Note on the analytical integration of circumterrestrial orbits

The paper presents a complete, explicit J2-only solution in the extended phase space using true-anomaly Delaunay-similar variables: it sets the gauge dt = r^2/Γ dτ and Hamiltonian in these variables (Eqs. 2 and 11), gives the first-order short-period generator W1 (Eq. 24) and long-period generator V1 with the divisor Δ and detuning terms δ, υ (Eqs. 26–28), provides the full second-order short-period generator W2 (Eq. 32), the O(J2^3) secular Hamiltonian term F3 written with inclination polynomials qk,i,j and the Δ−2 divisor (Eq. 35), and the second-order long-period generator V2 as a Δ−3-weighted series with tabulated bl,k,i,j (Eq. 36 and Table 2). These results are explicitly tabulated and internally consistent, and the paper also documents the practical timing-inversion errors in ephemeris evaluation (Figs. 3 and 6) that arise from mapping between physical and fictitious times, which is essential to accuracy assessment. The candidate solution reproduces the correct structural program (sequential Lie–Deprit eliminations; same variable set; same homological equations; same Δ, δ, υ), and its W1 and V1 expressions match the paper. However, it omits the explicit second-order coefficients, does not derive or present the tabulated qk,i,j and bl,k,i,j, and makes a material claim that extended-phase avoids time back-substitution truncation errors; the paper shows such timing inversion introduces additional errors in practice for ephemeris generation. Therefore, the paper’s argument is complete and correct, while the model is incomplete and contains a misleading claim about timing errors. Citations: the extended-phase gauge and Hamiltonian in Delaunay-similar variables are given in Eq. 2 and Eq. 11; W1 in Eq. 24; V1 and the Δ, δ, υ definitions in Eqs. 26–28; W2 structure in Eq. 32; F3 (with qk,i,j) in Eq. 35; V2 (with bl,k,i,j) in Eq. 36 and Table 2; figures illustrating timing-error impacts are discussed alongside Figs. 3 and 6.