Perturbation-Aware Admissible Regions: First-Order J2 Corrections to Too-Short-Arc Boundary Construction
Abstract
Abstract The admissible-region (AR) method for too-short-arc (TSA) initial orbit determination constructs the set of dynamically consistent (range, range-rate) pairs $(\rho,\dot\rho)$ consistent with an optical attributable, using two-body boundary curves: an energy constraint and a perigee constraint, both constructed in the existing literature from unperturbed two-body dynamics alone. We derive, verify symbolically and numerically, and scope the domain of validity of first-order corrections to both boundaries arising from the Earth's oblateness ($J_2$). The energy-constraint correction is a purely position-dependent shift, valid throughout the admissible region except near its edge, where it diverges as the inverse $3/2$ power of the unperturbed discriminant. The perigee-constraint correction requires the short-period response of the osculating semi-major axis and eccentricity to $J_2$, derived from the Lagrange planetary equations to first order in eccentricity $e$. Numerical integration confirms the correction to a few percent for $e \gtrsim 10\,J_2(R_\oplus/a)^2$ across LEO, MEO and GEO regimes, breaking down sharply below this threshold. Within its validated domain, the correction yields a near-constant safety margin of $10$--$12$~km at LEO, scaling as $J_2R_\oplus^2/a$ at higher altitudes. We extend the derivation to general argument of perigee $\omega$, finding the worst-case margin varies by a factor of $\sim1.7$ across $\omega$, so the $\omega=0$ figures are…
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