A critical approach to the concept of a polar, low-altitude LARES satellite

Lorenzo Iorio

Dipartimento di Fisica dell' Università di Bari, via Amendola 173, 70126 Bari, Italy

Received 16 July 2002
Published 16 August 2002

PACS number: 0480C

1. Introduction

In its weak-field and slow-motion approximation general relativity predicts that, among other things, the orbit of a test particle freely falling in the gravitational field of a central rotating body is affected by the so-called gravitomagnetic dragging of the inertial frames or the Lense-Thirring effect. More precisely, the longitude of the ascending node Ω and the argument of the perigee ω of the orbit undergo tiny secular precessions according to (Lense and Thirring 1918)

where G is the Newtonian gravitational constant, J is the proper angular momentum of the central body, c is the speed of light in vacuum, a, e and i are the semimajor axis, the eccentricity and the inclination, respectively, of the orbit of the test particle. The Lense-Thirring precessions for the LAGEOS satellites, in milliarcseconds per year (mas yr ^- 1 in the following), amount to

The first measurement of this effect in the gravitational field of the Earth has been obtained by analysing a suitable combination of the laser-ranged data from the existing passive geodetic satellites LAGEOS and LAGEOS II (Ciufolini et al 1998). The observable (Ciufolini 1996) is a linear trend with a slope of 60.2 mas yr ^- 1 and includes the residuals of the nodes of LAGEOS and LAGEOS II and the perigee of LAGEOS II^Note1. The total relative accuracy of the measurement of the solve-for parameter μ_LT, introduced in order to account for this general relativistic effect, is of the order of 2 × 10 ^- 1 (Ciufolini et al 1998).

In this kind of satellite-based space experiment, the major source of systematic errors is represented by the aliasing trends due to the classical secular precessions (Kaula 1966) of the node and the perigee induced by the mismodelling in the even (l =  2n) zonal (m =  0) harmonics of the geopotential J₂, J₄, J₆,  .... Indeed, according to the present knowledge of the Earth's gravity field based on the EGM96 model (Lemoine et al 1998), they amount to a large part of the gravitomagnetic precessions of interest, especially for the first two even zonal harmonics. In the performed LAGEOS-LAGEOS II Lense-Thirring experiment, the adopted observable allows for the cancellation of the static and dynamical effects of J₂ and J₄. The remaining higher-degree even zonal harmonics affect the measurement at a 13% level, according to the convariance matrix of EGM96 up to the degree l =  20.

In order to achieve a few per cent accuracy, in Ciufolini (1986) it was proposed to launch a passive geodetic laser-ranged satellite - the former LAGEOS III - with the same orbital parameters as LAGEOS apart from its inclination which should be supplementary to that of LAGEOS.

This orbital configuration would be able to cancel out exactly the classical nodal precessions, which are proportional to cos i, provided that the observable to be adopted is the sum of the residuals of the nodal precessions of LAGEOS III and LAGEOS

Later on the concept of the mission slightly changed. The area-to-mass ratio of LAGEOS III was reduced in order to make the impact of the non-gravitational perturbations less relevant and the eccentricity was enhanced in order to perform other general relativistic tests: LARES was born (Ciufolini 1998). The orbital parameters of LAGEOS, LAGEOS II and LARES are given in table 1.

Recent developments of the concept of the twin satellites in supplementary orbits have led to the discovery of new, possible independent gravitomagnetic observables based on the use of the perigees as well (Iorio 2002a) and of unexpected connections with the gravitomagnetic clock effect (Iorio and Lichtenegger 2002).

Unfortunately, at present we do not know if the LARES mission will be approved by any space agency. Although it is much cheaper than other proposed and/or approved space-based missions, funding is the major obstacle in implementing the LARES project. The most expensive part is the launching segment.

Very recently, the possibility of launching the LARES satellite into an orbit with a =  8378 km, i =  90°, e =  0.04 and using its node as an observable has been considered (Lucchesi and Paolozzi 2001). In the following, we will name POLARES the LARES satellite in such a proposed polar orbit. The Lense-Thirring secular rate of the POLARES node would amount to 96.9 mas yr ^- 1. Then, the observable would be

i.e., a linear secular trend with a slope of 96.9 mas yr ^- 1. The choice of such a low altitude is motivated by the need to use a cheap rocket launcher; so, it must certainly be considered as an admirable and important further effort towards the practical realization of the LARES project. The polar orbit would allow us to prevent the aliasing effects of the mismodelled classical secular precessions of the node induced by the even zonal coefficients of the multipolar expansion of the static part of the geopotential. The non-gravitational perturbations (Lucchesi 2001, 2002, Lucchesi and Paolozzi 2001) would represent a minor problem.

In this letter, we wish to analyse critically such important and interesting evolution of the LARES concept.

2. The POLARES

2.1. The node-only scenario

Here we will show that the proposed POLARES configuration and the use of the node only as gravitomagnetic observable should present some drawbacks due to the gravitational static and time-varying perturbations.

With regard to the classical secular precessions induced by the centrifugal oblateness of the Earth, it should be considered that the choice of the inclination is of crucial importance. Such an orbital element is affected neither by the secular perturbations induced by the even zonal harmonics of the geopotential (Kaula 1966) nor by the semisecular 18.6-year and 9.3-year tidal perturbations (Iorio 2001). On the other hand, due to possible orbital injection errors which are closely related to the quality of the rocket launcher to be used, the POLARES inclination would be far from being exactly 90°. The relatively low altitude of POLARES would enhance the impact of even small departures of i_PL from its projected nominal value on the systematic error δμ^{even zonals}_LT due to the classical static even zonal harmonics of the geopotential. This is clearly shown in figure 1 in which it has been calculated by means of the covariance matrix of the EGM96 Earth gravity model up to degree l =  20. It should be considered that, contrary to the LAGEOS satellites which are almost insensitive to the even zonal harmonics of degree higher than l =  20, this would not be the case for the POLARES, due to its projected altitude of 2000 km. Then, it might turn out that the estimates of figure 1 are optimistic.

If we consider that the impact of the mismodelled even zonal harmonics of the geopotential on the current LAGEOS-LAGEOS II Lense-Thirring experiment is of the order of 13%, according to EGM96, a certain weakness of the POLARES node-only scenario becomes apparent. With regard to the possibilities offered by the forthcoming new gravity models from the CHAMP and GRACE missions, it should be considered that the major improvements in the accuracy of the even zonal coefficients of the geopotential are expected especially for the first degrees. Now, in the LAGEOS-LAGEOS II Lense-Thirring experiment the first two even zonal harmonics are cancelled out and the terms of degree higher than l =  20 are not relevant because the LAGEOS and LAGEOS II orbits are not sensitive to them. So, the systematic error due to the geopotential, in this case, is due to the even zonal harmonics ranging from l =  6 to l =  20. Then, the CHAMP and GRACE missions will certainly improve such error. On the other hand, the POLARES configuration would be sensitive both to the first two even zonal harmonics, which should be greatly improved by the CHAMP and GRACE results, and to the terms of degree higher than l =  20, for which the improvements should be less relevant. So, even in this perspective, maybe a simple reanalysis of the LAGEOS and LAGEOS II data would be more fruitful than the node-only POLARES choice.

Such a conclusion is enforced also by a simple evaluation of the impact of some tidal perturbations^Note2 (Iorio 2001). Putting aside the fact that the 18.6-year lunar tide, which has a period of 18.6 years due to the lunar node-only motion and a large amplitude, is a l =  2, m =  0 constituent which would affect the POLARES node for i_PL≠ 90°, let us draw our attention to the l =  2, m =  1 K₁ tesseral tide. It is an important constituent which exerts a relevant perturbing action on the node of a satellite^Note3 and has the same period as just the node of the satellite. It turns out that for, say, i_PL =  89.8° the period of the POLARES node would amount to 26 801 days, i.e., 73.3 years. This means that, even without considering the even zonal perturbations at all, the mismodelled part of the K₁ tide itself would resemble a superimposed aliasing trend over a reasonable observational time span of a few years and would completely mask the Lense-Thirring trend.

2.2. The combined residuals scenario

Let us try to see what could happen by inserting the node and the perigee of POLARES in a suitable combination of orbital residuals with the nodes of LAGEOS and LAGEOS II and the perigee of LAGEOS II along the lines sketched in Ciufolini (1996) and Iorio (2002b). Recall that if N orbital elements are present in such combinations, the effects of the first N -  1 even zonal harmonics of the geopotential are cancelled out, irrespective of the orbital geometry of the employed satellites.

It turns out that by combining the nodes of LAGEOS, LAGEOS II and POLARES and the perigees of LAGEOS II and POLARES in

the inclination i_PL =  90° is singular in the sense that the slope of the relativistic effect diverges because the coefficient c₂ with which the node of POLARES enters the combination diverges. For values of i_PL close to 90°, figures 2 and 3 show that the systematic error due to the remaining δJ₁₀, δJ₁₂,  ..., according to the covariance matrix of EGM96 up to degree l =  20, would seriously affect the measurement of the Lense-Thirring linear trend. Moreover, the impact of all the gravitational and non-gravitational time-varying perturbations would be greatly enhanced by the large values of c₂. The situation would not improve even if we include only the perigee of POLARES in the combined residuals

as shown in figure 4.

3. Conclusions

In this paper, the proposal of putting the LARES satellite into a polar, elliptical orbit with an altitude of 2000 km, in order to look at its gravitomagnetic secular node shift, has been critically analysed.

The key point is that the mismodelled classical nodal secular precessions induced by the even zonal coefficients of the multipolar expansion of the terrestrial gravitational field vanish if and only if the inclination of the satellite is exactly 90°. Of course, mainly due to possible orbital injection errors, this could never happen. It turns out that the low altitude of the proposed orbital configuration, and the consequent high sensitivity to the higher even degree zonal terms of the geopotential, would greatly enhance the impact of even small departures of the real values of the inclination from the nominal value of 90°. For example, for just i_PL =  90 ± 0.2° the systematic gravitational error would be of the order of 5-10%, according to the EGM96 Earth gravity model up to l =  20, which is of the same order of magnitude as the present LAGEOS-LAGEOS II experiment. (Its total error, including various systematic gravitational and non-gravitational perturbations, is of the order of 20-30%.) Of course, such a situation would be further made critical by the possible use of a low-cost launcher which, unavoidably, would induce non-negligible orbital injection errors. Moreover, the tesseral K₁ tidal perturbation, which has the same period as the satellite's node, would induce a secular aliasing trend over an observational time span of a few years because its period would amount to several tens of years for near-polar orbits.

Another important point is that the proposed POLARES should not yield substantial improvements also in the context of the combined residuals approach which allows us to cancel out the contribution of the first even zonal coefficients of the geopotential irrespective of the inclination of the satellites. Indeed, it turns out that a combination including the nodes of LAGEOS, LAGEOS II, POLARES and the perigees of LAGEOS II and POLARES would not be defined for i_PL =  90° because the coefficient weighting the node of the POLARES would go to infinity. For very small deviations of i_PL from such a critical value the systematic error induced by the remaining even zonal harmonics would amount to 10-20%. Moreover, the coefficient with which the node of POLARES would enter the combination would be much larger than unity and would greatly enhance the impact of the gravitational and non-gravitational time-dependent perturbations. Even the use of the nodes of LAGEOS and LAGEOS II and the perigees of LAGEOS II and POLARES, in which case i_PL =  90° should not create problems, would not yield benefits because the systematic gravitational error should be of the order of 20-30%.

Moreover, the accuracy of the practical data reduction from such a version of LARES would be affected by the atmospheric drag.

Finally, we can conclude that the proposed low-cost version of the LARES mission would not yield any significant improvements in the measurement of the elusive Lense-Thirring effect, according to the present-day level of knowledge of the Earth's gravitational field. Perhaps, the situation might improve to a certain extent with the new, more accurate gravity models from the CHAMP and GRACE missions which should become available in the next few years. On the contrary, the original concept of the couple of supplementary satellites deserves greater attention, thanks to its much richer spectrum of high-accuracy relativistic observables.

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Notes

Note1

 The perigee of LAGEOS was not used because it introduces large observational errors due to the smallness of the LAGEOS eccentricity (Ciufolini 1996) which amounts to 0.0045.

Note2

 The amplitude of the tidal perturbations on the node is proportional to (1/sin i)dF_lmp/di where F_lmp(i) are the inclination functions (Kaula 1966). For the even zonal (m =  0), tesseral (m =  1) and sectorial (m =  2) terms they are and , respectively.

Note3

 For LAGEOS it has a nominal amplitude of the order of 10³ mas (Iorio 2001).