The perturbations of the hyperbolic motion of a test particle due to the general relativistic gravitoelectromagnetic Schwarzschild and LenseThirring components of the gravitational field of a rotating massive body are analytically worked out to the first postNewtonian level in terms of the osculating Keplerian orbital elements. To the Newtonian order the impact of the quadrupole mass moment of the source is calculated as well. The resulting analytical expressions are valid for a generic orientation in space of both the orbital plane of the probe and the spin axis of the primary and for arbitrary values of the eccentricity. They are applied to Oumuamua an interstellar asteroid which recently visited our solar system along an unbound heliocentric orbit and to the Near Earth Asteroid Rendezvous spacecraft during its flyby of the Earth. The calculational approach developed can be straightforwardly extended to any alternative models of gravity as well.
['General relativity', 'Celestial mechanics', 'Planetary probes', '641', '211', '1252', 'General Relativity and Quantum Cosmology', 'Astrophysics - Earth and Planetary Astrophysics', 'Physics - Space Physics']
PostKeplerian Perturbations of the Hyperbolic Motion in the Field of a Rotating Massive Object Analysis in Terms of Osculating and Nonosculating Contact Elements
{'Post-Keplerian perturbations of the hyperbolic motion in the field of a rotating massive object. Analysis in terms of osculating and nonosculating (contact) elements': "Lorenzo Iorio 1  \n1 Ministero dell' Istruzione e del Merito. Viale Unit'a di Italia 68, I-70125, Bari (BA), Italy", 'Abstract': "The perturbations of the hyperbolic motion of a test particle due to the general relativistic gravitoelectromagnetic Schwarzschild and Lense-Thirring components of the gravitational field of a rotating massive body are analytically worked out to the first post-Newtonian level in terms of the osculating Keplerian orbital elements. To the Newtonian order, the impact of the quadrupole mass moment of the source is calculated as well. The resulting analytical expressions are valid for a generic orientation in space of both the orbital plane of the probe and the spin axis of the primary, and for arbitrary values of the eccentricity. They are applied to ' Oumuamua, an interstellar asteroid which recently visited our solar system along an unbound heliocentric orbit, and to the Near Earth Asteroid Rendezvous (NEAR) spacecraft during its flyby of the Earth. The calculational approach developed can be straightforwardly extended to any alternative models of gravity as well.  \nKeywords: General relativity (641); Celestial mechanics (211); Planetary probes (1252)", '1. Introduction': "Let a localized gravitational source like, e.g., a planet, a natural satellite, a main sequence star or any astrophysical compact object endowed with mass M , equatorial radius R e, quadrupole mass moment J 2 and angular momentum J be considered. Let its external gravitational field be calculated in points far enough so that it is weak and the speeds of any moving test particles are small with respect to the speed of light in vacuum c . Then, in addition to the dominant Newtonian inverse-square mass monopole, also further post-Keplerian (pK) terms of both Newtonian and post-Newtonian (pN) origin come into play. The most relevant ones are the classical contribution of J 2 and, to the first post-newtonian (1pN) order, the so-called gravitoelectromagnetic Schwarzschild and Lense-Thirring (LT) components induced by M and J , respectively.  \nUntil now, their orbital e GLYPH<11> ects have been studied mainly in the case of bound, otherwise Keplerian elliptical trajectories (Brumberg 1991; So GLYPH<11> el 1989; Kopeikin et al. 2011; Gurfil and Seidelmann 2016; So GLYPH<11> el and Han 2019; O'Leary 2021; Iorio 2024), used as tools to perform tests of gravitational theories. The most famous case is represented by the then anomalous perihelion precession of Mercury of 42 : 98 arcseconds per century (arcsec cty GLYPH<0> 1 ) (Nobili and Will 1986) in the field of the Sun, known since the second half of the nineteenth century (Le Verrier 1859b,a), and its successive explanation by Einstein (Einstein 1915) in terms of his newborn General Theory of Relativity (GTR). For a historical overview, see, e.g., Roseveare (1982).  \nInstead, studies of pK perturbations of hyperbolic trajectories are comparatively much more rare, being mainly focussed on the e GLYPH<11> ects of the primary's oblateness for a particular orientation 1 of its spin axis ˆ J (Sauer 1963; Anderson and Giampieri 1999; Rappaport et al. 2001; Martinusi and Gurfil 2013; Kim and Park 2015). Other works investigated the hyperbolic motions of test particles and photons in the Schwarzschild spacetime at various levels of completeness (Morton 1921; Hagihara 1930; Leavitt 1939; Darwin 1959, 1961; Mielnik and Pleba'nski 1962; Davidson 1980; Hioe and Kuebel 2010; Chowdhuri et al. 2024). The case of the hyperbolic motion of a spinning particle in the Schwarzschild metric was treated by Bini and Geralico (2017), while Battista and Esposito (2022) dealt with geodesic motion in Euclidean Schwarzschild geometry. To the author's knowledge, the gravitomagnetic e GLYPH<11> ects of the rotation of the primary on hyperbolic trajectories have never been treated so far, apart from the study by Mummery and Balbus (2023) in the Kerr metric.  \nlorenzo.iorio@libero.it  \nFlybys of planets and natural satellites by artificial spacecraft traveling along patched hyperbolic conical sections are commonplace in current astrodynamics and planetary sciences (Flandro 1966; Anderson 1997; van Allen 2003; Anderson et al. 2007). Furthermore, they are often repeated several times within the same missions; su GLYPH<14> ce it to think about the grand tour of the Cassini probe in the Kronian system (Wolf and Smith 1995). Finally, also the Galactic Centre and the cluster of stars surrounding the supermassive black hole at Sgr A GLYPH<3> (Genzel et al. 2010) may be considered. Indeed, the star S111 is following a hyperbolic path (Trippe et al. 2008; Gillessen et al. 2009, 2017); other stars like that might be discovered in the future. Eventually, such kind of trajectories may represent, in principle, further opportunities to test gravitational theories in addition to the traditional bound ones.  \nHere, in order to make closer contact with observations in actually accessible astronomical scenarios, a perturbative approach is followed. It allows to analytically calculate the variations experienced by all the usual Keplerian orbital elements of a hyperbolic trajectory perturbed by the aforementioned pK components of the gravitational field of the primary. In this respect, the present work follows a similar strategy as that adopted in Sauer (1963); Anderson and Giampieri (1999); Rappaport et al. (2001); Kim and Park (2015). Nonetheless, the e GLYPH<11> ects of the Newtonian quadrupole mass moment J 2 and of the 1pN gravitomagnetic LT field are worked out in full generality, without any a priori simplifying assumptions about the orientations of both ˆ J and the orbit in space. Furthermore, all the formulas obtained are valid for any values of the eccentricity.  \nThe paper is organized as follows. In Section 2, the basics of the Keplerian hyperbolic motion is reviewed and the perturbative equations for the rates of change of the Keplerian orbital elements in the form of Lagrange are presented for such kind of unperturbed, reference trajectory. Furthermore, the way of calculating the disturbing function, to be used with the aforementioned equations, for the pK e GLYPH<11> ects considered is discussed as well. The 1pN gravitoelectric shifts induced solely by M are calculated in Section 3. The 1pN gravitomagnetic LT perturbations due to J is the subject of Section 4, while the impact of J 2 is worked out, to the Newtonian order, in Section 5; both e GLYPH<11> ects are calculated without any a priori assumptions on both ˆ J and the orientation of the orbital plane. Certain subtleties concerning the proper use of the Lagrange planetary equations in presence of velocitydependent disturbing functions are dealt with in Section 6. The results of the previous Sections are used for numerical calculation in Section 7 for two astronomical scenarios in our solar system: the interstellar asteroid ' Oumuamua and the Sun in Section 7.1, and the spacecraft Near Earth Asteroid Rendezvous (NEAR) approaching the Earth in Section 7.2. Section 8 summarizes the findings and o GLYPH<11> ers conclusions.", '2. Calculational overview': 'In the Keplerian hyperbolic motion (Rappaport et al. 2001; Roy 2005; Gurfil and Seidelmann 2016), a is the semimajor axis, e is the eccentricity, I is the inclination, GLYPH<10> is the longitude of the ascending node, ! is the argument of pericentre, and GLYPH<17> is the mean anomaly at epoch. The semimajor axis measures the distance between the vertex, namely the point Q of closest approach to the primary, and the centre O of the hyperbola; it is a < 0. For the eccentricity, which is at any time the constant ratio of the distance of the test particle at the point P( t ) on the hyperbola to the focus F where the primary resides to the distance of P( t ) itself to the directrix, it always holds e > 1; the larger it is, the straightest the hyperbola, while its asymptotes tend to get closer for e & 1. The inclination is the tilt of the orbital plane to the reference f x ; y g plane of the body-fixed reference frame adopted. The longitude of the ascending node is the angle, counted in the reference plane, from the reference x direction to the point N on the line of nodes crossed by the test particle from below; the line of nodes is the intersection between the orbital and the fundamental planes. The argument of pericentre is the angle, reckoned in the orbital plane, from N to Q. The mean anomaly at epoch is proportional to the time of closest approach t p; indeed, from the definition of the mean anomaly  \nM ( t ) = n K GLYPH<16> t GLYPH<0> t p GLYPH<17> = n K t + GLYPH<17>; (1)  \nit follows  \nIn Equations (1)-(2),  \nn K = r GLYPH<0> m a 3 (3)  \nis the Keplerian mean motion which, of course, has not the same meaning as for the elliptic orbits. Moreover,  \nm : = GM (4)  \nis the standard gravitational parameter of the source of the gravitational field given by the product of its mass by the Newtonian gravitational constant G . Instead, I ; GLYPH<10> and ! determine the orientation of the orbit in space and of the orbit itself within its orbital plane also for the hyperbolic motion.  \nGLYPH<17> : = GLYPH<0> n K t p : (2)  \nwhere  \np : = GLYPH<0> a GLYPH<16> e 2 GLYPH<0> 1 GLYPH<17> (14)  \nis the semilatus rectum. The hyperbolic excess velocity is defined as (Rappaport et al. 2001)  \nv 1 = GLYPH<0> n K a = r GLYPH<0> m a : (15)  \nEquations (6)-(7) allow to express Equations (11)-(13) in terms of f .  \nThe components of r and v can be referred to the primary-fixed reference frame by means of the rotation matrix (Montenbruck and Gill 2000)  \nR ( GLYPH<10> ; I ; ! ) = R z ( GLYPH<0> GLYPH<10> ) R x ( GLYPH<0> I ) R z ( GLYPH<0> ! ) ; (16)  \nwhere, for a generic angle GLYPH<30> , it is  \nR ( GLYPH<0> GLYPH<30> ) z = 0 B B B B B B B B B B B @ cos GLYPH<30> GLYPH<0> sin GLYPH<30> 0 sin GLYPH<30> cos GLYPH<30> 0 0 0 1 1 C C C C C C C C C C C A ; (17)  \nIn view of the forthcoming calculation, it is convenient to express the mean anomaly in terms of the hyperbolic eccentric anomaly H ( t ) as (Rappaport et al. 2001)  \nM = e sinh H GLYPH<0> H : (5)  \nFurthermore, it is (Rappaport et al. 2001)  \nsinh H = sin f p e 2 GLYPH<0> 1 1 + e cos f ; (6)  \ncosh H = e + cos f 1 + e cos f : (7)  \nIn Equations (6)-(7), f ( t ) is the true anomaly, counted from Q to P( t ) in such a way that f = 0 at the pericentre and  \nGLYPH<0> f 1 GLYPH<20> f GLYPH<20> f 1 ; (8)  \nwhere  \nf 1 = arccos GLYPH<0> 1 e ! : (9)  \nFrom Equation (1) and by using Equations (5)-(7), one gets  \nd t d f = GLYPH<16> e 2 GLYPH<0> 1 GLYPH<17> 3 = 2 n K (1 + e cos f ) 2 : (10)  \nThe instantaneous distance of the test particle from the primary can be expressed as (Rappaport et al. 2001)  \nr = a (1 GLYPH<0> e cos H ) : (11)  \nThe position and velocity vectors, referred to the orbital plane 2 f X ; Y g , are (Rappaport et al. 2001)  \nr = n a (cos H GLYPH<0> e ) ; p GLYPH<0> ap sin H ; 0 o ; (12)  \nv = ( p GLYPH<0> m a r sin H ; p m p r cos H ; 0 ) ; (13)  \nR ( GLYPH<0> GLYPH<30> ) x = 0 B B B B B B B B B B B @ 1 0 0 0 cos GLYPH<30> GLYPH<0> sin GLYPH<30> 0 sin GLYPH<30> cos GLYPH<30> 1 C C C C C C C C C C C A : (18)  \nEquation (16) allows to determine the orientation of the orbit in space and of the orbit itself within the orbital plane in full generality.  \nFor calculational purposes, it is convenient to introduce the following mutually orthogonal unit vectors (So GLYPH<11> el 1989; Brumberg 1991; So GLYPH<11> el and Han 2019)  \nˆ l : = f cos GLYPH<10> ; sin GLYPH<10> ; 0 g ; (19)  \nˆ m : = fGLYPH<0> cos I sin GLYPH<10> ; cos I cos GLYPH<10> ; sin I g ; (20)  \nˆ h : = f sin I sin GLYPH<10> ; GLYPH<0> sin I cos GLYPH<10> ; cos I g ; (21)  \n(22)  \nˆ l is directed along the line of nodes towards the ascending node, ˆ h is perpendicular to the orbital plane, being aligned with the orbital angular momentum, and ˆ m lies in the orbital plane so that  \nˆ l GLYPH<2> ˆ m = ˆ h (23)  \nholds.  \nThe planetary equations in the form of Lagrange which allow to calculate the perturbations of the Keplerian orbital elements in the case of the hyperbolic motion are (Rappaport et al. 2001)  \nd a d t = GLYPH<0> 2 n K a @ R @GLYPH<17> ; (24)  \nd e d t = p e 2 GLYPH<0> 1 n K a 2 e @ R @! + GLYPH<16> e 2 GLYPH<0> 1 GLYPH<17> n K a 2 e @ R @GLYPH<17> ; (25)  \nd I d t = GLYPH<0> 1 n K a 2 p e 2 GLYPH<0> 1 sin I @ R @ GLYPH<10> + cos I n K a 2 p e 2 GLYPH<0> 1 sin I @ R @! ; (26)  \nd GLYPH<10> d t = 1 n K a 2 p e 2 GLYPH<0> 1 sin I @ R @ I ; (27)  \nd ! d t = GLYPH<0> p e 2 GLYPH<0> 1 n K a 2 e @ R @ e GLYPH<0> cos I n K a 2 p e 2 GLYPH<0> 1 sin I @ R @ I ; (28)  \nd GLYPH<17> d t = 2 n K a @ R @ a GLYPH<0> GLYPH<16> e 2 GLYPH<0> 1 GLYPH<17> n K a 2 e @ R @ e ; (29)  \nwhere R is the disturbing function. R is given by the pK part L pK of the Lagrangian per unit mass L of the test particle which can be obtained from the spacetime metric tensor g GLYPH<22>GLYPH<23> ; GLYPH<22>; GLYPH<23> = 0 ; 1 ; 2 ; 3 as follows.  \nWritten in spatially isotropic or harmonic coordinates, the latter can be expressed, to the pN order, as  \ng 00 \' 1 + h 00 = 1 + 2 U ( r ) c 2 + 2 U 2 ( r ) c 4 + O GLYPH<16> 1 = c 6 GLYPH<17> ; (30)  \ng 0 i \' h 0 i = O GLYPH<16> 1 = c 3 GLYPH<17> ; i = 1 ; 2 ; 3 ; (31)  \ngij \' GLYPH<0> 1 + hij = GLYPH<0> " 1 GLYPH<0> 2 U ( r ) c 2 # GLYPH<14> i j + O GLYPH<16> 1 = c 4 GLYPH<17> ; i ; j = 1 ; 2 ; 3 : (32)  \nIn Equations (30)-(32), the coe GLYPH<14> cients h GLYPH<22>GLYPH<23> ; GLYPH<22>; GLYPH<23> = 0 ; 1 ; 2 ; 3 are the pN corrections to the constant components GLYPH<17>GLYPH<22>GLYPH<23> ; GLYPH<22>; GLYPH<23> = 0 ; 1 ; 2 ; 3 of the \'flat\' Minkowskian spacetime metric tensor,  \nGLYPH<14> i j : = 8 > > < > > : 1 for i = j 0 for i , j ; i ; j = 1 ; 2 ; 3 ; (33)  \nis the 3-dimensional Kronecker delta (Olver et al. 2010), and U ( r ) is the Newtonian potential of the source including also J 2  \nU ( r ) = GLYPH<0> m r " 1 GLYPH<0> J 2 GLYPH<18> R e r GLYPH<19> 2 P 2 GLYPH<16> ˆ J GLYPH<1> ˆ r GLYPH<17> # ; (34)  \nin which  \nP 2 ( GLYPH<24> ) = 3 GLYPH<24> 2 GLYPH<0> 1 2 (35)  \nis the Legendre polynomial of degree \' = 2 in the generic dimensionless argument GLYPH<24> . Furthermore,  \nh 0 i = 2 GJ GLYPH<15> i jk ˆ J j x k c 3 r 3 ; i = 1 ; 2 ; 3 (36)  \nwhere  \nGLYPH<15> i jk : == 8 > > > > > < > > > > > : + 1 if ( i ; j ; k ) is (1 ; 2 ; 3), (2 ; 3 ; 1), or (3 ; 1 ; 2) GLYPH<0> 1 if ( i ; j ; k ) is (3 ; 2 ; 1), (1 ; 3 ; 2), or (2 ; 1 ; 3) 0 if i = j , or j = k , or k = i (37)  \nis the 3-dimensional Levi-Civita symbol (Olver et al. 2010), are the components of the gravitomagnetic LT potential. In Equation (36), ˆ J i ; i = 1 ; 2 ; 3 are the components of the spin unit vector ˆ J , and x k ; k = 1 ; 2 ; 3 are the Cartesian coordinates x ; y ; z of the test particle. In Equation (36), the Einstein summation convention (Olver et al. 2010) is applied to the dummy summation indexes j and k .  \nTo the 1pN order, the Lagrangian per unit mass turns out to be (Brumberg 1991, p. 56, Equation (2.2.53))  \nL = L N + L 1pN ; (38)  \nwhere 3  \nL N = 1 2 v 2 GLYPH<0> 1 2 c 2 h ( 1 = c 2 ) 00 ; (39)  \nL 1pN = GLYPH<0> 1 2 c 2 h ( 1 = c 4 ) 00 + v 4 8 c 2 GLYPH<0> 1 4 h 00 v 2 + c 2 8 h 2 00 GLYPH<0> 1 2 hijv i v j GLYPH<0> ch 0 jv j ; (40)  \nwhere h GLYPH<11>GLYPH<12> ; GLYPH<11>; GLYPH<12> = 0 ; 1 ; 2 ; 3 are given by Equations (30)-(32); h ( 1 = c 2 ) 00 and h ( 1 = c 4 ) 00 denote the 1pN and second post-Newtonian (2pN) parts of h 00, respectively; both of them are needed to keep the Lagrangian to the 1pN level. To this aim, it is meant that only h ( 1 = c 2 ) 00 enters the third and fourth terms of Equation (40).  \nStrictly speaking, the Lagrange equations in the form of Equations (24)-(29) return either the osculating elements if R depends only on the position r of the test particle, or the nonosculating, contact elements, to the first 4 order in the perturbation, if R does depend also on the velocity v (Brumberg 1991; Kopeikin et al. 2011). The consequences of this fact, often overlooked, will be treated in detail in Section 6.  \nwhere  \nmin  \nGLYPH<0> f 1 < f min < 0 ; (44)  \n0 < f max < f 1 ; (45)  \none gets  \nGLYPH<1> a GE ( f min ; f max) = 0 ; (46)  \nGLYPH<1> e GE ( f min ; f max) = 0 ; (47)  \nGLYPH<1>  \nI  \n(  \nf  \n;  \nf  \nmax)  \n=  \n0  \n;  \n(48)  \nGLYPH<1>GLYPH<10> GE ( f min ; f max) = 0 ; (49)  \nGLYPH<1> ! GE ( f min ; f max) = 2 m c 2 ae 2 0 B B B B B B B @ GLYPH<0> 3 GLYPH<16> 1 + e 2 GLYPH<17> GLYPH<1> f e 2 GLYPH<0> 1 GLYPH<0> 2 GLYPH<16> 3 + e 2 GLYPH<17> p e 2 GLYPH<0> 1 8 > > > < > > > : arctanh 2 6 6 6 6 6 6 4 ( e GLYPH<0> 1) tan GLYPH<16> f max 2 GLYPH<17> p e 2 GLYPH<0> 1 3 7 7 7 7 7 7 5 GLYPH<0> arctanh 2 6 6 6 6 6 6 4 ( e GLYPH<0> 1) tan GLYPH<16> f min 2 GLYPH<17> p e 2 GLYPH<0> 1 3 7 7 7 7 7 7 5 9 > > > = > > > ;  \nGLYPH<0> e sin f max 1 + e cos f max + e sin f min 1 + e cos f min ! ; (50)  \nGLYPH<1> GLYPH<17> GE ( f min ; f max) = m 2 c 2 ae 2 0 B B B B B B B @ 2 GLYPH<16> 12 GLYPH<0> 11 e 2 GLYPH<17> 8 > > > < > > > : arctanh 2 6 6 6 6 6 6 4 ( e GLYPH<0> 1) tan GLYPH<16> f max 2 GLYPH<17> p e 2 GLYPH<0> 1 3 7 7 7 7 7 7 5 GLYPH<0> arctanh 2 6 6 6 6 6 6 4 ( e GLYPH<0> 1) tan GLYPH<16> f min 2 GLYPH<17> p e 2 GLYPH<0> 1 3 7 7 7 7 7 7 5 9 > > > = > > > ;  \nGLYPH<0> p e 2 GLYPH<0> 1 ( 12 GLYPH<1> f + e GLYPH<16> e 2 GLYPH<0> 4 GLYPH<17> " sin f max 1 + e cos f max GLYPH<0> sin f min 1 + e cos f min #)! ; (51)  \nGLYPH<1> f : = f max GLYPH<0> f min : (52)  \nGE  \nThe orbital shifts experienced by the test particle during the flyby can be explicitly worked out by integrating the right hand sides of Equations (24)-(29), calculated onto the unperturbed Keplerian hyperbola, by means of Equations (6)-(7) and Equations (10)(14) from f min to f max. As it will be shown, while for the 1pN LT and the Newtonian J 2 terms one can analytically work out shifts covering the whole motion by assuming  \nj f min j = f max = f 1 ; (41)  \nit is not possible for the 1pN gravitoelectric perturbations since they diverge when calculated with Equation (41); however, analytical expressions valid for restricted ranges of values of f including the passage at the pericentre can be obtained.', '3. The 1pN gravitoelectric shifts': 'The 1pN gravitoelectric disturbing function, due solely to M , can be extracted from Equation (40) by neglecting the last o GLYPH<11> -diagonal term and using Equation (30) and Equation (32) calculated for J 2 ! 0. It turns out to be  \nR GE = r 2 v 4 + 12 m rv 2 GLYPH<0> 4 m 2 8 c 2 r 2 : (42)  \nBy inserting Equation (42) in Equations (24)-(29) and integrating their right hand sides by means of Equation (10) within the range  \nf min GLYPH<20> f GLYPH<20> f max ; (43)  \nwith  \nand  \nAs anticipated in Section 2, Equations (50)-(51) turn out to be singular for  \nf min = GLYPH<0> f 1 ; (53)  \nf max = f 1 : (54)  \nIt may happen that observations are collected during a larger time interval before the passage at the point of closest approach than after it, or vice versa; thus, the condition  \nj f min j , j f max j (55)  \nshould be generally allowed. If, instead, data are taken during identical finite time spans before and after the flyby, it is  \nf min = GLYPH<0> f max ; (56)  \nso that Equations (50)-(51) become  \nGLYPH<1> ! GE ( f max) = GLYPH<0> 4 m c 2 ae 2 8 > > > > > > < > > > > > > : 3 GLYPH<16> 1 + e 2 GLYPH<17> f max e 2 GLYPH<0> 1 + 2 GLYPH<16> 3 + e 2 GLYPH<17> arctanh " ( e GLYPH<0> 1) tan GLYPH<16> f max 2 GLYPH<17> p e 2 GLYPH<0> 1 # p e 2 GLYPH<0> 1 + e sin f max 1 + e cos f max 9 > > > > > > = > > > > > > ; ; (57)  \nGLYPH<1> GLYPH<17> GE ( f max) = m c 2 ae 2 8 > > > < > > > : 2 GLYPH<16> 12 GLYPH<0> 11 e 2 GLYPH<17> arctanh 2 6 6 6 6 6 6 4 ( e GLYPH<0> 1) tan GLYPH<16> f max 2 GLYPH<17> p e 2 GLYPH<0> 1 3 7 7 7 7 7 7 5 GLYPH<0> p e 2 GLYPH<0> 1 2 6 6 6 6 6 6 4 12 f max + e GLYPH<16> e 2 GLYPH<0> 4 GLYPH<17> sin f max 1 + e cos f max 3 7 7 7 7 7 7 5 9 > > > = > > > ; (58)  \nExpressions valid for short time intervals symmetric with respect to the flyby can be obtained by expanding Equations (57)-(58) in powers of f max, assumed close to zero; indeed, f = 0 corresponds just to the passage at pericentre. Thus, one obtains  \nGLYPH<1> ! GE p ( f max) \' GLYPH<0> 4 m (2 + e ) c 2 ae ( e GLYPH<0> 1) f max + O GLYPH<16> f 2 max GLYPH<17> ; (59)  \nGLYPH<1> GLYPH<17> GE p ( f max) \' m GLYPH<16> 8 + 3 e GLYPH<0> 10 e 2 GLYPH<0> e 3 GLYPH<17> c 2 ae p e 2 GLYPH<0> 1 f max + O GLYPH<16> f 2 max GLYPH<17> : (60)  \nFrom Equation (15) and Equation (46), it can be straightforwardly inferred that v 1 is not changed by the 1pN gravitoelectric acceleration.', '4. The 1pN gravitomagnetic Lense-Thirring shifts': "The 1pN gravitomagnetic disturbing function, arising from the last term in Equation (40) calculated with Equation (36), turns out to be  \nR LT = GLYPH<0> 2 GJ c 2 r 3 GLYPH<16> ˆ J GLYPH<2> r GLYPH<17> GLYPH<1> v : (61)  \nIntegrating Equations (24)-(29), calculated with Equation (61), by means of Equations (8)-(10) finally yields  \nGLYPH<1> a LT 1 = 0 ; (62)  \nGLYPH<1> e LT 1 = 0 ; (63)  \nGLYPH<1> I LT 1 = GLYPH<0> 4 GJ h arcsec ( GLYPH<0> e ) + p e 2 GLYPH<0> 1 i Jl c 2 n K a 3 GLYPH<0> e 2 GLYPH<0> 1 GLYPH<1> 3 = 2 ; (64)  \nwhere  \nGLYPH<1>GLYPH<10> LT 1 = GLYPH<0> 4 GJ csc I h arcsec ( GLYPH<0> e ) + p e 2 GLYPH<0> 1 i Jm c 2 a 3 n K GLYPH<0> e 2 GLYPH<0> 1 GLYPH<1> 3 = 2 ; (65)  \nGLYPH<1> ! LT 1 = 4 GJ n e 2 cot I h arcsec ( GLYPH<0> e ) + p e 2 GLYPH<0> 1 i Jm + h 5 e 2 arcsec ( GLYPH<0> e ) + GLYPH<16> 3 + 2 e 2 GLYPH<17> p e 2 GLYPH<0> 1 i Jh o c 2 a 3 n K e 2 GLYPH<0> e 2 GLYPH<0> 1 GLYPH<1> 3 = 2 ; (66)  \nGLYPH<1> GLYPH<17> LT 1 = GLYPH<0> 12 GJ p e 2 GLYPH<0> 1 Jh c 2 a 3 n K e 2 ; (67)  \nJl : = ˆ J GLYPH<1> ˆ l = ˆ Jx cos GLYPH<10> + ˆ Jy sin GLYPH<10> ; (68)  \nJm : = ˆ J GLYPH<1> ˆ m = cos I GLYPH<16> GLYPH<0> ˆ Jx sin GLYPH<10> + ˆ Jy cos GLYPH<10> GLYPH<17> + ˆ Jz sin I ; (69)  \nJl : = ˆ J GLYPH<1> ˆ h = sin I GLYPH<16> ˆ Jx sin GLYPH<10> GLYPH<0> ˆ Jy cos GLYPH<10> GLYPH<17> + ˆ Jz cos I : (70)  \nEquations (62)-(67), which cover the full motion of the test particle, retain a general validity since they hold for arbitrary orientations in space of both the orbit and the primary's spin axis.  \nFrom Equations (62)-(67) and Equations (68)-(70) it turns out that the inclination and the node stay constant for equatorial orbits, characterized by  \nJh = GLYPH<6> 1 ; (71)  \nJl = Jm = 0 ; (72)  \nwhile the pericentre and the mean anomaly at epoch undergo nonvanishing net shifts. Instead, for polar orbits ( Jh = 0), the inclination, the node and the pericentre are, in general, shifted.  \nFrom Equation (15) and Equation (62), it can be straightforwardly inferred that v 1 is not changed by the LT acceleration.", '5. The Newtonian J 2 shifts': "The Newtonian disturbing function due to the primary's oblateness, obtained from the J 2-driven pK component of Equation (39) calculated with Equation (34), turns out to be  \nR J 2 = m J 2 R 2 e GLYPH<20> 1 GLYPH<0> 3 GLYPH<16> ˆ J GLYPH<1> ˆ r GLYPH<17> 2 GLYPH<21> 2 r 3 : (73)  \nThe resulting orbital shifts, integrated according to Equations (24)-(29) and Equations (8)-(10), are  \nGLYPH<1> a J 2 1 = 0 ; (74)  \nGLYPH<1> e J 2 1 = J 2 R 2 e p e 2 GLYPH<0> 1 a 2 e 3 6 X i = 1 E J 2 1 ; i b Ti ; (75)  \nGLYPH<1> I J 2 1 = J 2 R 2 e a 2 e 2 GLYPH<0> e 2 GLYPH<0> 1 GLYPH<1> 2 6 X i = 1 I J 2 1 ; i b Ti ; (76)  \nwhere  \nand  \n;  \nGLYPH<1>GLYPH<10> J 2 1 = J 2 R 2 e csc I a 2 e 2 GLYPH<0> e 2 GLYPH<0> 1 GLYPH<1> 2 6 X i = 1 N J 2 1 ; i b Ti ; (77)  \nGLYPH<1> ! J 2 1 = J 2 R 2 e 2 a 2 e 4 GLYPH<0> e 2 GLYPH<0> 1 GLYPH<1> 2 6 X i = 1 G J 2 1 ; i b Ti ; (78)  \nGLYPH<1> GLYPH<17> J 2 1 = 3 J 2 R 2 e 2 a 2 e 4 6 X i = 1 H J 2 1 ; i b Ti ; (79)  \nb T 1 : = 1 ; (80)  \nb T 2 : = Jl 2 + Jm 2 ; (81)  \nb T 3 : = Jl 2 GLYPH<0> Jm 2 ; (82)  \nb T 4 : = JhJl ; (83)  \nb T 5 : = JhJm ; (84)  \nb T 6 : = JlJm ; (85)  \nE J 2 1 ; 1 : = 0 ; (86)  \nE J 2 1 ; 2 : = 0 ; (87)  \nE  \nJ  \n2  \n1  \n3  \nE J 2 1 ; 4 : = 0 ; (89)  \nE J 2 1 ; 5 : = 0 ; (90)  \nE J 2 1 ; 6 : = GLYPH<0> 2 E J 2 1 ; 3 cot 2 !; (91)  \nI J 2 1 ; 1 : = 0 ; (92)  \nI J 2 1 ; 2 : = 0 ; (93)  \nI J 2 1 ; 3 : = 0 ; (94)  \n:  \n=  \nsin 2  \n(88)  \n!;  \n1  \n2  \nI J 2 1 ; 4 : = GLYPH<0> 3 e 2 arcsec ( GLYPH<0> e ) + p e 2 GLYPH<0> 1 h GLYPH<0> 3 e 2 GLYPH<0> GLYPH<16> e 2 GLYPH<0> 1 GLYPH<17> cos 2 ! i ; (95)  \nI J 2 1 ; 5 : = GLYPH<0> GLYPH<16> e 2 GLYPH<0> 1 GLYPH<17> 3 = 2 sin 2 !; (96)  \nI J 2 1 ; 6 : = 0 ; (97)  \nN  \nJ  \n2  \n1  \nN  \nJ  \n;  \n2  \n1  \n;  \nN J 2 1 ; 3 : = 0 ; (100)  \nN  \nJ  \n2  \n1  \n;  \n4  \nJ  \n2  \n1  \n;  \n5  \nN J 2 1 ; 5 : = GLYPH<0> 3 e 2 arcsec ( GLYPH<0> e ) + p e 2 GLYPH<0> 1 h GLYPH<0> 3 e 2 + GLYPH<16> e 2 GLYPH<0> 1 GLYPH<17> cos 2 ! i ; (102)  \nN J 2 1 ; 6 : = 0 ; (103)  \nG J 2 1 ; 1 : = 6 e 2 h p e 2 GLYPH<0> 1 GLYPH<16> 1 + e 2 GLYPH<17> + 2 e 2 arcsec ( GLYPH<0> e ) i ; (104)  \nG J 2 1 ; 2 : = GLYPH<0> 3 2 G J 2 1 ; 1 ; (105)  \nG J 2 1 ; 3 : = GLYPH<0> 3 p e 2 GLYPH<0> 1 GLYPH<16> 2 GLYPH<0> 3 e 2 + e 4 GLYPH<17> cos 2 !; (106)  \nG J 2 1 ; 4 : = 2 e 2 GLYPH<16> e 2 GLYPH<0> 1 GLYPH<17> 3 = 2 cot I sin 2 !; (107)  \nG J 2 1 ; 5 : = 2 e 2 n 3 e 2 arcsec ( GLYPH<0> e ) + p e 2 GLYPH<0> 1 h 3 e 2 GLYPH<0> GLYPH<16> e 2 GLYPH<0> 1 GLYPH<17> cos 2 ! io cot I ; (108)  \nG J 2 1 ; 6 : = GLYPH<0> 6 p e 2 GLYPH<0> 1 GLYPH<16> 2 GLYPH<0> 3 e 2 + e 4 GLYPH<17> sin 2 !; (109)  \nH J 2 1 ; 1 : = GLYPH<0> 2 e 2 ; (110)  \nH J 2 1 ; 2 : = GLYPH<0> 3 2 H J 2 1 ; 1 ; (111)  \nH J 2 1 ; 3 : = GLYPH<16> 2 + e 2 GLYPH<17> cos 2 !; (112)  \n:  \n=  \nI  \n(101)  \n:  \n=  \n0  \n:  \n=  \n0  \n;  \n;  \n(98)  \n(99)  \n;  \nH J 2 1 ; 4 : = 0 ; (113)  \nH J 2 1 ; 5 : = 0 ; (114)  \nH J 2 1 ; 6 : = 2 H J 2 1 ; 3 tan 2 !: (115)  \nAlso Equations (74)-(79), covering the whole motion, are valid for any spatial orientations of the primary's spin axis and the orbital plane.  \nFrom Equations (74)-(115) and Equations (68)-(70) it turns out that, for equatorial orbits, the eccentricity, the inclination and the node stay constant, while the pericentre and the mean anomaly at epoch do generally vary. Instead, for polar orbits, only the inclination and the node remain una GLYPH<11> ected.  \nFrom Equation (15) and Equation (74), it can be straightforwardly inferred that v 1 is not changed by the primary's oblateness.", '6. A subtlety about the results obtained: choosing the gauge of the Lagrange planetary equations': 'If the disturbing function depends also on the velocity v , as in the case of the general relativistic Equation (42) and Equation (61), the Lagrange planetary equations as given by Equations (24)-(29) actually provide, to the first order in the perturbation, the instantaneous rates of change of the so-called contact Keplerian orbital elements. They are not osculating, and parameterize confocal instantaneous conics which may be generally neither tangent nor coplanar to the actual trajectory. Thus, the correct position of the test particle is returned at each instant of time, but not its velocity since v , v K. Non-osculating Keplerian orbital elements were used also for studying the impact of, e.g., J 2 (Gurfil 2004) and the 1pN gravitoelectric field (Gurfil and Efroimsky 2022) on bound, quasi-elliptical orbits. For a thorough analysis of the subtle technicalities involved, see Efroimsky and Goldreich (2004); Efroimsky (2005b,a); Kopeikin et al. (2011). The physical and geometrical meaning of the contact elements is not as straightforward as for the osculating ones, and wrong interpretations 5 of (mathematically correct) results actually obtained in terms of nonosculating orbital elements can be found in the literature, as pointed out in Efroimsky and Goldreich (2004). Thus, it may be preferable to have expressions for the orbital shifts written in terms of the osculating Keplerian elements also for velocity-dependent disturbing functions. Indeed, in general, it is not guaranteed that, in the case of a hyperbolic motion, any variations of the osculating and contact elements coincide when they are integrated over some range for f .  \nIt turns out that the di GLYPH<11> erence between the instantaneous values of the contact elements C ct i and the osculating ones C os i is given by (Kopeikin et al. 2011, p. 74, Equation (1.323))  \nC ct i ( f ) GLYPH<0> C os i ( f ) = GLYPH<0> 6 X j = 1 n C os i ; C os j o @ r @ C os j GLYPH<1> @ R @ v : = Z i ( f ) ; i = a ; e ; I ; GLYPH<10> ; !; GLYPH<17>; (116)  \nwhere n C os i ; C os j o are the Poisson brackets 6  \nA straightforward calculation yields  \nfor the ith and jth elements.  \nZ a ( f ) = Z x a @ R @ vx + Z y a @ R @ vy + Z z a @ R @ vz ; (117)  \nZ e ( f ) = Z x e @ R @ vx + Z y e @ R @ vy + Z z e @ R @ vz ; (118)  \nZ I ( f ) = Z x I @ R @ vx + Z y I @ R @ vy + Z z I @ R @ vz ; (119)  \nZ GLYPH<10> ( f ) = Z x GLYPH<10> @ R @ vx + Z y GLYPH<10> @ R @ vy + Z z GLYPH<10> @ R @ vz ; (120)  \nwhere  \n;  \nL. Iorio  \nZ ! ( f ) = Z x ! @ R @ vx + Z y ! @ R @ vy + Z z ! @ R @ vz ; (121)  \nZ GLYPH<17> ( f ) = Z x GLYPH<17> @ R @ vx + Z y GLYPH<17> @ R @ vy + Z z GLYPH<17> @ R @ vz ; (122)  \n(123)  \nZ x a ( f ) = 0 ; (124)  \nZ y a ( f ) = 0 ; (125)  \nz  \na  \nZ  \n(  \nf  \n)  \n=  \n0  \n(126)  \nZ x e ( f ) = GLYPH<16> e 2 GLYPH<0> 1 GLYPH<17> 3 = 2 (cos GLYPH<10> sin u + cos I sin GLYPH<10> cos u ) aen K (1 + e cos f ) ; (127)  \nZ y e ( f ) = GLYPH<16> e 2 GLYPH<0> 1 GLYPH<17> 3 = 2 ( GLYPH<0> cos I cos GLYPH<10> cos u + sin GLYPH<10> sin u ) aen K (1 + e cos f ) ; (128)  \nZ z e ( f ) = GLYPH<0> GLYPH<16> e 2 GLYPH<0> 1 GLYPH<17> 3 = 2 sin I cos u aen K (1 + e cos f ) ; (129)  \nZ x I ( f ) = GLYPH<0> p e 2 GLYPH<0> 1 sin I sin GLYPH<10> cos u an K (1 + e cos f ) ; (130)  \nZ y I ( f ) = p e 2 GLYPH<0> 1 sin I cos GLYPH<10> cos u an K (1 + e cos f ) ; (131)  \nZ z I ( f ) = GLYPH<0> p e 2 GLYPH<0> 1 cos I cos u an K (1 + e cos f ) ; (132)  \nZ x GLYPH<10> ( f ) = GLYPH<0> p e 2 GLYPH<0> 1 sin GLYPH<10> sin u an K (1 + e cos f ) ; (133)  \nZ y GLYPH<10> ( f ) = p e 2 GLYPH<0> 1 cos GLYPH<10> sin u an K (1 + e cos f ) ; (134)  \nZ z GLYPH<10> ( f ) = GLYPH<0> p e 2 GLYPH<0> 1 cot I sin u an K (1 + e cos f ) ; (135)  \nZ x ! ( f ) = p e 2 GLYPH<0> 1 n cos GLYPH<10> h 2 e + GLYPH<16> 1 + e 2 GLYPH<17> cos f i cos u GLYPH<0> cos I sin GLYPH<10> ( e + cos f ) sin u o aen K (1 + e cos f ) ; (136)  \nZ y ! ( f ) = p e 2 GLYPH<0> 1 n cos I cos GLYPH<10> ( e + cos f ) sin u + sin GLYPH<10> h 2 e + GLYPH<16> 1 + e 2 GLYPH<17> cos f i cos u o aen K (1 + e cos f ) ; (137)  \nZ z ! ( f ) = p e 2 GLYPH<0> 1 csc I h 3 e + cos f + 2 e 2 cos f GLYPH<0> cos 2 I ( e + cos f ) i sin u aen K (1 + e cos f ) ; (138)  \nZ x GLYPH<17> ( f ) = GLYPH<0> GLYPH<16> e 2 GLYPH<0> 1 GLYPH<17> 2 (cos GLYPH<10> cos u GLYPH<0> cos I sin GLYPH<10> sin u ) cos f aen K (1 + e cos f ) ; (139)  \nZ y GLYPH<17> ( f ) = GLYPH<0> GLYPH<16> e 2 GLYPH<0> 1 GLYPH<17> 2 (cos I cos GLYPH<10> sin u + sin GLYPH<10> cos u ) cos f aen K (1 + e cos f ) ; (140)  \nZ z GLYPH<17> ( f ) = GLYPH<0> GLYPH<16> e 2 GLYPH<0> 1 GLYPH<17> 2 sin I cos f sin u aen K (1 + e cos f ) : (141)  \nIn Equations (124)-(141),  \nu : = ! + f (142)  \nis the argument of latitude. In general, also the components of the gradient of R with respect to v are time-dependent through f  \n.  \nThe next step is calculating the di GLYPH<11> erence of the integrated shifts of the contact and osculating orbital elements. To this aim, by taking the time derivative of both members of Equation (116), one gets  \nd C ct i ( f ) d t GLYPH<0> d C os i ( f ) d t = @ Z i ( f ) @ f d f d t ; i = a ; e ; I ; GLYPH<10> ; !; GLYPH<17>: (143)  \nThe analytical expressions for @ Z i ( f ) =@ f ; i = a ; e ; I ; GLYPH<10> ; !; GLYPH<17> , obtainable straightforwardly from Equations (117)-(141), are too cumbersome to be explicitly displayed. Integrating both members of Equation (143) from f min to f max finally yields  \nGLYPH<1> C os i ( f min ; f max) = GLYPH<1> C ct i ( f min ; f max) + GLYPH<4> i ( f min ; f max) ; i = a ; e ; I ; GLYPH<10> ; !; GLYPH<17>; (144)  \nwhere  \nGLYPH<4> i ( f min ; f max) : = GLYPH<0> Z f max f min @ Z i ( f ) @ f d f ; i = a ; e ; I ; GLYPH<10> ; !; GLYPH<17>: (145)  \nIn the case of the general relativistic disturbing functions of Equation (42) and Equation (61), the integrated shifts of the contact elements GLYPH<1> C ct i ( f min ; f max) ; i = a ; e ; I ; GLYPH<10> ; !; GLYPH<17> are given, to the 1pN order, by Equations (46)-(51) and Equations (62)-(67), respectively.  \nFrom the above considerations, it turns out that the oblateness-driven classical orbital shifts of Equations (74)-(79) are to be intended as written in terms of the osculating elements. Indeed, the disturbing function of Equation (73) depends only on the position vector r . In this case, the contact and the osculating elements coincide, as per Equation (116).  \nFor elliptical motions perturbed by Equation (42) and Equation (61), it can be shown that GLYPH<4> i (0 ; 2 p ) = 0 ; i = a ; e ; I ; GLYPH<10> ; !; GLYPH<17>; when Equation (145) is integrated from 0 to 2 p .', '6.1. The 1pN gravitoelectric corrections to the shifts of the contact Keplerian orbital elements': "The corrections GLYPH<4> GE i ( f min ; f max) ; i = a ; e ; I ; GLYPH<10> ; !; GLYPH<17> to the 1pN gravitoelectric integrated shifts of the contact Keplerian orbital elements are obtained in the following way.  \nThe calculation of the gradient of Equation (42) with respect to v returns  \n@ R GE @ vx = m n K GLYPH<16> 7 + e 2 + 8 e cos f GLYPH<17> [cos GLYPH<10> ( e sin ! + sin u ) + cos I sin GLYPH<10> ( e cos ! + cos u )] 2 c 2 GLYPH<0> e 2 GLYPH<0> 1 GLYPH<1> 3 = 2 ; (146)  \n@ R GE @ vy = GLYPH<0> m n K GLYPH<16> 7 + e 2 + 8 e cos f GLYPH<17> [cos I cos GLYPH<10> ( e cos ! + cos u ) GLYPH<0> sin GLYPH<10> ( e sin ! + sin u )] 2 c 2 GLYPH<0> e 2 GLYPH<0> 1 GLYPH<1> 3 = 2 ; (147)  \n@ R GE @ vz = GLYPH<0> m n K sin I GLYPH<16> 7 + e 2 + 8 e cos f GLYPH<17> ( e cos ! + cos u ) 2 c 2 GLYPH<0> e 2 GLYPH<0> 1 GLYPH<1> 3 = 2 : (148)  \nThe functions Z GE i ( f ) ; i = a ; e ; I ; GLYPH<10> ; !; GLYPH<17> entering Equation (116) can be calculated by inserting Equations (146)-(148) in Equations (124)-(141). One gets  \nZ GE a ( f ) = 0 ; (149)  \nZ GE e ( f ) = GLYPH<0> m GLYPH<16> 7 + e 2 + 8 e cos f GLYPH<17> 2 c 2 ae ; (150)  \nZ GE I ( f ) = 0 ; (151)  \nZ GE GLYPH<10> ( f ) = 0 ; (152)  \nZ GE ! ( f ) = m h 6 e GLYPH<16> 3 + e 2 GLYPH<17> + GLYPH<16> 7 + 24 e 2 + e 4 GLYPH<17> cos f + 4 e GLYPH<16> 1 + e 2 GLYPH<17> cos 2 f i sin f 2 c 2 a GLYPH<0> e 2 GLYPH<0> 1 GLYPH<1> (1 + e cos f ) 2 ; (153)  \nZ GE GLYPH<17> ( f ) = GLYPH<0> m p e 2 GLYPH<0> 1 GLYPH<16> 7 + e 2 + 8 e cos f GLYPH<17> sin 2 f 4 c 2 a (1 + e cos f ) 2 : (154)  \nAccording to Equations (149)-(154), the instantaneous osculating eccentricity, argument of pericenter and mean anomaly at epoch are generally di GLYPH<11> erent from their contact counterparts, while the osculating and contact semimajor axis, inclination and longitude of the ascending node always coincide.  \nBy plugging Equations (149)-(154) in Equation (145), the explicit expressions of GLYPH<4> GE i ( f min ; f max) ; i = a ; e ; I ; GLYPH<10> ; !; GLYPH<17> can be obtained. They are  \nGLYPH<4> GE a ( f min ; f max) = 0 ; (155)  \nGLYPH<4> GE e ( f min ; f max) = 4 m (cos f min GLYPH<0> cos f max) c 2 a ; (156)  \nGLYPH<4> GE I ( f min ; f max) = 0 ; (157)  \nGLYPH<4> GE GLYPH<10> ( f min ; f max) = 0 ; (158)  \nGLYPH<4> GE ! ( f min ; f max) = m 2 c 2 a GLYPH<0> e 2 GLYPH<0> 1 GLYPH<1> 8 > > > < > > > : h 6 e GLYPH<16> 3 + e 2 GLYPH<17> + GLYPH<16> 7 + 24 e 2 + e 4 GLYPH<17> cos f min + 4 e GLYPH<16> 1 + e 2 GLYPH<17> cos 2 f min i sin fmin (1 + e cos f min) 2  \nGLYPH<0> h 6 e GLYPH<16> 3 + e 2 GLYPH<17> + GLYPH<16> 7 + 24 e 2 + e 4 GLYPH<17> cos f max + 4 e GLYPH<16> 1 + e 2 GLYPH<17> cos 2 f max i sin fmax (1 + e cos f max) 2 9 > > > = > > > ; ; (159)  \nGLYPH<4> GE GLYPH<17> ( f min ; f max) = m p e 2 GLYPH<0> 1 2 c 2 a 2 6 6 6 6 6 6 4 GLYPH<16> 7 + e 2 + 8 e cos f max GLYPH<17> sin 2 f max 2 (1 + e cos f max) 2 GLYPH<0> GLYPH<16> 7 + e 2 + 8 e cos f min GLYPH<17> sin 2 f min 2 (1 + e cos f min) 2 3 7 7 7 7 7 7 5 : (160)  \nAlso Equations (155)-(160), as Equations (46)-(51), hold only for the range of values of f given by Equations (43)-(45).  \nBy using Equation (56), Equations (156)-(160) reduce to  \nGLYPH<4> GE a ( f max) = 0 ; (161)  \nGLYPH<4> GE e ( f max) = 0 ; (162)  \nGLYPH<4> GE I ( f max) = 0 ; (163)  \nGLYPH<4> GE GLYPH<10> ( f max) = 0 ; (164)  \nGLYPH<4> GE ! ( f max) = GLYPH<0> m h 6 e GLYPH<16> 3 + e 2 GLYPH<17> + GLYPH<16> 7 + 24 e 2 + e 4 GLYPH<17> cos f max + 4 e GLYPH<16> 1 + e 2 GLYPH<17> cos 2 f max i sin fmax c 2 a GLYPH<0> e 2 GLYPH<0> 1 GLYPH<1> (1 + e cos f max) 2 ; (165)  \nGLYPH<4> GE GLYPH<17> ( f max) = m p e 2 GLYPH<0> 1 GLYPH<16> 7 + e 2 + 8 e cos f max GLYPH<17> sin 2 f max 2 c 2 a (1 + e cos f max) 2 : (166)  \nBy expanding Equations (165)-(166) in powers of f max around 0, as done for Equations (59)-(60), yields  \nGLYPH<4> GE ! ( f max) ' m (7 + e ) c 2 a (1 GLYPH<0> e ) f max + O GLYPH<16> f 2 max GLYPH<17> ; (167)  \nGLYPH<4> GE GLYPH<17> ( f max) ' m (7 + e ) p e 2 GLYPH<0> 1 c 2 a ( e + 1) f max + O GLYPH<16> f 2 max GLYPH<17> : (168)  \n6.2. The Lense-Thirring corrections to the shifts of the contact Keplerian orbital elements  \nExplicit expressions for the corrections GLYPH<4> LT i ( f min ; f max) ; i = a ; e ; I ; GLYPH<10> ; !; GLYPH<17> to the LT integrated shifts of the contact Keplerian orbital elements can be obtained as follows.  \nCalculating the gradient of Equation (61) with respect to v yields  \n@ R LT @ vx = 2 GJ (1 + e cos f ) 2 hGLYPH<16> ˆ Jz cos I cos GLYPH<10> GLYPH<0> ˆ Jy sin I GLYPH<17> sin u + ˆ Jz sin GLYPH<10> cos u i c 2 a 2 GLYPH<0> e 2 GLYPH<0> 1 GLYPH<1> 2 ; (169)  \n@ R LT @ vy = GLYPH<0> 2 GJ (1 + e cos f ) 2 h ˆ Jz cos GLYPH<10> cos u GLYPH<0> GLYPH<16> ˆ Jx sin I + ˆ Jz cos I sin GLYPH<10> GLYPH<17> sin u i c 2 a 2 GLYPH<0> e 2 GLYPH<0> 1 GLYPH<1> 2 ; (170)  \n@ R LT @ vz = 2 GJ (1 + e cos f ) 2 hGLYPH<16> ˆ Jy cos GLYPH<10> GLYPH<0> ˆ Jx sin GLYPH<10> GLYPH<17> cos u GLYPH<0> cos I GLYPH<16> ˆ Jx cos GLYPH<10> + ˆ Jy sin GLYPH<10> GLYPH<17> sin u i c 2 a 2 GLYPH<0> e 2 GLYPH<0> 1 GLYPH<1> 2 : (171)  \nEquations (169)-(171), inserted in Equations (124)-(141), allow to calculate the functions Z LT i ( f ) ; i = a ; e ; I ; GLYPH<10> ; !; GLYPH<17> entering Equation (116). They turn out to be  \nZ LT a ( f ) = 0 ; (172)  \nZ LT e ( f ) = 2 GJ (1 + e cos f ) Jh c 2 a 3 n K e p e 2 GLYPH<0> 1 ; (173)  \nZ LT I ( f ) = GLYPH<0> 2 GJ (1 + e cos f ) cos u ( Jm cos u GLYPH<0> Jl sin u ) c 2 a 3 n K GLYPH<0> e 2 GLYPH<0> 1 GLYPH<1> 3 = 2 ; (174)  \nZ LT GLYPH<10> ( f ) = GLYPH<0> 2 GJ csc I (1 + e cos f ) sin u ( Jm cos u GLYPH<0> Jl sin u ) c 2 a 3 n K GLYPH<0> e 2 GLYPH<0> 1 GLYPH<1> 3 = 2 ; (175)  \nZ LT ! ( f ) = 2 GJ cot I (1 + e cos f ) sin u ( Jm cos u GLYPH<0> Jl sin u ) c 2 a 3 n K GLYPH<0> e 2 GLYPH<0> 1 GLYPH<1> 3 = 2 ; (176)  \nZ LT GLYPH<17> ( f ) = 0 : (177)  \nAccording to Equations (172)-(177), the instantaneous osculating eccentricity, inclination, longitude of ascending node and argument of pericenter are generally di GLYPH<11> erent from their contact counterparts, while the osculating and contact semimajor axis and mean anomaly at epoch always coincide.  \nBy means of Equation (145), calculated with Equations (172)-(177), one straightforwardly obtains the explicit expressions of GLYPH<4> LT i ( f min ; f max) ; i = a ; e ; I ; GLYPH<10> ; !; GLYPH<17> which, apart from the semimajor axis and the mean anomaly at epoch, turn out to be generally nonvanishing functions valid for any values of f min and f max. They are too cumbersome to be explicitly displayed for the general case of Equation (55). Instead, it is possible to write down manageable formulas if the condition of Equation (56) holds. The resulting nonvanishing shifts are  \nGLYPH<4> LT I ( f max) = 2 GJ (1 + e cos f max) sin 2 f max ( Jl cos 2 ! + Jm sin 2 ! ) c 2 a 3 n K GLYPH<0> e 2 GLYPH<0> 1 GLYPH<1> 3 = 2 ; (178)  \nGLYPH<4> LT GLYPH<10> ( f max) = 2 GJ csc I (1 + e cos f max) sin 2 f max ( Jl sin 2 ! GLYPH<0> Jm cos 2 ! ) c 2 a 3 n K GLYPH<0> e 2 GLYPH<0> 1 GLYPH<1> 3 = 2 ; (179)  \nGLYPH<4> LT ! ( f max) = 2 GJ cot I (1 + e cos f max) sin 2 f max ( GLYPH<0> Jl sin 2 ! + Jm cos 2 ! ) c 2 a 3 n K GLYPH<0> e 2 GLYPH<0> 1 GLYPH<1> 3 = 2 : (180)  \n(181)  \nEquations (178)-(180) vanish if  \nf max = f 1 : (182)  \nThus, the LT total shifts of the osculating orbital elements accumulated over the entire path are equal just to those given by Equations (62)-(67).", '7. Numerical evaluations for some natural and artificial bodies': "Here, the results obtained in the previous Sections are applied to some flybys occurred in our solar system.  \n7.1. ' Oumuamua in the field of the Sun  \nThe case of the interstellar, cigar-shaped asteroid 7 1I / 2017 U1 ( ' Oumuamua) (Meech et al. 2017), which briefly visited the inner regions of out solar system in 2017 along an unbound trajectory, is considered here. Its orbital parameters are listed in Table 1. It may be interesting to calculate the size of the pK e GLYPH<11> ects of gravitational origin treated in the previous Sections if only  \nTable 1. Orbital parameters of the heliocentric trajectory of the interstellar asteroid ' Oumuamua referred to the International Celestial Reference Frame (ICRF) at epoch J2000.0. retrieved from the HORIZONS WEB interface maintained by the Jet Propulsion Laboratory (JPL) for the epoch 23th November 2017. The distance of closest approach turns out to be r p = 0 : 38 au = 82 R GLYPH<12> e , while f 1 = 146 : 4 deg = 2 : 55 rad.  \nto get an idea of the potential o GLYPH<11> ered by this type of unusual objects (Jewitt 2024) whose number may increase in the future 8 . However, it should be made clear that using them as probes for tests of gravitational theories would be quite challenging non only because of the observational accuracy needed but also because of the heavy non-gravitational accelerations perturbing their motion (Micheli et al. 2018).  \nIn Table 2, the relevant physical parameters of the Sun are listed. The components of the Sun's spin axis, parameterized in  \nTable 2. Relevant physical parameters of the Sun (Pijpers 1998; Seidelmann et al. 2007; Emilio et al. 2012; Park et al. 2017; Mecheri and Meftah 2021; Park et al. 2021). R.A. GLYPH<11> GLYPH<12> and decl. GLYPH<14> GLYPH<12> of the north pole of rotation are equatorial coordinates referred to the International Celestial Reference Frame (ICRF) at epoch J2000.0.  \nterms of the right ascension (R.A.) GLYPH<11> GLYPH<12> and declination (decl.) GLYPH<14> GLYPH<12> of its north pole of rotation, are  \nˆ J GLYPH<12> x = cos GLYPH<11> GLYPH<12> cos GLYPH<14> GLYPH<12> ; (183)  \nˆ J GLYPH<12> y = sin GLYPH<11> GLYPH<12> cos GLYPH<14> GLYPH<12> ; (184)  \nˆ J GLYPH<12> z = sin GLYPH<14> GLYPH<12> ; (185)  \nthey are needed to calculate Equations (62)-(67) and Equations (74)-(79).  \nThe nominal values of the gravitational pK orbital shifts of ' Oumuamua are displayed in Table 3. They are exceedingly small. Su GLYPH<14> ce it to say that the e GLYPH<11> ects of the Sun's oblateness and angular momentum are at the microarcseconds ( GLYPH<22> as) level over the whole trajectory, while the 1pN gravitoelectric shifts, to be rescaled by f max & 0 since they are valid just around the flyby, are of the order of less than a hundred milliarcseconds (mas).  \nTable 3. Nominal values of the pK orbital shifts of ' Oumuamua calculated with the values of Table 2 and Table 1; f max entering the 1pN gravitoelectric shifts is assumed to be close to 0, and its value has to be given in rad. Here, mas and GLYPH<22> as stand for milliarcseconds and microarcseconds, respectively.", '7.2. NEAR in the field of the Earth': "Here, the case of the spacecraft NEAR 9 (Prockter et al. 2002) when it approached the Earth is treated.  \nTable 4 lists the orbital parameters of such a spacecraft for the flyby of the Earth occurred on 23th January 1998. Table 5  \nTable 4. Orbital parameters of the geocentric trajectory of the probe NEAR referred to the International Celestial Reference Frame (ICRF) at epoch J2000.0. retrieved from the HORIZONS WEB interface maintained by the Jet Propulsion Laboratory (JPL) for the epoch 23th January 1998. The distance of closest approach turns out to be r p = 6 : 90 GLYPH<2> 10 3 km = 1 : 08 R GLYPH<8> e , while f 1 = 123 : 4 deg = 2 : 15 rad.  \ndisplays the nominal values for the pK orbital shifts experienced by NEAR; the relevant physical parameters of the Earth needed to calculate them were retrieved from Petit and Luzum (2010). It turns out that the nominal orbital shifts due to the Earth's first even zonal harmonic, being as large as ' 10 6 GLYPH<0> 10 8 GLYPH<22> as, neatly overwhelm the pN ones; su GLYPH<14> ce it to say that the LT displacements are as little as . 10 GLYPH<22> as, while the gravitoelectric ones are less than a few mas. However, the present-day relative uncertainty in determining J GLYPH<8> 2 from several dedicated satellite missions is 10  \nGLYPH<27> J GLYPH<8> 2 J GLYPH<8> 2 ' 10 GLYPH<0> 8 ; (186)  \nas it can be inferred by inspecting the latest Earth's gravity models retrievable from, e.g., the webpage 11 of the International Centre for Global Earth Models (ICGEM) maintained by the GeoForschungsZentrum (GFZ). Thus, the mismodelled classical shifts would be smaller than the nominal pN ones by about one order of magnitude (LT), or more (Schwarzschild). Also in the case of artificial probes like NEAR, the impact of the non-gravitational accelerations during flybys should be carefully investigated.  \nGiven the low altitude  \nh p ' 532 km (187)  \n2  \nTable 5. Nominal values of the pK orbital shifts of the probe NEAR calculated with the values of Table 4; f max entering the 1pN gravitoelectric shifts is assumed to be close to 0, and its value has to be given in rad. Here, mas and GLYPH<22> as stand for milliarcseconds and microarcseconds, respectively.  \nreached by NEAR at the perigee of its Earth's flyby, one may wonder what could be the impact of, say, the mismodeling GLYPH<27> J GLYPH<8> 4 of the second even zonal harmonic J GLYPH<8> 4 of the geopotential. It can be naively evaluated by multiplying the nominal values of the shifts due to J 2 listed in Table 5 by the following scaling factor  \na J GLYPH<8> 4 ' GLYPH<18> R a GLYPH<19> 2 GLYPH<27> J GLYPH<8> 4 J GLYPH<8> 2 : (188)  \nThe square of the ratio of the Earth's equatorial radius to the semimajor axis of NEAR, listed in Table 4, is of the order of  \nGLYPH<18> R a GLYPH<19> 2 ' 0 : 5 : (189)  \nAccording to the most recent Earth's gravity models listed by ICGEM, the formal uncertainty in J GLYPH<8> 4 is as little as  \nGLYPH<27> J GLYPH<8> 4 ' 10 GLYPH<0> 13 : (190)  \nThe nominal value of the first even zonal harmonic of our planet is  \nJ GLYPH<8> 2 ' 10 GLYPH<0> 4 : (191)  \nThus, the bias due to the imperfect knowledge of the Earth's octupole mass moment seems somewhat negligible with respect to the pN e GLYPH<11> ects of interest, shown in Table 5, since  \na J GLYPH<8> 4 ' 10 GLYPH<0> 10 : (192)  \nFinally, as a further source of potential systematic bias, it may be mentioned the coe GLYPH<14> cient of degree ' = 2 and order m = 2 of the geopotential accounting for the Earth's dynamical triaxiality. Its impact on spacecraft's flybys was analytically investigated by, e.g., Rappaport et al. (2001).", '8. Summary and conclusions': "Analytical expressions of the variations of all the Keplerian orbital elements of an otherwise unperturbed hyperbolic trajectory are obtained in full generality for some known post-Keplerian perturbing accelerations of both Newtonian and postNewtonian origin: that due to the primary's quadrupole mass moment, and, to the first post-Newtonian level, the general relativistic Schwarzschild and Lense-Thirring ones. In the case of the classical perturbation, the resulting formulas describe the shifts of the osculating Keplerian orbital elements. Instead, the general relativistic e GLYPH<11> ects are to be intended as written in terms of the nonosculating, contact elements, to the first post-Newtonian order, because their disturbing functions depend on the velocity of the test particle. The corrections required to have final expressions in terms of the osculating elements are explicitly calculated. It turns out that, for a generally asymmetric range of values of the true anomaly smaller than the maximum allowed one,  \nthe gravitoelectric variations of the contact eccentricity, argument of pericentre and mean anomaly at epoch are di GLYPH<11> erent from their osculating counterparts. Instead, the gravitomagnetic orbital shifts, written in terms of either the contact or the osculating elements, coincide if they are calculated over the entire unbound trajectory.  \nThe resulting formulas for the Newtonian and post-Newtonian shifts, all expressed with the osculating elements and valid for any spatial orientations of both the orbital plane and the spin axis of the source, are applied to ' Oumuamua, the first asteroid of interstellar origin which recently entered the inner regions of our solar system. While the quadrupole mass moment of the Sun and its angular momentum induce shifts as little as a few microarcseconds throughout the whole path, the post-Newtonian gravitoelectric e GLYPH<11> ect due to the solar mass amounts to less than a hundred milliarcseconds around the passage at perihelion. Actual usage of such kind of natural bodies as possible probes to perform gravitational experiments is likely prevented from the usually large non-conservative accelerations heavily perturbing their motions. As far as the Earth's flyby by the NEAR spacecraft is concerned, the size of its post-Newtonian disturbances is close to those of ' Oumuamua within about one order of magnitude. Instead, the perturbations due to the terrestrial oblateness are nominally much larger. However, the present-day relative uncertainty in our knowledge of the first even zonal harmonic of the geopotential is small enough to make the mismodeled part of such classical orbital shifts smaller than the corresponding relativistic ones.  \nThe situation may become more favorable in the case of numerous planetary or satellite flybys by the many artificial probes currently travelling through the solar system. Furthermore, dedicated gravity experiments may be suitably designed relying upon the analytical results obtained.  \nThe calculational approach adopted can straightforwardly be extended to any modified model of gravity as well.", 'Data availability': 'No new data were generated or analysed in support of this research.', 'Conflict of interest statement': 'I declare no conflicts of interest.', 'References': "- J. D. Anderson. Gravity-Assist Navigation. In J. H. Shirley and R. W. Fairbridge, editors, Encyclopedia of Planetary Sciences , pages 287-289. Chapman & Hall, 1997.\n- J. 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