On the gravitomagnetic effects on the orbits of two counter orbiting satellites

Transcription

1 11th March 2008 BARI-TH/00 On the gravitomagnetic effects on the orbits of two counter orbiting satellites arxiv:gr-qc/ v1 10 Oct 2002 Lorenzo Iorio, Herbert Lichtenegger Dipartimento di Fisica dell Università di Bari, via Amendola 173, 70126, Bari, Italy Institut für Weltraumforschung, Österreichische Akademie der Wissenschaften, A-8042 Graz, Austria Abstract The effects of the general relativistic gravitomagnetic force on the orbits of a couple of counter orbiting test particles, denoted as + and, respectively, in the gravitational field of a central rotating mass are investigated. Based on the fact that, for identical orbital configurations, the secular Lense Thirring rates of the nodes Ω are the same, contrary to the secular rates of the perigees ω which are equal and opposite, two different, new observables for a couple of new, counter orbiting satellites in the gravitational field of the Earth are proposed and preliminary error budgets are presented. As a complementary consequence, also the gravitoelectric Einstein perigee precession could be measured by suitably combining the data of one of the two new satellites and of LAGEOS and LAGEOS II.

2 1 1 Introduction In the slow motion and weak field approximation of General Relativity the gravitomagnetic field due to the off diagonal components of the metric breaks the symmetry of orbital motions around its source in the sense that it discriminates between the directions of motion of test particles following close paths in opposite directions. One of its most interesting consequences is the time shift between the coordinate sidereal orbital periods amounting to [Cohen and Mashhoon, 1993; Mashhoon et al., 1999; Mashhoon et al., 2001; Iorio et al., 2002a] t + t = 4πJ Mc 2 (1) for identical, circular, equatorial geodesic orbits followed in opposite directions denoted conventionally with the + and signs. In eq. (1) J and M are the angular momentum and the mass, respectively, of the central rotating body and c is the speed of light in vacuum. Recently, many efforts have been devoted to preliminary investigations of the possibility of measuring such an effect in an Earth space based experiment [Gronwald et al., 1997; Iorio, 2001a; 2001b; Lichtenegger et al., 2000; 2001]. By neglecting terms of order O(c 4 ), the same result holds also for the difference of the proper periods τ + τ [Cohen and Mashhoon, 1993; Mashhoon et al., 1999]. Since the measurement of such time shifts would be, at present, troublesome, as suggested in [Iorio, 2001a; Lichtenegger et al., 2001], it could be helpful to investigate some other gravitomagnetic observables which, on one hand, are sensitive to the direction of motion of the test particles along their orbits and, on the other, can be expressed in terms of standard Keplerian orbital elements (or other orbital elements commonly used in practical orbit data reductions), so to try to setup a space based experiment exploiting the large experience gained, e.g., with the Satellite Laser Ranging (SLR) technique in the LAGEOS LAGEOS II Lense Thirring experiment [Ciufolini et al., 1998]. The paper is organized as follows. In Section 2 we present an orbital observable ξ, based on a suitable combination of some Keplerian orbital elements, whose gravitomagnetic secular rate turns out to be sensitive to the direction of motion of the test particle. In Section 3 we characterize the orbits of a couple of counter orbiting test particles and investigate the gravitomagnetic effect on the difference of the rates of ξ for such a system. Finally, we also

3 2 calculate the time required to ξ for passing from 0 to 2π and present a very preliminary error budget. In Section 4 we discuss the gravitomagnetic rates of the perigees ω for a couple of counter orbiting satellites and propose an observable based on the difference of the residual rates of the perigees for a couple of counter revolving satellites in the gravitational field of the Earth. A possibility for measuring also the gravitoelectric Einstein perigee advance is discussed and a very preliminary error budget is presented as well. Section 5 is devoted to the conclusions. 2 An alternative gravitomagnetic observable Let us define the following quantity ξ = Ω cosi + ω + M starting from the inclination i, the longitude of the ascending node Ω, the argument of pericenter ω and the mean anomaly M. The angular variable ξ could be usefully exploited in ordinary data reductions by means of orbital processors like GEODYN II or UTOPIA 1. For a general orbit, the rate of ξ is given by dξ where, according to the Lagrange planetary equations = Ω sin idi + dø dω cosi + + dm, (2) di dø dω dm = = cosi na 2 (1 e 2 ) 1 2 sin i 1 na 2 (1 e 2 ) 1 2 sin i R ω 1 na 2 (1 e 2 ) 1 2 sin i cosi R = na 2 (1 e 2 ) 1 2 sin i i + (1 e2 ) na 2 e = n 1 e2 R na 2 e e 2 na R Ø, (3) R i, (4) 1 2 R e, (5) R a. (6) In them R is the disturbing function accounting for any departure from sphericity of the gravitational potential of the central body, a and e are the semimajor axis and the eccentricity, respectively, of the test particle s unperturbed orbit and n = ( GM a 3 ) 1 2 is the Keplerian mean motion in which G is the Newtonian gravitational constant. 1 For orbits with small, but finite, values of e and i the use of the standard Keplerian orbital elements may create some problems because their computation from well-behaved cartesian components of the position and velocity vectors r and v leads to unphysical singularities in them. In such case, it is better to use the so called equinoctial orbital elements, as it will be shown later.

4 3 By inserting eqs. (3)-(6) into eq. (2) we find dξ = n Ω cosi na 2 (1 e 2 ) 1 2 R ω + Ω na 2 (1 e 2 ) 1 2 R Ø + (1 e2 ) 1 2 (1 e 2 ) na 2 e R e 2 na R a. (7) In order to obtain a general relativistic gravitomagnetic signature, the disturbing function for the gravitomagnetic field [Iorio, 2001c] must be inserted in eq. (7) R = 2GJ cosi du c 2 r cosi(1 + e cosf) du = 2GJ c 2 a(1 e 2 ), (8) where f is the true anomaly and u = ω + f is the argument of latitude. To work out the long term rate of ξ for a test particle along a general orbit, eq. (8) is averaged over an orbital revolution by assuming du = dω + df df and integrating it from 0 to 2π, one obtains 2GJn cosi R = c 2 a(1 e 2 ). (9) In [Iorio, 2001c] it has been shown that this approach yields the well known Lense Thirring formulas for the gravitomagnetic shifts of the node and the perigee. Now, by inserting eq. (9) into eq. (7) the secular Lense Thirring rate of the quantity ξ is found to be dξ 4GJ cosi =. (10) c 2 a 3 (1 e 2 ) 3 2 The same result can also be obtained by using the projections of the gravitomagnetic acceleration R, T, N onto the radial, transverse and out of plane directions in the Gaussian perturbative equations. The calculations yield (see Appendix A) { dξ Ω sin 2 i cosu [ = n 1 ǫ 0 2(1 + e cosf) 3 sin u + e(1 + e cosf) 2 sin f cosu ] (1 e 2 ) 3 cosi 2ǫ 0 (1 + e cosf) e2 (1 e 2 ) 3 2 (1 + e cosf) 2 cosf (1 e 2 ) 3 2e, (11) where and T (0) = 2π n orbital period by means of ǫ 0 = t+ t = 2nJ (12) T (0) c 2 M is the unperturbed Keplerian orbital period. By averaging eq. (11) over an = 1 n ( r a ) 2 df (1 e 2 ) 1 2 = 1 (1 e 2 ) 3 2 n (1 + e cosf) 2df, (13) it turns out that the second term in eq. (11), which is due to Ω sin i di, vanishes while the third term yields exactly the result of eq. (10).

5 3 The gravitomagnetic effect on a couple of counter orbiting satellites 3.1 The characterization of the orbits of a pair of counter orbiting satellites It can be shown that the Keplerian orbital elements of two bodies moving along identical orbits but in opposite directions are related via a = a +, (14) e = e +, (15) i = π i +, (16) Ø = Ø + π, (17) ω = π ω +, (18) M = M +, (19) f = f +, (20) u = π u +. (21) While eqs. (14)-(15) and eqs. (19)-(20) are obvious, the demonstration of eqs. (16)-(18), from which eq. (21) can be obtained, is less trivial and it is as follows. The inclination i. According to [Vinti, 1998] sin i = (h2 x + h2 y )1 2 h, (22) cosi = h z h, (23) where h is the orbital angular momentum per unit mass whose components change sign when v v. By reversing the sign of the velocity vector one obtains sin i = sin i +, (24) cosi = cosi +, (25) 4 from which it follows i = π i +. (26)

6 5 The longitude of the ascending node Ø. According to [Vinti, 1998] h y cos Ω = h sin i, (27) h x sin Ω = h sin i. (28) From them it turns out that, by reversing the sign of the velocity vector of the test particle, cos Ω = cos Ω +, (29) sin Ω = sin Ω +, (30) so that Ω = Ω + π. (31) The argument of perigee ω. According to [Vinti, 1998] e sin i cosω = hż GM + h xy h y x, (32) hr e sin i sin ω = h yẋ h x ẏ GM z (33) r and hence, from them, by reversing the sign of the velocity vector of the test particle, it follows cosω = cosω +, (34) sin ω = sin ω +, (35) so that ω = π ω +. (36) The same results could also be obtained in a more geometrical and intuitive fashion from the definitions of the Keplerian orbital elements [Milani et al., 1987]. 3.2 The gravitomagnetic effect on ξ for a pair of counter orbiting satellites Since the secular rate of ξ for the counter orbiting particle is simply eq. (10) with a minus sign (according to eq. (25)), the following quantity for a couple of counter revolving test particles along identical orbits (a + = a a, e + = e e) could be considered

7 6 dξ + dξ = 8GJ cosi. (37) c 2 a 3 (1 e 2 ) 3 2 In case of orbits with small, but finite, eccentricity and inclination, eq. (37) becomes to first order in e and i dξ + dξ 8GJ c 2 a. (38) 3 For an orbit around Earth with semimajor axis a = 12, 270 km, like LAGEOS, eq. (38) yields a secular shift of milliarcseconds per year (mas yr 1 in the following), while for a = 26, 578 km, like GPS, it reduces to 12.1 mas yr 1. We note that with regard to a major source of systematic errors, the secular effects of the even zonal harmonics of the geopotential, which would affect ξ ±, cancel out for the difference ξ ξ +. Indeed, upon inserting the disturbing function R geop for the long term l = 2p, m = q = 0 part of the geopotential [Kaula, 1966] into eq. (7), it turns out that only even powers of sin i would appear. Moreover, also the insidious 18.6 year lunar tide [Iorio, 2001d], which is a l = 2, m = 0, p = 1, q = 1 constituent, would be cancelled out. It should be emphasized, however, that such features are strictly valid only for identical orbits Preliminary error budget Eqs. (37)-(38) hold only for satellites with exactly the same orbital parameters, with particular care to the semimajor axes entering the expression of the mean motions which must cancel out. In reality, this situation would be quite unfeasible because of the unavoidable orbital injection errors. Therefore, let us investigate the difference of the observed residuals 2 δ dξ δ obs + dξ, (39) obs which contains n n +. If we define the difference of the semimajor axes as a a + d, where d/a 1 and a represents the nominal value of the semimajor axis of the two satellites, we find n n n + = (GM) 1 2 [ 1 a 3 2 ] 1 (a + d) 3 2 3d(GM)1 2 2a 5 2. (40) 2 The gravitomagnetic effect could be considered in the orbital data reduction as an unmodelled contribution so that the residuals would absorb it entirely.

8 7 Thus, the error in n, which would affect the measurement of the relativistic trend, would be due to the errors δ(gm), δa and δd in the Earth s GM, the satellite semimajor axis and the satellites separation d, respectively, i.e. 3d[δ(GM)] δ( n) 4(GM) 1 2a d(δa)(GM) 1 2 4a 7 + 3(δd)(GM) a 5. (41) 2 By assuming a = 12, 000 km, δa = 1 cm, δ(gm) = cm 3 s 2 [McCarthy, 1996], eq. (41) yields 3 δ( n) ( cm 1 s 1 ) d + ( cm 1 s 1 ) (δd). (42) Eq. (42) clearly shows that the errors due to the bad knowledge of the Earth s GM and a would be negligible, contrary to that due to the uncertainty δd on the difference d between the semimajor axes of the counter revolving satellites, even if it was at the centimeter level. It is interesting to notice that the level of accuracy in knowing the difference of the semimajor axes of the satellites which would allow for the determination of the differential gravitomagnetic shift should be δd cm, which is the same as that for the standard time shift of eq. (1) [Gronwald et al., 1997; Lichtenegger et al., 2000]. Although there are no long-term gravitational perturbations on the semimajor axis, δd = δa δa + would be affected in general by the non gravitational perturbations, both secular and time varying. Moreover, if the satellites had been constructed in the same way, the impact of the mismodelled non gravitational forces, which depend on the geometrical and physical properties of the satellites and of their orbital configurations, could be made quite similar for both. Finally, since we are interested in the mismodelled perturbations on the semimajor axes, it is also important to notice that the non gravitational effects on a have been extensively studied for the LAGEOS satellites and should be accurately modelled. Of course, the accurate data reduction of the perigees and the mean anomalies would not be at all a trivial task, especially due to the many non gravitational perturbations acting upon them. However, the large experience gained with the existing laser ranged satellites, in particular with LAGEOS and LAGEOS II, could be fully exploited. A detailed analysis of such topic is outside the scope of the present work. 3 The assumption δa = 1 cm is well justified by the fact that for LAGEOS the rms of the satellite s distance, after many revolutions, is of the order of just 1 cm.

9 8 3.3 The equatorial and circular orbit case The variable ξ is quite similar to the mean longitude l = + M Ω + ω + M, to which it reduces for small inclinations. For almost circular, equatorial orbits 4, it can be interpreted as the approximate right ascension of the satellite [Montenbruck and Gill, 2000]. The right ascension is nothing but the angular variable φ used in deriving the gravitomagnetic clock effect leading to eq. (1) [Cohen and Mashhoon, 1993; Iorio et al., 2002a; Mashhoon et al, 1999; Mashhoon et al, 2001]. With such particular orbital configurations l has to be computed, in practical orbit data reductions, from the expression l = F k sin F + h cosf, (43) where F = E + Ω + ω is the so called eccentric longitude (E is the usual eccentric anomaly) and h and k are two of the equinoctial elements [Broucke and Cefola, 1972] a = a, (44) h = e sin (ω + Ω), (45) k = e cos (ω + Ω), (46) ( i p = tan sin Ω, (47) 2) ( i q = tan cos Ω, (48) 2) l = n(t t 0 ) + l 0 = M + Ω + ω. (49) They are commonly used for orbits with e i 0. As shown at pag of [Montenbruck and Gill, 2000], they can be calculated from the cartesian components of the position and velocity vectors without giving rise to any singularities for such limiting orbital configuration, contrary to the case of the Keplerian orbital elements. Therefore, in the following we also calculate the time shift t + l t l arising from l. From eq. (49) and Table 2 of [Broucke and Cefola, 1972] for the Lagrange brackets [a, l 0 ] for general orbits one finds dl = n 2 R na a. (50) 4 Of course, in a practical satellite space based mission only small, but finite, values of e and i could be obtained. It should also be considered that, since most of the laser ranging stations are located in the Northern emisphere far from the equator, a rather large value for the inclination i of the satellites should be needed in order to assure a very precise tracking.

10 9 For the case of near circular orbits with small inclinations, by retaining only the terms linear in h, k, p and q in eqs. (17) (18) of [Broucke and Cefola, 1972], the gravitomagnetic disturbing function R can be expressed in terms of the equinoctial elements as 2G(J r) v R = c 2 r 3 2GJ(xẏ yẋ) = c 2 r 3 2GJn c 2 a(1 k cos F h sin F) 3. (51) By inserting eq. (51) into eq. (50) and by noticing that dl = df(1 k cos F h sin F) it is found dl n 4J df c 2 M (1 k cosf h sin F) dl 2 n 4J (1 + 2k cosf + 2h sin F)dF. (52) c 2 M By integrating eq. (52) with respect to F from 0 to 2π for the orbiting satellite and from 2π to 0 for the counter orbiting one, respectively, one obtains from which follows. t ± l T (0) ± 8πJ c 2 M, (53) t + l t l 16πJ c 2 M Let us now calculate the time required for passing ξ from 0 to 2π for generic orbits. From eq. (11) and by using which is well adequate over an orbital revolution, we get (54) dξ dm = (1 e2 ) 3 2 (1 + e cosf) 2df, (55) T ξ = T (0) 8πJ cosi + c 2 M(1 e 2 ) 3 2 (56) and hence T ξ + Tξ = 16πJ cos i. (57) c 2 M(1 e 2 ) 3 2 For linear order in e and i, eqs. (56)-(57) reduce to eqs. (53)-(54) Preliminary error budget Eq. (54) would be valid only if the Keplerian periods of the two satellites were exactly equal; this condition, however, cannot be achieved due to the unavoidable orbital injection errors. Then

11 10 eq. (54) has to account for the difference in the Keplerian periods induced by the difference in the semimajor axes of the two satellites d with t + l t l T (0) + 16πJ c 2 M, (58) T (0) = 2π[a3 2 (a + d) 3 2] (GM) 1 2 3πda1 2, (59) (GM) 1 2 where, as before, a represents the nominal value of the semimajor axis of the two satellites. In order to be able to measure the relativistic effect of interest, which accumulates during the orbital revolution, the error δ( T (0) ), which is present at every orbital revolution, too, and is due to the uncertainties in the Earth s GM, a and d, must be smaller than the gravitomagnetic time shift. This error is given by δ( T (0) 3πda 1 2[δ(GM)] ) 2(GM) 3 + 3πd(δa) 2 2(GMa) π(δd)a 1 2 (GM) 1 2 and by assuming a = 12, 000 km, δa = 1 cm and δ(gm) = cm 3 s 2 [McCarthy, 1996], eq. (60) yields (60) δ( T (0) ) ( cm 1 s) d + ( cm 1 s) (δd). (61) Eq. (61) tells us, similar as for δ ξ δ ξ +, that the error due to the uncertainty in a and GM is negligible, while δd should be at the level of 10 2 cm. However, it should be pointed out that for orbits with e i 0, for which eq. (58) holds, many non gravitational perturbations 5 affecting the semimajor axis vanish [Iorio, 2001b] or can be strongly constrained by constructing the two satellites very carefully with the same geometrical and physical properties. Of course, the same considerations hold also for eq. (57). 5 On the semimajor axis there are no long term gravitational perturbations.

12 11 4 A perigee only scenario 4.1 The gravitomagnetic effect on the perigees of a pair of counter orbiting satellites From eq. (5) and eq. (9) it turns out that the secular gravitomagnetic rate for the perigee is, as it is well known dω 6GJ cosi =. (62) c 2 a 3 (1 e 2 ) 3 2 Contrary to the node, eq. (62) depends on the inclination i through cosi, so that for a couple of counter orbiting satellites we have + dω dω =. (63) Then, a possible alternative to eq. (37) for a couple of counter revolving satellites in non circular identical orbits could be the following gravitomagnetic quantity dω + dω = 12GJ cos i. (64) c 2 a 3 (1 e 2 ) 3 2 It is interesting to notice that an analogous situation also holds for a couple of satellites in identical orbits with supplementary inclinations. This configuration has been extensively and quantitatively investigated in [Iorio and Lucchesi, 2002]. 4.2 Preliminary error budget The key point is that the proposed observable of eq. (64) is not sensitive to the cancellation of the mean motions, contrary to eq. (37). Moreover, the systematic errors in the classical secular precessions due to the mismodelled even zonal spherical harmonics coefficients of the geopotential do not affect eq. (64). Indeed, from the results of [Iorio, 2002] it turns out that dω even zonals + dω = 0, (65) even zonals because the classical even zonal perigee precessions depend on even powers of sin i and on cos 2 i. This means that the difference of the residuals of the perigee rates of a couple of counter orbiting satellites following identical orbits would not be affected by the mismodelled secular even zonal

13 12 precessions due to the geopotential which otherwise would have represented the major source of systematic errors. Let us suppose, for the sake of concreteness, to place a geodetic laser ranged satellite into orbit, called, e.g., LARES II, with the same orbital parameters as that of LAGEOS II (a = 12, 163 km, e = 0.014, i = deg, n = s 1 ) but with opposite direction of motion. Then one could consider δ ω LAGEOS II δ ω LARES II 115.2µ, (66) where δ ω LAGEOS II and δ ω LARES II are the orbital residuals of the perigee rates of the two satellites, is the slope, in mas yr 1, of the secular trend predicted by General Relativity and µ is the solve for least squares parameter which accounts for the gravitomagnetic effect, as in the LAGEOS LAGEOS II Lense Thirring experiment. In the following we make a comparison with the analogous analysis of the originally proposed LAGEOS LARES nodes only mission. First, because we assume e LAGEOS II = e LARES II, contrary to the LAGEOS LARES mission [Iorio et al., 2002b], the geopotential error falls down to zero just for i LARES II = i LAGEOS II = deg. Moreover, since the eccentricity of the second satellite could be chosen to be equal to that of LAGEOS II, the cancellation of the mismodelled part of the geopotential would be much more accurate than that of the LAGEOS LARES mission, which amounts to , according to the covariance matrix of the EGM96 gravity model up to degree l = 20. It turns out that the present proposal seems to be only slightly more sensitive than the LAGEOS LARES mission to the deviation from the nominal value of the inclination of the new satellite to be launched. On one hand, in regard to the time dependent perturbations, the impact of the non gravitational perturbations could turn out to be relevant. Indeed, cancellation of all these perturbing terms, some of which depend linearly on cos i [Lucchesi, 2001], in general does not occur and it is well known that the perigees, contrary to the nodes, are particularly sensitive to such kind of perturbations. We recall that the perigee of LAGEOS II is affected both by a tidal perturbation with a period of 5.07 years [Iorio, 2001d] and a direct solar radiation pressure harmonic with a period of 11.6 years [Lucchesi, 2001].

14 13 On the other hand, with a careful choice of the orbital geometry for an entirely new couple of twin satellites, it would be possible to make the periods of all the perturbations not too long, so that they would not resemble superimposed trends over observational time spans of a few years and could thus be fitted and removed from the signal, as in the LAGEOS LAGEOS II Lense Thirring experiment. An acceptable choice could be a couple of counter orbiting satellites placed in orbits with 6 a = 12, 000 km, e = 0.05, i = 63.4 deg, n = s 1. However, a detailed analysis of all the perturbing effects is beyond the scope of the present work. 18 δω ( ) δω (+) =Xµ fixed combination coefficients; a =12,000 km; e = /µ (%) even zonals δµ Inclination i of ( ) (deg) Figure 1: Influence of the injection errors in the inclination of the second supplementary satellite with frozen perigee on the zonal error of the perigees-only combination. In Figs. 1 and 2 we show how the systematic error due to the mismodelling in the even zonal harmoincs of the geopotential, according to the covariance matrix of the EGM96 gravity model 6 The value for the inclination would be of crucial importance in obtaining the so called frozen perigee configuration which allows for making the period of the classical perigee precession extremely long. It turns out that in this way many of the periods of the time varying perturbations affecting the perigee would become reasonably short.

15 δω ( ) δω (+) =Xµ constant combination coefficients = 1; i =63.4 deg; e = /µ (%) even zonals δµ Semimajor axis a of ( ) (km) x 10 4 Figure 2: Influence of the injection errors in the semimajor axis of the second supplementary satellite with frozen perigee on the zonal error of the perigees-only combination. up to l = 20, depends on the departures of the inclination and the semimajor axis, respectively, of the second satellite with respect to their nominal values. It is seen that the requirement for the concurrence of the orbital planes is rather stringent. It should be noticed that, of course, the sum δ Ω + + δ Ω could be measured, too. Indeed, since the classical nodal precessions depend on cos i [Iorio, 2002], for the counter-rotating configuration the sum of the nodes would cancel the systematic error due to the mismodelled classical even zonal precessions brought about by the geopotential. 4.3 Other possible relativistic tests Two new LAGEOS like satellites in orbit with the proposed orbital parameters would further allow to use the residuals of the node and the perigee of one of them to measure the gravitoelectric Einstein precession of the perigee by means of some suitable combinations of orbital

16 15 residuals 7 of the other existing satellites [Iorio et al., 2002c]. For example, the combination of eq. (67) below, in which one of the new satellites is denoted with the superscript i, would allow to obtain a relative accuracy in the systematic error due to the even zonal harmonics of the geopotential of the order of , according to the covariance matrix of EGM96 8 up to degree l = 20. The observable is given by δ ω i + c 1 δ Ω LAGEOS II + c 2 δ Ω LAGEOS + c 3 δ ω LAGEOS II + c 4 δ Ω i = ν GE X GE, (67) where ν GE is the least square fit parameter which accounts for the relativistic effect, X GE = 4, mas yr 1 is the slope of the secular trend predicted by General Relativity and c 1 = 1.55, (68) c 2 = 2.77, (69) c 3 = 0.348, (70) c 4 = (71) The perigee of LAGEOS II, which is a rather dirty element affected by some long period gravitational and non gravitational perturbations, is weighted by c 3 which would reduce their impact on the proposed measurement. This feature is particularly important in view of certain recent modifications of the reflectivity properties of the surface of LAGEOS II which have an impact on the response to the direct solar radiation pressure perturbation, as pointed out in [Iorio et al., 2002b]. Concerning the perigee of the new satellite, whose coefficient in the combination of eq. (67) is equal to one, the influence of the non gravitational perturbations could be greatly reduced by suitably constructing the satellite, as intended for LARES. Moreover, as already pointed out, thanks to the choice of the inclination, the periods of its time dependent perturbations would be short enough to allow for fitting and removing them from the signal over a reasonable observational time span of a few years. The perturbations on the nodes of LAGEOS and LAGEOS II would not yield particular problems, even if weighted by c 1 and c 2. 7 We note that the gravitoelectric Einstein perigee precession ω GE = 3nGM c 2 a(1 e 2 ) is independent of the inclination and therefore the same for a pair of counter revolving satellites. So, the difference of the perigee rates cannot be used to measure this relativistic effect. 8 These results will be improved when new, more accurate Earth gravity models based on the CHAMP and GRACE missions will be available.

17 16 Last but not least, it is interesting to note that the same combination of eq. (67) could be used for measuring also the Lense Thirring effect. It turns out that in such case, the slope of the relativistic trend amounts to -187 mas yr 1 and the systematic relative error due to the even zonal harmonics of the geopotential is , according to the covariance matrix of EGM96 up to degree l = Conclusions In this paper, in order to extend and generalize the gravitomagnetic clock effect concept, we have looked for some new orbital observables affected by the gravitomagnetic force and sensitive to the direction of motion of a test particle along its orbit. By defining ξ = Ω cosi+ω + M we have calculated the coordinate time (eq. (56)) required for ξ to pass from 0 to 2π both for general orbits and for orbits with small but finite values of eccentricity and inclination (eq. (53)). In the latter case we have found an expression of the gravitomagnetic time shift t + l t l four times larger than that given by eq. (1) which is based on the concept of azimuthal closure. The fact that the gravitomagnetic secular rates of ξ and of the perigee ω change their sign upon reversing the direction of motion suggests to consider the differences of the residuals of their rates, i.e. δ ξ δ ξ + and δ ω δ ω + to test the existence of the gravitomagnetic field. In view of a practical measurement based on the analysis of the orbits of laser ranged satellites, we have noticed that both observables are insensitive to the aliasing classical secular precessions induced by the mismodelled even zonal coefficients of the multipolar expansion of the Earth s gravitational field. On one hand, the measurement of δ ξ δ ξ + would be affected by the error in the difference of the Keplerian mean motions due to the uncertainty in the difference of the semimajor axes δd = δa + δa which should not be larger than 10 2 cm in order to allow for the detection of the relativistic effect. However, since a would be affected only by the long period non gravitational perturbations, δd could be strongly constrained by carefully constructing the satellites with the same geometrical and physical properties. On the other hand, δ ω δ ω + would not suffer from this limitation and could be measured with a good accuracy over long enough time spans. The optimal choice would be the launch of

18 17 an entirely new couple of twin LAGEOS like geodetic laser ranged, counter rotating satellites along identical, eccentric orbits with an inclination of 63.4 deg. The advantages of such a relatively low cost experiment are listed in the following. There are four orbital observables for accurate tests of General Relativity which could be performed at relatively low expense of time and money and which would benefit by the great experience obtained with the LAGEOS LAGEOS II Lense Thirring experiment. For the test we could adopt The sum of the nodes δ Ω + + δ Ω, as in the originally proposed LAGEOS LARES mission for the measurement of the Lense Thirring effect The difference of the perigees δ ω δ ω +, contrary to the originally proposed LAGEOS LARES mission; it would be independent of δ Ω + + δ Ω, although less precise due to the sensitivity of the perigee to many orbital perturbations of gravitational and non gravitational origin and to possible changes in the physical properties of the surfaces of the satellites, as it happened for LAGEOS II. A suitable combination of orbital residuals including both the node and the perigee of one of the two new satellites and also the node of LAGEOS and the node and the perigee of LAGEOS II for a further, complementary measurement of the Lense Thirring effect The same combination could be used to measure also the gravitoelectric Einstein perigee precession, which is independent of the direction of motion of the two satellites Since the two satellites would be placed in eccentric orbits with the same eccentricities, contrary to the LAGEOS LARES mission, the cancellation of the classical even zonal geopotential precessions should occur at a very accurate level. In the LAGEOS LARES case, since e LAGEOS = and e LARES = 0.04, it would occur at the level Even if the perigees, contrary to the nodes, are heavily affected by various gravitational and non gravitational perturbations, the choice of the frozen perigee configuration would allow for a reduction of the periods of many of them, so that an observational time

19 18 span of only a few years would be required. Moreover, by constructing the satellites with particular care and shrewdness, as proposed for LARES, the impact of the non gravitational perturbations could be greatly reduced. All the proposed observables would benefit from this fact, especially the sum of the nodes and the difference of the perigees which would involve only the new satellites. Thus, even the measurement of the sum of the nodes should be more precise than that obtainable with the LAGEOS LARES mission. For more detailed and quantitative analysis see [Iorio and Lucchesi, 2002] in which an analogous orbital configuration with supplementary LAGEOS like satellites is investigated The gravitomagnetic time shift T + ξ T ξ for generic orbits (or t + l t l for almost circular and equatorial orbits), could be measured as well A An alternative derivation of the gravitomagnetic effect on ξ Eq. (2) of [Mashhoon et al., 2001] can be rewritten as r + r r = ǫ 3(r Ĵ)(r v) + r2 (v Ĵ) 3 0, (72) r 5 provided that we measure all the distances and the times in units of a and T(0), respectively. In 2π eq. (72), ǫ 0 is given by eq. (12) and Ĵ is the unit vector along the proper angular momentum of the central source. It is worth noticing that eq. (72) has the form of eq. (B1) of Appendix B of [Chicone et al., 1999]. From eq. (72) it follows that the radial, transverse and out of plane components of the disturbing Lense Thirring acceleration can be written in terms of the Keplerian orbital elements as where R T N = ǫ 0 C cos i r 4, (73) = ǫ 0 e cosisin f Cr 3, (74) sin i = ǫ 0 [2(1 + e cosf) sinu + e sin f cosu], (75) Cr3 C = (1 e 2 ) 1 2, (76)

20 19 r = 1 e e cosf. (77) From eq. (B15), eq. (B17) and eq. (B21) of [Chicone et al., 1999] it follows di dω dω dm = rn cos u, (78) C rn = sin u, (79) C sin i rn cosi = C sin i sin u + C { [ ] } r R cos f + T 1 + sin f, (80) e a(1 e 2 ) = n + r [ R( 2e + cos f + e cos 2 f) T(2 + e cosf) sin f ]. (81) ea 1 2 The correspondence with the notation of [Chicone et al., 1999] is given by Ω h, (82) ω g, (83) M l, (84) f ˆv, (85) u ϕ, (86) n ω. (87) By inserting eqs. (73)-(75) into eqs. (78)-(81) yields eq. (11). Acknowledgements L.Iorio is grateful to L. Guerriero for his support to him while in Bari and to H. Lichtenegger for his kind invitation and hospitality at IWF in Graz. The authors are warmly grateful to B. Mashhoon for his important suggestions and contributions for the improvement of the work. References [Broucke and Cefola, 1972] Broucke, R.A., and P.J. Cefola, Equinoctial orbit elements, Celest. Mech., 5, , 1972.

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