The potential of IORIO as relativity probe; the impact of the mismodeling in the Jovian gravity field multipoles
Abstract
The IORIO (InOrbit Relativity Iuppiter Observatory) concept consists of a fast orbiter of Jupiter , endowed with a Junotype Doppler ranging device, accurate to , originally proposed to measure the postNewtonian rangerate signature proportional to . It could be made as large as , depending on the orientation of the orbital plane. Also other general relativistic effects could be measurable, including also the one proportional to , never put to the test so far. A major source of systematic error is our imperfect knowledge of the magnitude and orientation of the Newtonian multipolar expansion of the Jovian gravity field. We numerically investigate its biasing impact on the general relativistic signatures by determining the required level of improvement of each Newtonian multipole with respect to its presentday accuracy based on the analysis of just some perijove passages of the spacecraft Juno, currently orbiting Jupiter. If, on the one hand, the accuracy of most of the Jovian multipoles up to degree should be improved by a factor of about , with a peak of for , in order to reach the size of the effect, on the other hand, the spin pole position, impacting mostly the classical signal, would require a relatively more modest improvement by a factor of 100. The improvements needed to reach the size of the signals are smaller, amounting to a factor of or so. Remarkably, most of the competing Newtonian signals have quite different temporal signatures with respect to the postNewtonian ones, making, thus, potentially easier disentangling them.
keywords General relativity and gravitation; Experimental studies of gravity; Experimental tests of gravitational theories; Lunar, planetary, and deepspace probes; Satellite orbits
1 Introduction
Recently, Iorio (2018) proposed a spacecraftbased mission targeted to Jupiter, tentatively named IORIO (InOrbit Relativity Iuppiter Observatory, or IOvis Relativity Inorbit Observatory), aimed to measure a novel general relativistic effect, proportional to , induced by the postNewtonian gravitomagnetic spinoctupole moment (Panhans & Soffel, 2014) of the gaseous giant. By assuming a Junotype probe in a much faster () and almost circular jovicentric orbit, in Iorio (2018) it was shown that, by suitably choosing the orientation of its orbital plane in space, it would be possible to raise the relativistic rangerate signature to the level; the current accuracy of the Juno Doppler measurements is at the level over s (Iess et al., 2018).
In this paper, we want to dig more thoroughly into the potential of the IORIO concept as a tool to measure even more general relativistic features of motion ranging from the standard Schwarzschildlike one to the so far never tested gravitoelectric effect proportional to (Soffel et al., 1988; Soffel, 1989; Brumberg, 1991), including also the gravitomagnetic LenseThirring framedragging (Lense & Thirring, 1918). Moreover, we will investigate also the impact of the Newtonian part of the multipolar expansion of the Jovian gravity field, acting as a major source of systematic errors on the proposed relativistic measurements. We will work numerically by integrating the equations of motion of the Earth, Jupiter and the probe with respect to the International Celestial Reference Frame (ICRF) (Petit, Luzum & et al., 2010), and producing simulated time series of the Earthspacecraft rangerate signatures for each of the Newtonian and postNewtonian accelerations considered. In addition to a quantitative evaluation of the biasing level due to the size of the presentday uncertainties in the Jovian spin axis orientation and gravity field multipoles (Iess et al., 2018; Durante et al., 2018), a crucial feature which will be investigated in detail is the temporal patterns of both the Newtonian and the postNewtonian rangerate signatures in order to see if they are different or too similar to make difficult disentangling them in real data reductions. The analytical expressions of the longterm orbital precessions proportional to , valid for an arbitrary orientation of the spin axis of the primary and for quite general orbital configurations of the orbiter, can be found in Iorio (2011, 2015, 2018).
The paper is organized as follows. In Section 2, we investigate the consequences of the mismodeling in the Newtonian potential coefficients of Jupiter, while Section 3 is devoted to the impact of the uncertainty in the Jovian spin axis orientation. Our findings and conclusions are summarized in Section 4. Appendix A collects the symbols and definitions of the quantities used throughout the text, while the tables and the figures are displayed in Appendix B.
2 The impact of the mismodeling in the Jovian gravity field’s multipoles
Our most accurate knowledge of the Newtonian multipolar expansion of the Jovian gravity field comes from the analysis of the Doppler measurements of the ongoing Juno mission, accurate to over 1,000 s, collected at some selected perijove passages. At the time of writing, results were published in the peer reviewed literature (Iess et al., 2018, Tab. 1) just for two perijove passes (PJ03 and PJ06), while data from PJ08, PJ10, PJ11 are still under analysis (Durante et al., 2018). The total perijove passages of Juno dedicated to the gravity field determination should be
Figures 1 to 15 depict the numerically integrated Newtonian and postNewtonian rangerate time series for a given orbital configuration of the probe which, as it will be shown below, should make the detection of the relativistic signals more favorable. We numerically integrated the equations of motion of the Earth, Jupiter and the probe in Cartesian rectangular coordinates referred to the ICRF with and without the disturbing Newtonian or postNewtonian accelerations which, from time to time, are under investigation. For each acceleration, both the runs, spanning 1 d, shared the same set of initial conditions which, for the Earth and Jupiter, were retrieved from the WEB interface HORIZONS maintained by JPL, NASA, for a given initial epoch which, in the present case, is January 1, 2020. After each run, a numerical time series of the Earthprobe rangerate was produced; includes the perturbing acceleration, while is the purely Newtonian one due to only the monopoles of the Sun and Jupiter. In order to single out the effect of the perturbing acceleration considered, the difference of both the time series was computed obtaining the curves for displayed in Figures 1 to 15. In order to better visualize the temporal patterns of the various effects, the classical signatures were computed by using fictitious values of the Newtonian gravity field coefficients able to make their magnitudes roughly equal to those of the postNewtonian time series of interest. If such figures for the Jovian multipoles are smaller than their presentday uncertainties listed in Table 1, they can be interpreted as a measure of how much they should still be improved with respect to their current levels of accuracy in order to make the size of the Newtonian signatures at least equal to the relativistic ones. If, instead, are larger than their present mismodeling, they can be viewed as a measure of the relative accuracy with which a given relativistic signal would be impacted right now. See Table 2 for a complete list of such improvement factors for all the Newtonian multipoles considered here in connection with the various relativistic effects. It turns out that the largest improvementsof the order of , with a peak of for would be required to bring the Newtonian signals to the level of the postNewtonian gravitomagnetic effect proportional to . Interestingly, a much smaller improvement would be required to make the size of the classical multipole signatures comparable with the postNewtonian gravitoelectric and gravitomagnetic effects proportional to . As far as the Schwarzschildtype signature is concerned, the current level of accuracy in almost all the Jovian multipoles, with the exception of , would yield a bias at the level. A very important feature of all the curves displayed in Figures 1 to 15 is that the relativistic ones exhibit neatly different temporal patterns with respect to the Newtonian ones, making, thus, easier to detect them. It would not be so for different orbital geometries of the probe.
3 The impact of the uncertainty in the Jupiter’s pole position
The position of the Jovian spin axis, determined by its right ascension and declination with respect to the ICRF (Durante et al., 2018), enters the Newtonian accelerations induced by the gravity field multipoles in a nonlinear way. It can be easily realized, e.g., by inspecting the analytical expressions of the longterm precessions of the Keplerian orbital elements due to some even and odd zonal harmonics calculated by Iorio (2011); Renzetti (2013, 2014) for an arbitrary orientation of . Thus, the uncertainties has an impact on the general relativistic effects of interest through the Newtonian multipolar signatures. The latest determinations of along with the associated realistic uncertainties, of the order of (Durante et al., 2018), are listed in Table 1.
Figure 16 depicts the numerically simulated mismodeled rangerate signals due to the first four even zonals of Jupiter induced by the presentday errors . They were obtained as described in the previous Section by using the nominal values of the even zonals and taking the differences between the time series computed with (red curves) and (blue curves), respectively. It turns out that the largest residual signals are due to the uncertainty in the declination. The largest one occurs for , with an amplitude which can reach . The signatures of the odd zonals are completely negligible. It can be shown that an improvement of by a factor of 100 with respect to the current value of Table 1 would bring the size of the Newtonian induced rangerate time series to the same level of the postNewtonian one proportional to . Such an improvement seems to be quite feasible in view of the fact that it already occurred from the analysis of PJ03, PJ06 (Iess et al., 2018, Tab. 1) to that of PJ08, PJ10, PJ11 (Durante et al., 2018). In any case, as already noticed in the previous Section, the temporal pattern of the classical signal is different from the relativistic ones.
4 Summary and overview
We explored the potential of the recently proposed InOrbit Relativity Iuppiter Observatory (IORIO), which should orbit Jupiter along an almost circular, path, as an effective probe for measuring several general relativistic features of motion in the field of the gaseous giant with accurate Earthspacecraft Doppler measurements. In particular, we looked at the postNewtonian gravitoelectric and gravitomagnetic effects proportional to , which have never been put to the test so far, and at the standard LenseThirring and Schwarzschild signatures, proportional to , respectively.
The experimental uncertainties in the values of both the Newtonian coefficients of the multipolar expansion of the Jovian gravity field and in the orientation of the spin axis of Jupiter would induce mismodeled rangerate signatures in the Doppler measurements of the spacecraft acting as sources of competing systematic biases for the postNewtonian signals of interest. At present, just 5 of the planned 25 perijove passes dedicated to mapping the planet’s gravity field of the ongoing Juno mission, scheduled to end in July 2021, have been analyzed so far. Thus, if and when IORIO will be finally implemented, it will benefit of the analysis of the entire Juno data record yielding a much more accurate determination of the Jovian gravity field coefficients and pole position than now.
For a given orbital configuration of the spacecraft, we numerically simulated its mismodeled Newtonian rangerate signatures due to the gravity field coefficients and the spin axis position of Jupiter currently determined by Juno, and the predicted postNewtonian signals. We determined the level of improvement of the Jovian multipoles and pole position with respect to their presentday accuracies still required to bring the competing classical effects to the level of the various relativistic ones. It turned out that the most demanding requirements pertain the measurability of the signature, implying improvements by a factor of for most of the Jovian gravity coefficients considered, with a peak of for . The other relatively small postNewtonian effects, proportional to , require less demanding improvements by a factor of just or less. The Schwarzschild signature would be measurable right now at a level, apart from the impact of . As far as the Jupiter’s spin axis is concerned, an improvement by a factor of 100 would be required for its declination to make the size of the induced signature to the same level of the postNewtonian one. The uncertainty in the declination is less important. The rangerate signals due to the odd zonals are affected by the errors in the pole position at a negligible level. A fundamental outcome of our analysis consists of the fact that the temporal patterns of the relativistic signatures turned out to be quite different from the classical ones, making, thus, easier, in principle, to separate the postNewtonian from the Newtonian effects.
Acknowledgements
I am grateful to D. Durante for useful information.
Appendix A Notations and definitions
Here, some basic notations and definitions used throughout the text are presented (Brumberg, 1991; Bertotti, Farinella & Vokrouhlický, 2003; Kopeikin, Efroimsky & Kaplan, 2011; Poisson & Will, 2014).

Newtonian constant of gravitation

speed of light in vacuum

mass of Jupiter

gravitational parameter of Jupiter

magnitude of the angular momentum of Jupiter

spin axis of Jupiter with respect to the ICRF

right ascension (RA) of the Jovian spin axis

declination (DEC) of the Jovian spin axis

component of the Jovian spin axis with respect to the ICRF

component of the Jovian spin axis with respect to the ICRF

component of the Jovian spin axis with respect to the ICRF

equatorial radius of Jupiter

Newtonian zonal multipole mass moments of Jupiter

tesseral and sectorial multipole mass moments of degree of Jupiter

true anomaly of the spacecraft

semimajor axis of the spacecraft

Keplerian mean motion of the spacecraft

orbital period of the spacecraft

eccentricity of the spacecraft

inclination of the orbital plane of the spacecraft to the Earth’s mean equator at the epoch J2000.0

longitude of the ascending node of the spacecraft with respect to the ICRF

argument of pericenter of the spacecraft with respect to the ICRF
Appendix B Tables and figures
Parameter  Units  Numerical value 

deg  
deg  
km  
Multipole  

Footnotes
 affiliationmark:
 D. Durante, private communication.
 See https://www.jpl.nasa.gov/missions/juno/ on the Internet.
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