Eur. Phys. J. C (2019) 79:816 https://doi.org/10.1140/epjc/s10052-019-7337-8

Regular Article - Theoretical Physics

On the mean anomaly and the mean longitude in tests of post-Newtonian gravity

Lorenzo Iorioa Ministero dell’Istruzione, dell’Università e della Ricerca (M.I.U.R.)-Istruzione, Viale Unità di Italia 68, 70125 Bari, BA, Italy

Received: 2 July 2019 / Accepted: 21 September 2019 / Published online: 4 October 2019 © The Author(s) 2019

Abstract The distinction between the mean anomaly M(t) with, e.g., Earth’s artificial satellites, Solar system’s plan- and the mean anomaly at epoch η, and the mean longitude ets and other astrophysical binaries, there is a certain con- l(t) and the mean longitude at epoch  is clarified in the fusion in the literature about the possible use of the mean context of a their possible use in post-Keplerian tests of anomaly M (t) as potential observable in addition to the gravity, both Newtonian and post-Newtonian. In particular, widely inspected argument of pericentre ω and, to a lesser the perturbations induced on M(t), η, l(t),  by the post- extent, longitude of the ascending node . Indeed, it is as Newtonian Schwarzschild and Lense–Thirring fields, and the if some researchers, including the present author, who have classical accelerations due to the atmospheric drag and the tried to compute perturbatively the mean rate of change of oblateness J2 of the central body are calculated for an arbi- the mean anomaly in excess with respect to the Keplerian trary orbital configuration of the test particle and a generic case due to some pN accelerations were either unaware of orientation of the primary’s spin axis Sˆ. They provide us the fact that what they, actually, calculated was the secular with further observables which could be fruitfully used, e.g., precession of the mean anomaly at the epoch η, or they sys- in better characterizing astrophysical binary systems and in tematically neglected a potentially non-negligible contribu- more accurate satellite-based tests around major bodies of tion to the overall change of the mean anomaly induced indi- the Solar System. Some erroneous claims by Ciufolini and rectly by the semimajor axis a through the mean motion nb. Pavlis appeared in the literature are confuted. In particular, Such a confusion has produced so far some misunderstand- it is shown that there are no net perturbations of the Lense– ing which led, e.g., to unfounded criticisms about alleged Thirring acceleration on either the semimajor axis a and the proposals of using the mean anomaly, especially in the case mean motion nb. Furthermore, the quadratic signatures on of man-made spacecraft orbiting the Earth, or even uncor- M(t) and l(t) due to certain disturbing non-gravitational rect evaluations of the total pN effects sought. An example accelerations like the atmospheric drag can be effectively that sums up well the aforementioned confusion and mis- disentangled from the post-Newtonian linear trends of inter- understanding, even in the peer-reviewed literature, is the est provided that a sufficiently long temporal interval for the following one. Ciufolini and Pavlis [6] wrote “[…] one of data analysis is assumed. A possible use of η along with the the most profound mistakes and misunderstandings of Iorio longitudes of the ascending node  in tests of general rela- (2005) is the proposed use of the mean anomaly of a satel- tivity with the existing LAGEOS and LAGEOS II satellites lite to measure the Lense–Thirring effect […] This is simply is suggested. a nonsense statement: let us, for example, consider a satel- lite at the LAGEOS altitude, the Lense–Thirring effect on its mean longitude is of the order of 2 m/year, however, the 1 Introduction mean longitude change is about 1.8 × 1011 m/year. Thus, from Kepler’s law, the Lense–Thirring effect corresponds In regard to possible tests of post-Newtonian (pN) features to a change of the LAGEOS semi-major of less than 0.009 of general relativity1 and of alternative models of gravity cm! Since, even a high altitude satellite such as LAGEOS showed a semimajor axis change of the order of 1 mm/day, 1 For a recent overview of the current status and challenges of the due to atmospheric drag and to the Yarkoski–Rubincam effect Einsteinian theory of gravitation, see, e.g., Debono and Smoot [8], and references therein. (because of atmospheric drag, the change of semimajor axis and mean motion is obviously much larger for lower altitude a e-mail: [email protected] 123 816 Page 2 of 14 Eur. Phys. J. C (2019) 79 :816 satellites), and since the present day precision of satellite anomaly f the other two anomalies. In the unperturbed Kep- laser ranging is, even in the case of the best SLR stations, lerian case, the mean anomaly is defined as of several millimeters, it is a clear nonsense to propose a . test of the Lense–Thirring effect based on using the mean M(t) = η + nb (t − t0) , (1) anomaly of any satellite, mean anomaly largely affected by 2 non-conservative forces.” It is difficult to understand what where η is the mean anomaly at the reference epoch t0, and is the target of the arrows by Ciufolini and Pavlis [6] since  the mean anomaly is not even mentioned in the published μ nb = (2) version of the criticized paper by the present author, not to a3 mention any explicitly detailed proposal to use it. Be that as . μ = it may, in the following, we will show that, actually, using is the Keplerian mean motion. In Eq. (2), GM is the the mean anomaly, or the mean longitude l (t),inpNtests gravitational parameter of the primary having mass M, while with artificial Earth’s satellites may be feasible, provided that G is the Newtonian constant of gravitation; in the following, μ = η certain non-gravitational perturbations are compensated by we will assume const. The mean anomaly at epoch some active drag-free mechanism. However, even in case of is one of the six Keplerian orbital elements parameterizing passive, geodetic satellites, we will show that, under certain the orbit of a test particle in space. In the unperturbed case, M( ) η conditions, it is possible to separate the relativistic linear t is a linear function of time t because both a and trends of interest from the unwanted parabolic signatures are constants of motion. If a relatively small perturbing pK η of non-conservative origin. Furthermore, the arguments pro- acceleration A is present, both a and are, in general, affected vided by Ciufolini and Pavlis [6] about the Lense–Thirring by it, becoming time-dependent. As a result, also the mean effect and the mean longitude are erroneous. Finally, the use motion is, in general, modified so that of the mean anomaly at epoch or of the mean longitude at → pert = +  ( ). epoch  is, in principle, possible even with passive, geodetic nb nb nb nb t (3) spacecraft like those of the LAGEOS family because they are, by construction, free from the aforementioned potential Thus, the perturbed mean anomaly is the sum of the now η( ) drawbacks exhibited by the mean anomaly and the mean lon- time-dependent mean anomaly at epoch t and a function ( ) gitude themselves, which was completely ignored or unrec- of time t whose derivative is equal to the (perturbed) mean ognized by Ciufolini and Pavlis [6]. motion, i.e.,  The paper is organized as follows. In Sect.2, we will t   Mpert( ) = η( ) + ( ) = η( ) + pert   review the basics of the mean anomaly, the mean anomaly at t t t t nb t dt epoch, the mean longitude, and the mean longitude at epoch t0 along with the calculation of their perturbations with respect = η + η(t)  to the purely Keplerian case in presence of a generic disturb- t   + ( − ) +   . ing post-Keplerian (pK) acceleration. Section 3 is devoted nb t t0 nb t dt (4) t0 to the calculation of the effects of some well-known pN accelerations (Schwarzschild and Lense–Thirring), while the The resulting change M of the mean anomaly with respect impact of the atmospheric drag and the oblateness of the pri- to the unperturbed case is, thus, mary are treated in Sect.4. The potential of a possible use of the mean anomaly at epoch in the ongoing tests with the M(t) = Mpert(t) − M(t) = η(t) + (t), (5) satellites LAGEOS and LAGEOS II is discussed in Sect. 5. Section6 summarizes our findings, and offers our conclu- where we defined sions.  t   .   (t) = nb t dt . (6) t0 2 The mean anomaly and the mean longitude as a function whose derivative yields the perturbation of the mean motion. In Eq. (6), the instantaneous shift of the mean 2.1 The mean anomaly motion due to the time-varying semimajor axis3 a is In the restricted two-body problem, the mean anomaly M(t) 2 The symbol η is used for the mean anomaly at epoch by Milani et al. is one of the three time-dependent fast angular variables [15]. In the notation by Brumberg [3], the mean anomaly is l,whilethe which, in celestial mechanics, can be used to characterize mean anomaly at epoch is l0. Kopeikin et al. [9] denote η as M0, while the instantaneous position of a test particle along its Kep- Bertotti et al. [2] adopt . lerian ellipse, being the eccentric anomaly E and the true 3 It should be recalled that we kept μ constant. 123 Eur. Phys. J. C (2019) 79 :816 Page 3 of 14 816

3 nb in non-trivial scenarios in which several perturbing accelera- nb(t) =− a(t) 2 a  tions of different nature act simultaneously on the test particle f =−3 nb da dt  inducing non-vanishing long-term effects on the semimajor  d f 2 a f0 dt d f axis a. Actually, we will show that it may not be the case 3 nb in practical satellite data reductions if certain conditions are =− a ( f0, f ) , (7) η 2 a fulfilled. On the contrary, the mean anomaly at the epoch , which is one of the six osculating Keplerian orbital elements so that in the perturbed restricted two-body problem, is not affected  by such drawbacks. As such, it can be safely used, at least in f   3 n  dt  (t) =− b a f , f d f =  ( f , f ) . principle, as an additional piece of information to improve 0  0 ω  2 a f0 d f some tests of pN gravity on the same foot of and .This (8) fact seems to have gone unnoticed so far in the literature, as in the case of Ciufolini and Pavlis [6]. The shifts a(t) and η(t) can be perturbatively calculated by evaluating the right-hand-sides of the Gauss equations for 2.2 The mean longitude their rates of change [2]    Similar considerations hold for the mean longitude l defined da 2 p = √ eAR sin f + AT , (9) as dt n 1 − e2 r b   dη 2 r . =− AR l(t) = + M(t), (13) dt n a a  b  1 − e2 r − −AR cos f + AT 1 + sin f , where nb ae p (10) . =  + ω (14) onto the unperturbed Keplerian.  ellipse. In Eqs. (9) and (10), = − 2 e is the eccentricity, p a 1 e is the semilatus rectum, is the longitude of pericentre. If a disturbing acceleration A = / ( + ) r p 1 e cos f is the (unperturbed) distance of the test is present, it can be expressed in terms of the mean longitude particle from the primary, and AR, AT are the projections of at epoch4  [2,3,15,19]as the perturbing pK acceleration A onto the radial and trans-  verse directions, respectively. The derivative of t with respect t   pert   to f entering Eqs. (7) and (8) is, up to terms of the first order l (t) = (t) + nb t dt in the perturbing acceleration A, t0  t   =  + ( ) +   , 2 t nb t dt (15) dt r t  √ + O (A) . (11) 0 d f μ p so that its shift is Depending on the disturbing acceleration, (t) is linear in time if the average over an orbital period P of its rate of b l(t) = (t) + (t). (16) change    + π ( ) · 3 n2 f0 2   dt The shift t of the mean longitude at epoch can be worked (t) =− b a f , f  f  π 0  d (12) out by means of the Gauss equation for its variation [2] 4 a f0 d f   is constant. Otherwise, it may exhibit a more complex tempo- d 2 r e2 d =− AR + √ ral pattern, as when the semimajor axis a undergoes a secular + − 2 dt nb a a 1 1 e dt change due to, e.g., some non-gravitational perturbing accel-   2 I d erations as in artificial satellites’ dynamics. In general, the + 2 1 − e2 sin , (17) 2 dt calculation of (t) is rather cumbersome since it involves two integrations. Moreover, it depends on f . 0 where Bertotti et al. [2] From such considerations it follows that, at first sight, using the mean anomaly M(t) may not be a wise choice because of the disturbances introduced by (t), especially 4 It is more suited than η at low orbital inclinations [2]. 123 816 Page 4 of 14 Eur. Phys. J. C (2019) 79 :816

   d 1 r 3.1.1 The shift (t) due to the variation of the mean motion = √ AN sin u, (18) dt n a 1 − e2 sin I a √b d 1 − e2 r By using Eq. (21)inEq.(7) yields = −AR cos f + AT 1 + sin f dt nb ae p 3 e μ n (cos f − cos f ) n ( f , f ) =− b   0  b 0 2 2 I d 4 c2 a 1 − e2 + 2sin . (19)   2 dt × 4 −7 + 3 ζ + e2 (−3 + 4 ζ ) In Eq. (18), AN is the projection of the perturbing accelera- +e [e ζ cos 2 f + 4 (−5 + 4 ζ ) cos f0 tion A onto the out-of-plane direction, while +2 cos f (−10 + 8 ζ + e ζ cos f0) .  = ω + u f (20) +e ζ cos 2 f0 . (25) is the argument of latitude. From it, the rate of change of (t) averaged over one orbital period Pb can be straightforwardly worked out as η( ), ( ), ( ) 3 The secular rates of change of t t t for    + · 1 t0 Pb d(t) some pN accelerations (t) = dt Pb t0 dt  + π Here, we will preliminarily look at the effects due to the n f0 2 dt = b n ( f , f ) f, standard general relativistic pN accelerations induced by the π b 0 d (26) 2 f0 d f static, gravitoelectric (Schwarzschild, Sect. 3.1) and station- ary, gravitomagnetic (Lense–Thirring, Sect.3.2) components where of the spacetime of an isolated rotating body. We will not   restrict to almost circular orbits; furthermore, we will allow nb dt nb d dt 1 d nb = = the primary’s spin axis Sˆ, entering the Lense–Thirring accel- 2π d f 2π dt d f Pb d f e μ n ( f − f ) eration, to assume any orientation in space. =− 3 √b cos cos 0 π 2 − 2 ( + )2 8 c a 1 e 1 e cos f 3.1 The 1pN gravitoelectric Schwarzschild-like × 4 −7 + 3 ζ + e2 (−3 + 4 ζ ) acceleration +e [e ζ cos 2 f + 4 (−5 + 4 ζ ) cos f0 To the first pN order (1pN), the relative acceleration for two +2 cos f (−10 + 8 ζ + e ζ cos f )  0 pointlike bodies of masses mA, mB separated by a distance r and moving with relative velocity v is [7,19] +e ζ cos 2 f0 . (27)  μ μ 3 A1pN = tot ( + ζ ) tot − ( + ζ ) 2 + ζ 2 rˆ From the analytical expression of the right-hand-side of 2 2 4 2 1 3 v vr c r  r 2 Eq. (27), it turns out that the true anomaly f , and, thus, also the time t, appears only in trigonometric functions. This + (4 − 2 ζ ) v v , (21) r implies that, in this case, (t) does not exhibit a polyno- mial temporal pattern, being, at most, linear in t provided where c is the speed of light in vacuum, that Eq. (26) is not vanishing. Note also the dependence of Eq. (27)on f0. We are not able to analytically calcu- μ = ( + ) tot G mA mB (22) late Eq. (26) unless a power expansion in e of Eq. (27) is made. Nonetheless, it is possible to perform a numeri- is the total gravitational parameter of the binary system, cal integration of Eq. (26) for given values of the physi- . cal and orbital parameters entering it without any restric- v = v · rˆ (23) r tion on e. We successfully tested it for a fictitious can- nonball geodetic satellite moving along an eccentric orbit, is the the radial velocity of the relative orbital motion, and whose arbitrarily chosen physical and orbital parameters are . displayed in Table1, by numerically integrating its equa- ζ = mAmB , ≤ ζ ≤ 1. 2 0 (24) tions of motion in rectangular Cartesian coordinates, and (mA + mB) 4 by numerically performing the integral of Eq. (26) with In Sects.3.1.1 and 3.1.2, we will work out the effect of Eq. (27). Figure1 displays the plot of Eq. (27), in mil- Eq. (21)on, and η and , respectively. liarcseconds per year mas year−1 , for the orbital param- 123 Eur. Phys. J. C (2019) 79 :816 Page 5 of 14 816

Table 1 Orbital and physical configuration of a fictitious terrestrial −16 −3 geodetic satellite. Since it is ρLARES = 5.96 × 10 kg m [16], −18 −3 and ρLAGEOS = 6.579 × 10 kg m [14], we, first, used them −1 in ρ(h) = ρ0 exp − (h − h0) ,whereρ0 and h0 are, in gen- eral, referred to some reference height, to determine in the case h0 = hLARES, h = hLAGEOS. Then, we used the so obtained char- acteristic length LR/L = 999.51 km, valid in the range hLARES = 1, 442.06 km < h < hLAGEOS = 5, 891.96 km, to calculate ρmax for our orbital geometry. Instead, the value ρmin is just a guess which may be even conservative. The values of the satellite’s physical parameters were taken from [16](m, ,CD). Orbital and physical Numerical value Units parameter

Mass (LARES) m 386.8 kg Area-to-mass ratio 2.69 × 10−4 m2 kg−1 (LARES)

Neutral drag coefficient CD 3.5 – (LARES) Semimajor axis a 12,500 km

Orbital period Pb 3.86 h Orbital eccentricity e 0.36 –

Perigee height hmin 1,621.86 km

Apogee height hmax 10,621.9 km Orbital inclination I 63.43 ◦ Argument of perigee ω 0 ◦ Period of the node P −1.76 year Period of the perigee Pω −2903.62 year Neutral atmospheric density 4.71 × 10−16 kg m−3 Fig. 1 Upper panel: plot of Eq. (27), computed for the orbital con- ◦ at perigee ρmax figuration of Table1 and with f0 = 228 , over a full orbital cycle of −1 Neutral atmospheric density 1 × 10−20 kg m−3 the true anomaly f . Its area, giving Eq. (26) in mas year , amounts to 2, 326.6 mas year−1. Lower panel: numerically produced time series, in at apogee ρmin mas, of (t) over 1 year obtained by integrating the equations of motion Characteristic atmospheric 836.34 km in rectangular Cartesian coordinates for the fictitious Earth’s satellite

length scale of Table 1. The 1pN gravitoelectric Schwarschild-like acceleration was added to the Newtonian monopole. As initial value for the true anomaly, ◦ f0 = 228 was adopted. The slope of the linear trend amounts just to the area under the curve in the upper panel eters of Table1, and the numerically produced time series of (t), in mas, over 1 year for the same orbital config- to either the duration of the typical data analyses or to the uration; the agreement between the slope of (t) and the observational accuracy. Indeed, in all such cases, the per- area under the curve of Eq. (27) is remarkable. From Fig. 1, turbed evolution of the mean anomaly can, in principle, be (˙ ) it can be noted that, as expected, the 1pN Schwarzschild- monitored as well, and t may represent an important M( ) like acceleration induces a secular variation on (t) which contribution to the overall long-term rate of change of t . has to be added to those affecting η and  displayed in Suffice it to say that, in the case of Mercury and the Sun, it Sect. 3.1.2. is The opportunity offered by the exact expression of   · (˙ ) − Eq. (27) to calculate t as per Eq. (26) is important also  = 210.3arcseccty 1. (28) in astronomical and astrophysical scenarios, like the almost circular orbital motions of the major bodies of our solar sys- tem and the much more eccentric ones of various types of 3.1.2 The mean anomaly at epoch η and the mean binary systems (extrasolar planets, binary stars, binary pul- longitude at epoch  sars hosting at least one emitting neutron star, stellar systems revolving around supermassive galactic black holes, etc.), in The Gauss equations for the variation of η and  (Eqs. (10) which secular variations of the semimajor axis a-or even and (17)) allow to straightforwardly work out their secular of the masses involved-are absent or negligible with respect rates of change which turn out to be 123 816 Page 6 of 14 Eur. Phys. J. C (2019) 79 :816

 √  √    μ − + − 2 + − − 2 ζ is the unit vector directed along the orbital angular momen- · nb 15 6 1 e 9 7 1 e η = √ , (29) tum along the out-of-plane direction, while c2 a 1 − e2  √  √    μ − + − 2 + 2 ( − ζ) + − − 2 ζ · nb 9 15 1 e e 6 7 7 9 1 e  =−   . mˆ = {− cos I sin , cos I cos , sin I } (39) c2 a 1 − e2 (30) is the unit vector directed transversely to the line of the nodes in the orbital plane. In the case of an Earth’s satellite, by They were confirmed by a numerical integration of the equa- assuming, as usual, an equatorial coordinate system with its tions of motion in the case of the satellite’s orbital config- reference z axis directed along Sˆ,Eq.(36) reduces to uration of Table1 which returned linear times series whose   · ( − ) slopes agree with Eqs. (29) and (30). In the case of Mercury  = 2GS 1 3 cos I .   / (40) and the Sun, Eqs. (29) and (30) yield c2a3 1 − e2 3 2   · − η =−127.986 arcsec cty 1, (31) Equations (35) and (40) show that the claim by Ciufolini and   Pavlis [6] “[…] let us, for example, consider a satellite at · −  =−85.004 arcsec cty 1. (32) the LAGEOS altitude, the Lense–Thirring effect on its mean longitude is of the order of 2 m/year, […]" is wrong. Indeed, 3.2 The 1pN gravitomagnetic Lense–Thirring acceleration the gravitomagnetic linear shift corresponding to Eq. (40) amounts to 3.68 m year−1 for LAGEOS; it is an enormous In the case of the 1pN gravitomagnetic Lense–Thirring accel- discrepancy with respect to the statement by Ciufolini and eration [19] induced by the spin dipole moment of the central Pavlis [6] since the present-day accuracy in reconstructing mass, i.e. its proper angular momentum S, on a test particle the orbits of the laser-ranged satellites of the LAGEOS type  − . orbiting it with velocity v is notoriously at the 1 0 5cmlevel.  2 GS ALT = 3 ξ rˆ × v + v × Sˆ , (33) η( ), ( ), ( ) c2 r 3 4 The secular rates of change of t t t for some Newtonian perturbing accelerations it turns out that Here, we will deal with the impact of the oblateness of the primary (Sect. 4.1), whose spin axis Sˆ is assumed arbitrarily nb ( f0, f ) = 0(34) oriented in space, and of the atmospheric drag (Sect. 4.2). The small eccentricity approximation for the satellite’s orbit for an arbitrary orientation of the body’s spin axis Sˆ in space. will not be adopted. Such classical accelerations represent Thus, it is two of the most important sources of systematic errors in accurate tests of pN gravity with artificial satellites. On the ( ) = . t 0 (35) other hand, they can be considered interesting in themselves if one is interested in better characterizing the shape and the It implies that the claims by Ciufolini and Pavlis [6] about an inner mass distribution of the primary like, e.g., a star, at alleged non-vanishing perturbing effect of the gravitomag- hand, and the properties of the atmosphere of the orbited netic field of the Earth on both the semimajor axis a and the planet. mean motion nb of a satellite are, in fact, erroneous for any spacecraft. 4.1 The quadrupole mass moment J2 Moreover, it is also   To the Newtonian level, the external potential of an oblate · η = 0, (36) body at the outside position r is    ˆ   2G S · −2h + (csc I − cot I ) mˆ μ R 2 · ( ) = +  =− − e P (ξ) ,  =   . (37) U r U0 U2 1 J2 2 (41) 3/2 r r c2a3 1 − e2 where J is the first even zonal harmonic coefficient of the ˆ 2 for any S as well. In Eq. (36), multipolar expansion of its classical gravitational potential, . hˆ = {sin I sin , − sin I cos , cos I } (38) ξ = Sˆ · rˆ (42) 123 Eur. Phys. J. C (2019) 79 :816 Page 7 of 14 816 is the cosine of the angle between the primary’s spin axis and the particle’s position, and

3ξ 2 − 1 P (ξ) = (43) 2 2 is the Legendre polynomial of degree 2. The Newtonian acceleration due to J2 experienced by a test particle orbit- ing the distorted axisymmetric primary is

NJ2 =−∇ A UJ2   3 μ R2 J = e 2 5 ξ 2 − 1 rˆ − 2 ξ Sˆ . (44) 2 r 4 In Sects.4.1.1 and 4.1.2, we will work out its impact on (t), and η and , respectively.

4.1.1 The shift (t) due to the variation of the mean motion

It turns out that   ·  = 0, (45) so that (t), which depends on f0, is linear in time. It is not possible to explicitly display the analytical expression which we obtained for (1/Pb) d/d f in the case of an arbitrary ˆ Fig. 2 Upper panel: plot of Eq. (46), computed for the orbital con- orientation of S in space because of its cumbersomeness. ◦ figuration of Table1 and with f0 = 228  , over a full orbital cycle of However, it can be fruitfully used with, e.g., any astronom- · −1 ical binary systems since, in general, their spin axes are not thetrueanomaly f . Its area, giving  in mas year , turns out to aligned with the line of sight which, usually, is assumed as be equal to 3.8 × 107 mas year−1. Lower panel: numerically produced reference z axis of the coordinate systems adopted. In regard time series, in mas, of (t) over 1 year obtained by integrating the equa- to an Earth’s satellite, whose motion is customarily studied tions of motion in rectangular Cartesian coordinates for the fictitious Earth’s satellite of Table 1. The Newtonian acceleration of Eq. (44) due in an equatorial coordinate system whose reference z axis is to J2 was added to the Newtonian monopole. As initial value for the ˆ ◦ aligned with S,wehave true anomaly, f0 = 228 was adopted. The slope of the linear trend amounts just to the area under the curve in the upper panel nb dt nb d dt 1 d nb = = 2π d f 2π dt d f Pb d f 2 − 3 nb Re J2 J , + 3 3 2   / 24 e cos f cos 2u sin I 64 π a2 1 − e2 3 2 (1 + e cos f )2    + 3 cos f e 4 + e2 (1 + 3 cos 2I ) (46) +24 e sin2 I cos 2u with     + 3 −4 2 + 3 e2 + 3 e2 cos 2 f sin 2 f J =−e 12 cos f0 + e (−6 cos 2 f − e cos 3 f  + ( + )3 2 ω. 3 8 1 e cos f0 sin 2 f0 sin I sin 2 (47) +4 e cos f0 + 6 cos 2 f0 (1 + 3 cos 2I )    2 − 3 −4 2 + 3 e cos 2 f + 8 cos 2 f0  The numerical value of the area under the plot of Eq. (46), +e 12 (cos f0 + cos 3 f0)    depicted in the upper panel of Fig. 2, is confirmed by the 3 ( ) + e −6 cos 4 f + 4 3 + 2 e cos f0 cos 2 f0 time series for t produced by numerically integrating the  equations of motion of the fictitious satellite of Table 1, and 2 +6 cos 4 f0 sin I cos 2ω displayed in the lower panel of Fig.2. 123 816 Page 8 of 14 Eur. Phys. J. C (2019) 79 :816

4.1.2 The mean anomaly at epoch η and the mean where  is the Earth’s angular velocity. We will model the longitude at epoch  atmospheric density as

The Gauss equations for the variations of η and  (Eqs. (10) (r − r0) ρ(r) = ρ0 exp − , (54) and (17)) allow to straightforwardly obtain

      ρ 2 2 where 0 refers to some reference distance r0, while is the 2 ˆ ˆ ˆ   3 nb R J2 2 − 3 S · l + S · mˆ · e characteristic scale length. By assuming η = ,   / (48) 4 a2 1 − e2 3 2 ⎧     r = r = a (1 − e) , (55) ⎪ 2 2 0 min   ⎨⎪ 2 − 3 Sˆ · lˆ + Sˆ · mˆ · 3 n R2 J  = b e 2   2 ⎪ 3/2 can be determined as 4 a ⎩⎪ 1 − e2        ⎫ 2 ae 2 2 ⎪ =−  , 2 − 3 Sˆ · lˆ + Sˆ · mˆ − 2 Sˆ · hˆ Sˆ · mˆ (1 − cot I ) ⎬⎪ (56) ρmin +   , ln ρ 2 ⎪ max 1 − e2 ⎭⎪ (49) where

ρmin = ρ(rmax), (57) where lˆ = {cos , sin , 0} is the unit vector directed along ρmax = ρ(rmin) (58) the line of the nodes such that lˆ × mˆ = hˆ.AlsoEqs.(48) and (49) can be used with any astronomical binary system in are the values of the atmospheric density at the apogee and view of their generality. In the case of a coordinate system perigee heights, respectively. Table 1 shows the neutral atmo- with its reference z axis aligned with the body’s spin axis, spheric density at the perigee height chosen as inferred from as in the case of an Earth’s satellite referred to an equatorial existing data on LAGEOS and LARES. On the other hand, coordinate system, Eqs. (48) and (49) reduce to the values reported for the apogee are purely speculative and   should be regarded as subjected to huge uncertainties. Actu- · 3 n R2 J (1 + 3cos2I ) η = b e 2  , ally, even the density at a given height may not be regarded as 3/2 (50) 8 a2 1 − e2 truly constant because of a variety of geophysical phenomena  √  √    2 + − 2 − + + − 2 characterized by quite different time scales. Anyway, in order · 3 nb Re J2 3 1 e 4cosI 5 3 1 e cos 2I  =   . to have an order-of-magnitude evaluation of the perturbing 2 − 2 2 8 a 1 e action of Eq. (52) on the motion of the fictitious satellite of (51) Table1, we will make our calculation by keeping ρ0 fixed during one orbital period Pb. An exact analytical calcula- 4.2 The atmospheric drag tion. without recurring to any approximation in both e and ν = /nb is difficult. The atmospheric drag induces, among other things, a secular In Sects.4.2.1 and 4.2.2, we will calculate the impact of decrease of the semimajor axis a which, in turn, has an impact Eq. (52)on(t), and η and , respectively. on nb(t) and (t). For a cannonball geodetic satellite, the drag acceleration 4.2.1 The shift (t) due to the variation of the mean motion can be expressed as n (t) Let us, now, start to look at b by means of. Eq. (7). We 1 will show that it is linear in time because nb = 0. The AD =− CD ρV V. (52) 2 analytical expression of 1/Pb dnb/d f is   , ,ρ, In Eq. (52), CD V are the dimensionless drag coef- nb −3 nb da ficient of the satellite, its area-to-mass ratio, the atmospheric 2π 2 a d f √ density at its height, and its velocity with respect to the atmo- 3 C ρ( f ) n2 1 − e2 V ( f ) = D b sphere, respectively. In the following, we will assume that the π ( + )2  4 1 e cos f  atmosphere co-rotates with the Earth. Thus, V is   3/2 × 1 + 2 e cos f + e2 − ν 1 − e2 cos I , (59) V = v −  × r, (53) 123 Eur. Phys. J. C (2019) 79 :816 Page 9 of 14 816

Fig. 4 Numerically produced time series, in mas, of (t) over 1 year obtained by integrating the equations of motion in rectangular Cartesian coordinates for the fictitious Earth’s satellite of Table 1. The drag accel- eration of Eq. (52) was added to the Newtonian monopole. As initial ◦ valueforthetrueanomaly, f0 = 228 was adopted. The quadratic sig- nature is apparent, and its final value is in agreement with what expected from Fig. 3

 .  5 The fact that nb = 0 implies that nb is linear in time and, thus, (t) is quadratic. It is explicitly shown in Fig. 4 by the time series calculated for Eq. (8) from the same Fig. 3 Upper panel: plot of Eq. (59), computed for the orbital config- integration of the satellite’s equations of motion. uration of Table1 and with f = 228◦, over a full orbital cycle of the 0  .  It is an important feature because it allows to accurately true anomaly f . Its area, giving n in mas year−2, turns out to be b separate the unwanted parabolic signature due to the atmo- equal to 107, 217 mas year−2. Lower panel: numerically produced time −1  ( ) spheric drag from the relativistic trend of interest affecting series, in mas year ,of nb t over 1 year obtained by integrating M( ) ( ) the equations of motion in rectangular Cartesian coordinates for the the time series of t or l t , provided that a sufficiently fictitious Earth’s satellite of Table 1. The drag acceleration of Eq. (52) long time span is chosen for the data analysis. The same was added to the Newtonian monopole. As initial value for the true holds, in principle, also for any other perturbing acceleration = ◦ anomaly, f0 228 was adopted. The linear trend is apparent, and its of non-gravitational origin inducing a secular trend in the slope amounts just to the area under the curve in the upper panel satellite’s semimajor axis like, e.g., the Yarkovsky-Rubincam thermal effect. We numerically confirmed that by integrating the equations of motion of the fictitious satellite of Table 1 where including the 1pN Schwarzschild-like and the atmospheric   3/2 drag accelerations, and fitting a linear plus quadratic model 2 1 − e2 cos I V2 ( f ) = 1 − ν to the resulting time series of (t) over, say, 5 year for a given 1 + e2 + 2 e cos f     value of f0. As a result, we were able to accurately recover 3 1 − e2 3 + cos 2I + 2sin2 I cos 2u the slope of the relativistic secular signal. We successfully + ν2   . 4 (1 + e cos f )2 1 + e2 + 2 e cos f repeated it for different values of f0 as well. It turns out that (60) the longer the data span is, the more accurate the recovery of the linear signal. This suggests that, actually, also the mean M( ) ( ) Since it is not possible to analytically integrate Eq. (59) with anomaly t and the mean longitude l t may be fruitfully Eq. (60) in the most general case without recurring to approx- used in tests of pN gravity in the field of the Earth even with imations in e and ν, we will plot it as a function of f over a passive artificial satellites, contrary to the claims by Ciufolini ( ) full orbital cycle and integrate it numerically for the physical and Pavlis [6]. The dependence of t on f0 may even repre- and orbital parameters of Table 1. The upper panel of Fig. 3 sent an advantage to enhance the signal-to-noise ratio since, depicts Eq. (59), while the lower panel displays the time in principle, one can choose f0 in order to maximize the rel- series for nb(t) calculated from a numerical integration of the satellite’s equations of motion in rectangular Cartesian coordinates over 1 year. 5 Strictly speaking, it is, in general, true only for fast satellites orbiting in much less than a day, so that the term proportional to ν2 in Eq. (60), which contains ω, can be neglected. However, in the particular case of the fictitious satellite of Table 1, ω stays essentially constant because of the frozen perigee configuration. 123 816 Page 10 of 14 Eur. Phys. J. C (2019) 79 :816   · ativistic rate for  to be added to the further contribution     · · due to η ,  .

4.2.2 The mean anomaly at epoch η and the mean longitude at epoch 

About the secular rates of η and , the Gauss equations for their variations allow to obtain   2 η C ρ ( f ) n V ( f ) 1 − e2 sin f nb d = D b 2π d f 4π e (1 + e cos f )4   × 2 + 3e2 + 2e 2 + e2 cos f + e2 cos 2 f   3/2 −ν 1 − e2 (2 + e cos f ) cos I , (61)   2 n d CD ρ ( f ) nb V ( f ) 1 − e b =−  √  π 2 d f 8π 1 + 1 − e2 (1 + e cos f )4      × 4 e (1 + e cos f ) −1 + e2 1 + 1 − e2  +e 1 − e2 cos f sin f     Fig. 5 Plots of Eqs. (61)and(62), computed for the orbital configura- 2  = ◦ 2 2 tion of Table 1 and with f0 228 , over a full orbital cycle of the true −ν 1 − e 1 + 1 − e (1 − cos I ) sin 2u · · −  anomaly f . Their areas give η ,  in mas year 1. In this case, they −2 e cos I (2 + e cos f ) sin f . (62) vanish, as confirmed also by a numerical integration of the satellite’s equations of motion for the same physical and orbital parameters Since it is not possible to analytically integrate Eqs. (61) and (62) in an exact form, we, first, plot them as functions of f (much) worse than the mere formal, statistical sigmas of the over a full orbital cycle in Fig. 5 for the orbital configuration various global gravity field solutions6 releasing the experi- of Table 1, and, then, numerically  calculate  the areas under mentally estimated values of the geopotential’s parameters, · · their curves in order to obtain η ,  . does not yet allow to use the residuals of a single orbital ele- Also in this case, a numerical integration of the satellite’s ment separately. To circumvent such an issue, some strategies equations of motion turns out to confirm such results. involving the simultaneous use of more than one orbital ele- ment have been devised so far over the years: for a general overview, see, e.g., Renzetti [17], and references therein. To 5 Some possible uses with the LAGEOS and LAGEOS the benefit of the reader, we review here the linear combi- II satellites nation approach, which is an extension of the one proposed by Ciufolini [5] to test the gravitomagnetic field of the Earth As an illustrative example, here we will look at the possibility with artificial satellites of the LAGEOS family. In turn, it is of using the nodes  and the mean anomalies at epoch η of, a generalization of the strategy put forth, for the first time, say, the existing satellites LAGEOS and LAGEOS II in order by I.I. Shapiro [18] who, at that time, wanted to separate the to propose an accurate test of the 1pN Lense–Thirring effect Sun-induced 1pN gravitoelectric perihelion precession from exploiting their multidecadal data records. that due to the solar quadrupole mass moment J2 by using The availability of η in addition to  may be particularly other planets or highly eccentric asteroids. 7 κ(i), = , ,... important in view of the fact that the competing classical sec- By looking at N orbital elements i 1 2 N ular precessions due to the even zonals of low degree, which experiencing, among other things, classical secular preces- have just the same time signature of the gravitomagnetic ones 6 They are freely available on the Internet at the webpage of the Inter- of interest, are nominally several orders of magnitude larger national Centre for Global Earth Models (ICGEM), currently located than them; thus, the signal-to-noise ratio must be somehow at http://icgem.gfz-potsdam.de/tom_longtime. enhanced. The present-day level of actual mismodeling in 7 At least one of them must be affected also by the 1pN effect one the geopotential coefficients, which should be considered as is looking for. The N orbital elements κ(i) may be different from one 123 Eur. Phys. J. C (2019) 79 :816 Page 11 of 14 816 sions due to the even zonals of the geopotential, the following while the constant terms are the N orbital residuals N linear combinations can be written down   · (i) · (i) δκ , i = 1, 2,...N. (68)   N−1 ∂ κ · (i) " J2s μ1pN κ + δ J2s, i = 1, 2,...N. (63) 1pN ∂ J s=1 2s It turns out that, after some algebraic manipulations, the   dimensionless 1pN scaling parameter can be expressed · (i) They involve the 1pN averaged precessions κ as pre- as 1pN dicted by General Relativity and scaled by a multiplicative C parameter8 μ , and the errors in the computed secular node μ = δ . 1pN 1pN C (69) precessions due to the uncertainties in the first N − 1even 1pN zonals J2s, s = 1, 2,...N − 1, assumed as mismodeled through δ J2s, s = 1, 2,...N − 1. In the following and in In Eq. (69), the combination of the N orbital residuals AppendixA, we will use the shorthand   · N"−1 ∂ κ . · (1) · ( j+1) · . Cδ = δκ + c j δκ (70) κ = J . (64) j=1 ∂ J

for the partial derivative of the classical averaged precession − · is, by construction, independent of the first N 1 even zonals, κ with respect to the generic even zonal J of degree . being, instead, impacted by the other ones of degree > J Then, the N combinations of Eq. (63) are posed equal to the 2(N − 1) along with the non-gravitational perturbations and · (i) other possible orbital perturbations which cannot be reduced experimental residuals δκ , i = 1, 2,...N of each of the to the same formal expressions of the first N − 1 even zonal N orbital elements considered getting rates. On the other hand,

  N−1 · (i) · (i) " · (i) δκ = μ κ + κ δ , = , ,... .   N−1   1pN .2s J2s i 1 2 N (65) ( ) " ( j+ ) 1pN . · 1 · 1 s=1 C1pN = κ + c j κ (71) 1pN 1pN j=1 · (i) It should be recalled that, in principle, the residuals δκ account for the purposely unmodelled 1pN effect, the mis- combines the N 1pN orbital precessions as predicted by modelling of the static and time-varying parts of the geopo- General Relativity. The dimensionless coefficients c j , j = tential, and the non-gravitational forces. If we look at the 1pN 1, 2,...N − 1inEqs.(70) and (71) depend only on some scaling parameter μ1pN and the mismodeling in the even zon- of the orbital parameters of the satellite(s) involved in such als δ J2s, s = 1, 2,...N − 1 as unknowns, we can interpret a way that, by construction, Cδ = 0ifEq.(70) is calculated Eq. (65) as an inhomogenous linear system of N algebraic by posing equations in the N unknowns · (i) · (i) μ ,δJ ,δJ ...δJ ( − ), (66) δκ = κ. δ J, i = 1, 2,...N (72) # 1pN 2 $%4 2 N 1& N for any of the first N − 1 even zonals, independently of the whose coefficients are value assumed for its uncertainty δ J.   · (i) · (i) As far as the Lense–Thirring effect and the satellites κ , κ , = , ,... , = , ,... − , .2s i 1 2 N s 1 2 N 1 (67) LAGEOS and LAGEOS II are concerned, the linear combina- 1pN tion of the four experimental residuals δL,δLII,δηL,δηLII of the satellites’s nodes and mean anomalies at epoch suitably another belonging to the same satellite, or some of them may be iden- tical belonging to different spacecraft (e.g., the nodes of two different designed to cancel out the secular precessions due to the first vehicles). three even zonal harmonics J2, J4, J6 of the geopotential is 8 It is equal to 1 in the Einstein’s theory of gravitation, and 0 in the Newtonian one. In general, μ1pN is not necessarily one of the parameters of the parameterized post-Newtonian (PPN) formalism, being possibly L LII L LII a combination of some of them. Cδ = δ + c1 δ + c2 δη + c3 δη (73) 123 816 Page 12 of 14 Eur. Phys. J. C (2019) 79 :816 whose coefficients c1, c2, c3 are purposely constructed with instead, the difference C8,0 between the values of C8,0 from the results of Sect. A.1. They turn out to be Tongji-Grace02s and the zero-tide model ITU_GRACE16 [1], whose formal errors are comparable, is adopted as a mea- · L · L · LII · L · L · LII · L · LII · L sure of the actual uncertainty in the even zonal of degree 8, Dc =  η η − η  η −  η η 1 .2 .4 .6 .2 .4 .6 .2 .4 .6 the resulting mismodeled signal amounts to 2.1 mas year−1 · L · L · L · LII · L · L · LII · LII · L corresponding to a percent error in the Lense–Thirring com- + η. . η. + η. η. . − η. η. . , (74) 2 4 6 2 4 6 2 4 6 bined signature of 1.8%. · L · LII · LII · LII · L · LII · L · LII · LII =−  η +   η +  η  In fact, an accurate investigation, both analytical and Dc2 .2 .4 .6 .2 .4 .6 .2 .4 .6 numerical, of the perturbations on η induced by the main · LII · L · LII · LII · LII · L · LII · LII · L − η   −  η  + η   , .2 .4 .6 .2 .4 .6 .2 .4 .6 non-gravitational accelerations acting on the LAGEOS-type (75) satellites like, e.g., the direct solar radiation pressure, the · L · L · LII · L · L · LII · L · LII · L Earth’s albedo, the Earth’s direct infrared radiation pres- =− η  + η   +   η Dc3 .2 .4 .6 .2 .4 .6 .2 .4 .6 sure, the Earth’s Yarkovsky-Rubincam and Solar Yarkovsky- · LII · L · L · L · LII · L · LII · L · L Schach thermal effects, possible anisotropic reflectivity, etc. −   η − η   +  η  , .2 .4 .6 .2 .4 .6 .2 .4 .6 [10–13,16,20] is required to realistically assess the overall (76) error budget of the promising combination of Eq. (73). This is outside the scopes of the present paper. where the common denominator is

· L · LII · LII · LII · L · LII · L · LII · LII =η  η −  η η − η η  D .2 .4 .6 .2 .4 .6 .2 .4 .6 6 Summary and overview · LII · L · LII · LII · LII · L · LII · LII · L + η η  +  η η − η  η . .2 .4 .6 .2 .4 .6 .2 .4 .6 In presence of Newtonian, general relativistic 1pN or mod- (77) ified gravity-induced disturbing accelerations, the shifts M(t) and l(t) of the mean anomaly M(t) and the mean Their numerical values, computed with the satellites’ orbital longitude l(t) with respect to their Keplerian linear trends elements inserted in Eqs. (A6)–(A11), are are, in general, due to the perturbations η(t) and (t) of the mean anomaly at epoch η and mean longitude at epoch c = 2.77536, (78) 1 , and the change nb(t) in the mean motion nb which, in c2 =−2.46439, (79) some cases, can induce a quadratic shift (t) in M(t) and l(t) depending on the true anomaly at epoch f . c3 = 10.9532. (80) 0 In the case of an Earth’s artificial satellite, the atmo- ( ) Thus, the predicted combined Lense–Thirring signature is spheric drag affects t quadratically; nonetheless, the non- Newtonian linear trends of interest may be effectively sepa- · L · LII · L · LII rated from such a potentially competing aliasing effect if a C =  +  + η + η LT LT c1 LT c2 LT c3 LT sufficiently long time span for the data analysis is adopted. − = 118.04 mas year 1. (81) Thus, also M(t) and l(t) can, in principle, be employed in gravity tests even with passive geodetic satellites, not to men- The combination of Eq. (73) is mainly affected by the orbital tion the use of drag-free apparatuses. If, instead, η and  are precessions induced by the fourth even zonal harmonic J8 of adopted, such an issue is a-priori circumvented because they the geopotential. The resulting mismodeled combined signal are not impacted by the possible change in the mean motion can be evaluated by means of Eqs. (A12) and (A13) along nb. Since η and  undergo secular precessions due to the , = , ,... with some measure of the uncertainty in J8. If one were to rely even zonal harmonics J 2 4 of the geopoten- upon on the formal sigmas of the latest global Earth’s gravity tial, it is possible, in principle, to use them in combination field models by the dedicated GRACE and GOCE missions, with, say, the nodes  to reduce the impact of the mismod- the resulting impact on Eq. (81) would be much smaller than eled even zonals in experiments of fundamental physics with 1%. Indeed, from, e.g., the zero-tide model Tongji-Grace02s existing satellites. In an actual test, a detailed analysis of the [4], it is9 σ = 1.3×10−14. It implies a combined mismod- perturbations affecting η and  by all the most relevant non- C8,0 eled precessions as little as 0.01 mas year−1, correspond- gravitational accelerations should be performed. There are ingto0.01% of the combined Lense–Thirring effect. If, no net Lense–Thirring rates of change of the semimajor axis a and nb. In astronomical binary systems, not affected by non- 9 The zonal harmonics J of the geopotential are connected with gravitational perturbations, using η may provide a further its√ fully normalized Stokes coefficients C,0 by the relation J = − 2 + 1 C,0,= 2, 3, 4,... valuable observable in addition to the usual periastron pre- 123 Eur. Phys. J. C (2019) 79 :816 Page 13 of 14 816 cession to put to the test general relativity and, say, modified integer multiple of that of perigee ω. In the calculation, the models of gravity, or to better characterize the physical prop- Earth’s symmetry axis Sˆ is assumed to be aligned with the erties of the bodies like, e.g., their oblateness J2 and their reference z axis; moreover, no a-priori simplifying assump- orbital configurations as well. Indeed, the 1pN effects on η tions concerning the orbital geometry of the satellite were are often larger than the corresponding pericenter rates. made at all.

Data Availibility Statement This manuscript has no associated data or the data will not be deposited. [Authors’ comment: This is a theoretical A.1 Secular effects study and no experimental data has been listed.]

· s 3 n R2 cos I Open Access This article is distributed under the terms of the Creative  =− b  e  , .2 2 (A6) Commons Attribution 4.0 International License (http://creativecomm 2 a2 1 − e2 ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, · s 3 n R2 (1 + 3cos2I ) η = b e   , and reproduction in any medium, provided you give appropriate credit .2 3/2 (A7) 8 a2 1 − e2 to the original author(s) and the source, provide a link to the Creative   · s 15 n R4 2 + 3 e2 (9cosI + 7cos3I ) Commons license, and indicate if changes were made.  = b e , 3 .4   (A8) Funded by SCOAP . 128 a4 1 − e2 4

· s 4 2 ( + + ) η =−45 nb Re e 9 20 cos 2I 35 cos 4I , .4   / (A9) 1, 024 a4 1 − e2 7 2  η   Appendix A Mean orbital precessions of and due to · s 105 n R6 8 + 40 e2 + 15 e4 (50 cos I + 45 cos 3I + 33 cos 5I )  =− b e   , the even zonal harmonics of the geopotential .6 6 16, 384 a6 1 − e2 (A10)   Here, we analytically calculate the coefficients · s 6 η = 35 nb Re − + 2 + 4 .6   / 8 20 e 15 e   , 6 − 2 11 2 · 65 536 a 1 e ∂ κ × (50 + 105 cos 2I + 126 cos 4I + 231 cos 6I ) , (A11) · . J κ. = ,= 2, 4, 6, 8,κ= , η (A1) · s 8 ∂ J  = 315 nb Re .8   2, 097, 152 a8 1 − e2 8      of the precessions × 16 + 7 e2 24 + 5 e2 6 + e2   · × 1, 225 cos I + 11 (105 cos 3I κ ,= , , , ,κ= , η, [ 2 4 6 8 (A2) + + ) , J 91 cos 5I 65 cos 7I ] (A12) · s 8 η =− 315 nb Re  η .8   / of the node and of the mean anomaly at epoch aver- 33, 554, 432 a8 1 − e2 15 2    aged over one full orbital period Pb, induced by the first four × −32 + 35 e4 4 + e2 even zonal harmonics J. To this aim, we use the standard Lagrange planetary equations [2] × (1, 225 + 2, 520 cos 2I + 2, 772 cos 4I + , + , ) .   3 432 cos 6I 6 435 cos 8I (A13) · 1 ∂U  =− √ , (A3) 2 − 2 ∂ I nb a sin I 1 e    2 References · ∂  1 − e ∂  η = 2 U + U . 2 (A4) nb a ∂a nb a e ∂e 1. O. Akyilmaz et al., ITU_GRACE16 The global gravity field model including GRACE data up to degree and order 180 of ITU and other In them, the correction of degree  collaborating institutions (2016). Accessed 16 Oct 2018 2. B. Bertotti, P. Farinella, D. Vokrouhlický, Physics of the Solar Sys-  tem (Kluwer Academic Press, Dordrecht, 2003) μ Re U (r) = J P (ξ) ,= 2, 4,...8(A5)3. V.A. Brumberg, Essential Relativistic Celestial Mechanics (Adam r r Hilger, Bristol, 1991) 4. Q. Chen, Y. Shen, O. Francis, W. Chen, X. Zhang, H. Hsu, J. Geo- to the Newtonian monopole is straightforwardly averaged phys. Res. 123, 6111 (2018) over one full orbital revolution by using the Keplerian ellipse 5. I. Ciufolini, Il Nuovo Cimento A 109, 1709 (1996) 10 P (ξ) 6. I. Ciufolini, E. Pavlis, New Astron. , 636 (2005) as reference unperturbed orbit. In Eq. (A5),  is the Leg- 7. T. Damour, N. Deruelle, Ann. Inst. Henri Poincaré Phys. Théor. endre polynomial of degree . As a result, two kind of aver- 43, 107 (1985) aged, long-term effects occur: secular precessions, explic- 8. I. Debono, G.F. Smoot, Universe 2, 23 (2016) itly displayed in Sect. A.1 and labelled with a superscript 9. S. Kopeikin, M. Efroimsky, G. Kaplan, Relativistic Celestial Mechanics of the Solar System (Wiley-VCH, Weinheim, 2011) “s", and long-periodic signatures, not shown here, having a 10. D.M. Lucchesi, Planet. Space Sci. 49, 447 (2001) harmonic pattern characterized by a frequency which is an 11. D.M. Lucchesi, Planet. Space Sci. 50, 1067 (2002) 123 816 Page 14 of 14 Eur. Phys. J. C (2019) 79 :816

12. D.M. Lucchesi, Geophys. Res. Lett. 30, 1957 (2003) 17. G. Renzetti, Open Phys. 11, 531 (2013) 13. D.M. Lucchesi, L. Anselmo, M. Bassan, C. Magnafico, C. Pardini, 18. I.I. Shapiro, in General Relativity and Gravitation, 1989, ed. by R. Peron, G. Pucacco, M. Visco, Universe 5, 141 (2019) N. Ashby, D.F. Bartlett, W. Wyss (Cambridge University Press, 14. D.M. Lucchesi, L. Anselmo, M. Bassan, C. Pardini, R. Peron, G. Cambridge, 1990), pp. 313–330 Pucacco, M. Visco, Class. Quantum Gravity 32, 155012 (2015) 19. M.H. Soffel, Relativity in Astrometry, Celestial Mechanics and 15. A. Milani, A. Nobili, P. Farinella, Non-gravitational Perturbations Geodesy (Springer, Heidelberg, 1989) and Satellite Geodesy (Adam Hilger, Bristol, 1987) 20. M. Visco, D.M. Lucchesi, Phys. Rev. D 98, 044034 (2018) 16. C. Pardini, L. Anselmo, D.M. Lucchesi, R. Peron, Acta Astronaut. 140, 469 (2017)

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