation axis in order to leave some space in the surroundings of the MPO nominal center-of-mass position. This alternative solution will be useful for the location of the fuel tank and probably for a better determination of the MPO center-of-mass movements, due to the fuel consumption and sloshing. These aspects have been analysed (together with the HGA movements) in detail in Iafolla et al. (2006). The importance of a refined attenuation of the thermal effects on the accelerometer has been described in the case of the gravimetry experiment. The attenuation of the perturbing thermal effects, both at the MPO orbital period and at half of Mercury’s sidereal period, has been obtained
Fig. 7. Thermal simulation, frequency response. In the simulation, the input signal was kept equal to 2 C at all frequencies, seen in the blue line. The temperature of the Accelerometer Set (AS) is reported in green while the temperature of the electronics is reported in red. The ratio between the blue line and the red line gives the attenuation factor for the simulation. However, the results from the simulation are not reliable in the case of the attenuation factor estimate, but they give very confident estimates for the power dissipation, useful to implement the thermal control. The ISA mechanical part, which is more sensitive to the temperature, is allocated separately inside a thermal shielded box. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this paper.)
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through an active thermal control that guarantees an attenuation larger than a factor 700. The future work devoted to the implementation of a space version has already started. In particular, the ISA error budget and the requirements imposed on the angular rates and angular accelerations necessary to guarantee the MPO nadir pointing are under investigation. References Fuligni, F., Iafolla, V. Measurement of small forces in the physics of gravitation and geophysics. Il Nuovo Cimento 20C (5), 619–628, 1997. Fuligni, F., Iafolla, V., Milyukov, V., Nozzoli, S. Experimental gravitation and geophysics. Il Nuovo Cimento 20C (5), 637–642, 1997.
Iafolla,V., Lucchesi, D.M., Nozzoli, S., On the ISA accelerometer positioning inside the Mercury planetary orbiter. Planet. Space Sci., in press, 2006. Iafolla, V., Nozzoli, S. Italian spring accelerometer (ISA) a high sensitive accelerometer for ‘’BepiColombo‘’ ESA CORNERSTONE. Planet. Space Sci. 49, 1609–1617, 2001. Iess, L., Boscagli, G. Advanced radio science instrumentation for the mission BepiColombo to Mercury. Planet. Space Sci. 49, 1597–1608, 2001. Milani, A., Rossi, A., Vokrouhlicky´, D., Villani, D., Bonanno, C. Gravity field and rotation state of Mercury from the BepiColombo Radio Science Experiments. Planet. Space Sci. 49, 1579–1596, 2001. Milani A., Rossi, A., Villani, D., The BepiColombo Radio Science Simulations, Version 2, 11 April 2003. Milani, A., Vokrouhlicky, D., Villani, D., Bonanno, C., Rossi, A. Testing general relativity with the Bepicolombo radio science experiment. Phys. Rev. D 66, 082001, 2002.
ISA (Italian Spring Accelerometer): A new configuration for the Radio Science Experiment V. Iafolla, D.M. Lucchesi *, S. Nozzoli Istituto di Fisica dello Spazio Interplanetario (IFSI/INAF), Via Fosso del Cavaliere 100, 00133 Roma, Italy Received 11 October 2004; received in revised form 14 March 2005; accepted 2 March 2006
Abstract We describe the accelerometer to be used in the Radio Science Experiment conceived for the BepiColombo space mission to Mercury. The key roˆle of the ISA is outlined in order to remove from the list of unknowns the strong non-gravitational perturbations acting on the satellite. The main characteristics of the three-axis accelerometer are given, with particular emphasis on its configuration inside the planned spacecraft and to the attenuation of the thermal perturbing effects that represent one of the major difficulties in a gravimetry experiment. 2006 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Planetology; Accelerometer; Space instrumentation, Non-gravitational perturbations; Thermal analysis; General relativity
1. Introduction The Radio Science Experiment (RSE) is one of the main experiments considered for the BepiColombo ESA cornerstone mission to Mercury. The RSE is devoted to the planet study and to test the General Relativity (GR) theory to an unprecedented level of accuracy. For such measurements, it is necessary to allocate a set of dedicated instrumentation on-board the Mercury Planetary Orbiter (MPO). These instruments allow the determination of the MPO orbit around the planet and the accurate determination of Mercury’s center-of-mass around the Sun (Milani et al., 2001, 2002), by which one can estimate several parameters related to the planet structure and verify the theory of GR. In particular, the scientific goals to achieve are: (1) the rotation state of Mercury, in order to constrain the size and physical state of the core of the planet, (2) the global gravity field of Mercury and its temporal variations due to solar tides, in order to constrain the internal structure of the planet, (3) the local gravity anomalies, in order to *
Corresponding author. Tel.: +39 06 49934367. E-mail address: [email protected] (D.M. Lucchesi).
constrain the mantle structure of the planet and the interface between mantle and crust, and (4) the orbit of Mercury’s center-of-mass around the Sun and the propagation of electromagnetic waves between the Earth and the spacecraft, in order to improve the determination of the parameterized post-newtonian (PPN) parameters of GR. These goals can be achieved by: (1) range and range-rate derivations of the MPO position with respect to Earthbound radar station(s) (Iess and Boscagli, 2001), (2) the determination of the non-gravitational signals on the MPO by means of an on-board accelerometer (Iafolla and Nozzoli, 2001), (3) the determination of the MPO absolute attitude by means of a Star-Tracker, and (4) the determination of angular displacements of reference points on the solid surface of the planet by means of a High Resolution Camera. The MPO spacecraft will be nadir pointing and threeaxis stabilised in a 400 · 1500 km polar orbit around Mercury. In Table 1 we give the spacecraft orbital parameters used in our numerical simulations. The measurement of the non-gravitational signals will use a high sensitivity accelerometer. The accelerometer allows to remove the perturbing non-gravitational accelerations from the list of unknowns and to transform, a posteriori, the spacecraft
0273-1177/$30 2006 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2006.03.006
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Table 1 The MPO orbital parameters Orbital parameter
Symbol
Numerical value
Semimajor axis Eccentricity Inclination Orbital period Ascending node longitude Argument of pericenter Nodal rate Pericenter rate
a e I P X x X_ x_
3389 km 0.162 90 8355 s 0 deg 0.7 deg 0 deg/day 0.0915 deg/day
The values used for the spacecraft initial ascending node longitude and argument of pericenter are only indicative.
into a drag-free satellite. The Italian Spring Accelerometer (ISA) is a three-axis instrument with an intrinsic noise level p of 1010 g¯/ Hz (g¯@ 9.8 m/s2) in the frequency band of 3 Æ 105–101 Hz. In order to reach the ambitious goals of the RSE, we need to determine the MPO orbit with an accuracy of about 1 m over the spacecraft orbital period (about 2.3 h). This corresponds to an accuracy of about 108 m/s2 for the component of the acceleration necessary to reconstruct the MPO orbit. Therefore ISA meets the BepiColombo science requirements. The rest of the paper is organised as follows. In Section 2, we describe the accelerometer characteristics and performances. We demonstrate the superiority of the accelerometer readings with respect to the modelling of the nongravitational accelerations. In Section 3, we analyse the problem of the accelerometer positioning inside the MPO, showing a new configuration for the three sensitive elements that minimises the inertial and gravitational perturbing effects. In Section 4, we address the problem of the impact of the thermal signals on the accelerometer and the way to attenuate or remove such effects. Finally, in Section 5, the conclusions are given and the future work is outlined. 2. The Italian Spring Accelerometer The Italian Spring Accelerometer (ISA), developed at IFSI/CNR (Fuligni and Iafolla, 1997; Fuligni et al., 1997), has been considered to fly on-board the MPO. ISA is a three-axis room temperature accelerometerpand its final intrinsic noise will be equal to 5 Æ 1013 g¯/ Hz. In Fig. 1, one of the mechanical sensitive units is shown. An accuracy of 109 g¯ in the frequency band 3 Æ 105– 1 10 Hz has already been obtained for an accelerometer located in the Gran Sasso underground laboratories of the INFN (Istituto Nazionale di Fisica Nucleare), a result which fits very well the requirements imposed by the RSE. In Table 2, we report the accelerometer characteristics necessary to fulfil the RSE. The key roˆle of the accelerometer is to remove, from the list of unknowns, the non-gravitational perturbations, i.e., the strong non-conservative accelerations acting on the MPO produced by the Sun’s visible radiation and by
Fig. 1. One of the three ISA sensitive elements and its control capacitors. The sensitive unit has been built by working a single piece of Aluminum 5056. The proof-mass is connected to an external rigid frame by a crankshaped suspension, i.e., by two torsional springs; the sensitive axis is perpendicular to the proof-mass face.
Table 2 ISA characteristics for the BepiColombo RSE Intrinsic noise Frequency range Resonance frequency Dynamic range Mass Volume mechanical part Volume electronic part Power dissipation Supply voltage A/D converter Bit rate Sensitivity to temperature
p 109 g¯/ Hz 1 (10 –3 · 105) Hz 3.5 Hz 3000 4.6 kg (310 · 170 · 130) mm (170 · 130 · 50) mm 7.4 W 5V 16 bit 9600 bit/s 5 Æ 108 g¯/C
Mercury’s infrared and visible radiation. To fulfil the RSE, the non-gravitational accelerations must be considered to a level of 108 m/s2, which has been taken as the accelerometer accuracy. The largest non-gravitational acceleration is due to direct solar radiation pressure, which has a maximum value of about 106 m/s2 in the strong radiation environment of Mercury (i.e., about two orders-of-magnitude larger than the accuracy requested for the space mission). The acceleration due to solar radiation is defined as: 2 A USun R ^ ~ aSun ¼ C R S ð1Þ m c R where CR represents the satellite radiation coefficient, A/m the area-to-mass ratio of the satellite (about 1.9 Æ 102 m2/
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kg), c is the speed of light, and USun is the solar irradiance at the average distance R of Mercury from the Sun. The last term accounts for the modulation due to Mercury’s eccentric orbit around the Sun (em @ 0.206). In our numerical simulations of the solar radiation we assumed equal to 1 the radiation coefficient. Of course, this is not true, but no matter this value is because we are only interested to the main behaviours of the solar radiation acceleration and to prove the superiority of the accelerometer readings with respect to the modelling approach. In Fig. 2, we plot the transversal component of the solar radiation pressure over a time span of about three orbital revolutions of the MPO spacecraft around Mercury, while
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Fig. 2. Transversal component of the solar radiation pressure over three orbital revolutions of the MPO around Mercury. The numerical integration has been performed by integrating Eq. (1) with a 1 degree step-size over three consecutive orbits of the satellite.
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in Fig. 3, we show the corresponding Fourier analysis. The MPO orbital period around Mercury is about 2.3 h, and three orbital revolutions correspond to the typical arc length that will be used for the spacecraft orbit determination during the RSE. From Fig. 2 we see that the peaks of the transversal acceleration are about 106 m/s2 in magnitude, while in Fig. 3, the maximum value for the disturbing acceleration component corresponds to the spectral line at the MPO orbital period. Therefore, with the ISA we are able to reduce the impact of the solar radiation by two orders-of-magnitude. Assuming a 10% modelling of the solar radiation, we are able to gain at least a factor 10 with respect to the modelling approach. In reality, under these conditions, we are able to gain more than a factor 10 because, with an on-board accelerometer, we are able to detect the variations of the solar radiation pressure during the MPO eclipses by Mercury, i.e., the penumbra effects, which are very difficult to model correctly. However, the 10% modelling assumption will be possible probably only during a small fraction of the BepiColombo 1 year nominal mission. The reason is that the MPO will be a three axes satellite of complex shape (giving rise to intricate shadow effects among the various surfaces) and characterised by a degradation of its optical properties because of the strong radiation environment of Mercury (the solar irradiance will vary between 14,448 W/m2 and 6272 W/m2, respectively when Mercury at perihelion and aphelion) Therefore, the radiation coefficient will degrade very quickly. A forthcoming paper is devoted to the study of the impact of both the solar radiation pressure and Mercury’s albedo on the MPO orbit.
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Fig. 4. Simplified scheme of an ISA single axis element. The mechanical oscillator with the pick-up system is shown together with the amplifier and the transformer. The two sensing capacitors, C1 and C2, and the two fixed external capacitors, Ca and Cb, used in a bridge configuration, provide the extraction of the signal.
The fundamental components of the accelerometer are three mechanical oscillators with a low resonance frequency (f0 = 3.5 Hz). In Fig. 4, we show a simplified scheme of an ISA single-axis mechanical oscillator plus the pick-up system. The displacement of the proof-mass due to a perturbing acceleration is detected by means of capacitive transducer in a bridge configuration, followed by a low noise amplifier. The capacitive bridge is biased at high frequency (fp = 10 kHz), so that accelerations at frequencies fs, acting on the proof-mass, produce the bridge unbalance. At the output of the capacitive bridge, the signal is seen as a modulation of the bias voltage at the two side bands f± = fp ± fs in the frequency domain. The bridge is driven by the transformer with an alternate voltage. The mechanical signal produces the variation of the two detector capacitors (unbalance of the bridge) as well as a residual output voltage that is modulated at the signal frequency. The output of the capacitive bridge is sent to a low-noise amplifier, characterised by very high input impedance Zi, an equivalent generator of voltage noise en, and an equivalent generator of current noise in, see Fig. 4 for a circuit diagram of the ISA. 3. ISA positioning inside the MPO In order to reduce the gravity-gradient forces on the ISA structure, and the effects of the apparent forces on its sensing masses due to the MPO rotation, the ideal location for the accelerometer inside the MPO is with its center-of-mass coincident with that of the MPO spacecraft. The ISA is a three-axis accelerometer, with three sensitive units, of which only one can be positioned on the spacecraft center-of-mass. Therefore, we need to determine the best configuration of the accelerometer structure with respect to the MPO center-of-mass (by which we mean the nominal position for the satellite center-of-mass, not considering its displacements due to the fuel consumption-and-sloshing and to the High Gain Antenna (HGA) movements).
Once this best configuration is determined it is also necessary to determine the maximum tolerable distance between the accelerometer center-of-mass and that of the spacecraft (Iafolla et al., 2006). Such information will also be useful for the location of the fuel tank around the MPO nominal center-of-mass. Eq. (2) gives the resultant acceleration on a point close to the MPO center-of-mass: _ € ~ ~ ~ ~ ð~ RÞ ~ ANGP ; R ¼ ðr gÞ~ Rx x~ RÞ ~ x_ ~ R 2ð~ x~ ð2Þ where ~ R is the displacement vector between the MPO center-of-mass and one of the ISA proof-mass center-of-mass, ~ and x ~_ are, respective~ g represents Mercury gravity field, x ly, the MPO angular rate and angular acceleration, and finally, ~ ANGP represents the linear acceleration due to the non-gravitational perturbations. From Eq. (2), we can recognise the contribution due to the gravity-gradients of Mercury, that due to the centrifugal and tangential accelerations, that due to the Coriolis acceleration, and finally, the component due to the non-conservative accelerations of non-gravitational origin. Eq. (3) gives the resultant perturbing acceleration on the ISA proof-masses due to the gravity-gradient tensor plus the contribution of the apparent forces for every axis: 0
1 0 2 1 n ½ð3 þ 5e2 þ 10ecos MÞDX ð2esin M þ 5e2 sin 2M ÞDY aX 1 2 C B C B 2 2 5 2 ~ a ¼ @ aY A ¼ @ n ð2e sin M þ 5e sin 2M ÞDX þ 2 e þ ecos M þ 2 e cos 2M DY A: n2 1 þ 32 e2 þ 3ecos M þ 92 e2 cos2M DZ aZ ð3Þ
The XYZ reference frame is the MPO Gauss co-moving frame, i.e., X is along the radial direction from Mercury’s center-of-mass (zenith pointing), Z is in the out-ofplane direction (along the MPO rotation axis), Y is the transversal direction perpendicular to the other two (in the orbital plane). The other quantities are, respectively, n the satellite mean motion (7.52 Æ 104 rad/s), e and M the satellite eccentricity and mean anomaly, finally, (DX, DY, DZ) are the components of the vector ~ R previously introduced.
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The apparent accelerations have been computed considering the main contribution arising from the nominal angu~ ¼ ð0; 0; x0 Þ and lar rate and angular acceleration, i.e., x ~_ ¼ ð0; 0; x_ 0 Þ, while there is no contribution from the x Coriolis acceleration because the proof-masses act as onedimensional harmonic oscillators (no movements along the direction orthogonal to the sensitive axis). For the full expressions of the gravity field and apparent accelerations ~ and x ~_ , we refer to Iafolla tensor, as well as for those of x et al. (2006). Neglecting the MPO center-of-mass movements (which are considered when we estimate the angular rate and angular acceleration corrections necessary to guarantee nadir pointing (Iafolla et al., 2006)), in order to compute the accuracy in the accelerometer location we need only to consider the contribution of the periodic terms inside Eq. (3). As we can see from Eq. (3), the amplitude of the acceleration along the MPO rotation axis is unaffected by the displacements along the radial and transversal directions. At the same time, the amplitudes of the accelerations along these two directions do not depend on the shifting along the rotation axis. Therefore, for the best configuration of the accelerometer, this result suggests a configuration of the three sensitive masses aligned along the rotation axis of the MPO, with ISA center-of-mass coincident with the MPO one, see Fig. 5. When comparing the perturbing acceleration of Eq. (3) with the requested accelerometer accuracy (109 g¯), we have been able to obtain the following
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position-and-error matrix for (units are in meters): 0 1 0 ~ D~ R0X þ D~ Rt RtX R0X þ ~ B C B B~ C B ~ ~ ~ @ R0Y þ Rt A þ @ DR0Y þ DRtY ~ R0Z þ ~ Rt 0 0 þ Xt B ¼ @0 þ Xt
D~ R0Z þ D~ RtZ
the ISA inside the MPO 1 C C A 1
0 þ Yt
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0 þ Yt
C Y 0Y þ Z t A
0 þ X t 0 þ Y t Z 0Z þ Z t 0 1 5 103 7:5 103 15 103 B C 3 þB 7:5 103 15 103 C @ 5 10 A; 3 3 3 5 10 7:5 10 15 10
ð4Þ
where we have also included the contribution ~ Rt ¼ ðX t ; Y t ; Z t Þ due to the spacecraft center-of-mass movements. The three vectors ð~ R0X ; ~ R0Y ; ~ R0Z Þ define the position of the center-of-mass of the proof-masses with sensitive axis respectively along the radial direction, the transversal direction and the out-of-plane direction (i.e., along the MPO rotation axis), with X0X = Y0Y = 5 cm and Z0Z = 0 the displacements along the rotation axis. These vectors define a time-invariant contribution to the accelerometer position matrix, while ~ Rt is responsible of the timevariant contribution. In our analyses we estimated for the components of ~ Rt a value of a few cm, but the industrial contractor will be responsible for the final estimate of such numbers. The other numbers represent the uncertainties in
Fig. 5. Configuration of the ISA. The three sensitive units are aligned along the MPO rotation axis in order to reduce the perturbative effects of Mercury gravity-gradients and the apparent forces (nominal case). The center-of-mass of the central unit, the one with sensitive axis along the rotation axis, is coincident with the MPO center-of-mass. The three sensitive axes X, Y and Z of the accelerometer are aligned, respectively, along the radial direction (i.e., along the nadir-zenith direction), along the transversal direction (in the orbital plane) and along the out-of-plane direction (i.e., along the MPO rotation axis).
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consider are: (1) the temperature change over one orbital period (about 2.3 h) of the MPO around Mercury, (2) the temperature change over half of the sidereal period (about 88 days) of Mercury around the Sun (the thermal effects at half of the sidereal period of Mercury around the Sun plays a significant roˆle). The exact impact of the cited thermal disturbances on the accelerometer structure is dependent on the accelerometer thermal stability to the temperature variations. In the case of ISA, this thermal stability is about 5 Æ 108 g¯/C. The simulations of the gravimetry and general relativity experiments have been performed following three different assumptions (Milani et al., 2003): (1) no modelling of the 44 days thermal effect is performed, i.e., 100% of the thermal signal considered equal to 11.8 C peak to peak, (2) the thermal signal at half of the sidereal period is decreased to 10% of its value, (3) the thermal signal at half of the sidereal period is decreased to 1% of its value. In Fig. 6, we show the results of the simulations in the case of the gravimetry experiment (courtesy of Prof. A. Milani). The error is given as an RMS value for all the normalized harmonic coefficients of a given degree ‘. When passing from 100% to 10% calibration, the error decreases by a factor 10 over the entire spectrum up to ‘ = 25. This linearity proves that the 44 days thermal effect is by far the largest source of systematic effect.
the positions of the masses. In reality, the accuracy in the positioning of each mass along the rotation axis is higher, but we refer to Eq. (4) as a conservative way to estimate the combinations of the errors due to the location of the sensitive masses in the accelerometer box, plus the error in positioning this box on the MPO mounting plate, plus the errors in the MPO center-of-mass knowledge. In particular, the vectors D~ R0 (one for each axis of the ISA) represent the time-invariant contribution of the position errors, while the vectors D~ Rt (again one for each axis of the ISA) represent the time-variant contribution of the position errors. We can also choose a configuration with Z0Z 6¼ 0, i.e., with the ISA configuration of Fig. 5 shifted along the rotation axis, useful if the positioning of the accelerometer in the center-of-mass of the MPO must be avoided. The limits of the maximum shift along the Z axis of the accelerometer center-of-mass with respect to the spacecraft center-ofmass are reported in Iafolla et al. (2006). 4. Attenuation of the thermal effects A good attenuation (or elimination) of the thermal effects acting on the accelerometer are an essential prerequisite in order to fulfil both the gravimetry and the general relativity experiments. The main temperature variations to
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Fig. 6. Degree variance as a function of the degree ‘. The red line gives the simulated field while the lower pink line gives the formal uncertainty. The other lines give the error when we consider the thermal effects, especially the long-term thermal effect at half of the sidereal period of Mercury around the Sun. With no attenuation (blue line), i.e., 100% of the thermal effect, the signal-to-noise ratio becomes 1 at degree ‘ = 20. On the other hand, with a 10% attenuation (black line), we are able to reach the degree ‘ = 25 with a signal-to-noise ratio still above 1 (and about 10 to the degree ‘ = 20). Finally, the 99% attenuation (i.e., only 1% residual thermal effect at half of the sidereal period (upper pink line)) will tune toward the degree ‘ = 30 of the expansion in spherical harmonics of the planet gravity field with a signal-to-noise ratio still greater than 1 (and larger than 10 to the degree ‘ = 20). The higher the degree ‘ the higher the accuracy in constraining the planet internal structure. Moreover, the higher the degree, the more accurate will be the resolution of the gravity field. With the 10% calibration, we will reach a spatial resolution of about 400 km (‘ @ 20) and about 300 km (‘ @ 25) with a 1% calibration. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this paper.)
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To reduce the effects of the temperature below the required precision, it is necessary to reduce the thermal variations on the internal part of the accelerometer. The thermal attenuation can be obtained using an active thermal control. A passive system has been considered, but it has not been able to attenuate the thermal variation at the long periods, such as half of the revolution period of Mercury. Therefore, the thermal variation must be reduced by filtering the data out of the required frequency band, or correcting them by measuring the temperature with a precision of 0.1 C. The active control has the advantage to attenuate the temperature variations at very low frequency, but it requires additional power dissipation. In Fig. 7, we report the main results of the thermal analysis simulation for the active thermal control. Of course, the attenuation factor obtained from the simulation (about 10,000) is not easy to achieve experimentally, hence we use a factor 700 obtained from the experimental activity through an active thermal control of the accelerometer. On the contrary, the power dissipation obtained in the simulation is retained as an acceptable value. These results demonstrate that for the Accelerometer Set (AS) – the more sensitive part of the accelerometer to the temperature – an attenuation factor larger than 700 has been obtained experimentally both at the revolution period of Mercury around the Sun and at the orbital period of the MPO around Mercury. The power dissipation necessary for the thermal stabilization equals 3.69 W.
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5. Conclusions The ISA characteristics and roˆle in the case of the RSE has been described in the present paper. In particular, we have demonstrated that the accelerometer accuracy (about 109 g¯) is enough to remove the perturbing effects due to the solar radiation pressure, the largest non-gravitational perturbation. It has been shown that the best geometrical configuration of the accelerometer is with the three sensitive proof-masses aligned along the spacecraft rotation axis and with the center-of-masses of the accelerometer and the MPO coincident; in this way the perturbing effects due to Mercury’s gravity-gradients and to the MPO apparent accelerations are less than the instrument accuracy. However, we have examined the possibility of shifting the accelerometer along the spacecraft rotation axis in order to leave some space in the surroundings of the MPO nominal center-of-mass position. This alternative solution will be useful for the location of the fuel tank and probably for a better determination of the MPO center-of-mass movements, due to the fuel consumption and sloshing. These aspects have been analysed (together with the HGA movements) in detail in Iafolla et al. (2006). The importance of a refined attenuation of the thermal effects on the accelerometer has been described in the case of the gravimetry experiment. The attenuation of the perturbing thermal effects, both at the MPO orbital period and at half of Mercury’s sidereal period, has been obtained
Fig. 7. Thermal simulation, frequency response. In the simulation, the input signal was kept equal to 2 C at all frequencies, seen in the blue line. The temperature of the Accelerometer Set (AS) is reported in green while the temperature of the electronics is reported in red. The ratio between the blue line and the red line gives the attenuation factor for the simulation. However, the results from the simulation are not reliable in the case of the attenuation factor estimate, but they give very confident estimates for the power dissipation, useful to implement the thermal control. The ISA mechanical part, which is more sensitive to the temperature, is allocated separately inside a thermal shielded box. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this paper.)
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through an active thermal control that guarantees an attenuation larger than a factor 700. The future work devoted to the implementation of a space version has already started. In particular, the ISA error budget and the requirements imposed on the angular rates and angular accelerations necessary to guarantee the MPO nadir pointing are under investigation. References Fuligni, F., Iafolla, V. Measurement of small forces in the physics of gravitation and geophysics. Il Nuovo Cimento 20C (5), 619–628, 1997. Fuligni, F., Iafolla, V., Milyukov, V., Nozzoli, S. Experimental gravitation and geophysics. Il Nuovo Cimento 20C (5), 637–642, 1997.
Iafolla,V., Lucchesi, D.M., Nozzoli, S., On the ISA accelerometer positioning inside the Mercury planetary orbiter. Planet. Space Sci., in press, 2006. Iafolla, V., Nozzoli, S. Italian spring accelerometer (ISA) a high sensitive accelerometer for ‘’BepiColombo‘’ ESA CORNERSTONE. Planet. Space Sci. 49, 1609–1617, 2001. Iess, L., Boscagli, G. Advanced radio science instrumentation for the mission BepiColombo to Mercury. Planet. Space Sci. 49, 1597–1608, 2001. Milani, A., Rossi, A., Vokrouhlicky´, D., Villani, D., Bonanno, C. Gravity field and rotation state of Mercury from the BepiColombo Radio Science Experiments. Planet. Space Sci. 49, 1579–1596, 2001. Milani A., Rossi, A., Villani, D., The BepiColombo Radio Science Simulations, Version 2, 11 April 2003. Milani, A., Vokrouhlicky, D., Villani, D., Bonanno, C., Rossi, A. Testing general relativity with the Bepicolombo radio science experiment. Phys. Rev. D 66, 082001, 2002.