Measuring the De Sitter precession with a new Earth's satellite to the $\mathbf{\simeq 10^{-5}}$ level: a proposal

The inclination $I$ of an Earth's satellite in polar orbit undergoes a secular De Sitter precession of $-7.6$ milliarcseconds per year for a suitable choice of the initial value of its non-circulating node $\Omega$. The competing long-periodic harmonic rates of change of $I$ due to the even and odd zonal harmonics of the geopotential vanish for either a circular or polar orbit, while no secular rates occur at all. This may open up, in principle, the possibility of measuring the geodesic precession in the weak-field limit with an accurately tracked satellite by improving the current bound of $9\times 10^{-4}$ from Lunar Laser Ranging, which, on the other hand, may be even rather optimistic, by one order of magnitude, or, perhaps, even better. The most insidious competing effects are due to the solid and ocean components of the $K_1$ tide since their perturbations have nominal huge amplitudes and the same temporal pattern of the De Sitter signature. They vanish for polar orbits. Departures of $\simeq 10^{-5}-10^{-3}~\textrm{deg}$ from the ideal polar geometry allow to keep the $K_1$ tidal perturbations to a sufficiently small level. Most of the other gravitational and non-gravitational perturbations vanish for the proposed orbital configuration, while the non-vanishing ones either have different temporal signatures with respect to the De Sitter effect or can be modeled with sufficient accuracy. In order to meet the proposed goal, the measurement accuracy of $I$ should be better than $\simeq 35~\textrm{microarcseconds}=0.034~\textrm{milliarcseconds}$ over, say, 5 yr.


Introduction
According to general relativity 1 (Iorio 2015), when a spinning gyroscope follows a geodesic trajectory in the spacetime describing the gravitational field of a static body, its spin axis, viewed in the gyro's rest frame, experiences a change in its orientation with respect to some fixed reference direction pointing to distant stars. Such a phenomenon, known as geodetic or De Sitter precession, was described for the first time by de Sitter (1916) and, later, by Schouten (1918) and Fokker (1920). For other, more recent derivations, see, e.g., Barker & O'Connell (1970); Boerner, Ehlers & Rudolph (1975); Barker & O'Connell (1979); Damour, Soffel & Xu (1994).
The geodetic precession plays a role in the binary systems hosting at least one emitting radiopulsar. Indeed, soon after the discovery of PSR B1913+16 by Hulse & Taylor (1975), Damour & Ruffini (1974) realized that studying the measured pulse shape, in particular the profile width, would allow to reveal the De Sitter effect. The first successful, although qualitative, detections were obtained partly by Weisberg, Romani & Taylor (1989) and, with more confidence, by Kramer (1998) with the PSR B1913+16 system. Subsequent studies were performed by Weisberg & Taylor (2002). Later, the geodetic precession was revealed also in other binary pulsars such as PSR B1534+12 (Konacki, Wolszczan & Stairs 2003), PSR J1141-6545 (Hotan, Bailes & Ord 2005) and PSR J1906+0746 (Lorimer et al. 2006), although with a modest accuracy; see Kramer (2012) for a recent overview. The most recent and accurate measurement was performed by Breton et al. (2008) with the double pulsar PSR J0737-3039A/B (Burgay et al. 2003;Lyne et al. 2004); the accuracy level reached is of the order of ≃ 13%.
Until now, the most accurate direct measurements of the geodetic precession have been performed in the weak-field scenario of our solar system by using both the orbital angular momentum of the Earth-Moon system as a giant gyroscope moving in the external field of the Sun (Bertotti, Ciufolini & Bender 1987;Shapiro et al. 1988;Dickey, Newhall & Williams 1989;Mueller et al. 1991;Williams, Newhall & Dickey 1996;Williams, Turyshev & Boggs 2004;Hofmann & Müller 2018) and the anthropogenic gyroscopes carried onboard the Gravity Probe B (GP-B) spacecraft orbiting the Earth (Everitt et al. 2011(Everitt et al. , 2015. While GP-B reached a relative accuracy of 3 × 10 −3 (Everitt et al. 2011(Everitt et al. , 2015, the Lunar Laser Ranging (LLR) technique (Dickey et al. 1994) recently allowed to obtain the most accurate measurement of such a relativistic effect with a relative accuracy of 9 × 10 −4 (Hofmann & Müller 2018).
In the present work, we show that, with a new accurately tracked Earth's satellite in circular polar orbit, it should be possible to improve the constraint by (Hofmann & Müller 2018) by about one order of magnitude, or, perhaps, even better, by measuring the De Sitter effect on the spacecraft's orbital inclination. It is assumed that we will adopt a kinematically rotating and dynamically non-rotating (Brumberg & Kopeikin 1989;Damour, Soffel & Xu 1994) geocentric equatorial coordinate system throughout the paper. In Iorio (2018), it is shown that, in addition to the De Sitter precession to ≃ 10 −5 , also the Lense-Thirring effect (Lense & Thirring 1918) could be measured at a some percent level if an ecliptic coordinate system is used for the data analysis. An appropriate name for the proposed satellite would, thus, be ELXIS, from ἕλξις meaning 'dragging', 'trailing'.
The plan of the paper is as follows. In Section 2, the De Sitter rate of change of the inclination of a test body orbiting its primary which, in turn, moves in the external gravitational field of another massive object is analytically worked out. A non-vanishing, long-term effect with a magnitude of 7.6 mas yr −1 is found for the Earth-Sun scenario. Depending on the temporal behaviour of the satellite's node, it can be either a sinusoidal signal or a secular trend. The next three Sections are devoted to the main perturbations of gravitational origin on the satellite's inclination. Section 3 deals with the long-term signatures induced by the even and odd zonal harmonics of the Earth's geopotential. It turns out that they all vanish if the satellite follows a circular path, or if its orbital plane is perpendicular to the Earth's equator. In Section 4, the aliasing due to the Earth's solid and ocean tides is discussed. Both the solid and the ocean components of the K 1 tidal constituent, whose key parameters are rather poorly known at present, induce long-term rates of change on the inclination which have nominally huge amplitudes and the same temporal pattern of the De Sitter effect. Luckily, they vanish for polar orbits. The impact of deviations from such an ideal orbital configuration is discussed by finding that departures up to ≃ 100 times larger than those characterizing GP-B at its launch are able to reduce the nominal tidal perturbations of K 1 to a sufficiently small level. The 3rd-body perturbations due to the Sun and the Moon are worked out in Section 5. While the heliocentric gravitational parameter is determined with an accuracy which allows to deem the Sun-induced effect as negligible, the lunar one is more effective in potentially impacting the satellite's inclination. However, the present-day level of accuracy of the selenocentric gravitational parameter allows to fulfil our requirements. Section 6 treats the non-gravitational perturbations by assuming a LAGEOS-type cannonball geodetic satellite. It turns out that none of them should pose a threat to our goals since most of them vanish for a circular polar orbit, or have temporal signatures which are distinctively different from the De Sitter one. The geomagnetic field may affect the inclination of an electrically charged satellite in a circular polar orbit with a secular trend whose residual effect, however, should be small enough in view of the current level of accuracy in our knowledge of the Earth's magnetic dipole moment. The issue of the actual observability of a change in the inclination of the order of the De Sitter one is tackled in Section 7. It appears that reaching a measurement accuracy for the satellite's inclination better than ≃ 30 µas = 0.03 mas does not seem completely unrealistic in a near future. Section 8 resumes our findings and offers our conclusions. A list of definitions of all the physical and orbital parameters used in the text can be found in Appendix A, while the numerical values of most of them are in Appendix B along with the figures.

The De Sitter orbital precessions
The perturbing De Sitter potential per unit mass of a satellite orbiting the Earth which, in turns, moves in the external field of the Sun is (Barker & O'Connell 1970) (1) Its doubly averaged expression, obtained by using the Keplerian ellipses as unperturbed reference orbits for both the geocentric satellite motion and the heliocentric trajectory of the Earth, turns out to be The standard Lagrange equation for the rate of change of the inclination induced by a perturbing potential U pert (Bertotti, Farinella & Vokrouhlický 2003) dI dt = 1 applied to Equation (2), allows to straightforwardly obtain the long-term, doubly averaged De Sitter rate of change of the satellite's inclination: It can be shown that Equation (4) can be obtained also within the standard radial-transversenormal perturbative scheme by doubly averaging the right-hand-side of the Gauss equation for the variation of the inclination (Bertotti, Farinella & Vokrouhlický 2003) Eq. (6) is obtained by taking the third term of Eq. (10.12) in Petit, Luzum & et al. (2010), which describes the De Sitter acceleration, to the case of the Earth-satellite system orbiting the Sun, and projecting it onto the normal direction spanned by the out-of-plane unit vectorŵ. The trigonometric term sin (Ω − Ω ⊕ ) entering Equation (4) tells us that the De Sitter rate of change of the inclination can be viewed either as an essentially secular precession or as a long-periodic, harmonic signal depending on the frequencyΩ of the satellite's node and of its initial value Ω 0 . Indeed, given that the node of the heliocentric Earth's orbit stays constant over any conceivable time span devoted to the data analysis since its period amounts to T Ω ⊕ = −149, 229.87 yr in such a way that Ω ⊕ (t) = Ω 0 ⊕ +Ω ⊕ t ≃ Ω 0 ⊕ , if the satellite's node circulates as Ω (t) = Ω 0 +Ωt and its period fulfils the condition T Ω ≪ T Ω ⊕ , then the frequency of the harmonic term in Equation (4) In this case, the De Sitter effect is a harmonic one. Instead, if the satellite's node is locked in a fixed position in view of its peculiar orbital geometry which makesΩ ≃ 0, it is, thus, possible to obtain an essentially secular precessions for the De Sitter effect on the inclination by choosing Ω 0 = Ω ⊕ + 90 deg. From Equation (4), the magnitude of the De Sitter inclination rate is where mas yr −1 stands for milliarcseconds per year.
For the sake of completeness, we explicitly show also De Sitter rates of change of the satellite's node Ω and perigee ω which can be directly obtained from Equation (2) with the appropriate Lagrange perturbing equations: It is important to note that Equation (4) and Equations (8) to (9) are valid for any orbital configuration of both the satellite about its primary and of the motion of the latter one with respect to the third body.

The geopotential perturbations
A major source of systematic bias is represented, in principle, by the competing long-term classical orbital variations induced by the even and odd zonal multipoles in terms of which the departures from spherical symmetry of the Newtonian part of the Earth's gravity field are expressed (Heiskanen & Moritz 1967;Kaula 2000). In particular, the node and the perigee of an Earth's satellite undergo, among other things, secular precessions due to the even zonal harmonics J ℓ , ℓ = 2, 4, 6, . . . of the geopotential (Heiskanen & Moritz 1967;Kaula 2000). As such, their mismodeled components pose a major threat to a clean measurement of the relativistic signatures of interest depending on the level of uncertainty in our knowledge of J ℓ .
On the other hand, the satellite's inclination does not suffer from such an important drawback, as we will show below. Here, we look at the orbital motion of a spacecraft around the Earth, assumed non spherically symmetric, and analytically calculate the rates of change of I averaged over one full orbital period P b induced by the first five zonal harmonics J ℓ of the geopotential. To this aim, we use Equation (3) where the correction of degree ℓ to the Newtonian monopole which replaces U pert , is straightforwardly averaged over one full orbital revolution by using the Keplerian ellipse as reference unperturbed orbit. As a result, no secular precessions occur for the inclination. Indeed, only long-periodic effects having harmonic patterns characterized by integer multiples of the frequency of the perigee motion are obtained. They turns out to be dI dt dI dt dI dt 2048a 5 1 − e 2 5 4 + 3e 2 (58 cos I + 49 cos 3I + 21 cos 5I) cos ω+ + 14e 2 (23 cos I + 9 cos 3I) sin 2 I cos 3ω , dI dt 32768a 6 1 − e 2 6 −5(2 + e 2 ) (37 sin 2I + 60 sin 4I + 33 sin 6I) sin 2ω− − 24e 2 (29 cos I + 11 cos 3I) sin 3 I sin 4ω .
In the calculation, the Earth's symmetry axisŜ ⊕ was assumed to be aligned with the reference z axis; moreover, no a-priori simplifying assumptions concerning the orbital geometry of the satellite were made. It is important to note that the largest zonal harmonic, i.e. J 2 , does not contribute at all to the long-term variation of I, as per Equation (11). Moreover, Equations (12) to (15) vanish for either circular (e = 0) or polar (I = 90 deg) orbits.

The solid and ocean tidal perturbations
A further class of competing long-term gravitational orbital perturbations is represented by the solid and ocean tides (Iorio 2001;Kudryavtsev 2002).
Among them, the tesseral (m = 1) K 1 tide, with Doodson number (165.555), is the most insidious one since it induces, among other things, long-periodic, harmonic orbital perturbations having large nominal amplitudes and the same frequency of the satellite's node. In the case of the inclination, the largest contribution to the long-term rate of change of the inclination induced by both the solid and the ocean components of it vanishes for strictly polar orbits. The complete expressions for the tidal rates of change of I can be obtained by applying Equation (3) to Eq. (18) and Eq. (46) of Iorio (2001) with the minus sign because of a different sign convention for U pert adopted there; they are dI dt Note that, for an a-priori established satellite's node rate Ω , which is largely determined by the first even zonal harmonic according to Equation (16) is nearly independent of the semimajor axis a.
The largest effect comes from the solid component, whose perturbation varies just as sin Ω − δ 2,1,K 1 ; see the upper row of Figure 1 for a plot of the nominal amplitudes of the rate of change of I as a function of a for different values of the inclination within the broad range 80 deg ≤ I ≤ 100 deg. The uncertainty in the Love number of degree ℓ = 2 and order m = 1 entering the amplitude of the K 1 -induced perturbation is still of the order of 2 ≃ 10 −3 (Iorio 2001). The ocean prograde perturbation, proportional to cos Ω − ε + 2,1,K 1 , has a smaller amplitude, as shown by the lower row of Figure 1. On the other hand, the relative mismodeling in the C + 2,1,K 1 ocean tidal height coefficient entering Equation (18) is 4 × 10 −2 (Lemoine et al. 1998), or even at the ≃ 10 −3 level if some more recent global ocean models like TPXO.6.2 (Egbert & Erofeeva 2002), GOT99 (Ray 1999) and FES2004 (Lyard et al. 2006) are compared each other.
A strict polar orbital configuration can bring the nominal K 1 tidal perturbations significantly below Equation (4), so that their currently assumed mismodeling, or even worse, is quite able to fulfil our requirements. Figure 2 shows the case of a circular polar orbit with the same departures from the ideal polar geometry of GP-B at its launch (Kahn 2007, p. 141), i.e. I = 90 ± 5 × 10 −5 deg. However, also less tight constraints on I may be adequate for our goals, especially if orbits with a 10, 000 km are considered. Figure 3 depicts a scenario for a circular and nearly polar orbit with I = 90 ± 5 × 10 −3 deg.

The 3rd-body perturbations: the Sun and the Moon
Another source of potential systematic uncertainty of gravitational origin is represented by the 3rd-body perturbations induced by a distant mass X. Its doubly averaged effect on the satellite's inclination can be worked out by averaging Eq. (7) of Iorio (2012) over the orbital period P X of X. The general result is dI dt For e = 0, I = 90 deg, Equation (20) reduces to dI dt For a terrestrial satellite, the most important contributions to Equations (20) to (21) are due to the Moon and the Sun.
The heliocentric gravitational parameter µ ⊙ is known with a relative accuracy of 7 × 10 −11 (Pitjeva 2015); since the nominal value of Equation (20) varies within ≃ 10 4 − 10 5 mas yr −1 for a satellite's circular polar orbit with a ranging from, say, 10, 000 km to 30, 000 km, the systematic bias due to the 3rd-body solar perturbation can be deemed as negligible with respect to Equation (4); the same holds, a fortiori, for lower altitudes.
In the case of the Moon, the situation is subtler because of the relatively less accurate determination of its gravitational parameter µ . It should be noted that, when referred to the Earth's equator, the lunar node oscillates around zero with a period T Ω = 18.6 yr (Roncoli 2005 ,  Fig. (2.4)), while the lunar inclination has a periodicity of about 20 yr (Roncoli 2005 , Fig. (2.4)); thus, for a satellite with a fixed node, Equation (21) represents essentially a secular trend. According to Petit, Luzum & et al. (2010), which rely upon Pitjeva & Standish (2009), the relative uncertainty µ can be assumed of the order of 3 3 × 10 −8 . It turns out that, for e = 0, I = 90 deg, the variability of the Moon's inclination and node, as referred to the Earth's equator, within their natural bounds (Roncoli 2005) 18 deg I 29 deg, −14 deg Ω 14 deg couples to the Moon's gravitational parameter uncertainty yielding a bias on Equation (4) of the order of ≃ 3 × 10 −5 − 5 × 10 −4 for a ranging from 8, 000 km to 30, 000 km; see Figure 4. Future, likely advances in determining µ will improve such evaluations.

The non-gravitational perturbations
The impact of the non-gravitational perturbations (Sehnal 1975;Milani, Nobili & Farinella 1987b;de Moraes 1994) is, in general, more difficult to be assessed because they depend, among other things, on the actual satellite's composition, shape, physical properties, rotational state. For the sake of definiteness, in the following we will consider a LAGEOS-type cannonball geodetic satellite covered by retroreflectors for Earth-based laser tracking with the Satellite Laser Ranging (SLR) technique (Combrinck 2010).
As far as the direct solar radiation pressure is concerned, Eq. (15) of Lucchesi (2001) shows that, if the eclipses are neglected, the perturbation induced by it on the inclination vanish for circular orbits. If, instead, the effect of shadow is considered, non-vanishing perturbations with frequenciesΩ, 2Ω would occur at zero order in the eccentricity, as shown by Tab. (5) of Lucchesi (2001).
According to Eq. (32) of Lucchesi (2001), the perturbation induced by the Earth's albedo on the satellite's inclination vanishes for circular orbits if the effect of the eclipses are neglected. Instead, if the satellite enters the Earth's shadow, zero-order perturbations in e, some of which with frequenciesΩ, occur (Lucchesi 2001, p.456).
Eq. (20) to (22) of Lucchesi (2002) show that the perturbation of the satellite's inclination due to the terrestrial thermal Yarkovsky-Rubincam effect consists of three long-term components: a secular one, which vanishes for a polar orbit or if the thermal lag angle is θ = 0, and two long-periodic harmonic signals with frequenciesΩ, 2Ω which vanish if the orientation of the satellite's spin axisσ is σ z = ±1, σ x = σ y = 0 or, for a polar orbit, if θ = 90 deg.
In the case of the solar thermal Yarkovsky-Schach effect, Eq. (35) of Lucchesi (2002), which includes the effect of the eclipses, tells us that, luckily, there are no long-periodic harmonic perturbations on I with frequencies multiple of the satellite's nodal one.
Eq. (43) of Lucchesi (2002) tells us that the perturbation induced by a hypothetical asymmetry in the reflectivity of the satellite's surface on the inclination vanishes for circular orbits.
According to Sehnal (1981, p. 176), the rate of change of the inclination due to the terrestrial infrared radiation pressure is proportional to e 2 , so that it vanishes for circular orbits.
The atmospheric drag causes a long-term variation of the satellite's inclination to the zero order in the eccentricity which, among other things, is proportional to the atmospheric density, as, e.g., per Eq. (6.17) of Milani, Nobili & Farinella (1987a). Thus, if it experiences marked seasonal or stochastic temporal variations during the data analysis due to some physical phenomena like, e.g., the solar activity, the resulting temporal pattern may no longer be deemed as a regular trend.
The interaction between the Earth's magnetic field, assumed here dipolar and with its dipole moment m ⊕ aligned with the rotational axis, and the possible surface electric charge Q of the satellite induce long-term orbital perturbations (Abdel-Aziz & Khalil 2014). By means of Eq. (24) in Abdel-Aziz & Khalil (2014), with 1/ sin f in its first term corrected to sin f and B 0 → (µ 0 /4π)m ⊕ , for e = 0, I = 90 deg it is possible to obtain dI dt Since the Earth's magnetic dipole is currently known with a relative accuracy of the order of 6 × 10 −4 (Durand-Manterola 2009, Tab. 1), Equation (22) impacts Equation (4) at a ≃ 10 −5 level, as depicted by Figure 5 obtained for the mass of the existing LAGEOS satellite and by varying the satellite's electric charge within −100 × 10 −11 C ≤ Q ≤ −1 × 10 −11 C (Vokrouhlický 1989). (2014) shows that, for polar orbits, the inclination is not affected by electric forces of dipolar origin.
As a consequence of such an analysis, it turns out that, in presence of eclipses, the solar radiation pressure and the albedo induce perturbations on I having essentially the same temporal pattern of the De Sitter signal of Equation (4). Actually, as it can be inferred from Fig. (2) and Fig. (3) of Ismail et al. (2015) and with the aid of the expressions forŵ,ŝ of Lucchesi (2001, p. 450), it turns out that, for I = 90 deg, Ω = Ω ⊕ + 90 deg, it is not possible to avoid the entrance of the satellite into the Earth's shadow during the yearly cycle of the solar longitude λ ⊙ since −1 ≤ cos i ⊙ ≤ 1. This would suggest to adopt a sun-synchronous orbit which, by construction, avoids the eclipses. Indeed, in this case, the satellite's node circulates with the same period of the apparent geocentric motion of the Sun, i.e. 1 yr (Capderou 2014). In order to meet such a condition, the orbital plane should be no longer polar, with an inclination depending on the adopted value of the semimajor axis. Abandoning the polar orbital configuration does not affect the previously outlined error budget, at least as far as the static part of the geopotential is concerned, provided that the orbit is still kept circular. Indeed, Equations (12) to (15) tell us that they vanish for e = 0 independently of I. If Ω were not constant, the De Sitter signature of Equation (4) would look like a long-periodic, harmonic effect with the yearly period of the node. Such a choice would have the advantage of avoiding any possible competing perturbations of non-gravitational origin characterized by the same peculiar temporal pattern. Indeed, the only non-vanishing non-gravitational rates of change of I, i.e. the Yarkovsky-Rubincam effect and the atmospheric drag, are secular and, perhaps, stochastic or seasonal. Furthermore, a time-varying periodic signal with a definite frequency can be measured much more accurately. On the other hand, it is an unfortunate circumstance that a sun-syncronous orbital configuration would leave very large tidal perturbations due to the solid and ocean components of the K 1 tide, as shown by Equations (17) to (18) and Equation (19). It turns out that their nominal amplitudes would amount to 4576 mas yr −1 , −412 mas yr −1 , respectively. The current level of mismodeling in them would not allow to meet our ≃ 10 −5 accuracy goal. Thus, a strict polar orbital configuration has to be finally deemed as preferable, although at the price of introducing potential non-gravitational effects due to the eclipses. However, an analytical calculation of the rate of change of I under the action of the direct solar radiation pressure, performed, to the zero order in e, by using the first term in the series of Eq. (2) and Eq. (4) in Ferraz Mello (1972) for the shadow function, shows that that, for I = 90 deg and Ω fixed to some given value Ω, no secular effects occur. Indeed, the resulting general expression is dI dt √ µ ⊕ a 4 cos ǫ cos 2 Ω sin 2λ ⊙ − (3 + cos 2ǫ) cos 2λ ⊙ + 2 sin 2 ǫ sin 2 Ω , (24) which is a harmonic signal with the yearly period of the solar longitude. The same feature holds also for the effect of the eclipses on the perturbations induced by the Earth's albedo. Indeed, they can be calculated in the same way as for the direct solar radiation pressure, apart from the modification introduced by Eq. (36) in Lucchesi (2001) which does not change the frequencies of the resulting signature: cos ǫ cos 2Ω sin I sin 2λ ⊙ − − 4 cos I cos Ω sin ǫ sin 2λ ⊙ + sin 2ǫ sin 2 λ ⊙ sin Ω − − sin I (3 + cos 2ǫ) cos 2λ ⊙ + 2 sin 2 ǫ sin 2Ω .

Accuracy in determining the inclination
From an observational point of view, reaching the present-day LLR-based relative accuracy level of 9 × 10 −4 (Hofmann & Müller 2018) in measuring the shift corresponding to Equation (4) over, say, 5 yr would imply an ability to determine the satellite's inclination with an accuracy of σ I ≃ 34 µas = 0.034 mas. Ciufolini et al. (2009) claimed they were able to determine the inclinations of LAGEOS and LAGEOS II, respectively, to the ≃ 30 − 10 µas level over ≃ 1 − 3 yr. As far as the "instantaneous" errors are concerned 4 , they are about ≃ 10.8 − 18 µas for 3-day solutions of GPS satellites. The spacecraft of the Global Navigation Satellite System (GNSS) have higher orbits than the LAGEOS' ones, and their orbits are based on continuous observations. Therefore, the angular Keplerian orbital parameters are well determined for these satellites. Although undoubtedly challenging, it should not be, perhaps, unrealistic to expect further improvements which would allow to reach the ≃ 10 −5 level of the De Sitter effect in a foreseeable future.

Summary and conclusions
The present-day best measurement of the geodetic precession has been obtained by continuously monitoring the motion of the Earth-Moon system in the field of the Sun with the Lunar Laser ranging technique; its relative accuracy is 9 × 10 −4 . In this paper, we showed that measuring the long-term De Sitter effect on the inclination of a dedicated terrestrial artificial satellite to a ≃ 1 × 10 −4 − 5 × 10 −5 level should be feasible in a foreseeable future.
By adopting a circular trajectory in an orbital plane perpendicular to the Earth's equator and suitably oriented in space has several important advantages.
First, it is possible to transform the otherwise harmonic De Sitter signal having the satellite's node frequency into an essentially secular precession of −7.6 mas yr −1 .
Moreover, all the competing long-term perturbations induced by the even and odd zonals of the geopotential vanish, although they have a temporal signature different from the relativistic one since their frequencies are multiple of that of the satellite's perigee.
Furthermore, also the competing long-term perturbations due to the solid and ocean components of the K 1 tide, which are characterized by huge nominal amplitudes and the same temporal pattern of the De Sitter signature, vanish. It is quite important since the current accuracy in knowing their key parameters is relatively modest. In order to bring their nominal signatures significantly below the threshold of the relativistic one, departures from the ideal polar configuration as little as 5 × 10 −5 deg are required, especially for relatively small values of the satellite's semimajor axis. However, even relaxing such a tight requirement by two orders of magnitude should not compromise our goal if altitudes over 3, 600 km are considered.
The 3rd-body perturbations due to the Sun are far negligible since the heliocentric gravitational parameter is known with high accuracy. As far as the Moon is concerned, its impact is potentially more important; however, the present-day level of accuracy of its gravitational parameter is adequate to meet our goal for most of the satellite's altitudes considered. It is entirely plausible to assume that the continuous laser tracking of our natural satellite will further improve the determination of its gravitational parameter in the foreseeable future.
Most of the non-gravitational perturbations vanish for the orbital geometry proposed here. The remaining ones either have temporal signatures other than the De Sitter one or are modeled with a sufficiently high accuracy for our purposes.
The measurement accuracy required to improve the current level of 9 × 10 −4 , obtained with the Lunar Laser Ranging technique, over, say, 5 yr is below ≃ 30 µas. Depending on the actual tracking techniques which will be finally adopted, it should not be a prohibitive task to be accomplished in a not too distant future in view of the currently available results for different types of existing spacecraft.