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How accurate is the cancelation of the first even zonal harmonic of the geopotential in the present and future LAGEOS-based Lense-Thirring tests?

The strategy followed so far in the performed or proposed tests of the general relativistic Lense-Thirring effect in the gravitational field of the Earth with laser-ranged satellites of LAGEOS type relies upon the cancelation of the disturbing huge precessions induced by the first even zonal harmonic coefficient J_2 of the multipolar expansion of the Newtonian part of the terrestrial gravitational potential by means of suitably designed linear combinations of the nodes Ωof more than one spacecraft. Actually, such a removal does depend on the accuracy with which the coefficients of the combinations adopted can be realistically known. Uncertainties of the order of 2 cm in the semimajor axes a and 0.5 milliarcseconds in the inclinations I of LAGEOS and LAGEOS II, entering the expression of the coefficient c_1 of the combination of their nodes used so far, yield an uncertainty δc_1 = 1.30 10^-8. It gives an imperfectly canceled J_2 signal of 10.8 milliarcseconds per year corresponding to 23% of the Lense-Thirring signature. Uncertainties of the order of 10-30 microarcseconds in the inclinations yield δc_1=7.9 10^-9 which corresponds to an uncanceled J_2 signature of 6.5 milliarcseconds per year, i.e. 14% of the Lense-Thirring signal. Concerning a future LAGEOS-LAGEOS II-LARES combination with coefficients k_1 and k_2, the same uncertainties in a and the less accurate uncertainties in I as before yield δk_1=1.1 10^-8, δk_2=2 10^-9; they imply a residual J_2 combined precession of 14.7 milliarcseconds per year corresponding to 29% of the Lense-Thirring trend. Uncertainties in the inclinations at about 10 microarcseconds level give δk_1=5 10^-9, δk_2 = 2 10^-9; the uncanceled J_2 effect is 7.9 milliarcseconds per year, i.e. 16% of the relativistic effect.