Post-Newtonian orbital effects induced by the mass quadrupole and spin octupole moments of an axisymmetric body

The post-Newtonian orbital effects induced by the mass quadrupole and spin octupole moments of an isolated, oblate spheroid of constant density that is rigidly and uniformly rotating on the motion of a test particle are analytically worked out for an arbitrary orbital configuration and without any preferred orientation of the body's spin axis. The resulting expressions are specialized to the cases of (a) equatorial and (b) polar orbits. The opportunity offered by a hypothetical new spacecraft moving around Jupiter along a Juno-like highly elliptical, polar orbit to measure them is preliminarily studied. Although more difficult to be practically implemented, also the case of a less elliptical orbit is considered since it yields much larger figures for the relativistic effects of interest. The possibility of using the S stars orbiting the supermassive black hole in Sgr A$^\ast$ at the Galactic Center as probes to potentially constrain some parameters of the predicted extended mass distribution surrounding the hole by means of the aforementioned orbital effects is briefly examined.


Introduction
The most known post-Newtonian (pN) orbital effects, which have been extensively tested so far in different astronomical scenarios, are the gravitoelectric and gravitomagnetic precessions due to the mass monopole and the spin-dipole moments of the central body which acts as source of the gravitational field, respectively.The former is responsible for the well known, previously anomalous perihelion precession of Mercury in the Sun's field (Le Verrier 1859) of 42.98 ′′ per century (arcsec cty −1 ) (Nobili and Will 1986), whose explanation by Einstein (1915) was the first empirical success of his general theory of relativity (GTR).Such a feature of motion was later repeatedly measured with radar measurements of Mercury itself (Shapiro et al. 1972;Shapiro 1990), of other inner planets (Anderson et al. 1978(Anderson et al. , 1993)), and of the asteroid Icarus (Shapiro et al. 1968(Shapiro et al. , 1971) ) as well.In more recent times, stars in the Galactic Center (GRAVITY Collaboration et al. 2020), binary pulsars (Kramer et al. 2006) and Earth's artificial satellites (Lucchesi andPeron 2010, 2014) were also used.The latter is the so-called Lense-Thirring effect (Lense and Thirring 1918;Mashhoon et al. 1984), which is induced by the angular momentum of the central spinning body.It is currently being experimentally investigated around the Earth with the geodetic satellites of the LAGEOS family; see, for example, Renzetti (2013); Iorio et al. (2011); Lucchesi et al. (2020), and references therein.The gravitomagnetic precessions of the spins of some orbiting gyroscopes (Pugh 1959;Schiff 1960) were detected in the field of the rotating Earth with the dedicated Gravity Probe B spaceborne mission (Everitt 1974) to a 19% accuracy level (Everitt et al. 2011(Everitt et al. , 2015)), although the originally expected error was ≃ 1% (Everitt et al. 2001).
Less known features of motion are the pN gravitoelectric and gravitomagnetic effects associated with the asphericity of a central body induced by its mass quadrupole and spin octupole moments, respectively (Soffel et al. 1987;Soffel 1989;Heimberger et al. 1989;Brumberg 1991;Huang and Liu 1992;Panhans and Soffel 2014;Will 2014;Iorio 2015;Meichsner and Soffel 2015;Frutos-Alfaro and Soffel 2018;Schanner and Soffel 2018).To date, they have never been measured; a detailed study for a proposed satellite-based mission in the field of the Earth, dubbed Highly Elliptical Relativity Orbiter, can be found in Iorio (2019c).
We will analytically work out, to the first pN (1pN) order, the net rates of change per orbit induced by the aforementioned departures from spherical symmetry of the source of the gravitational field without recurring to any simplifying assumption pertaining both the orbital configuration of the test particle and the orientation of the primary's spin axis in space.We will present lorenzo.iorio@libero.itour results in a form that should make their interpretation and use in specific situations at hand simple and direct.Then, we will look at some astronomical scenarios that could be favorable for their measurement.Such a task would enlarge the empirical basis of GTR extending it to phenomena, although just in the 1pN regime, not yet tested.Conversely, they may provide further means to dynamically constrain, at least in principle, some key physical parameters of astronomical and astrophysical systems of interest.We will look neither at the directly measurable quantities in real spacecraft-based missions such as range and range rates, nor at the actual data reduction procedure (Moyer 2003).Instead, our goal is just to gain meaningful insights about the potential offered by the considered scenario(s) by performing a preliminary sensitivity analysis.To this aim, we will use the usual osculating Keplerian orbital elements (Soffel 1989;Brumberg 1991;Kopeikin et al. 2011;Soffel and Han 2019), which are easy to visualize in view of their clear geometric meaning.
The paper is organized as follows.In Section 2, the calculation scheme adopted to work out the long-term orbital features of motion under investigation is reviewed.The case of the pN gravitoelectric static field of a massive, oblate body is treated in Section 3, while the consequences of the gravitomagnetic spin octupole moment of a spinning primary is the subject of Section 4. Section 5 deals with some particular orbital configurations; in Section 5.1, the scenario where the satellite's orbital plane and the body's equatorial plane coincide (equatorial orbit) is considered, while Section 5.2 investigates the case where the orbital plane contains the body's spin axis (polar orbit).The results of Section 5.2 are applied in Section 6 to a Jovian scenario characterized by a putative spacecraft orbiting the fifth planet of our solar system along a highly elliptical polar orbit.A possible application of our results to the highly elliptical stellar motions in Sgr A * at the center of the Galaxy is outlined in Section 7. Section 8 summarizes our findings and offers concluding remarks.Appendix A contains a list of the symbols and definitions used throughout the paper.

Computational outline
The long-term orbital effects of interest are calculated by averaging over one satellite's orbital period the right-hand-sides of the equations for the variations of the Keplerian osculating elements in the Euler-Gauss form (Soffel 1989;Brumberg 1991;Kopeikin et al. 2011;Soffel and Han 2019) evaluated onto the Keplerian ellipse by means of dt It turns out to be computationally convenient to express the position and velocity vectors as In the following, for the sake of simplicity, the angular brackets dκ/dt denoting the average over one orbital period of the rate of change of any one of the orbital elements κ will be omitted.
3. The 1pN acceleration induced by the oblateness of the central body The pN gravitoelectric acceleration A obl pN experienced by a test particle orbiting an extended, oblate body is (Will 2014) where A obl pN was derived earlier by Soffel et al. (1987); Soffel (1989); Brumberg (1991); Huang and Liu (1992) in a reference frame whose z-axis is aligned with k.
By following the computational scheme outlined in Section 2, one obtains the following expressions for the long-term rates of change of the Keplerian osculating elements induced by the sum of Equations ( 12)-( 14).
+ 4 e 2 T 4 cot I sin 2ω + T 5 cot I 6 − e 2 cos 2ω + 7 T 6 sin 2ω , L. Iorio The coefficients T j , j = 1, 2, . . .6 are explicitly displayed in Appendix B. It should be remarked that Equations ( 15)-( 20) are valid for any orbital configuration and for an arbitrary orientation of the body's symmetry axis in space.They were calculated in Soffel et al. (1987); Brumberg (1991); Huang and Liu (1992) by orienting k along the z axis of the reference frame chosen.Iorio (2015) worked out the pN oblateness-driven net shifts per orbit ∆κ of κ = {p, e, I, Ω, ω} for an arbitrary orientation of k, but the resulting expressions are much more cumbersome than Equations ( 15)-( 20) and a comparison with them is difficult.
The mixed effects due to the simultaneous presence of the 1pN gravitoelectric acceleration due to the mass monopole of the central body and the Newtonian acceleration induced by J 2 in the equations of motion (Heimberger et al. 1989;Huang and Liu 1992;Will 2014;Iorio 2015) will not be calculated here; they can be found, worked out in their full generality, in Iorio (2023).From an empirical point of view, it can be expected that they would affect post-fit residuals just with tiny mismodelled signatures since both the aforementioned accelerations are routinely modeled in any software used to reduce astronomical observations.In principle, also orbital variations of order O J 2 2 /c 2 , arising from the mixed action of Equation ( 11) and the Newtonian oblatenessdriven acceleration, should occur.Nonetheless, they are completely negligible since they would bring about in Equations ( 15)-(20) the scaling factor J 2 (R e /a) 2 .

The gravitomagnetic acceleration induced by the spin octupole moment of a rotating body
To the 1pN order, the gravitomagnetic Panhans-Soffel (PS) spin octupole1 acceleration A oct PS experienced by a test particle orbiting an oblate spheroid of constant density that is rigidly and uniformly rotating is (Panhans and Soffel 2014) where the gravitomagnetic field B oct can be calculated as Panhans and Soffel ( 2014) with the gravitomagnetic octupolar potential φ oct given by Panhans and Soffel ( 2014) From Equations ( 21)-( 23), the spin octupole PS acceleration can finally be cast into the form According to the computational scheme outlined in Section 2, the long-term rates of change of the osculating Keplerian orbital elements induced by Equation ( 24) turn out to be 56 c 2 a 5 1 − e 2 7/2 4 3 + 2 e 2 k• ĥ −2 T 1 + 5 T 2 +   + 2 2 + 3 e 2 k• m −4 T 1 + 5 T 2 cot I + 5 2 1 + 2 e 2 k• ĥ T 3 It must be remarked that Equations ( 25)-( 30) retain their validity for any orientation of k in space, and for arbitrary orbital configurations.The gravitomagnetic spin octupole orbital precessions were explicitly calculated by Iorio (2019a,b) in the special case of k aligned with the z axis of the reference frame adopted.In fact, also general formulas for them, valid for arbitrary orbital geometries and orientations of k in space, were obtained by Iorio (2019a,b); nonetheless, they are much more cumbersome and less compact than Equations ( 25)-( 30), so that a straightforward comparison is difficult.The interplay of the Newtonian acceleration due to J 2 and the pN one of Equation ( 24) would induce, in principle, mixed effects of order O S ε2 J 2 /c 2 .Nonetheless, also in this case, they would be negligible because of the multiplicative factor J 2 (R e /a) 2 by which Equations ( 25)-( 30) would be scaled down.

Some special orbital configurations
Here, two particular orbital configurations are considered: (a) equatorial (Section 5.1) and (b) polar (Section 5.2) orbits.By parameterizing k in terms of the polar angles 2 α, δ as in Appendix A, one has k• ĥ = cos I sin δ − sin I cos δ sin (α − Ω) . (33)

Polar orbit
Let us assume that the body's spin axis, irrespectively of its orientation in the adopted coordinate system, i.e., for generic values of α, δ, lies somewhere in the satellite's orbital plane between ˆl and m.Such an orbital geometry is widely adopted in L. Iorio spacecraft-based missions to solar system planets like, e.g., Juno at Jupiter (Bolton et al. 2017).In such a scenario, it is Equation ( 52) implies that the orbital angular momentum is perpendicular to the body's spin axis.Equations ( 31)-( 33) tell us that the conditions of Equations ( 50)-( 52) are satisfied for indeed, with Equations ( 53)-( 54), one has just As a consequence, Equations ( 15)-( 30) reduce to while Equations ( 25)-( 30) become L. Iorio Equations ( 58)-( 59), Equations ( 62)-( 63), and Equations ( 66)-( 67), in addition to secular trends, when present, include also longperiod signatures due to the evolution of pericentre which is mainly driven by the zonal harmonics of the Newtonian component of the multipolar field of the central body (Capderou 2005).
It can be shown that, for the orbital configuration of Equations ( 53)-( 54), the classical precessions of I and Ω due to the zonal harmonics of the primary's gravitational potential vanish; see Appendix A of Iorio et al. (2023).It is important since such classical effects are usually regarded as a major source of systematic bias in assessing the error budget of a mission dedicated to measure certain general relativistic features of motion.
On the other hand, Equation ( 15) has not a classical counterpart acting as a systematic bias: it is true for any orbital configuration and primary's spin axis orientation, being the Newtonian oblateness-driven rate of change of a exactly zero.As far as the eccentricity is concerned, in general, the odd zonal harmonics of the classical gravity field of the source induce net variations per orbit which do not vanish for the polar geometry of Equations ( 53)-( 54).It turns out that they are harmonic signatures oscillating with odd multiples of the pericentre frequency, while Equation ( 59) varies just at twice the frequency of ω.More specifically, by following the approach outlined in Appendix A of Iorio et al. (2023), one has and so on.Either the even and the odd zonals make the pericentre to vary, as per Equations (A10) to (A16) of Iorio et al. (2023).Furthermore, if the condition is selected, it turns out that the pericentre precessions due to the odd zonals vanish (Iorio et al. 2023).On the other hand, if the condition of Equation ( 73) holds, Equations ( 58)-( 59) vanish as well.

The Jovian scenario for a dedicated mission
In view of its size, Jupiter, whose relevant physical parameters are listed in Table 1, seems to be the ideal 3 candidate, at least in principle, to try to measure the general relativistic effects treated in the previous sections with a dedicated spacecraft-based mission.With a pinch of irony, we preliminarily dub it IORIO, acronym of In-Orbit Relativity Iuppiter 4 Observatory, or, equally well, of IOvis 5 Relativity In-orbit Observatory (Iorio 2019a,b).Moreover, both for typical planetological goals 6 and to preserve the onboard scientific instrumentation from harmful radiation 7 , the orbits of the spacecraft targeted to the fifth planet of our solar system are usually just polar and widely elliptical; thus, we can rely upon the results of Section 5.2.A notable existing example of such a peculiar orbital geometry is provided by the ongoing mission Juno (Bolton et al. 2017).It is currently orbiting Jupiter along a wide 53 days orbit characterized by a pericenter height of and an apocenter height of corresponding to a semimajor axis a = 4.1 × 10 6 km = 57.7 R e (76) and an eccentricity e = 0.981.
Originally, a 13 days orbit, corresponding to a lower apocenter height of and eccentricity e = 0.954 (79) was planned for Juno (Matousek 2007), but problems with two helium valves in its propulsion system in 2016 October prevented to meet such a goal 8 .For the sake of concreteness, in the following we will consider a putative spacecraft having the same pericenter height as Juno, given by Equation (74), while its apocenter height is allowed to vary from, say, to the current Juno value of Equation ( 75), corresponding to a range variation for e of ≃ 0.92 − 0.98.Obtaining Equation ( 80) would be a demanding task since more fuel would be needed, and the spacecraft would be exposed to a larger amount of radiation.
4 Iuppȋtȇr is one of the forms of the Latin noun of the god Jupiter. 5 In Latin, Iȏvis means "of Jupiter". 6Polar orbits, taking the spacecraft over the poles of the planet, allow the probe to traverse the latter in a north-south direction; they are optimal for mapping and monitoring it (Montenbruck and Gill 2000). 7Radiation belts are the regions of a magnetosphere where high energy charged particles, such as electrons, protons, and heavier ions, are trapped in large amounts.
All planets in our solar system having a sufficiently intense magnetic field, such as Earth, Jupiter, Saturn, Uranus, and Neptune, host radiation belts (Mauk and Fox 2010;Mauk 2014).Jupiter has the most complex and energetic radiation belts in our solar system and one of the most challenging space environments to measure and characterize in-depth (Roussos et al. 2022). 8See https://www.science.org/content/article/avoid-risk-misfire-nasas-juno-probe-will-keep-its-distance-jupiter .58)-( 59), calculated for the polar orbital configuration of Equations ( 53)-( 54), as functions of the apocenter height h apo for a fixed value of the pericenter height h peri = 4200 km.The dashed vertical line corresponds to h apo = 3.2 × 10 6 km which was the originally planned apocenter height of Juno before the problems encountered by its propulsion system.
According to Figure 1, the amplitude of the J 2 /c 2 long-period signature of a ranges from 500 m yr −1 = 0.01 mm s −1 to 1100 m yr −1 = 0.03 mm s −1 , a seemingly unexpected feature due to the impact of e in Equation ( 58 62)-( 63), in mas yr −1 , calculated for the polar orbital configuration of Equations ( 53)-( 54) with the condition of Equation ( 73), as functions of the apocenter height h apo for a fixed value of the pericenter height h peri = 4200 km.The dashed vertical line corresponds to h apo = 3.2×10 6 km which was the originally planned apocenter height of Juno before the problems encountered by its propulsion system.
whose ranges of variation amount to 0.050 mm s −1 , with a root-mean-square value of 0.015 mm s −1 .However, caution is in order when such risky comparisons are made since the directly measurable range-rate and the theoretically computed semimajor axis are quite distinct quantities.As far as the J 2 /c 2 effect on e is concerned, it turns out that, according to Table 1, the nominal value of Equation ( 70), which is a potentially major source of systematic bias, is smaller than it by about one order of magnitude.The J 2 /c 2 pericentre precession, shown in Figure 2, is in the range ≃ 2 − 12 milliarcseconds per year (mas yr −1 ).It would be overwhelmed by the classical secular trend due to J 2 .Furthermore, both the latter and the 1pN signature are proportional to  66)-( 67), in mas yr −1 , calculated for the polar orbital configuration of Equations ( 53)-( 54) with the condition of Equation ( 73), as functions of the apocenter height h apo for a fixed value of the pericenter height h peri = 4200 km.The dashed vertical line corresponds to h apo = 3.2×10 6 km which was the originally planned apocenter height of Juno before the problems encountered by its propulsion system.
J 2 itself; their ratio is independent of it, so that one may not invoke any future improvement in our knowledge of the first even zonal of the Jovian gravity field to reduce the impact of the Newtonian effect on the 1pN one.A further potentially major source of systematic bias is represented by the competing classical N-body perturbations due to the Galilean moons of Jupiter, which orbit in its equatorial plane.It preliminarily turns out that 9 , if on the one hand, they leave the semimajor axis unaffected, on the other hand, for a polar orbit, they induce nonvanishing doubly-averaged orbital disturbances on e and ω.According to the present-day level of uncertainty10 in their masses as per the Planetary Satellite Ephemeris JUP365 (Jacobson 2021), the resulting mismodelled signatures of e and ω would be roughly one order of magnitude larger than, or about of the same order of magnitude of, the J 2 /c 2 ones, apart for Callisto for which they are smaller than the pN ones.In the near future, the masses of the three outer Galilean satellites will be accurately determined by the JUpiter ICy moons Explorer (Grasset et al. 2013, JUICE; ) and Clipper missions (Korth et al. 2022), while the flybys of Io by Juno should allow to improve the mass of Io as well.In particular, according to Tables 1-3 of Magnanini (2021), the masses of Europa, Ganymede, and Callisto should be determined by JUICE with an improvement of about 1-2 orders of magnitude with respect to the errors by Jacobson (2021).
Figure 3 tells us that the nonvanishing gravitomagnetic spin octupole orbital effects amount to ≃ 0.1 − 2 mas yr −1 .The impact of the Galilean moons of Jupiter would be of no concern for Equation (67) if viewed in a frame11 aligned with the Jovian equator since the N−body node rate of change of a polar orbit acted upon by an equatorial perturber vanishes, as per Equation (4) of Iorio (2020).If, instead, the ICRF is adopted, the nonvanishing inclination of the orbits of the Galilean satellites to the Celestial Equator would make that the amplitudes of their mismodelled perturbations on I and Ω (Iorio 2020, Equations ( 4)-( 5)) may be up to about one order of magnitude larger than Equations ( 66)-( 67).The Sun does not represent an issue since it turns out that, by assuming σ µ ⊙ = 1 × 10 10 m 3 s −2 (Petit and Luzum 2010), its mismodelled classical perturbations are several orders of magnitude smaller than the pN effects of interest.
Despite inserting a spacecraft into a moderately elliptical orbits around Jupiter is a very daunting task because of the exceedingly amount of fuel required, we deem the study of the relativistic effects even in such an unlikely scenario worthy of investigation.By exploring the eccentricity range e ≃ 0.05 − 0.9, with the pericenter height fixed to the value of Equation ( 74 58)-( 59), calculated for the polar orbital configuration of Equations ( 53)-( 54), as functions of the apocenter height h apo for a fixed value of the pericenter height h peri = 4200 km.
In this case, the J 2 /c 2 precessions can reach even the arcsec yr −1 level, while the gravitomagnetic signatures can be as large as a few hundred mas yr −1 .As far as the semimajor axis is concerned, the amplitude of its J 2 /c 2 signature ranges from about 3 m yr −1 = 3 × 10 −4 mm s −1 to 500 m yr −1 = 0.05 mm s −1 .Now, the mismodelled N− body orbital perturbations due to all the Galilean moons turn out to be negligible.
Such results demonstrate that, perhaps, it would be worth trying to actually carry out such a mission, however challenging it is.After all, it should not be neglected that, in addition to the relatively tiny pN effects induced by the multipoles of Jupiter, it could measure also the traditional and much larger pN mass monopole and spin-dipole effects.62)-( 63), in mas yr −1 , calculated for the polar orbital configuration of Equations ( 53)-( 54) with the condition of Equation ( 73), as functions of the apocenter height h apo for a fixed value of the pericenter height h peri = 4200 km.
Incidentally, for, say, a Jovicentric Juno-like orbit, the aforementioned mixed effects of order O J 2 2 /c 2 and O S ε 2 J 2 /c 2 would be smaller than the direct ones retrievable in Figures 1 to 3 by a factor ≃ J 2 (R e /a) 2 = 4 × 10 −6 , as per Equation ( 76) and Table 1.
About the possibility of looking at Saturn as well, its relevant physical parameters are listed in Table 2. From them and from Table 1, it turns out that, despite the Kronian oblateness and ellipticity are slightly larger than the Jovian ones, the relativistic effects considered so far, with the same orbit, are larger for Jupiter by about one order of magnitude.Indeed, it is  66)-( 67), in mas yr −1 , calculated for the polar orbital configuration of Equations ( 53)-( 54) with the condition of Equation ( 73), as functions of the apocenter height h apo for a fixed value of the pericenter height h peri = 4200 km.The general validity of Equations ( 15)-( 20) and of Equations ( 25)-( 30) allows one to apply them, in principle, also to the highly elliptical paths of the S stars (Ali et al. 2020) orbiting the supermassive black hole (SMBH) in Sgr A * at the Galactic Center (Ghez et al. 2008;Genzel et al. 2010) in order, e.g., to characterize the predicted mass component enclosed by the stellar orbits 12 (Gondolo and Silk 1999;Mouawad et al. 2005;Gillessen et al. 2009;Chan et al. 2022).Such an extended dark mass distribution surrounding Sgr A * might be made of faint S stars, neutron stars, stellar mass black holes, faint accretion gas clouds, stellar remnants and nonbaryonic dark matter as well.Should it has departures from spherically symmetry, they may be dynamically probed, in principle, with the pN orbital effects investigated here.

Summary and conclusions
We analytically worked out the pN orbital effects induced on a test particle by the quadrupole mass moment and the gravitomagnetic spin octupole moment of an axisymmetric rotating body.The resulting explicit expressions of the long-term rates of change of the satellite's osculating Keplerian orbital elements, averaged over one orbital period, retain a general validity since they hold for any orbital configuration and for an arbitrary orientation of the body's spin axis in space.In general, they are all nonzero, with the exception of the gravitomagnetic variation of the semimajor axis which vanishes over an orbital revolution.
Our results were subsequently specialized to two particular orbital geometries: (a) an equatorial orbit; (b) a polar orbit.In the scenario a), only the pericenter and the mean anomaly at epoch undergo nonzero net effects for both the pN accelerations considered; such orbital features of motion turn out to be genuine secular trends.The scenario b) is more complex.Indeed, in the case of the pN oblateness, only the inclination and the node stay constant.While the semimajor axis and the eccentricity experience purely long-period variations because of the generally varying pericenter entering the expressions of their averaged rates of change, the pericenter and the mean anomaly at epoch are affected also by secular trends in addition to long-period signals.It should be recalled that, in a realistic scenario, the pericenter does change mainly because of the zonal harmonics of the Newtonian part of the multipolar expansion of the gravitational potential of the central body.As far as the gravitomagnetic octupolar field is concerned, the only orbital elements that vary, on average, are the inclination and the node; they experience both secular and long-period effects.
The case b) was applied to a hypothetical scenario around Jupiter by first adopting the same pericenter height of the current Juno spacecraft and varying the apocenter height in such a way that the eccentricity ranged from e = 0.92 to the Juno's present value e = 0.98.The resulting amplitude of the long-period pN signature of the semimajor axis due to the Jovian oblateness is comprised within 500 m yr −1 = 0.01 mm s −1 (e = 0.92) and 1100 m yr −1 = 0.03 mm s −1 (e = 0.98).Although the following improper comparison is potentially misleading, it may be interesting to cautiously noting that the present-day accuracy level in measuring the range-rate shift of Juno is just at the ≃ 0.01 mm s −1 level.The other nonvanishing pN oblateness-driven effects on the pericenter and the mean anomaly at epoch amount to about 2 − 12 mas yr −1 , while the gravitomagnetic rates of the inclination and the node are at the ≃ 0.1−2 mas yr −1 level.Then, much less elliptical orbits with the same pericenter of Juno were considered, although they are at present almost impossible to achieve practically.In this case, the pN gravitoelectric rates can be as large as ≃ 1 − 3 arcsec yr −1 , while the spin octupole effects can reach the level of a few hundred mas yr −1 .Given the same orbital configuration, it turns out that the aforementioned effects around Saturn would be about an order of magnitude smaller.
Another potentially viable scenario for our results is represented by the highly elliptical orbits of the S stars moving around the supermassive black hole in Sgr A * at the Galactic Center.Indeed, it is likely surrounded by an extended matter distribution which can be made either of nonbaryonic dark matter or by the remnants of tidally disrupted stars, pulsars, etc.Such a halo is not necessarily spherically symmetric, and, in principle, the orbital effects calculated here may be useful to get more information on it.

Figure 1 .
Figure1.Plot of the amplitudes, in m yr −1 and 10 −8 yr −1 , respectively, of Equations (58)-(59), calculated for the polar orbital configuration of Equations (53)-(54), as functions of the apocenter height h apo for a fixed value of the pericenter height h peri = 4200 km.The dashed vertical line corresponds to h apo = 3.2 × 10 6 km which was the originally planned apocenter height of Juno before the problems encountered by its propulsion system.
Figures 1 to 2 deal with the pN gravitoelectric effects of Equations (58)-(63), while Figure3depicts the gravitomagnetic precessions of Equations (64)-(69).The conditions of Equations (53)-(54) and of Equation (73) were adopted in the calculation in order to obtain Figures2 to 3.According to Figure1, the amplitude of the J 2 /c 2 long-period signature of a ranges from 500 m yr −1 = 0.01 mm s −1 to 1100 m yr −1 = 0.03 mm s −1 , a seemingly unexpected feature due to the impact of e in Equation (58) for increasingly larger values of it.Such figures are not far from the two-way Ka-band range rate residuals ∆ ρ of Juno displayed inIess et al. (2018)

Figure 2 .
Figure 2. Plot of Equations (62)-(63), in mas yr −1 , calculated for the polar orbital configuration of Equations (53)-(54) with the condition of Equation (73), as functions of the apocenter height h apo for a fixed value of the pericenter height h peri = 4200 km.The dashed vertical line corresponds to h apo = 3.2×10 6 km which was the originally planned apocenter height of Juno before the problems encountered by its propulsion system.

Figure 3 .
Figure 3. Plot of Equations (66)-(67), in mas yr −1 , calculated for the polar orbital configuration of Equations (53)-(54) with the condition of Equation (73), as functions of the apocenter height h apo for a fixed value of the pericenter height h peri = 4200 km.The dashed vertical line corresponds to h apo = 3.2×10 6 km which was the originally planned apocenter height of Juno before the problems encountered by its propulsion system.

Figure 5 .
Figure 5. Plot of Equations (62)-(63), in mas yr −1 , calculated for the polar orbital configuration of Equations (53)-(54) with the condition of Equation (73), as functions of the apocenter height h apo for a fixed value of the pericenter height h peri = 4200 km.

Figure 6 .
Figure 6.Plot of Equations (66)-(67), in mas yr −1 , calculated for the polar orbital configuration of Equations (53)-(54) with the condition of Equation (73), as functions of the apocenter height h apo for a fixed value of the pericenter height h peri = 4200 km.
the extended mass component in Sgr A *

G
: Newtonian constant of gravitation c : speed of light in vacuum M : mass of the central body M ⊙ : mass of the Sun µ := GM : gravitational parameter of the central body S : angular momentum of the central body S : magnitude of the angular momentum of central body χ g : dimensionless spin parameter of a Kerr black hole: it is χ g ≤ 1. k = kx , ky , kz : spin axis of the central body with respect to some reference frame α : longitude of the spin axis of the central body in some reference frame δ : latitude of the spin axis of the central body in some reference frame kx = cos α cos δ : x component of the spin axis of the central body with respect to some reference frame ky = sin α cos δ : y component of the spin axis of the central body with respect to some reference frame kz = sin δ : z component of the spin axis of the central body with respect to some reference frame L. Iorio R e : equatorial radius of the central body R p : polar radius of the central body ε :the central body J 2 : dimensionless zonal harmonic coefficient of degree ℓ = 2 of the non spherically symmetric gravitational potential of the central body Q 2 : dimensional mass quadrupole moment of the non spherically symmetric gravitational potential of the central body B oct : gravitomagnetic spin octupole field in the empty space surrounding the rotating central body φ oct : gravitomagnetic spin octupole potential in the empty space surrounding the rotating central body A : perturbing acceleration experienced by the test particle r : position vector of the test particle with respect to the central body r : distance of the test particle from the central body r := r/r : radial unit vector ξ := k•r : cosine of the angle between the central body's spin axis and the position vector of the test particle v : velocity vector of the test particle v r := v•r : radial velocity of the test particle λ := k•v : projection of the velocity of the test particle onto the direction of the spin axis of the central body P ℓ (• • • ) : Legendre polynomial of degree ℓ a : semimajor axis of the test particle n b := µ/a 3 : Keplerian mean motion of the test particle P b := 2π/n b : orbital period of the test particle e : eccentricity of the test particle p := a 1 − e 2 : semilatus rectum of the orbit of the test particle I : inclination of the orbital plane of the test particle to the plane {x, y} of some reference frame Ω : longitude of the ascending node of the test particle ω : argument of pericenter of the test particle η : mean anomaly at epoch f : true anomaly of the test particle u := ω + f argument of latitude of the test particle l := {cos Ω, sin Ω, 0} : unit vector directed along the line of the nodes toward the ascending node m := {− cos I sin Ω, cos I cos Ω, sin I} : unit vector directed transversely to the line of the nodes in the orbital plane ĥ := {sin I sin Ω, − sin I cos Ω, cos I} : normal unit vector such that l× m = ĥ τ = ĥ×r : transverse unit vector A R := A•r : radial component of the perturbing acceleration A A T := A•τ : transverse component of the perturbing acceleration A A N := A• ĥ : normal component of the perturbing acceleration A B. Coefficients T j of the orbital rates of changes