## F Jacobian for Mean to Osculating Elements

This appendix contains the Jacobian for the mean to osculating transformation. The variables $ε1,ε2,ε3$ were defined in Eqs. (E.25)(E.27). The Jacobian $D$ is defined as

$D=\frac{\partial \text{œ}}{\partial \overline{\text{œ}}}=I-{J}_{2}{R}_{e}^{2}\left({D}^{\left(lp\right)}+{D}^{\left(sp1\right)}+{D}^{\left(sp2\right)}\right)$

(F.1)

${D}_{11}^{\left(lp\right)}=-\left(\frac{1}{a}\right){a}^{\left(lp\right)},{D}_{12}^{\left(lp\right)}={D}_{13}^{\left(lp\right)}={D}_{14}^{\left(lp\right)}={D}_{15}^{\left(lp\right)}={D}_{16}^{\left(lp\right)}=0$

(F.2)

${D}_{21}^{\left(lp\right)}=-\left(\frac{2}{a}\right){\theta }^{\left(lp\right)}$

(F.3)

${D}_{22}^{\left(lp}$

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