The post-Newtonian gravitomagnetic spin-octupole moment of an oblate rotating body and its effects on an orbiting test particle; are they measurable in the Solar System?
TThe post-Newtonian gravitomagnetic spin-octupole moment of an oblaterotating body and its e ff ects on an orbiting test particle; are they measurablein the Solar system? Lorenzo Iorio Ministero dell’Istruzione, dell’Universit`a e della Ricerca (M.I.U.R.)-IstruzionePermanent address for correspondence: Viale Unit`a di Italia 68, 70125, Bari (BA), Italy [email protected]
Received ; accepted a r X i v : . [ g r- q c ] J u l Abstract
We analytically work out the orbital e ff ects induced by the post-Newtonian grav-itomagnetic spin-octupole moment of an extended spheroidal rotating body endowedwith angular momentum S and quadrupole mass moment J . Our results, propor-tional to GS J c − , hold for an arbitrary orientation of the body’s symmetry axis ˆ S anda generic orbital configuration of the test particle. Such e ff ects may be measurable,in principle, with a dedicated spacecraft-based mission to Jupiter. For a moderatelyeccentric and fast path, the gravitomagnetic precessions of the node and the pericen-tre of a dedicated orbiter could be as large as 400 milliarcseconds per year or even1 , − ,
000 milliarcseconds per year depending on the orientation of its orbital planein space. Numerical simulations of the Earth-probe range-rate signal confirm such ex-pectations since its magnitude reaches the (cid:39) . − . (cid:39) .
003 millimetre per secondafter 1 ,
000 seconds. Other general relativistic e ff ects might be measurable, includingalso those proportional to GM J c − , never put to the test so far. Most of the compet-ing Newtonian signals due to the classical multipoles of the planet’s gravity field havequite di ff erent temporal signatures with respect to the post-Newtonian ones, making,thus, potentially easier disentangling them.keywords gravitation − celestial mechanics − space vehicles − planets and satellites:individual: Jupiter
1. Introduction
The Einstein’s General Theory of Relativity (see, e.g., Iorio (2015a) and references therein)is currently the best description of the gravitational interaction at our disposal. It has successfullypassed all the experimental and observational checks with which it has been put to the test sofar (Will 2014) at di ff erent scales ranging from the Earth’s surrounding (Everitt et al. 2011) andour Solar System (Nordtvedt 2001) to extragalactic realms (Collett et al. 2018), including alsocompact stellar corpses (Kramer 2018; Archibald et al. 2018) and the main sequence stars orbitingthe supermassive black hole in our Galaxy (Gravity Collaboration et al. 2018), not to mentionthe recent direct discovery of the gravitational waves with Earth-based laser interferometers(Cervantes-Cota, Galindo-Uribarri & Smoot 2016). None the less, the still unexplained issues ofthe dark matter in galaxies and clusters of galaxies along with the observed accelerated expansionof the Universe may pose challenges to it (Debono & Smoot 2016; Vishwakarma 2016).Given its nature of fundamental pillar of our knowledge of the natural world, it is of theutmost importance to always submit under empirical scrutiny new parts of the theoretical structureof general relativity even where violations are, perhaps, least expected, as in the weak-field and 3 –slow-motion regime. Ginzburg (1959) wrote: “[. . . ] the history of physics has seen no end of casesin which the certain has turned out to be false. A theory so fundamental to modern science mustbe rigorously verified if it is to be applied with complete confidence to the further developmentof cosmology and other areas of physics.”. To this aim, in this paper we will show that it may bepossible, at least in principle, to bring an aspect of the post-Newtonian approximation (Poisson& Will 2014) which has never been tested so far into the detectability domain within the SolarSystem.In the first post-Newtonian approximation, the metric tensor g σν , σ, ν = , , , N gravitationally interacting rotating bodies of arbitrary shape and composition is parameterizedin terms of the so-called gravitoelectric potential φ , which is a generalization of the Newtonianpotential U , denoted sometimes by w , and the gravitomagnetic vector potential w (Brumberg &Kopeikin 1989; Damour, So ff el & Xu 1991; So ff el et al. 2003). The latter one, on which wewill focus, is generated by matter current densities proportional to the o ff -diagonal components T j , j = , , T σν , σ, ν = , , , ff el & Xu 1994; Meichsner & So ff el 2015) A gm = v c × B gm . (1)In the empty space outside the spinning body, its gravitomagnetic field B gm can be convenientlyexpressed in terms of a gravitomagnetic potential function φ gm as (Panhans & So ff el 2014,Eq. (30)) B gm = − ∇ × w = − ∇ φ gm . (2)By assuming a uniformly rotating homogeneous oblate spheroid at rest, φ gm can be expanded interms of its spin-multipole moments as (Panhans & So ff el 2014, Eqs. (31)-(32)) φ gm = − GSr ∞ (cid:88) i = ( − i (2 i +
3) (2 i + (cid:18) R e ε r (cid:19) i P i + ( ξ ) = − GSr (cid:34) ξ − (cid:18) R e ε r (cid:19) P ( ξ ) + . . . (cid:35) . (3)According to Panhans & So ff el (2014, Eq. (27)), the relation connecting the body’s ellipticity ε with the Newtonian even zonal harmonics is J i = − i (2 i +
1) (2 i + ε i , (4)so that, for i =
1, one has J = − ε . (5) 4 –The term with i = ff ects arising from Equation (1) evaluatedfor the spin-octupole moment in Equation (3), corresponding to i =
1. Then, we will show that,in principle, they could be detectable with a dedicated spacecraft-based mission to Jupiter. To thepresent author’s best knowledge, should his proposal be eventually successful, it would be the firsttime that a general relativistic higher spin multipole moment would be measured.The paper is organized as follows. In Section 2, we analytically work out the rates of change,averaged over one orbital period, of the Keplerian orbital elements of a test particle a ff ectedby the post-Newtonian acceleration imparted by the gravitomagnetic spin-octupole moment ofits primary, assumed uniformly rotating, homogeneous and spheroidal in shape. We treat itperturbatively by using the standard decomposition in radial, transverse and normal componentsand the Gauss equations for the variation of the osculating Keplerian orbital elements. We restricta priori neither to any peculiar spatial orientation of ˆ S nor to particular orbital configurations ofthe orbiter. We subsequently confirm the resulting analytical results by numerically integratingthe equations of motion. The perspectives for measuring such e ff ects around Jupiter, which isthe fastest spinning and most oblate major body of the Solar System, are treated in Section 3.While Juno (Section 3.1), currently orbiting the gaseous giant along a 53-day, highly eccentricorbit, is unsuitable because its expected post-Newtonian e ff ects are too small, a hypothetical newJovian probe, provisionally dubbed, with a touch of irony, IORIO (In-Orbit Relativity Iuppiter Observatory, or IOvis Relativity In-orbit Observatory), moving along a much faster, moderatelyeccentric orbit could, in principle, be successfully used (Section 3.2). Numerical simulationsof the Earth-spacecraft range-rate measurements, which are the actual observable quantities insuch an astronomical scenario, preliminarily confirm such expectations. We investigate also othergeneral relativistic features of motion impacting the probe’s range-rate. In Section 3.2.1, weassess the consequences of the mismodeling in the Newtonian potential coe ffi cients of Jupiter,while Section 3.2.2 is devoted to the impact of the uncertainty in the Jovian spin axis orientation.For each of such sources of systematic errors, we quantitatively evaluate the level of improvementwith respect to their current accuracies required to bring their range-rate signatures at least tothe same level of the various post-Newtonian signatures of interest. Furthermore, we look alsoat the temporal patterns of the competing Newtonian signals with respect to the relativistic ones.We summarize our findings and o ff er our conclusions in Section 4. For the benefit of the reader,Appendix A contains a list of symbols and definitions of the quantities used throughout the text,while tables and figures are grouped in Appendix B. Finally, it is appropriate to note that thepresent work should be regarded just as a concept study preliminarily investigating to a certain Iupp˘ıt˘er is one of the forms of the Latin noun of the god Jupiter. I ˘ovis means “of Jupiter” in Latin. 5 –level of detail a scenario which may be potentially able to measure the investigated e ff ect, not as aformal mission proposal.
2. The long-term orbital precessions
For the sake of simplicity, let us, first, assume a coordinate system whose fundamental { x , y } plane coincides with the body’s equator, so that its symmetry axis ˆ S is aligned with the reference z axis. The long-term rates of change of the osculating Keplerian orbital elements of the testparticle, obtained by averaging over one orbital revolution the right-hand-sides of the standardGauss equations (Kopeikin, Efroimsky & Kaplan 2011; Poisson & Will 2014) evaluated onto theKeplerian ellipse as reference trajectory, turn out to be˙ a gm = , (6)˙ e gm = eGS R J cos I sin I sin 2 ω c a (cid:0) − e (cid:1) / , (7)˙ I gm = − e GS R J cos I sin I sin 2 ω c a (cid:0) − e (cid:1) / , (8)˙ Ω gm = GS R J c a (cid:0) − e (cid:1) / (cid:104) − (cid:16) + e (cid:17) (3 + I ) + e (1 + I ) cos 2 ω (cid:105) , (9)˙ ω gm = − GS R J cos I c a (cid:0) − e (cid:1) / (cid:110) − e − (cid:16) + e (cid:17) cos 2 I + (cid:104) − − e + (cid:16) + e (cid:17) cos 2 I (cid:105) cos 2 ω (cid:111) . (10)For the sake of simplicity, here and in the following we omit the brackets (cid:104) . . . (cid:105) denoting theaverage over one orbital period.Let us, now, remove the limitation on the orientation of the primary’s spin axis allowing it tobe arbitrarily directed in space. The resulting long-term rates of change of the Keplerian orbitalelements are ˙ a gm = , (11)˙ e gm = eGJ R S a c (cid:0) − e (cid:1) / E (cid:16) I , Ω , ω ; ˆ S (cid:17) , (12) 6 –˙ I gm = GJ R S a c (cid:0) − e (cid:1) / I (cid:16) I , Ω , ω ; ˆ S (cid:17) , (13)˙ Ω gm = GJ R S a c (cid:0) − e (cid:1) / N (cid:16) I , Ω , ω ; ˆ S (cid:17) , (14)˙ ω gm = GJ R S c a (cid:0) − e (cid:1) / P (cid:16) I , Ω , ω ; ˆ S (cid:17) . (15)with E = (cid:16) ˆ S · ˆ k (cid:17) (cid:16) ˆ S · ˆ P (cid:17) (cid:16) ˆ S · ˆ Q (cid:17) , (16) I = e cos 3 I sin 2 ω (cid:16) ˆ S y cos Ω − ˆ S x sin Ω (cid:17) (cid:104) − S z + (cid:16) − ˆ S x + ˆ S y (cid:17) cos 2 Ω − S x ˆ S y sin 2 Ω (cid:105) −− I (cid:16) + e + e cos 2 ω (cid:17) (cid:16) ˆ S · ˆ l (cid:17) (cid:104) − + S z + (cid:16) ˆ S x − ˆ S y (cid:17) cos 2 Ω + S x ˆ S y sin 2 Ω (cid:105) −− e ˆ S z sin 3 I sin 2 ω (cid:104) − + S z + (cid:16) ˆ S x − ˆ S y (cid:17) cos 2 Ω + S x ˆ S y sin 2 Ω (cid:105) ++ e ˆ S z sin I sin 2 ω (cid:104) − S z + (cid:16) ˆ S x − ˆ S y (cid:17) cos 2 Ω +
10 ˆ S x ˆ S y sin 2 Ω (cid:105) −− e cos I sin 2 ω (cid:16) − ˆ S y cos Ω + ˆ S x sin Ω (cid:17) (cid:104) − S z + (cid:16) ˆ S x − ˆ S y (cid:17) cos 2 Ω +
10 ˆ S x ˆ S y sin 2 Ω (cid:105) −−
10 ˆ S z (cid:16) + e + e cos 2 ω (cid:17) sin 2 I (cid:104) − S x ˆ S y cos 2 Ω + (cid:16) ˆ S x − ˆ S y (cid:17) sin 2 Ω (cid:105) ++ (cid:16) ˆ S · ˆ l (cid:17) (cid:110) e cos 2 ω (cid:104) − − S z + (cid:16) ˆ S x − ˆ S y (cid:17) cos 2 Ω +
10 ˆ S x ˆ S y sin 2 Ω (cid:105) ++ (cid:16) + e (cid:17) (cid:104) − − S z + (cid:16) ˆ S x − ˆ S y (cid:17) cos 2 Ω +
10 ˆ S x ˆ S y sin 2 Ω (cid:105)(cid:111) , (17) N = − I cot I (cid:16) − − e + e cos 2 ω (cid:17) (cid:16) ˆ S y cos Ω − ˆ S x sin Ω (cid:17) ++ I cot I (cid:16) ˆ S y cos Ω − ˆ S x sin Ω (cid:17) (cid:104) S z (cid:16) + e − e cos 2 ω (cid:17) sin I + e (cid:16) ˆ S · ˆ l (cid:17) sin 2 ω (cid:105) + + e csc I sin 2 ω (cid:16) ˆ S · ˆ l (cid:17) (cid:20) S z sin I + (cid:16) ˆ S · ˆ l (cid:17) (cid:21) ++ S z cos ω (cid:104)(cid:16) + e (cid:17) ˆ S x cos Ω + (cid:16) + e (cid:17) ˆ S z sin I + (cid:16) + e (cid:17) ˆ S y (cid:16) ˆ S x sin 2 Ω + ˆ S y sin Ω (cid:17)(cid:105) ++ S z sin ω (cid:104)(cid:16) + e (cid:17) ˆ S x cos Ω + (cid:16) + e (cid:17) ˆ S z sin I + (cid:16) + e (cid:17) ˆ S y (cid:16) ˆ S x sin 2 Ω + ˆ S y sin Ω (cid:17)(cid:105) −− I (cid:110) S z sin I − cos I (cid:16) − − e + e cos 2 ω (cid:17) (cid:16) ˆ S y cos Ω − ˆ S x sin Ω (cid:17) ++ e (cid:104) ˆ S z sin I (12 − ω ) + (cid:16) ˆ S · ˆ l (cid:17) sin 2 ω (cid:105)(cid:111) ++ I (cid:16) ˆ S y cos Ω − ˆ S x sin Ω (cid:17) (cid:110) e ˆ S z (cid:16) ˆ S · ˆ l (cid:17) sin I sin 2 ω ++ cos ω (cid:104)(cid:16) + e (cid:17) ˆ S x cos Ω + (cid:16) + e (cid:17) ˆ S z sin I + (cid:16) + e (cid:17) ˆ S y (cid:16) ˆ S x sin 2 Ω + ˆ S y sin Ω (cid:17)(cid:105) ++ sin ω (cid:104)(cid:16) + e (cid:17) ˆ S x cos Ω + (cid:16) + e (cid:17) ˆ S z sin I + (cid:16) + e (cid:17) ˆ S y (cid:16) ˆ S x sin 2 Ω + ˆ S y sin Ω (cid:17)(cid:105)(cid:111) , (18) P = − e ˆ S z (cid:16) ˆ S · ˆ l (cid:17) cos I sin 2 ω (cid:16) ˆ S y cos Ω − ˆ S x sin Ω (cid:17) − (cid:16) + e (cid:17) (cid:16) ˆ S · ˆ l (cid:17) (cid:16) ˆ S · ˆ m (cid:17) (cid:16) ˆ S · ˆ k (cid:17) sin 2 ω −− I cot I (cid:16) ˆ S y cos Ω − ˆ S x sin Ω (cid:17) (cid:104) S z (cid:16) + e − e cos 2 ω (cid:17) sin I + e (cid:16) ˆ S · ˆ l (cid:17) sin 2 ω (cid:105) −− e cot I sin 2 ω (cid:16) ˆ S · ˆ l (cid:17) (cid:18) S z sin I + (cid:16) ˆ S · ˆ l (cid:17) (cid:19) − I cot I cos ω (cid:16) ˆ S y cos Ω − ˆ S x sin Ω (cid:17) ×× (cid:104)(cid:16) + e (cid:17) ˆ S x cos Ω + (cid:16) + e (cid:17) ˆ S z sin I + (cid:16) + e (cid:17) ˆ S y sin Ω (cid:16) S x cos Ω + ˆ S y sin Ω (cid:17)(cid:105) −− S z cos I sin ω (cid:104)(cid:16) + e (cid:17) ˆ S x cos Ω++ (cid:16) + e (cid:17) ˆ S z sin I + (cid:16) + e (cid:17) ˆ S y (cid:16) ˆ S x sin 2 Ω + ˆ S y sin Ω (cid:17)(cid:105) − − I cot I sin ω (cid:16) ˆ S y cos Ω − ˆ S x sin Ω (cid:17) (cid:104)(cid:16) + e (cid:17) ˆ S x cos Ω + (cid:16) + e (cid:17) ˆ S z sin I ++ (cid:16) + e (cid:17) ˆ S y (cid:16) ˆ S x sin 2 Ω + ˆ S y sin Ω (cid:17)(cid:105) −− (cid:16) ˆ S · ˆ k (cid:17) cos ω (cid:110)(cid:104)(cid:16) + e (cid:17) ˆ S x + (cid:16) + e (cid:17) ˆ S y cos I (cid:105) cos Ω + (cid:16) + e (cid:17) ˆ S z sin I ++ S y cos Ω (cid:110)(cid:16) + e (cid:17) ˆ S z cos I sin I + ˆ S x (cid:104) + e − (cid:16) + e (cid:17) cos I (cid:105) sin Ω (cid:111) ++ sin Ω (cid:104) − (cid:16) + e (cid:17) ˆ S x ˆ S z sin 2 I + (cid:16)(cid:16) + e (cid:17) ˆ S y + (cid:16) + e (cid:17) ˆ S x cos I (cid:17) sin Ω (cid:105)(cid:111) −− (cid:16) ˆ S · ˆ k (cid:17) sin ω (cid:110)(cid:104)(cid:16) + e (cid:17) ˆ S x + (cid:16) + e (cid:17) ˆ S y cos I (cid:105) cos Ω + (cid:16) + e (cid:17) ˆ S z sin I ++ S y cos Ω (cid:110)(cid:16) + e (cid:17) ˆ S z cos I sin I + ˆ S x (cid:104) + e − (cid:16) + e (cid:17) cos I (cid:105) sin Ω (cid:111) ++ sin Ω (cid:104) − (cid:16) + e (cid:17) ˆ S x ˆ S z sin 2 I + (cid:16)(cid:16) + e (cid:17) ˆ S y + (cid:16) + e (cid:17) ˆ S x cos I (cid:17) sin Ω (cid:105)(cid:111) ++ I (cid:110) − cos I (cid:16) − − e + e cos 2 ω (cid:17) (cid:16) ˆ S y cos Ω − ˆ S x sin Ω (cid:17) ++ (cid:16) + e (cid:17) sin I (cid:16) − ˆ S y cos Ω + ˆ S x sin Ω (cid:17) ++ I (cid:104) ˆ S z (cid:16) + e − e cos 2 ω (cid:17) sin I + e (cid:16) ˆ S · ˆ l (cid:17) sin 2 ω (cid:105)(cid:111) −− I (cid:26) − cos I (cid:16) − − e + e cos 2 ω (cid:17) (cid:16) ˆ S y cos Ω − ˆ S x sin Ω (cid:17) ++ ˆ S z sin I cos ω (cid:104)(cid:16) + e (cid:17) ˆ S x cos Ω + (cid:16) + e (cid:17) ˆ S z sin I ++ (cid:16) + e (cid:17) ˆ S y sin Ω (cid:16) S x cos Ω + ˆ S y sin Ω (cid:17)(cid:105)(cid:111) . (19)It can be noted that Equations (11) to (15), along with Equations (16) to (19), reduce toEquations (6) to (10) for ˆ S x = ˆ S y = , ˆ S z = i = ff ects proportional to GS J c − (Iorio 2015b) arise also fromthe interplay between the well known Newtonian quadrupolar acceleration due to J and thepost-Newtonian Lense-Thirring acceleration proportional to GS c − . Their order of magnitude isthe same of the direct rates of change treated in the present Section. None the less, such indirect,mixed e ff ects are likely unmeasurable in actual data reductions since they cannot be expressedin terms of a dedicated, solve-for scaling parameter which could be explicitly estimated. It is sobecause, contrary to the direct e ff ects derived from Equation (1), they do not come from a distinctacceleration which can be suitably parameterized.
3. Perspectives of measuring the post-Newtonian gravitomagnetic orbital precessions dueto the spin-octupole moment of Jupiter3.1. Juno
The spacecraft Juno is currently orbiting Jupiter, whose relevant physical parameters arereported in Table 1, along a highly elliptical trajectory characterized by the orbital parameterslisted in Table 2. The huge oblateness of the gaseous giant and the large eccentricity of theprobe may suggest, at first sight, to look at such a system as a unique opportunity, in principle,to put to the test for the first time the gravitomagnetic e ff ects due to the spin-octupole momentof an extended body. Unfortunately, the resulting orbital precessions of Juno turn out to be toosmall, as shown by Table 2 and Figure 2 displaying the simulated Earth-spacecraft range-ratesignatures at the perijove passages PJ03, PJ06. In fact, the directly observable quantity of Junoused to mapping the Jovian gravity field is the two-way Ka-band Doppler shift. The frequentmaneuvers required to keep the alignment of the transmitting antenna with the Earth tend todestroy the dynamical coherence of the orbit, not allowing to obtain steady time series of thespacecraft’s orbital elements. Thus, the analytical calculation based on them should be regardedjust as useful and easily understandable tools to perform a-priori sensitivity analyses. The sameconsiderations hold, in principle, also for any other spacecraft orbiting Jupiter and communicatingwith the Earth. The signatures in Figure 2 were obtained as follows. For each perijove passes,we numerically integrated the equations of motion of the Earth, Jupiter and Juno in Cartesianrectangular coordinates referred to the International Celestial Reference Frame (ICRF) with andwithout the disturbing post-Newtonian acceleration under investigation. More specifically, in oursimplified model the Earth is subjected to the Newtonian acceleration due to the Sun, while Jupiterfeels only the Newtonian acceleration of the Sun; the equations of motions of both the planets 10 –were integrated in a Solar System barycentric coordinate system. The equations of motion ofJuno were integrated in a Jovicentric coordinate system; they include the Newtonian accelerationsof Jupiter and the Sun and the post-Newtonian acceleration of Equation (1). For each perijovepasses, both the runs shared the same set of initial conditions which were retrieved from the WEBinterface HORIZONS maintained by JPL, NASA, for given initial epochs which, in the presentcase, are December 11, 2016, h: 13:00 (PJ03) and May 19, 2017, h: 02:00 (PJ06), respectively.After each run, a numerical time series of the Earth-probe range-rate ˙ ρ ( t ) was produced byprojecting the Juno’s velocity vector onto the Earth-Jupiter unit vector; ˙ ρ pert ( t ) includes also thee ff ect of the perturbing gravitomagnetic acceleration, while ˙ ρ N ( t ) is the purely classical one dueto only the Newtonian monopoles of the Sun and Jupiter. In order to single out the e ff ect of thepost-Newtonian acceleration of interest, the di ff erences of both the time series were computedobtaining the curves for ∆ ˙ ρ ( t ) = ˙ ρ pert ( t ) − ˙ ρ N ( t ) displayed in Figure 4. Our method , which will beused also in Section 3.2 for other Newtonian and post-Newtonian accelerations, was successfullytested by reproducing the Newtonian range-rate signatures due to the odd zonals J , J , J , J atPJ03, PJ06 displayed in Iess et al. (2018, Extended Data Fig. 3). However, Jupiter can still be considered as a viable scenario to try to measure its post-Newtonian gravitomagnetic spin-octupole e ff ects. Indeed, by keeping a hypothetical newspacecraft at about the same distance from it along a much faster orbit, it is possible to selectsuitable values for I , Ω , ω allowing for quite large precessions. Tables 3 to 4, which refer toa Jovian equatorial coordinate system, deal with two di ff erent orbital configurations yieldingnominal precessions for the node and the pericentre as large as (cid:39) − mas yr − , which areremarkably large values. More specifically, for a mildly eccentric orbit with r (cid:39) . R with I = ω =
90 deg, the gravitomagnetic node precession would be as large as ˙ Ω gm =
400 mas yr − ,while for I =
360 deg , ω =
270 deg and the same orbit radius as before one has even˙ Ω gm = − ,
600 mas yr − , ˙ ω gm = ,
000 mas yr − . Such an insight is confirmed by some numericalsimulations of the Earth-probe range-rate signature. Indeed, by adopting the ICRF and a Juno-likespatial orientation for the previously considered almost circular, fast jovicentric orbit of theproposed spacecraft, Figure 3 shows that the size of its relativistic signature would reach the (cid:39) .
03 mm s − level after just 1 d. It should be recalled that the Doppler measurement accuracyof Juno is (cid:39) .
003 mm s − after 1 ,
000 s. Figure 4 preliminarily investigates the sensitivity tothe individual orbital elements. It turns out that, while the gravitomagnetic range-rate signature In actual data reductions, the appropriate time and spatial coordinates transformations betweenthe Solar System Barycentric coordinate system and the suitably constructed planetocentric coor-dinate systems for Jupiter and the Earth (Brumberg & Kopeikin 1989) are fully modeled and imple-mented, among other things, according to the most recent IAU resolutions (Kopeikin, Efroimsky& Kaplan 2011). 11 –is rather insensitive to the eccentricity, at least for small values of it, the pericentre and thetrue anomaly, the size of the orbit and the orientation of its orbital plane in space have a majorimpact. Indeed, if, on the one hand, a su ffi ciently low orbit is mandatory to increase the signal ofinterest, on the other hand, certain values of the inclination and the node may push it up to the (cid:39) . − level for a = . R , e = . and thepotentially quite large ∆ v required to implement a successful orbit insertion. Another crucial issueis represented by the impact of other competing dynamical e ff ects, which would act as source ofsystematic errors potentially biasing the recovery of the relativistic e ff ect of interest. In this regard,we remark that the proposed scenario would benefit of the notable improvement of our knowledgeabout both the Jupiter’s spin pole position and the Newtonian part of its gravity field arising fromthe analysis of the full data record of Juno, which is scheduled to deorbit into the planet on July2021. Su ffi ce it to say that, until now, just 2 (PJ03 and PJ06) out of a total of expected 25 perijovepassages dedicated to gravity field determination have been fully analyzed (Iess et al. 2018), whilethe results from PJ08, PJ10, PJ11 should be publicly released soon (Durante et al. 2018). Table 1displays, among other things, the best estimates and the associated realistic uncertainties for theeven and odd zonal coe ffi cients J (cid:96) , (cid:96) = , , , . . . ,
12, and the tesseral and sectorial multipoles C , , S , , C , , S , . The RA and Dec. of ˆ S are currently known to an accuracy of (cid:39)
100 mas,as shown in Table 1, while their rates of change are accurate to (cid:39)
50 mas yr − (Durante et al.2018). As far as the first even zonal harmonic of the Jovian gravity field, from Fig. 2 of theposter presented by Durante et al. (2018) it seems that its most recent accuracy is (cid:39) × − ,corresponding to a relative accuracy of (cid:39) × − . Moreover, it is not unrealistic to assume thatthe measurement accuracy σ ˙ ρ may be better than that of Juno, whose measurements are mostlytaken only at its perijove passages, because of the comparatively much larger number N of datapoints due to the higher orbital frequency and lower eccentricity. Indeed, σ ˙ ρ scales as 1 / √ N. Inthe following, we want to quantitatively assess such issues in connection with the full potentialof the proposed mission concept as a tool to measure even more general relativistic featuresof motion ranging from the standard Schwarzschild-like one proportional to
GMc − , to the sofar never tested gravitoelectric e ff ect proportional to GM J c − (So ff el et al. 1988; So ff el 1989;Brumberg 1991), including also the gravitomagnetic Lense-Thirring frame-dragging (Lense &Thirring 1918) proportional to GS c − . The intense volcanic activity of Io, which is the dominant source of plasma at Jupiter, poursmaterial into Io’s atmosphere which is lost to the Jovian magnetosphere near Io. Such a material isthen ionized and trapped by the magnetic field forming a torus of plasma around Jupiter. The torusconsists of di ff erent regions extending from (cid:39) R to (cid:39) R (Hinton, Bagenal & Bodisch 2017). See https: // / missions / juno / on the Internet. 12 – Figures 5 to 19, obtained with the same computational method previously outlined inSection 3.1, depict the numerically simulated Newtonian (blue dashed curves) and post-Newtonian (red continuous curves) range-rate time series for a given orbital configuration of theprobe which, as it will be shown below, should make the detection of the relativistic signals morefavorable. In order to better visualize the temporal patterns of the various e ff ects, the classicalsignatures were produced by using fictitious values C ∗ of the Newtonian gravity field coe ffi cientsable to make their magnitudes roughly equal to those of the post-Newtonian time series of interest.If such figures C ∗ for the Jovian multipoles are smaller than their present-day uncertainties listedin Table 1, they can be interpreted as a measure of how much they should still be improved withrespect to their current levels of accuracy in order to make the size of the Newtonian signatures atleast equal to the relativistic ones. If, instead, C ∗ are larger than their present mismodeling, theycan be viewed as a measure of the relative accuracy with which a given relativistic signal wouldbe impacted right now. See Table 5 for a complete list of such improvement factors for all theNewtonian multipoles considered here in connection with the various relativistic e ff ects. It turnsout that the largest improvements-of the order of (cid:39) − ,
000 for J -wouldbe required to bring the Newtonian signals to the level of the post-Newtonian gravitomagnetice ff ect proportional to GS J c − . A much smaller improvement would be required to make thesize of the classical multipole signatures comparable with the post-Newtonian gravitoelectricand gravitomagnetic e ff ects proportional to GM J c − , GS c − . As far as the Schwarzschild-typesignature is concerned, the current level of accuracy in almost all the Jovian multipoles, with theexception of J , J , J , S , , S , , would yield a bias at the (cid:39) −
10 per cent level. A veryimportant feature of all the curves displayed in Figures 5 to 19 is that the relativistic ones exhibitneatly di ff erent temporal patterns with respect to the Newtonian ones, making, thus, easier todetect them. It would not be so for di ff erent orbital geometries of the probe. The position of the Jovian spin axis, determined by its right ascension α and declination δ with respect to the ICRF (Durante et al. 2018), enters the Newtonian accelerations induced bythe gravity field multipoles in a nonlinear way. It can be easily realized, e.g., by inspecting theanalytical expressions of the long-term precessions of the Keplerian orbital elements due to someeven and odd zonal harmonics calculated by Iorio (2011); Renzetti (2013b, 2014) for an arbitraryorientation of ˆ S . Thus, the uncertainties σ α , σ δ have an impact on the general relativistic e ff ects ofinterest through the Newtonian multipolar signatures. The latest determinations of α, δ along withthe associated realistic uncertainties, of the order of σ α , σ δ (cid:39) . σ α , σ δ . They were obtained as 13 –described in the previous Section by using the nominal values of the even zonals and taking thedi ff erences between the time series computed with δ max = δ + σ δ , δ min = δ − σ δ (green dashedcurves) and α max = α + σ α , α min = α − σ α (orange continuous curves), respectively. It turns outthat the largest residual signals are due to the uncertainty in the declination. The largest one occursfor J , with an amplitude which can reach ∆ ˙ ρ J σ δ (cid:46)
60 mm s − . The signatures of the odd zonals arecompletely negligible. It can be shown that an improvement of σ δ by a factor of 100 with respectto the current value of Table 1 would bring the size of the Newtonian J -induced range-rate timeseries to the same level of the post-Newtonian one proportional to GS J c − . Such an improvementseems 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, asalready noticed in the previous Section, the temporal pattern of the classical J signal is di ff erentfrom the relativistic ones.
4. Summary and overview
We analytically worked out the long-term rates of change of the Keplerian orbital elementsof a test particle orbiting an extended spheroidal rotating body induced by its general relativisticgravitomagnetic spin-octupole moment to the first post-Newtonian order. We neither assumed apreferred orientation for the body’s symmetry axis nor adopted a particular orbital configurationfor the test particle. Thus, our results have a general validity, being applicable, in principle, towhatsoever astronomical and astrophysical scenario of interest. We successfully checked themnumerically by integrating the equations of motion.We applied them to Jupiter, which is the fastest spinning and most oblate major body ofthe Solar System, and some existing or hypothetical spacecraft orbiting it. While for Juno thegravitomagnetic precessions, of the order of (cid:39) . − , are too small to be detectable, for aputative new probe orbiting the gaseous giant in, say, 0 .
12 d along a moderately eccentric orbitwith r (cid:39) . R , the spin-quadrupole e ff ects may be as large as 400 − ,
000 mas yr − dependingon the orbital geometry, within the measurability threshold with the current tracking technologies.We confirmed such expectations by numerically calculating in the ICRF the signature inducedby the general relativistic spin-octupole moment of Jupiter on the Earth-satellite range-ratemeasurements which, in a real data analysis, would represent the actual observable quantity.Indeed, by conservatively assuming a range-rate experimental precision of (cid:39) .
003 mm s − over1,000 s, as for Juno, it turns out that the post-Newtonian e ff ect of interest could overcome sucha level after just 1 full orbital revolution reaching, say, 0 . − . − after 1 d dependingmainly on the orientation of the orbital plane in space. Furthermore, in order to explore the fullpotential of the proposed mission concept, we looked also at the post-Newtonian gravitoelectrice ff ects proportional to GM J c − , which have never been put to the test so far, and at the standardLense-Thirring and Schwarzschild signatures, proportional to GS c − , GMc − , respectively.The experimental uncertainties in the values of both the Newtonian coe ffi cients of the 14 –multipolar expansion of the Jovian gravity field and in the orientation of the spin axis of Jupiterwould induce mismodeled range-rate signatures in the Doppler measurements of the spacecraftacting as sources of competing systematic biases for the post-Newtonian signals of interest. Atpresent, just 5 of the planned 25 perijove passes dedicated to mapping the planet’s gravity fieldof the ongoing Juno mission, scheduled to end in July 2021, have been analyzed so far. Thus, ifand when the proposed mission will be finally implemented, it will benefit of the analysis of theentire Juno data record yielding a much more accurate determination of the Jovian gravity fieldcoe ffi cients and pole position than now.For a given orbital configuration of the spacecraft, we numerically simulated its mismodeledNewtonian range-rate signatures due to the gravity field coe ffi cients and the spin axis position ofJupiter currently determined by Juno, and the predicted post-Newtonian signals. We determinedthe level of improvement of the Jovian multipoles and pole position with respect to their present-day accuracies still required to bring the competing classical e ff ects to the level of the variousrelativistic ones. It turned out that the most demanding requirements pertain the measurabilityof the GS J c − signature, implying improvements by a factor of (cid:39) −
500 for most of theJovian gravity coe ffi cients considered, with a peak of 1 ,
000 for J . The other relatively smallpost-Newtonian e ff ects, proportional to GM J c − , GS c − , require less demanding improvementsby a factor of just (cid:39) −
50 or less. The Schwarzschild signature would be measurable right nowat a (cid:39) −
10% level, apart from the impact of J , J , J , S , , S , . As far as the Jupiter’sspin axis is concerned, an improvement by a factor of 100 would be required for its declination δ to make the size of the J -induced signature to the same level of the post-Newtonian GS J c − one. The uncertainty in the declination α is less important. The range-rate signals due to the oddzonals are a ff ected by the errors in the pole position at a negligible level. A fundamental outcomeof our analysis consists of the fact that the temporal patterns of the relativistic signatures turnedout to be quite di ff erent from the classical ones, making, thus, easier, in principle, to separate thepost-Newtonian from the Newtonian e ff ects.Finally, we remark once more that the present work is not a formal mission proposal; instead,it should be regarded just as a sort of expanded mission concept which need further, dedicatedstudies concerning, e.g., the practical feasibility of the suggested scenario taking into accountseveral important technological and engineering issues. Acknowledgements
I am grateful to D. Durante for useful information. 15 –
Appendix A Notations and definitions
Here, some basic notations and definitions used throughout the text are presented (Brumberg1991; Bertotti, Farinella & Vokrouhlick´y 2003; Kopeikin, Efroimsky & Kaplan 2011; Poisson &Will 2014). G : Newtonian constant of gravitation c : speed of light in vacuum g σν : spacetime metric tensor φ, w : gravitoelectric potential U : Newtonian gravitational potential w : gravitomagnetic potential T σν : energy-momentum tensor of the source M : mass of the primary µ (cid:17) GM : gravitational parameter of the primary S : magnitude of the angular momentum of the primary ˆ S = (cid:110) ˆ S x , ˆ S y , ˆ S z (cid:111) : spin axis of the primary in some coordinate system α : right ascension (RA) of the primary’s spin axis with respect to the Earth’s mean equator atepoch J2000.0 δ : declination (DEC) of the primary’s spin axis with respect to the Earth’s mean equator atepoch J2000.0ˆ S x = cos δ cos α : x component of the primary’s spin axis with respect to the Earth’s meanequator at epoch J2000.0ˆ S y = cos δ sin α : y component of the primary’s spin axis with respect to the Earth’s mean equatorat epoch J2000.0ˆ S z = sin δ : z component of the primary’s spin axis with respect to the Earth’s mean equator atepoch J2000.0 R e : equatorial radius of the primary R p : polar radius radius of the primary 16 – ε (cid:17) (cid:113) − (cid:16) R p R e (cid:17) : ellipticity of the oblate primary J (cid:96) , (cid:96) = , , , . . . : Newtonian zonal multipole mass moments of the primary’s gravity field C , , S , , C , , S , : tesseral ( m =
1) and sectorial ( m =
2) multipole mass moments of degree (cid:96) = B gm : post-Newtonian gravitomagnetic field in the empty space surrounding the rotating primary φ gm : gravitomagnetic potential function in the empty space surrounding the rotating primary A gm : post-Newtonian gravitomagnetic acceleration experienced by the test particle r : instantaneous position vector of the test particle with respect to the primary r min : pericentre distance of the test particle with respect to the primary r min : apocentre distance of the test particle with respect to the primary r : instantaneous distance of the test particle from the primary ˆ r (cid:17) r / r : versor of the position vector of the test particle ξ (cid:17) ˆ S · ˆ r : cosine of the angle between the primary’s spin axis and the position vector of the testparticle P i + ( ξ ) : Legendre polynomial of degree 2 i + v : velocity vector of the test particle f : true anomaly of the test particle’s orbit a : semimajor axis of the test particle’s orbit n b (cid:17) (cid:112) µ/ a : Keplerian mean motion of the test particle’s orbit P b (cid:17) π / n b : orbital period of the test particle’s orbit e : eccentricity of the test particle’s orbit I : inclination of the orbital plane of the test particle’s orbit to the reference { x , y } plane of somecoordinate system Ω : longitude of the ascending node of the test particle’s orbit referred to the reference { x , y } plane of some coordinate system ω : argument of pericentre of the test particle’s orbit referred to the reference { x , y } plane ofsome coordinate system 17 – ˆ l (cid:17) { cos Ω , sin Ω , } : unit vector directed along the line of the nodes toward the ascending node ˆ m (cid:17) {− cos I sin Ω , cos I cos Ω , sin I } : unit vector directed transversely to the line of the nodesin the orbital plane ˆ k (cid:17) { sin I sin Ω , − sin I cos Ω , cos I } : unit vector perpendicular to the orbital plane directedalong the orbital angular momentum ˆ P (cid:17) ˆ l cos ω + ˆ m sin ω : unit vector in the orbital plane directed along the line of apsides towardsthe pericentre ˆ Q (cid:17) − ˆ l sin ω + ˆ m cos ω : unit vector in the orbital plane directed transversely to the line ofapsides Appendix B Tables and figures
18 –Table 1: Relevant physical parameters of Jupiter. Most of the reported values come from So ff elet al. (2003); Petit, Luzum & et al. (2010); Iess et al. (2018); Durante et al. (2018) and referencestherein. In particular, the values and the uncertainties of α, δ determining the Jovian pole positionat the epoch J2017.0 come from Durante et al. (2018), while the multipoles of the gravity potentialare retrieved from Iess et al. (2018, Tab. 1).Parameter Units Numerical value µ m s − . × S kg m s − . × α deg 268 . ± . δ deg 64 . ± . R km 71 , J (cid:16) × − (cid:17) , . ± . J (cid:16) × − (cid:17) − . ± . J (cid:16) × − (cid:17) − . ± . J (cid:16) × − (cid:17) − . ± . J (cid:16) × − (cid:17) . ± . J (cid:16) × − (cid:17) . ± . J (cid:16) × − (cid:17) − . ± . J (cid:16) × − (cid:17) − . ± . J (cid:16) × − (cid:17) . ± . J (cid:16) × − (cid:17) . ± . J (cid:16) × − (cid:17) . ± . C , (cid:16) × − (cid:17) − . ± . S , (cid:16) × − (cid:17) − . ± . C , (cid:16) × − (cid:17) . ± . S , (cid:16) × − (cid:17) . ± .
011 19 –Table 2: Relevant orbital parameters of the spacecraft Juno currently orbiting Jupiter. Here, R is meant as the equatorial radius R e of Jupiter. The source for the orbital elements of Juno, re-ferred to the Jovian equator, is the freely consultable database JPL HORIZONS on the Internet athttps: // ssd.jpl.nasa.gov / ?horizons from which they were retrieved by choosing the time of writingthis paper as input epoch. The values of the post-Newtonian gravitomagnetic precessions of Junodue to the spin-octupole moment of Jupiter, calculated by means of Equations (7) to (10), are listedas well. Parameter Units Numerical value a R . e − . r min R . r max R . I deg 98 . Ω deg 270 . ω deg 163 . P b d 52 . e gm yr − × − ˙ I gm mas yr − . Ω gm mas yr − . ω gm mas yr − . a R . e − . r min R . r max R . P b d 0 . I deg 90 ω deg 90˙ e gm yr − . I gm mas yr − . Ω gm mas yr − ω gm mas yr − . a R . e − . r min R . r max R . P b d 0 . I deg 360 ω deg 270˙ e gm yr − . I gm mas yr − . Ω gm mas yr − − , ω gm mas yr − , . κ (if >
1) required to each of the Jovian multipolecoe ffi cients with respect to their current accuracy levels (see Iess et al. (2018, Tab. 1) and Table 1)to make the size of the corresponding Newtonian range-rate signatures equal to the magnitudeof the general relativistic ones; see Figures 5 to 19. If, in a given row, κ <
1, the current levelof accuracy in the multipole of that row would allow right now to measure the correspondingrelativistic e ff ects with the relative accuracies as good as κ themselves. For example, in the secondrow corresponding to J , there are two figures smaller than 1; it means that the present-day accuracyin J would yield a mismodeled Newtonian signal impacting, say, the Schwarzschild-like one at1 . J should be improved by a factor of 12 . ff ect proportional to GS J c − . From Table 1, it should be noted that the values of J , J , C , , S , , C , , S , are statistically compatible with zero.Multipole GS J c − GM J c − GS c − GMc − J
70 5 2 . . J . . . . J
33 3 . . . J
10 0 . .
58 0 . J
100 5 2 . . J
50 4 . . . J
33 4 2 . . J
100 10 3 . . J ,
000 33 33 0 . J
500 28 . . J
500 50 28 . . C ,
50 4 2 . . S ,
500 20 12 . C ,
333 28 . . . S ,
500 33 20 1 22 – t ( yr ) ( × - ) pN spin - octupole shift of the eccentricity e t ( yr ) m a s pN spin - octupole shift of the inclination I - - - - - - - t ( yr ) m a s pN spin - octupole shift of the node Ω t ( yr ) m a s pN spin - octupole shift of the pericentre ω Fig. 1.— Numerically computed time series of the post-Newtonian shifts experienced by theeccentricity e , inclination I , node Ω and pericentre ω of a fictitious test particle induced bythe gravitomagnetic spin-octupole moment of a putative central body characterized by the samephysical properties of Jupiter (see Table 1). They were obtained by numerically integrating theequations of motion of the orbiter in Cartesian rectangular coordinates referred to the Earth’smean equator at the epoch J2000.0 with and without the acceleration of Equation (1) calcu-lated with Equation (3) for i =
1. Both runs shared the same set of arbitrary initial conditions a = . R , e = . , I =
45 deg , Ω =
30 deg , ω =
50 deg , f =
45 deg. For eachKeplerian orbital element, its time series calculated from the purely Newtonian run was sub-tracted from that obtained from the post-Newtonian integration in order to obtain the signaturesdisplayed here. The resulting rates of change, in yr − and mas yr − , turn out to agree withthe analytically computed ones in Equations (12) to (15) with Equations (16) to (19) which are˙ e gm = . × − yr − , ˙ I gm = .
05 mas yr − , ˙ Ω gm = − .
89 mas yr − , ˙ ω gm = .
74 mas yr − . 23 – - - - - - - - - - - Date ( UTC ) Δ ρ ( mm / s ) Earth - Juno post - Newtonian SJ c - range - rate at PJ03 - - - - - - - - - - Date ( UTC ) Δ ρ ( mm / s ) Earth - Juno post - Newtonian SJ c - range - rate at PJ06 Fig. 2.— Simulated range-rate signatures ∆ ˙ ρ , in mm s − , of Juno due to the post-Newtonian grav-itomagnetic spin-octupole moment of Jupiter at the first two perijove passages PJ03 (December11, 2016) and PJ06 (May 19, 2017) dedicated to gravity science. They were obtained by numer-ically integrating the equations of motion of the Earth, Jupiter and Juno in Cartesian rectangularcoordinates referred to the ICRF with and without the general relativistic acceleration of Equa-tion (1), calculated for Equation (3) with i =
1, starting from the same set of initial conditionsretrieved from the Web interface HORIZONS maintained by JPL. Then, for each perijove passage,the range-rate time series computed from the purely Newtonian run was subtracted from that ob-tained from the post-Newtonian integration in order to yield the curves displayed here. Cfr. withthe two-way Ka-band range-rate residuals of Juno for the same perijove passes displayed in theExtended Data Figure 1 of Iess et al. (2018) whose ranges of variation amount to (cid:39) .
050 mm s − ,with a root-mean-square value of (cid:39) .
015 mm s − . 24 – - - t ( s ) Δ ρ ( mm / s ) Earth - probe post - Newtonian SJ c - range - rate Fig. 3.— Simulated range-rate signature ∆ ˙ ρ , in mm s − , of a hypothetical Jovian orbiter charac-terized by the ICRF-related orbital configuration a = . R , e = . , I = .
63 deg , Ω = .
85 deg , ω = .
43 deg , f = .
32 deg induced by the post-Newtonian gravitomagneticspin-octupole moment of Jupiter after 1 d. It was obtained as described in the caption of Figure 2. 25 – a = Ra = Ra = Ra = R - - - t ( s ) Δ ρ ( mm / s ) Earth - probe GSJ c - range - rate e = e = e = e = - - - t ( s ) Δ ρ ( mm / s ) Earth - probe GSJ c - range - rate I = I =
50 deg I =
95 deg I =
140 deg - - - t ( s ) Δ ρ ( mm / s ) Earth - probe GSJ c - range - rate Ω = Ω =
50 deg Ω =
95 deg Ω =
140 deg - - t ( s ) Δ ρ ( mm / s ) Earth - probe GSJ c - range - rate ω = ω =
50 deg ω =
95 deg ω =
140 deg - - - t ( s ) Δ ρ ( mm / s ) Earth - probe GSJ c - range - rate f = f =
50 deg f =
95 deg f =
140 deg - - - t ( s ) Δ ρ ( mm / s ) Earth - probe GSJ c - range - rate Fig. 4.— Simulated range-rate signatures ∆ ˙ ρ , in mm s − , of a hypothetical Jovian orbiter inducedby the post-Newtonian gravitomagnetic spin-octupole moment of Jupiter after 1 d. They wereobtained as described in the caption of Figure 2 by allowing the orbital elements of the spacecraftto vary within certain ranges of values with respect to the reference orbital configuration used toproduce Figure 3. 26 – J ⨯ GSJ c - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate J ⨯ GMJ c - - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate J ⨯ GSc - - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate J ⨯ GMc - - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate Fig. 5.— Simulated range-rate signatures ∆ ˙ ρ , in mm s − , of a hypothetical Jovian orbiter in-duced by the nominal post-Newtonian accelerations considered in the text and by the Newto-nian first even zonal harmonic J of Jupiter after 1 d. In each panel, a fictitious value J ∗ isused in the Newtonian signature just for illustrative and comparative purposes. Indeed, it issuitably tuned from time to time in order to bring the associated classical signature to the levelof the nominal post-Newtonian e ff ect of interest, for which the actual value of J is, instead,used, so to inspect the mutual (de)correlations of their temporal patterns more easily. Upper-left corner: post-Newtonian gravitomagnetic spin-octupole moment (cid:16) GS J c − ; J ∗ = . × − (cid:17) .Upper-right corner: post-Newtonian gravitoelectric moment (cid:16) GM J c − ; J ∗ = . × − (cid:17) . Lower-left corner: Lense-Thirring e ff ect (cid:16) GS c − ; J ∗ = . × − (cid:17) . Lower-right corner: Schwarzschild (cid:16) GMc − ; J ∗ = . × − (cid:17) . The present-day actual uncertainty in the Jovian first even zonal is σ J = . × − (Iess et al. 2018, Tab. 1). The adopted orbital configuration for the probe is a = . R , e = . , I =
50 deg , Ω =
140 deg , ω = .
43 deg , f = .
32 deg 27 – J ⨯ GSJ c - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate J ⨯ GMJ c - - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate J ⨯ GSc - - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate J ⨯ GMc - - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate Fig. 6.— Simulated range-rate signatures ∆ ˙ ρ , in mm s − , of a hypothetical Jovian orbiter inducedby the nominal post-Newtonian accelerations considered in the text and by the Newtonian firstodd zonal harmonic J of Jupiter after 1 d. In each panel, a fictitious value J ∗ is used in theNewtonian signature just for illustrative and comparative purposes. Indeed, it is suitably tunedfrom time to time in order to bring the associated classical signature to the level of the nomi-nal post-Newtonian e ff ect of interest, so to inspect the mutual (de)correlations of their tempo-ral patterns more easily. Upper-left corner: post-Newtonian gravitomagnetic spin-octupole mo-ment (cid:16) GS J c − ; J ∗ = . × − (cid:17) . Upper-right corner: post-Newtonian gravitoelectric moment (cid:16) GM J c − ; J ∗ = . × − (cid:17) . Lower-left corner: Lense-Thirring e ff ect (cid:16) GS c − ; J ∗ = . × − (cid:17) .Lower-right corner: Schwarzschild (cid:16) GMc − ; J ∗ = . × − (cid:17) . The present-day actual uncertaintyin the Jovian first odd zonal is σ J = . × − (Iess et al. 2018, Tab. 1). The adopted orbitalconfiguration for the probe is a = . R , e = . , I =
50 deg , Ω =
140 deg , ω = .
43 deg , f = .
32 deg 28 – J ⨯ GSJ c - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate J ⨯ GMJ c - - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate J ⨯ GSc - - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate J ⨯ GMc - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate Fig. 7.— Simulated range-rate signatures ∆ ˙ ρ , in mm s − , of a hypothetical Jovian orbiter inducedby the nominal post-Newtonian accelerations considered in the text and by the Newtonian secondeven zonal harmonic J of Jupiter after 1 d. In each panel, a fictitious value J ∗ is used in theNewtonian signature just for illustrative and comparative purposes. Indeed, it is suitably tunedfrom time to time in order to bring the associated classical signature to the level of the nomi-nal post-Newtonian e ff ect of interest, so to inspect the mutual (de)correlations of their tempo-ral patterns more easily. Upper-left corner: post-Newtonian gravitomagnetic spin-octupole mo-ment (cid:16) GS J c − ; J ∗ = . × − (cid:17) . Upper-right corner: post-Newtonian gravitoelectric moment (cid:16) GM J c − ; J ∗ = . × − (cid:17) . Lower-left corner: Lense-Thirring e ff ect (cid:16) GS c − ; J ∗ = . × − (cid:17) .Lower-right corner: Schwarzschild (cid:16) GMc − ; J ∗ = . × − (cid:17) . The present-day actual uncertaintyin the Jovian second even zonal is σ J = × − (Iess et al. 2018, Tab. 1). The adopted orbitalconfiguration for the probe is a = . R , e = . , I =
50 deg , Ω =
140 deg , ω = .
43 deg , f = .
32 deg 29 – J ⨯ GSJ c - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate J ⨯ GMJ c - - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate J ⨯ GSc - - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate J ⨯ GMc - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate Fig. 8.— Simulated range-rate signatures ∆ ˙ ρ , in mm s − , of a hypothetical Jovian orbiter inducedby the nominal post-Newtonian accelerations considered in the text and by the Newtonian sec-ond odd zonal harmonic J of Jupiter after 1 d. In each panel, a fictitious value J ∗ is used in theNewtonian signature just for illustrative and comparative purposes. Indeed, it is suitably tunedfrom time to time in order to bring the associated classical signature to the level of the nomi-nal post-Newtonian e ff ect of interest, so to inspect the mutual (de)correlations of their tempo-ral patterns more easily. Upper-left corner: post-Newtonian gravitomagnetic spin-octupole mo-ment (cid:16) GS J c − ; J ∗ = . × − (cid:17) . Upper-right corner: post-Newtonian gravitoelectric moment (cid:16) GM J c − ; J ∗ = . × − (cid:17) . Lower-left corner: Lense-Thirring e ff ect (cid:16) GS c − ; J ∗ = . × − (cid:17) .Lower-right corner: Schwarzschild (cid:16) GMc − ; J ∗ = . × − (cid:17) . The present-day actual uncertaintyin the Jovian second odd zonal is σ J = × − (Iess et al. 2018, Tab. 1). The adopted orbitalconfiguration for the probe is a = . R , e = . , I =
50 deg , Ω =
140 deg , ω = .
43 deg , f = .
32 deg 30 – J ⨯ GSJ c - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate J ⨯ GMJ c - - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate J ⨯ GSc - - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate J ⨯ GMc - - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate Fig. 9.— Simulated range-rate signatures ∆ ˙ ρ , in mm s − , of a hypothetical Jovian orbiter inducedby the nominal post-Newtonian accelerations considered in the text and by the Newtonian thirdeven zonal harmonic J of Jupiter after 1 d. In each panel, a fictitious value J ∗ is used in theNewtonian signature just for illustrative and comparative purposes. Indeed, it is suitably tunedfrom time to time in order to bring the associated classical signature to the level of the nomi-nal post-Newtonian e ff ect of interest, so to inspect the mutual (de)correlations of their tempo-ral patterns more easily. Upper-left corner: post-Newtonian gravitomagnetic spin-octupole mo-ment (cid:16) GS J c − ; J ∗ = . × − (cid:17) . Upper-right corner: post-Newtonian gravitoelectric moment (cid:16) GM J c − ; J ∗ = . × − (cid:17) . Lower-left corner: Lense-Thirring e ff ect (cid:16) GS c − ; J ∗ = . × − (cid:17) .Lower-right corner: Schwarzschild (cid:16) GMc − ; J ∗ = . × − (cid:17) . The present-day actual uncertaintyin the Jovian third even zonal is σ J = × − (Iess et al. 2018, Tab. 1). The adopted orbitalconfiguration for the probe is a = . R , e = . , I =
50 deg , Ω =
140 deg , ω = .
43 deg , f = .
32 deg 31 – J ⨯ GSJ c - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate J ⨯ GMJ c - - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate J ⨯ GSc - - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate J ⨯ GMc - - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate Fig. 10.— Simulated range-rate signatures ∆ ˙ ρ , in mm s − , of a hypothetical Jovian orbiter in-duced by the nominal post-Newtonian accelerations considered in the text and by the Newtonianthird odd zonal harmonic J of Jupiter after 1 d. In each panel, a fictitious value J ∗ is used inthe Newtonian signature just for illustrative and comparative purposes. Indeed, it is suitably tunedfrom time to time in order to bring the associated classical signature to the level of the nomi-nal post-Newtonian e ff ect of interest, so to inspect the mutual (de)correlations of their tempo-ral patterns more easily. Upper-left corner: post-Newtonian gravitomagnetic spin-octupole mo-ment (cid:16) GS J c − ; J ∗ = . × − (cid:17) . Upper-right corner: post-Newtonian gravitoelectric moment (cid:16) GM J c − ; J ∗ = . × − (cid:17) . Lower-left corner: Lense-Thirring e ff ect (cid:16) GS c − ; J ∗ = . × − (cid:17) .Lower-right corner: Schwarzschild (cid:16) GMc − ; J ∗ = . × − (cid:17) . The present-day actual uncer-tainty in the Jovian third odd zonal is σ J = . × − (Iess et al. 2018, Tab. 1). The adoptedorbital configuration for the probe is a = . R , e = . , I =
50 deg , Ω =
140 deg , ω = .
43 deg , f = .
32 deg 32 – J ⨯ GSJ c - - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate J ⨯ GMJ c - - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate J ⨯ GSc - - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate J ⨯ GMc - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate Fig. 11.— Simulated range-rate signatures ∆ ˙ ρ , in mm s − , of a hypothetical Jovian orbiter in-duced by the nominal post-Newtonian accelerations considered in the text and by the Newto-nian fourth even zonal harmonic J of Jupiter after 1 d. In each panel, a fictitious value J ∗ is used in the Newtonian signature just for illustrative and comparative purposes. Indeed, itis suitably tuned from time to time in order to bring the associated classical signature to thelevel of the nominal post-Newtonian e ff ect of interest, so to inspect the mutual (de)correlationsof their temporal patterns more easily. Upper-left corner: post-Newtonian gravitomagneticspin-octupole moment (cid:16) GS J c − ; J ∗ = . × − (cid:17) . Upper-right corner: post-Newtonian grav-itoelectric moment (cid:16) GM J c − ; J ∗ = . × − (cid:17) . Lower-left corner: Lense-Thirring e ff ect (cid:16) GS c − ; J ∗ = . × − (cid:17) . Lower-right corner: Schwarzschild (cid:16) GMc − ; J ∗ = . × − (cid:17) . Thepresent-day actual uncertainty in the Jovian fourth even zonal is σ J = . × − (Iess et al. 2018,Tab. 1). The adopted orbital configuration for the probe is a = . R , e = . , I =
50 deg , Ω =
140 deg , ω = .
43 deg , f = .
32 deg 33 – J ⨯ GSJ c - - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate J ⨯ GMJ c - - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate J ⨯ GSc - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate J ⨯ GMc - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate Fig. 12.— Simulated range-rate signatures ∆ ˙ ρ , in mm s − , of a hypothetical Jovian orbiter in-duced by the nominal post-Newtonian accelerations considered in the text and by the Newtonianfourth odd zonal harmonic J of Jupiter after 1 d. In each panel, a fictitious value J ∗ is usedin the Newtonian signature just for illustrative and comparative purposes. Indeed, it is suitablytuned from time to time in order to bring the associated classical signature to the level of thenominal post-Newtonian e ff ect of interest, so to inspect the mutual (de)correlations of their tem-poral patterns more easily. Upper-left corner: post-Newtonian gravitomagnetic spin-octupole mo-ment (cid:16) GS J c − ; J ∗ = . × − (cid:17) . Upper-right corner: post-Newtonian gravitoelectric moment (cid:16) GM J c − ; J ∗ = . × − (cid:17) . Lower-left corner: Lense-Thirring e ff ect (cid:16) GS c − ; J ∗ = . × − (cid:17) .Lower-right corner: Schwarzschild (cid:16) GMc − ; J ∗ = . × − (cid:17) . The present-day actual uncertaintyin the Jovian fourth odd zonal is σ J = . × − (Iess et al. 2018, Tab. 1). The adopted orbitalconfiguration for the probe is a = . R , e = . , I =
50 deg , Ω =
140 deg , ω = .
43 deg , f = .
32 deg 34 – J ⨯ GSJ c - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate J ⨯ GMJ c - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate J ⨯ GSc - - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate J ⨯ GMc - - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate Fig. 13.— Simulated range-rate signatures ∆ ˙ ρ , in mm s − , of a hypothetical Jovian orbiter in-duced by the nominal post-Newtonian accelerations considered in the text and by the Newto-nian fifth even zonal harmonic J of Jupiter after 1 d. In each panel, a fictitious value J ∗ is used in the Newtonian signature just for illustrative and comparative purposes. Indeed, itis suitably tuned from time to time in order to bring the associated classical signature to thelevel of the nominal post-Newtonian e ff ect of interest, so to inspect the mutual (de)correlationsof their temporal patterns more easily. Upper-left corner: post-Newtonian gravitomagneticspin-octupole moment (cid:16) GS J c − ; J ∗ = . × − (cid:17) . Upper-right corner: post-Newtonian grav-itoelectric moment (cid:16) GM J c − ; J ∗ = . × − (cid:17) . Lower-left corner: Lense-Thirring e ff ect (cid:16) GS c − ; J ∗ = . × − (cid:17) . Lower-right corner: Schwarzschild (cid:16) GMc − ; J ∗ = × − (cid:17) . Thepresent-day actual uncertainty in the Jovian fifth even zonal is σ J = . × − (Iess et al. 2018,Tab. 1). The adopted orbital configuration for the probe is a = . R , e = . , I =
50 deg , Ω =
140 deg , ω = .
43 deg , f = .
32 deg 35 – J ⨯ GSJ c - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate J ⨯ GMJ c - - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate J ⨯ GSc - - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate J ⨯ GMc - - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate Fig. 14.— Simulated range-rate signatures ∆ ˙ ρ , in mm s − , of a hypothetical Jovian orbiter in-duced by the nominal post-Newtonian accelerations considered in the text and by the Newto-nian fifth odd zonal harmonic J of Jupiter after 1 d. In each panel, a fictitious value J ∗ is used in the Newtonian signature just for illustrative and comparative purposes. Indeed, itis suitably tuned from time to time in order to bring the associated classical signature to thelevel of the nominal post-Newtonian e ff ect of interest, so to inspect the mutual (de)correlationsof their temporal patterns more easily. Upper-left corner: post-Newtonian gravitomagneticspin-octupole moment (cid:16) GS J c − ; J ∗ = . × − (cid:17) . Upper-right corner: post-Newtonian grav-itoelectric moment (cid:16) GM J c − ; J ∗ = . × − (cid:17) . Lower-left corner: Lense-Thirring e ff ect (cid:16) GS c − ; J ∗ = . × − (cid:17) . Lower-right corner: Schwarzschild (cid:16) GMc − ; J ∗ = . × − (cid:17) . Thepresent-day actual uncertainty in the Jovian fifth odd zonal is σ J = . × − (Iess et al. 2018,Tab. 1). The adopted orbital configuration for the probe is a = . R , e = . , I =
50 deg , Ω =
140 deg , ω = .
43 deg , f = .
32 deg 36 – J ⨯ GSJ c - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate J ⨯ GMJ c - - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate J ⨯ GSc - - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate J ⨯ GMc - - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate Fig. 15.— Simulated range-rate signatures ∆ ˙ ρ , in mm s − , of a hypothetical Jovian orbiter in-duced by the nominal post-Newtonian accelerations considered in the text and by the Newto-nian fifth odd zonal harmonic J of Jupiter after 1 d. In each panel, a fictitious value J ∗ is used in the Newtonian signature just for illustrative and comparative purposes. Indeed, itis suitably tuned from time to time in order to bring the associated classical signature to thelevel of the nominal post-Newtonian e ff ect of interest, so to inspect the mutual (de)correlationsof their temporal patterns more easily. Upper-left corner: post-Newtonian gravitomagneticspin-octupole moment (cid:16) GS J c − ; J ∗ = . × − (cid:17) . Upper-right corner: post-Newtonian grav-itoelectric moment (cid:16) GM J c − ; J ∗ = . × − (cid:17) . Lower-left corner: Lense-Thirring e ff ect (cid:16) GS c − ; J ∗ = . × − (cid:17) . Lower-right corner: Schwarzschild (cid:16) GMc − ; J ∗ = . × − (cid:17) .The present-day actual uncertainty in the Jovian fifth odd zonal is σ J = . × − (Iess et al.2018, Tab. 1). The adopted orbital configuration for the probe is a = . R , e = . , I =
50 deg , Ω =
140 deg , ω = .
43 deg , f = .
32 deg 37 – C ⨯ GSJ c - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate C ⨯ GMJ c - - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate C ⨯ GSc - - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate C ⨯ GMc - - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate Fig. 16.— Simulated range-rate signatures ∆ ˙ ρ , in mm s − , of a hypothetical Jovian orbiter in-duced by the nominal post-Newtonian accelerations considered in the text and by the Newto-nian tesseral coe ffi cient C , of Jupiter after 1 d. In each panel, a fictitious value C ∗ , is usedin the Newtonian signature just for illustrative and comparative purposes. Indeed, it is suit-ably tuned from time to time in order to bring the associated classical signature to the levelof the nominal post-Newtonian e ff ect of interest, so to inspect the mutual (de)correlations oftheir temporal patterns more easily. Upper-left corner: post-Newtonian gravitomagnetic spin-octupole moment (cid:16) GS J c − ; C ∗ , = . × − (cid:17) . Upper-right corner: post-Newtonian gravi-toelectric moment (cid:16) GM J c − ; C ∗ , = . × − (cid:17) . Lower-left corner: Lense-Thirring e ff ect (cid:16) GS c − ; C ∗ , = . × − (cid:17) . Lower-right corner: Schwarzschild (cid:16) GMc − ; C ∗ , = . × − (cid:17) .The present-day actual uncertainty in the Jovian tessreral coe ffi cient is σ C , = . × − (Iess et al.2018, Tab. 1). The adopted orbital configuration for the probe is a = . R , e = . , I =
50 deg , Ω =
140 deg , ω = .
43 deg , f = .
32 deg 38 – S ⨯ GSJ c - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate S ⨯ GMJ c - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate S ⨯ GSc - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate S ⨯ GMc - - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate Fig. 17.— Simulated range-rate signatures ∆ ˙ ρ , in mm s − , of a hypothetical Jovian orbiter in-duced by the nominal post-Newtonian accelerations considered in the text and by the Newto-nian tesseral coe ffi cient S , of Jupiter after 1 d. In each panel, a fictitious value S ∗ , is usedin the Newtonian signature just for illustrative and comparative purposes. Indeed, it is suit-ably tuned from time to time in order to bring the associated classical signature to the levelof the nominal post-Newtonian e ff ect of interest, so to inspect the mutual (de)correlations oftheir temporal patterns more easily. Upper-left corner: post-Newtonian gravitomagnetic spin-octupole moment (cid:16) GS J c − ; S ∗ , = . × − (cid:17) . Upper-right corner: post-Newtonian grav-itoelectric moment (cid:16) GM J c − ; S ∗ , = . × − (cid:17) . Lower-left corner: Lense-Thirring e ff ect (cid:16) GS c − ; S ∗ , = . × − (cid:17) . Lower-right corner: Schwarzschild (cid:16) GMc − ; S ∗ , = . × − (cid:17) .The present-day actual uncertainty in the Jovian tessreral coe ffi cient is σ S , = . × − (Iess et al.2018, Tab. 1). The adopted orbital configuration for the probe is a = . R , e = . , I =
50 deg , Ω =
140 deg , ω = .
43 deg , f = .
32 deg 39 – C ⨯ GSJ c - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate C ⨯ GMJ c - - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate C ⨯ GSc - - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate C ⨯ GMc - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate Fig. 18.— Simulated range-rate signatures ∆ ˙ ρ , in mm s − , of a hypothetical Jovian orbiter in-duced by the nominal post-Newtonian accelerations considered in the text and by the Newto-nian sectorial coe ffi cient C , of Jupiter after 1 d. In each panel, a fictitious value C ∗ , is usedin the Newtonian signature just for illustrative and comparative purposes. Indeed, it is suit-ably tuned from time to time in order to bring the associated classical signature to the levelof the nominal post-Newtonian e ff ect of interest, so to inspect the mutual (de)correlations oftheir temporal patterns more easily. Upper-left corner: post-Newtonian gravitomagnetic spin-octupole moment (cid:16) GS J c − ; C ∗ , = . × − (cid:17) . Upper-right corner: post-Newtonian grav-itoelectric moment (cid:16) GM J c − ; C ∗ , = . × − (cid:17) . Lower-left corner: Lense-Thirring e ff ect (cid:16) GS c − ; C ∗ , = . × − (cid:17) . Lower-right corner: Schwarzschild (cid:16) GMc − ; C ∗ , = . × − (cid:17) . Thepresent-day actual uncertainty in the Jovian sectorial coe ffi cient is σ C , = . × − (Iess et al.2018, Tab. 1). The adopted orbital configuration for the probe is a = . R , e = . , I =
50 deg , Ω =
140 deg , ω = .
43 deg , f = .
32 deg 40 – S ⨯ GSJ c - - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate S ⨯ GMJ c - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate S ⨯ GSc - - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate S ⨯ GMc - - - - t ( s ) Δ ρ ( mm / s ) Earth - probe range - rate Fig. 19.— Simulated range-rate signatures ∆ ˙ ρ , in mm s − , of a hypothetical Jovian orbiter in-duced by the nominal post-Newtonian accelerations considered in the text and by the Newto-nian sectorial coe ffi cient S , of Jupiter after 1 d. In each panel, a fictitious value S ∗ , is usedin the Newtonian signature just for illustrative and comparative purposes. Indeed, it is suit-ably tuned from time to time in order to bring the associated classical signature to the levelof the nominal post-Newtonian e ff ect of interest, so to inspect the mutual (de)correlations oftheir temporal patterns more easily. Upper-left corner: post-Newtonian gravitomagnetic spin-octupole moment (cid:16) GS J c − ; S ∗ , = . × − (cid:17) . Upper-right corner: post-Newtonian grav-itoelectric moment (cid:16) GM J c − ; S ∗ , = . × − (cid:17) . Lower-left corner: Lense-Thirring e ff ect (cid:16) GS c − ; S ∗ , = . × − (cid:17) . Lower-right corner: Schwarzschild (cid:16) GMc − ; S ∗ , = . × − (cid:17) . Thepresent-day actual uncertainty in the Jovian sectorial coe ffi cient is σ S , = . × − (Iess et al.2018, Tab. 1). The adopted orbital configuration for the probe is a = . R , e = . , I =
50 deg , Ω =
140 deg , ω = .
43 deg , f = .
32 deg 41 – σ δ = σ α = - - - t ( s ) Δ ρ ( mm / s ) Earth - probe J range - rate σ δ = σ α = - - t ( s ) Δ ρ ( mm / s ) Earth - probe J range - rate σ δ = σ α = - - t ( s ) Δ ρ ( mm / s ) Earth - probe J range - rate σ δ = σ α = - - - t ( s ) Δ ρ ( mm / s ) Earth - probe J range - rate Fig. 20.— Numerically simulated impact of the present-day errors σ α = .
13 arcsec , σ δ = .
16 arcsec (Durante et al. 2018) in the position of the spin axis of Jupiter on the range-rate sig-natures ∆ ˙ ρ , in mm s − , of a hypothetical Jovian orbiter induced by the Newtonian accelerationsdue to the first four even zonals J , J , J , J after 1 d. It turns out that the uncertainties in theJupiter’s spin axis a ff ect the odd zonals signatures in a completely negligible way. The adoptedorbital configuration for the probe is a = . R , e = . , I =
50 deg , Ω =
140 deg , ω = .
43 deg , f = .
32 deg 42 –
Erratum:The post-Newtonian gravitomagnetic spin-octupole moment of an oblate rotatingbody and its e ff ects on an orbiting test particle; are they measurable in the Solar system? In the published version (Iorio 2019) of this paper, due to the unfortunate and misleadingdefinition J = − ε / J entering the analytically computed post-Newtonian spin-octupole orbital precessionsof Equations (6) to (15) di ff ers from the even zonal harmonic J usually determined instandard spacecraft-based geodetic and geophysical data reductions which is, indeed, positive.Moreover, also their magnitudes are di ff erent, as can be straightforwardly noted in thecase of Jupiter. Indeed, as per the IAU 2015 Resolution B3 on Recommended NominalConversion Constants for Selected Solar and Planetary Properties available on the Internetat https: // / administration / resolutions / general assemblies / , the nominal polar andequatorial radii of Jupiter amount to 66 ,
854 km and 71 ,
492 km, respectively, yielding − ε / = − . J = . . J = . e gm , ˙ I gm , ˙ Ω gm , ˙ ω gm quotedin Tables 2 to 4 and mentioned throughout the paper, and in producing Figs 1 to 4 and thepost-Newtonian spin-octupole curves in the upper-left panels of Figs 5 to 19 instead of − ε / e gm , ˙ I gm , ˙ Ω gm , ˙ ω gm of Tables 2 to 4, and inthe post-Newtonian gravitomagnetic spin-octupole signatures displayed in Figs 1 to 4 and in theupper-left panels of Figs 5 to 19. Moreover, the magnitudes of ˙ e gm , ˙ I gm , ˙ Ω gm , ˙ ω gm in Tables 2 to 4,and the amplitudes in Figs 1 to 4 and of both the Newtonian and post-Newtonian curves in theupper-left panels of Figs 5 to 19 are smaller than the correct ones by a factor of 1 . ff ect in Figs 1 to 4 and in the upper-left panelsof Figs 5 to 19 should be flipped, and their sizes, along with those of the Newtonian signatures inthe upper-left panels of Figs 5 to 19, rescaled by a factor of 1 . ff ect.The mutual (de)correlations of the post-Newtonian spin-octupole signatures with the classicalones in the upper-left panels of Figs 5 to 19 change accordingly. In the captions of Figs 5 to 19,the values of J ∗ (cid:96) , (cid:96) = , , . . .
12 and C ∗ , , C ∗ , , S ∗ , , S ∗ , associated with the post-Newtonianspin-octupole e ff ects should be rescaled by a factor of 1 . It comes from equation 27 of Panhans & So ff el (2014) for n =
1. It reproduces incorrectlyequation (56) of Klioner (2003) which, in fact, contains ( − n + instead of ( − n entering equation27 of Panhans & So ff el (2014). However, it is the post-Newtonian spin-octupole accelerationrelying upon φ gm of equation (32) of Panhans & So ff el (2014) that matters; it is independent of allsuch unnecessary definitions. In case of Figure 1, it is not relevant since its purpose was just confirming the analyticalcalculation of Equations (6) to (15) with a numerical integration of the equations of motion in thecase of a fictitious astronomical scenario: the numerical values actually adopted for the primary’sphysical properties are unimportant. 43 –from the left of Table 5 should be reduced by a factor of 1 . GS J c − with, say, GS ε c − throughout the paper to avoid further misunderstandings; inparticular, it would be better to replace J in the analytical precessions of Equations (6) to (15)with − ε /
5. The conclusions pertaining the other post-Newtonian e ff ects remain unchanged. 44 – REFERENCES