A general relativistic gravitomagnetic effect on the orbital motion of a test particle around its primary induced by the spin of a distant third body
Abstract
We study a general relativistic gravitomagnetic 3rdbody effect induced by the spin angular momentum of a rotating mass orbited at distance by a local gravitationally bound restricted twobody system of size consisting of a test particle revolving around a massive body . We analytically work out the doubly averaged rates of change of the Keplerian orbital elements of the test particle by finding nonvanishing longterm effects for the inclination , the node and the pericenter . We numerically calculate their magnitudes for some astronomical scenarios in our solar system. For putative manmade orbiters of the natural moons Enceladus and Europa in the external fields of Saturn and Jupiter, the relativistic precessions due to the angular momenta of the gaseous giant planets can be as large as . The effects induced by the Sun’s angular momentum on artificial probes of Mercury and the Earth are at the level of .
keywords Experimental studies of gravity; Experimental tests of gravitational theories; Satellite orbits
1 Introduction
Let us consider a local gravitationally bound restricted twobody system composed by a test particle orbiting a planet^{1}^{1}1In general, it may be endowed with its own gravitational multipole moments and spin affecting the satellite’s motion with the standard Newtonian and postNewtonian orbital effects. of mass at distance , and a distant 3rd body X with mass and proper spin around which revolves at distance . In a kinematically rotating and dynamically nonrotating coordinate system (Brumberg & Kopeikin, 1989; Damour, Soffel & Xu, 1994) attached to in geodesic motion through the external spacetime deformed by the massenergy currents of X, assumed stationary in a kinematically and dynamically nonrotating coordinate system whose axes point towards the distant quasars (Brumberg & Kopeikin, 1989; Damour, Soffel & Xu, 1994), the planetocentric motion of the test particle is affected two peculiar postNewtonian 3rdbody effects: the timehonored De Sitter precession due to solely the mass (de Sitter, 1916; Schouten, 1918; Fokker, 1920), and a further gravitomagnetic shift due to . Our purpose is to analytically work out the latter effect in full generality, without any apriori simplifying assumptions concerning both the orbital configurations of the planetocentric satellite’s motion and the astrocentric motion of the planetsatellite system, and for an arbitrary orientation of in space. For previous, approximate calculation restricted to the orbital angular momentum of the Moon orbiting the Earth in the field of the rotating Sun, see Gill et al. (1992, Sec. 3.3.3).
The plan of the paper is as follows. In Section 2, we analytically work out the longterm rates of change of the Keplerian orbital elements of the test particle. Section 3 is devoted to the application of the obtained results to some astronomical scenarios in our solar system. We summarize our results and offer our conclusions in Section 4.
2 The doubly averaged satellite’s orbital precessions
The 3rdbody gravitomagnetic potential induced by the angular momentum of the external spinning object X on the planetary satellite is
(1) 
It comes from Eq. (2.19) of Barker & O’Connell (1979, p. 155) for the interaction potential energy of two spins of masses separated by a distance and moving with relative speed in the limit , and by assuming that the spin is the orbital angular momentum of the planetocentric satellite’s motion while is the spin angular momentum of the distant 3rd body X. Thus, in Eq. (2.19) of Barker & O’Connell (1979, p. 155) has to be identified with , and is nothing but the orbital angular momentum of the motion of around . It is interesting to note that, with the same identifications, and of Eqs. (2.17)(2.18) in Barker & O’Connell (1979, p. 155) yield the gravitoelectric De Sitter orbital precession for the planetocentric motion of the satellite and the gravitomagnetic LenseThirring effect for the astrocentric orbit of , respectively.
The perturbing potential to be inserted into the Lagrange planetary equations for the rates of change of the osculating Keplerian orbital elements of the test particle (Bertotti, Farinella & Vokrouhlický, 2003; Kopeikin, Efroimsky & Kaplan, 2011), obtained by doubly averaging Equation (1) with respect to for arbitrary orbital configurations of both the external body X and the test particle and for a generic orientation of in space, is
(2) 
with
(3) 
The resulting doubly averaged rates of change of the relevant Keplerian orbital elements turn out to be
(4)  
(5)  
(6)  
(7)  
(8) 
with
(9)  
(10)  
(11) 
We remark that Equations (4) to (11) are exact in both and in the sense that the loweccentricity approximation was not adopted in the calculation.
A more computationally cumbersome approach to obtain the same longterm rates of change of Equations (4) to (11) consists, first of all, in deriving a perturbing acceleration from Equation (1). By writing the Lagrangian per unit mass of a gravitationally bound restricted twobody system affected by a generic perturbing potential as
(12) 
the conjugate momentum per unit mass is, by definition,
(13) 
Thus,
(14) 
The Hamiltonian per unit mass is
(15) 
From the Hamilton equations of motion, it is
(16) 
Since , by comparing Equation (14) and Equation (16), it turns out that the perturbing acceleration is just
(17) 
In our specific case, since , we have
(18) 
Then, Equation (18) must be decomposed into its radial (), transverse () and outofplane () components, which are
(19)  
(20)  
(21) 
They have to be inserted into the righthandsides of the standard Gauss equations for the variation of the orbital elements (Poisson & Will, 2014) which, finally, are doubly averaged with respect to .
It should be noted that the averaging procedures previously outlined for both the Lagrange and Gauss equations hold if the orbital elements of the test particle can be considered as approximately constant during one orbital period . Indeed, the Keplerian ellipses were used as unperturbed, reference orbits for both the test particle and itself. This is usually the case in several scenarios of interests, some of which are treated in Section 3. Indeed, the frequencies of the longterm variations of the test particle’s orbital elements induced by the usual internal dynamics of , determined by, e.g., the quadrupole mass moments of and the standard postNewtonian effects like the gravitoelectric Einstein pericenter precession and the LenseThirring effect due to the planetary angular momentum , are often much smaller than the mean motion characterizing the orbital revolution of itself around X. However, there may be cases in which the average with respect to must take into account also for the relatively fast precessing ellipse of the test particle. For example, some lowaltitude Earth’s satellites are impacted by the Earth’s oblateness in such a way that the frequency of their node is comparable to, or even larger than the mean motion of the heliocentric terrestrial revolution.
3 Some potentially interesting astronomical scenarios
For the sake of simplicity, we will consider a circular orbit () for the test particle motion around .
In the case of a hypothetical orbiter of the Kronian natural satellite Enceladus in the external field of Saturn, Equations (6) to (7) and Equations (9) to (10), referred to the mean Earth’s equator at the reference epoch J2000.0 as reference plane, yield
(22)  
(23) 
with
(24)  
(25) 
Instead, if the mean ecliptic at the reference epoch J2000.0 is adopted as reference plane, we have
(26)  
(27) 
with
(28)  
(29) 
By looking at a putative orbiter of the Jovian natural satellite Europa in the external field of Jupiter, we have
(30)  
(31) 
with
(32)  
(33) 
and
(35)  
(36) 
with
(37)  
(38) 
We considered just Enceladus and Europa because they are of great planetological interest in view of the possible habitability of their oceans beneath their icy crusts (Lunine, 2017). As a consequence, they are the natural targets of several concept studies and proposals for dedicated missions to them, including also orbiters (Razzaghi et al., 2008; Spencer & Niebur, 2010; MacKenzie et al., 2016; Verma & Margot, 2018; Sherwood et al., 2018).
In the case of an artificial satellite orbiting a planet in the field of the Sun, the effects are much smaller. For an Earth’s spacecraft, we have
(39)  
(40) 
with
(41)  
(42) 
and
(43)  
(44) 
with
(45)  
(46) 
For a probe orbiting Mercury one gets
(47)  
(48) 
with
(49)  
(50) 
and
(51)  
(52) 
with
(53)  
(54) 
4 Summary and overview
We analytically worked out the general relativistic gravitomagnetic longterm rates of change of the relevant Keplerian orbital elements of a test particle orbiting a primary at distance from it which, in turn, moves in the external spacetime deformed by the spin angular momentum of a distant () 3rd body X with mass . We did not assume any preferred orientation for the spin axis of the external body; moreover, we did not make simplifying assumptions pertaining the orbital configurations of both the ’s satellite and of itself in its motion around . Thus, our calculation have a general validity, being applicable to arbitrary astronomical systems of potential interest. It turns out that, by doubly averaging the perturbing potential employed in the calculation with respect to the orbital periods of both and , the semimajor axis and the eccentricity do not experience longterm variations, contrary to the inclination of the orbital plane, the longitude of the ascending node and the argument of pericenter . While the gravitomagnetic rates and are harmonic signatures characterized by the frequency of the possible variation of the node , induced by other dominant perturbations like, e.g., the Newtonian quadrupole mass moment of the satellite’s primary , the gravitomagnetic node rate exhibits also a secular trend in addition to a harmonic component with the frequency of the node itself.
The Sun’s angular momentum exerts very small effects on spacecraft orbiting Mercury () and the Earth (). Instead, the angular momenta of the gaseous giant planets like Jupiter and Saturn may induce much larger perturbations of the orbital motions of hypothetical anthropogenic orbiters of some of their major natural moons like, e.g., Europa () and Enceladus (). Such natural satellites have preeminent interest in planetology making them ideal targets for future, dedicated spacecraftbased missions which may be opportunistically exploited to attempt to measure such relativistic effects as well.
References
 Barker & O’Connell (1979) Barker B. M., O’Connell R. F., 1979, Gen. Relat. Gravit., 11, 149
 Bertotti, Farinella & Vokrouhlický (2003) Bertotti B., Farinella P., Vokrouhlický D., 2003, Physics of the Solar System. Kluwer Academic Press, Dordrecht
 Brumberg & Kopeikin (1989) Brumberg V. A., Kopeikin S. M., 1989, Nuovo Cimento B, 103, 63
 Damour, Soffel & Xu (1994) Damour T., Soffel M., Xu C., 1994, Phys. Rev. D, 49, 618
 de Sitter (1916) de Sitter W., 1916, Mon. Not. Roy. Astron. Soc., 77, 155
 Fokker (1920) Fokker A. D., 1920, Versl. Kon. Ak. Wet., 29, 611
 Gill et al. (1992) Gill E., Soffel M., Ruder H., Schneider M., 1992, Relativistic Motion of Gyroscopes and Space Gradiometry. Deutsche Geodätische Kommission, München
 Kopeikin, Efroimsky & Kaplan (2011) Kopeikin S., Efroimsky M., Kaplan G., 2011, Relativistic Celestial Mechanics of the Solar System. WileyVCH, Weinheim
 Lunine (2017) Lunine J. I., 2017, Acta Astronaut., 131, 123
 MacKenzie et al. (2016) MacKenzie S. M. et al., 2016, Adv. Space Res., 58, 1117
 Petit, Luzum & et al. (2010) Petit G., Luzum B., et al., 2010, IERS Technical Note, 36, 1
 Poisson & Will (2014) Poisson E., Will C. M., 2014, Gravity. Cambridge University Press, Cambridge
 Razzaghi et al. (2008) Razzaghi A. I., di Pietro D. A., Quinn D. A., SimonMiller A. A., Tompkins S. D., 2008, in American Institute of Physics Conference Series, Vol. 969, Space Technology and Applications International ForumSTAIF 2008, ElGenk M. S., ed., pp. 388–395
 Schouten (1918) Schouten W. J. A., 1918, Versl. Kon. Ak. Wet., 27, 214
 Seidelmann et al. (2007) Seidelmann P. K. et al., 2007, Celestial Mechanics and Dynamical Astronomy, 98, 155
 Sherwood et al. (2018) Sherwood B., Lunine J., Sotin C., Cwik T., Naderi F., 2018, Acta Astronaut., 143, 285
 Soffel et al. (2003) Soffel M. et al., 2003, AJ, 126, 2687
 Spencer & Niebur (2010) Spencer J., Niebur C., 2010, Planetary Science Decadal Survey. Enceladus Orbiter. National Aeronautics and Space Administration
 Verma & Margot (2018) Verma A. K., Margot J.L., 2018, Icarus, 314, 35
Appendix A Notations and definitions
Here, some basic notations and definitions pertaining the restricted twobody system moving in the external field of the distant 3rd body X considered in the text are presented. For the numerical values of some of them, see Tables 1 to 2.

Newtonian constant of gravitation

speed of light in vacuum

mean obliquity

mass of the distant 3rd body X (a star like the Sun or a planet like, e.g., Jupiter or Saturn)

gravitational parameter of the 3rd body X

magnitude of the angular momentum of the 3rd body X

spin axis of the 3rd body X in some coordinate system

right ascension (RA) of the 3rd body’s spin axis

declination (DEC) of the 3rd body’s spin axis

component of the 3rd body’s spin axis w.r.t. the reference axis of an equatorial coordinate system

component of the 3rd body’s spin axis w.r.t. the reference axis of an equatorial coordinate system

component of the 3rd body’s spin axis w.r.t. the reference axis of an equatorial coordinate system

position vector towards the 3rd body X

distance of to the 3rd body X

versor of the position vector towards the 3rd body X

semimajor axis of the orbit about the 3rd body X

mean motion of the orbit about the 3rd body X

orbital period of the orbit about the 3rd body X

eccentricity of the orbit about the 3rd body X

inclination of the orbital plane of orbit about the 3rd body X to the reference plane of some coordinate system

longitude of the ascending node of the orbit about the 3rd body X referred to the reference plane of some coordinate system

mass of the primary (planet or planetary natural satellite) orbited by the test particle and moving in the external field of the 3rd body X

gravitational parameter of the primary orbited by the test particle and moving in the external field of the 3rd body X

angular momentum of the primary

zonal multipole moments of the classical gravitational potential of the primary

position vector of the test particle with respect to its primary

magnitude of the position vector of the test particle

velocity vector of the test particle

orbital angular momentum per unit mass of the test particle

semimajor axis of the test particle’s orbit

Keplerian mean motion of the test particle’s orbit

orbital period of the test particle’s orbit

eccentricity of the test particle’s orbit

inclination of the orbital plane of the test particle’s orbit to the reference plane of some coordinate system

longitude of the ascending node of the test particle’s orbit referred to the reference plane of some coordinate system
Appendix B Tables
Parameter  Units  Numerical value 

deg  
deg  
km  
deg  
deg  
deg  
deg  
d  
deg  
deg  
km  
deg  
deg  
deg  
deg  
d 
Parameter  Units  Numerical value 

deg  
deg  
au  
deg  
deg  
deg  
deg  
yr  
au  
deg  
deg  
deg  
deg  
yr 