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A satellite relative motion model including \(J_2\) and \(J_3\) via Vinti’s intermediary

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Abstract

Vinti’s potential is revisited for analytical propagation of the main satellite problem, this time in the context of relative motion. A particular version of Vinti’s spheroidal method is chosen that is valid for arbitrary elliptical orbits, encapsulating \(J_2\), \(J_3\), and generally a partial \(J_4\) in an orbit propagation theory without recourse to perturbation methods. As a child of Vinti’s solution, the proposed relative motion model inherits these properties. Furthermore, the problem is solved in oblate spheroidal elements, leading to large regions of validity for the linearization approximation. After offering several enhancements to Vinti’s solution, including boosts in accuracy and removal of some singularities, the proposed model is derived and subsequently reformulated so that Vinti’s solution is piecewise differentiable. While the model is valid for the critical inclination and nonsingular in the element space, singularities remain in the linear transformation from Earth-centered inertial coordinates to spheroidal elements when the eccentricity is zero or for nearly equatorial orbits. The new state transition matrix is evaluated against numerical solutions including the \(J_2\) through \(J_5\) terms for a wide range of chief orbits and separation distances. The solution is also compared with side-by-side simulations of the original Gim–Alfriend state transition matrix, which considers the \(J_2\) perturbation. Code for computing the resulting state transition matrix and associated reference frame and coordinate transformations is provided online as supplementary material.

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Notes

  1. The \(F(\rho )\) quartic is also degenerate in the equatorial Vinti problem.

  2. Code updates will be available from this website: http://russell.ae.utexas.edu/index_files/vinti.html.

  3. Although the Vinti problem is unperturbed, when describing motion in the LVLH frame, the nonspherical part of the Vinti acceleration must be viewed as a perturbation because the LVLH frame is defined by the osculating Keplerian orbit.

  4. D’Errico, J., “HPF - A Big Decimal Class,” https://www.mathworks.com/matlabcentral/fileexchange/36534-hpf-a-big-decimal-class.

  5. Advanpix, Multiprecision Computing Toolbox, http://www.advanpix.com.

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Acknowledgements

This material is based upon work supported in part by the Space Vehicles Directorate of the Air Force Research Laboratory (AFRL) under Contract \(\#\)FA9453-13-C-0201. The authors are grateful for their interest and support. The authors also thank Gim Der for multiple fruitful discussions.

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Correspondence to Ashley D. Biria.

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Appendices

Appendix A: Analytical inverse transformation

Equation (26) establishes the structure of the Jacobian mapping VOEs to the time-varying elements. The analytical inverse transformation is defined here. The first three rows of \({}^{S}T^V\) are equal to the first three rows of \(({}^{S}T^V)^{-1}\) and the sixth column of \({}^{S}T^V\) is equal to the sixth column of \(({}^{S}T^V)^{-1}\). Thus, similar to Eq. (26), the inverse transformation can be expressed as

$$\begin{aligned} ({}^{S}T^{V})^{-1} = \left[ \begin{array}{cccccc} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ T^{-1}_{41} &{} T^{-1}_{42} &{} T^{-1}_{43} &{} T^{-1}_{44} &{} T^{-1}_{45} &{} 0 \\ T^{-1}_{51} &{} T^{-1}_{52} &{} T^{-1}_{53} &{} T^{-1}_{54} &{} T^{-1}_{55} &{} 0 \\ T^{-1}_{61} &{} T^{-1}_{62} &{} T^{-1}_{63} &{} T^{-1}_{64} &{} T^{-1}_{65} &{} 1 \end{array} \right] . \end{aligned}$$
(49)

The remaining 15 elements of the bottom three rows of \(({}^{S}T^V)^{-1}\) are determined as follows. First, define the common denominator D used in the elements of \(({}^{S}T^V)^{-1}\) as

$$\begin{aligned} D = T_{55} T_{44} - T_{45} T_{54} . \end{aligned}$$

Then, the elements of the matrix inverse can be expressed as

$$\begin{aligned} T^{-1}_{41}= & {} -\frac{T_{41} T_{55} - T_{45} T_{51}}{D}; \quad T^{-1}_{42} = -\frac{T_{42} T_{55} - T_{45} T_{52}}{D}; \quad T^{-1}_{43} = -\frac{T_{43} T_{55} - T_{45} T_{53}}{D};\\ T^{-1}_{44}= & {} \frac{T_{55}}{D}; \quad T^{-1}_{45} = -\frac{T_{45}}{D};\\ T^{-1}_{51}= & {} \frac{T_{54} T_{41} - T_{44} T_{51}}{D}; \quad T^{-1}_{52} = \frac{T_{54} T_{42} - T_{44} T_{52}}{D}; \quad T^{-1}_{53} = \frac{T_{54} T_{43} - T_{44} T_{53}}{D};\\ T^{-1}_{54}= & {} -\frac{T_{54}}{D}; \quad T^{-1}_{55} = \frac{T_{44}}{D};\\ T^{-1}_{61}= & {} \frac{(T_{41} T_{55} - T_{45} T_{51}) T_{64} - (T_{54} T_{41} - T_{44} T_{51}) T_{65}}{D} - T_{61};\\ T^{-1}_{62}= & {} \frac{(T_{42} T_{55} - T_{45} T_{52}) T_{64} - (T_{54} T_{42} - T_{44} T_{52}) T_{65}}{D} - T_{62};\\ T^{-1}_{63}= & {} \frac{(T_{43} T_{55} - T_{45} T_{53}) T_{64} - (T_{54} T_{43} - T_{44} T_{53}) T_{65}}{D} - T_{63};\\ T^{-1}_{64}= & {} \frac{T_{54} T_{65} - T_{55} T_{64}}{D}; \quad T^{-1}_{65} = -\frac{T_{44} T_{65} - T_{45} T_{64}}{D} . \end{aligned}$$

Appendix B: Fundamental frequencies

Vinti (1966a) developed analytical expressions for the first two mean frequencies assuming the 1966 potential (\(J_3 \ne 0\)), but an expression for the third has not been previously published. For comparison to the results of Wiesel (2015), it is useful to have an analytical expression for the third mean frequency.

From Eq. (7.14) in Vinti (1961), the third mean frequency can be expressed as

$$\begin{aligned} 2\pi \nu _3 = -\nu _1 j_{13} - \nu _2 j_{23} , \end{aligned}$$
(50)

where \(\nu _m\) for \(m = 1,2,3\) are the mean frequencies associated with \(\rho \), \(\eta \), and \(\phi \), respectively, and \(j_{mn}\) are the partials of the action variable \(j_m\) with respect to \(\alpha _n\) for \(n = 1,2,3\). The expression for \(j_{13}\) is unchanged from Eq. (7.18) in Vinti (1961), given as

$$\begin{aligned} j_{13} = 2\pi c^2 \alpha _3 (-2\alpha _1)^{-\frac{1}{2}} A_3 , \end{aligned}$$
(51)

but \(j_{23}\) under the 1966 Vinti potential is not the same. First, it can be shown that the generic form for \(j_{23}\) becomes

$$\begin{aligned} j_{23} = -2\alpha _3 N_3(\eta _0) = -2\alpha _3 N_3(\psi = \pi /2) . \end{aligned}$$
(52)

After much algebra, the expression for \(j_{23}\) reduces to

$$\begin{aligned} j_{23} = -\frac{2\pi \alpha _3 \sqrt{u}}{\alpha _2} \left[ B_3 + \frac{1}{\sqrt{1-S}} \left( \frac{h_1}{\sqrt{1-2\zeta }} + \frac{h_2}{\sqrt{1+2\zeta }} \right) \right] . \end{aligned}$$
(53)

The mean frequency \(\nu _3\) is finally obtained by substituting Eqs. (51) and (53), along with Eq. (122) from Vinti (1966a) for \(\nu _1\) and \(\nu _2\), into Eq. (50). After simplifying the result into a form similar to the expressions for \(\nu _1\) and \(\nu _2\), \(\nu _3\) can be determined as

$$\begin{aligned} 2\pi \nu _3 = \frac{\alpha _3}{a_0 + A_1 + c^2 A_2 B_1^\prime B_2^{-1}} \left\{ -c^2 A_3 + \frac{A_2}{B_2} \left[ B_3 + \frac{1}{\tilde{S}} \left( \frac{h_1}{\sqrt{1-2\zeta }} + \frac{h_2}{\sqrt{1+2\zeta }} \right) \right] \right\} . \end{aligned}$$
(54)

The expression for \(\nu _3\) in Eq. (54) is exact. Note that the singularity for polar orbits is expected here because the right ascension, \(\phi \), is discontinuous for polar orbits. Neglecting the rotation of the Earth, the notational relationship between the frequencies derived by Wiesel (2015), which he denoted as \(\mathrm {\Omega }_m\), and those of Vinti is

$$\begin{aligned} \mathrm {\Omega }_m = 2\pi \nu _m . \end{aligned}$$
(55)

Appendix C: New partial derivatives

1.1 C.1 Nonsingular partials related to the constant \(b_2\)

The following relation for \(b_2 \cdot \partial b_2 / \partial \sigma _j\) should be used in all partial derivative expressions where it appears:

$$\begin{aligned} b_2 \frac{\partial b_2 }{ \partial \sigma _j} = \frac{1}{2 a} \left[ (ap-c^2) \left( \frac{b_1}{a} \delta _{1j} - \frac{\partial b_1 }{ \partial \sigma _j} \right) - b_1 \left( p\delta _{1j} + a\frac{\partial p }{ \partial \sigma _j} \right) \right] . \end{aligned}$$
(56)

The term appears in equations for partials of \(p_0\), \(L_m\), \(A_{jk}\), \(\dot{\rho }\), and \(\dot{v}\). The expressions containing \(\partial /\partial \sigma _j (b_2/p)\) and \(\partial /\partial \sigma _j (b_1/b_2)\) in the equations for \(\partial A_k/\partial \sigma _j\) are replaced with the following. First, define

$$\begin{aligned} L_m \equiv \left( \frac{b_2}{p} \right) ^m P_m \left( \frac{b_1}{b_2} \right) = \frac{1}{(2p)^m} \sum _{k=0}^{[m/2]} \frac{(-1)^k (2m-2k)!}{k! (m-k)! (m-2k)!} b_1^{m-2k} b_2^{2k} , \end{aligned}$$
(57)

where

$$\begin{aligned}{}[m/2] = {\left\{ \begin{array}{ll} m/2, &{} \text {if }\,m\, \text {even} \\ (m-1)/2, &{} \text {if }\,m\, \text {odd} \end{array}\right. } . \end{aligned}$$

For \(j = 1,2,3\), the partials of \(L_m\) can be expressed as

$$\begin{aligned} \begin{aligned} \frac{\partial L_m}{\partial \sigma _j} =&-\frac{m L_m}{p} \frac{\partial p}{\partial \sigma _j} + \frac{1}{(2p)^m} \sum _{k=0}^{[m/2]} \frac{(-1)^k (2m-2k)!}{k! (m-k)! (m-2k)!} \\&\times \,\left[ (m-2k) b_1^{m-2k-1} b_2^{2k} \frac{\partial b_1}{\partial \sigma _j} + (2k) b_1^{m-2k} b_2^{2k-1} \frac{\partial b_2}{\partial \sigma _j} \right] . \end{aligned} \end{aligned}$$
(58)

Then, for \(m > 0\), define

$$\begin{aligned} \hat{L}_m \equiv \frac{1}{(2p)^m} \sum _{k=0}^{[m/2]} \frac{(-1)^k (2m-2k)!}{k! (m-k)! (m-2k-1)!} b_1^{m-2k-1} b_2^{2k} . \end{aligned}$$
(59)

With this definition, it can be shown that the recursive relationship

$$\begin{aligned} \frac{\partial L_m}{\partial \sigma _j} = -\frac{m L_m}{p} \frac{\partial p}{\partial \sigma _j} + \hat{L}_m \frac{\partial b_1}{\partial \sigma _j} - \hat{L}_{m-1} b_2 \frac{\partial b_2}{\partial \sigma _j} \end{aligned}$$
(60)

is equivalent to Eq. (58), with \(\hat{L}_0 = 0\). Accordingly, the new expressions for \(\partial A_k/\partial \sigma _j\) are now presented. For \(j = 1,2,3\),

$$\begin{aligned} \begin{aligned} \frac{\partial A_1}{\partial \sigma _j} =&\frac{A_1}{p} \frac{\partial p}{\partial \sigma _j} - \delta _{2j} A_1 \frac{e}{1-e^2} + p(1-e^2)^{\frac{1}{2}} \sum _{n=2}^{\infty } \frac{\partial L_n}{\partial \sigma _j} R_{n-2} \left[ (1-e^2)^{\frac{1}{2}} \right] \\&+\,\delta _{2j} pe \left\{ (1-e^2)^{-1} \sum _{n=3}^{\infty } \left[ (1-e^2)^{\frac{1}{2}} \right] ^{n-2} L_n P_{n-2}^\prime \left[ (1-e^2)^{-\frac{1}{2}} \right] \right. \\&-\,\left. (1-e^2)^{-\frac{1}{2}} \sum _{n=3}^{\infty } (n-2) L_n R_{n-2} \left[ (1-e^2)^{-\frac{1}{2}} \right] \right\} , \end{aligned} \end{aligned}$$
(61)
$$\begin{aligned} \begin{aligned} \frac{\partial A_2}{\partial \sigma _j} =&-\frac{A_2}{p} \frac{\partial p}{\partial \sigma _j} - \delta _{2j} A_2 \frac{e}{1-e^2} + \frac{(1-e^2)^{\frac{1}{2}}}{p} \sum _{n=1}^{\infty } \frac{\partial L_n}{\partial \sigma _j} R_{n} \left[ (1-e^2)^{\frac{1}{2}} \right] \\&+\,\delta _{2j} \frac{e}{p} \left\{ (1-e^2)^{-1} \sum _{n=1}^{\infty } \left[ (1-e^2)^{\frac{1}{2}} \right] ^{n} L_n P_{n}^\prime \left[ (1-e^2)^{-\frac{1}{2}} \right] \right. \\&-\,\left. (1-e^2)^{-\frac{1}{2}} \sum _{n=1}^{\infty } n L_n R_{n} \left[ (1-e^2)^{-\frac{1}{2}} \right] \right\} , \end{aligned} \end{aligned}$$
(62)

and

$$\begin{aligned} \begin{aligned} \frac{\partial A_3}{\partial \sigma _j} =&-\frac{3 A_3}{p} \frac{\partial p}{\partial \sigma _j} - \delta _{2j} A_3 \frac{e}{1-e^2} + \frac{(1-e^2)^{\frac{1}{2}}}{p^3} \sum _{n=1}^{\infty } \frac{\partial D_n}{\partial \sigma _j} R_{n+2} \left[ (1-e^2)^{\frac{1}{2}} \right] \\&+ \delta _{2j} \frac{e}{p^3} \left\{ (1-e^2)^{-1} \sum _{n=0}^{\infty } \left[ (1-e^2)^{\frac{1}{2}} \right] ^{n+2} D_n P_{n+2}^\prime \left[ (1-e^2)^{-\frac{1}{2}} \right] \right. \\&- \left. (1-e^2)^{-\frac{1}{2}} \sum _{n=0}^{\infty } (n+2) D_n R_{n+2} \left[ (1-e^2)^{-\frac{1}{2}} \right] \right\} , \end{aligned} \end{aligned}$$
(63)

where \(P_m^\prime (x)\) is the derivative of the Legendre polynomial with respect to the argument and

$$\begin{aligned} D_n = {\left\{ \begin{array}{ll} D_{2k} = \sum _{m=0}^{k} (-1)^{k-m} \left( \frac{c}{p} \right) ^{2(k-m)} L_{2m} &{} \text {if } m \text { even} \\ D_{2k+1} = \sum _{m=0}^{k} (-1)^{k-m} \left( \frac{c}{p} \right) ^{2(k-m)} L_{2m+1} &{} \text {if } m \text { odd} \end{array}\right. } . \end{aligned}$$
(64)

A recursive option for computing the derivative of \(P_{m+1}^\prime (x)\) is given by

$$\begin{aligned} P_{m+1}^\prime (x) = (m+1)P_m(x) + xP_m^\prime (x) . \end{aligned}$$
(65)

Notice that in the third line of each of Eqs. (61), (62), and (63), the expression has been simplified from its original form through the use of the \(R_m(x)\) function. The new partials of \(D_n\), for \(j = 1,2,3\), become

$$\begin{aligned} \begin{aligned} \frac{\partial D_n}{\partial \sigma _j} = \frac{\partial D_{2k}}{\partial \sigma _j} =&-\,2 \left[ \frac{1}{p} \sum _{m=0}^{k} (-1)^{k-m} (k-m) \left( \frac{c}{p} \right) ^{2(k-m)} L_{2m} \right] \frac{\partial p}{\partial \sigma _j} \\&+ \sum _{m=0}^{k} (-1)^{k-m} \left( \frac{c}{p} \right) ^{2(k-m)} \frac{\partial L_{2m}}{\partial \sigma _j} \end{aligned} \end{aligned}$$
(66)

for even n and

$$\begin{aligned} \begin{aligned} \frac{\partial D_n}{\partial \sigma _j} = \frac{\partial D_{2k+1}}{\partial \sigma _j} =&-\,2 \left[ \frac{1}{p} \sum _{m=0}^{k} (-1)^{k-m} (k-m) \left( \frac{c}{p} \right) ^{2(k-m)} L_{2m+1} \right] \frac{\partial p}{\partial \sigma _j} \\&+ \sum _{m=0}^{k} (-1)^{k-m} \left( \frac{c}{p} \right) ^{2(k-m)} \frac{\partial L_{2m+1}}{\partial \sigma _j} \end{aligned} \end{aligned}$$
(67)

for odd n.

1.2 C.2 Removing artificial singularities in the partials of true anomaly

The partial derivatives that Walden and Watson (1967) derived for the periodic terms of the true anomaly, \(v_0\), \(v_1\), and \(v_2\), contain singularities when the associated true anomaly, \(v^\prime \), \(v^{\prime \prime }\), or v, is 0 or \(\pi \). New partial derivatives are presented here that avoid the singularity of dividing the sine of eccentric anomaly by the sine of true anomaly.

The partials of the zeroth-order term, for \(j = 1,2,3\), are determined as

$$\begin{aligned} \frac{\partial v_0}{\partial \sigma _j} = \frac{\sqrt{1-e^2}}{1-e\cos E^\prime } \frac{\partial E^\prime }{\partial \sigma _j} + \delta _{2j} \frac{\sin v^\prime }{1-e^2} - \frac{\partial M_s}{\partial \sigma _j} \end{aligned}$$
(68)

and, for \(j = 1,2\),

$$\begin{aligned} \frac{\partial v_0}{\partial \lambda _j} = \frac{\sqrt{1-e^2}}{1-e\cos E^\prime } \frac{\partial E^\prime }{\partial \lambda _j} - \frac{\partial M_s}{\partial \lambda _j} , \end{aligned}$$
(69)

where

$$\begin{aligned} E^\prime = M_s + E_0 . \end{aligned}$$
(70)

Next, the partials of the first-order term, for \(j = 1,2,3\), are computed as

$$\begin{aligned} \frac{\partial v_1}{\partial \sigma _j} = \frac{\sqrt{1-e^2}}{1-e\cos E^{\prime \prime }} \frac{\partial E^{\prime \prime }}{\partial \sigma _j} + \delta _{2j} \frac{\sin v^{\prime \prime }}{1-e^2} - \frac{\partial v^\prime }{\partial \sigma _j} \end{aligned}$$
(71)

and, for \(j = 1,2\),

$$\begin{aligned} \frac{\partial v_1}{\partial \lambda _j} = \frac{\sqrt{1-e^2}}{1-e\cos E^{\prime \prime }} \frac{\partial E^{\prime \prime }}{\partial \lambda _j} - \frac{\partial v^\prime }{\partial \lambda _j} , \end{aligned}$$
(72)

where

$$\begin{aligned} E^{\prime \prime } = M_s + E_0 + E_1 . \end{aligned}$$
(73)

Finally, the partials of the second-order term, for \(j = 1,2,3\), are determined as

$$\begin{aligned} \frac{\partial v_2}{\partial \sigma _j} = \frac{\sqrt{1-e^2}}{1-e\cos E} \frac{\partial E}{\partial \sigma _j} + \delta _{2j} \frac{\sin v}{1-e^2} - \left( \frac{\partial v^\prime }{\partial \sigma _j} + \frac{\partial v_1}{\partial \sigma _j} \right) \end{aligned}$$
(74)

and, for \(j = 1,2\),

$$\begin{aligned} \frac{\partial v_2}{\partial \lambda _j} = \frac{\sqrt{1-e^2}}{1-e\cos E} \frac{\partial E}{\partial \lambda _j} - \left( \frac{\partial v^\prime }{\partial \lambda _j} + \frac{\partial v_1}{\partial \lambda _j} \right) , \end{aligned}$$
(75)

where

$$\begin{aligned} E = M_s + E_0 + E_1 + E_2 . \end{aligned}$$
(76)

The partials of \(v_0\), \(v_1\), and \(v_2\) with respect to \(\beta _3\) vanish.

1.3 C.3 Other partials required for the spheroidal element solution

Walden and Watson developed partial derivatives using the singular oblate spheroidal coordinates (Walden and Watson 1967; Walden 1968). They took partials of equations that suffer from singularities for polar orbits. Vinti (1969) developed a new transformation from oblate spheroidal orbital elements to Cartesian coordinates that avoids the singularities associated with polar orbits by introducing a slowly varying element, \(\mathrm {\Omega }^\prime \).

If the STM is desired in the spheroidal element space, then the only additional partials required to propagate a relative Vinti trajectory are those of \(\mathrm {\Omega }^\prime \). The partials of \(\mathrm {\Omega }^\prime \) are obtained from Eq. (51d) in Vinti (1969) as follows. For \(j = 1,2,3\),

$$\begin{aligned} \begin{aligned} \frac{\partial \mathrm {\Omega }^\prime }{\partial \sigma _j} =&-\,c^2 \left[ (-2 \alpha _1)^{-\frac{1}{2}} \frac{\partial \alpha _3}{\partial \sigma _j} + \alpha _3 (-2 \alpha _1)^{-\frac{3}{2}} \frac{\partial \alpha _1}{\partial \sigma _j} \right] \left( A_3 v + \sum _{k=1}^4 A_{3k}\sin kv \right) \\&-\,c^2 \alpha _3(-2 \alpha _1)^{-\frac{1}{2}} \left[ v \frac{\partial A_3}{\partial \sigma _j} + \sum _{k=1}^4 \sin kv \frac{\partial A_{3k}}{\partial \sigma _j} + \left( A_3 + \sum _{k=1}^4 kA_{3k} \cos kv \right) \frac{\partial v}{\partial \sigma _j} \right] \\&+\,\frac{u^{\frac{1}{2}}}{\alpha _2} \left( \frac{\partial \alpha _3}{\partial \sigma _j} - \frac{\alpha _3}{\alpha _2} \frac{\partial \alpha _2}{\partial \sigma _j} + \frac{\alpha _3}{2u} \frac{\partial u}{\partial \sigma _j} \right) \left( B_3 \psi - \frac{3}{4} C_1C_2Q \cos \psi + \frac{3}{32} C_2^2 Q^2 \sin 2\psi \right) \\&+\,\frac{\alpha _3 u^{\frac{1}{2}}}{\alpha _2} \left[ \psi \frac{\partial B_3}{\partial \sigma _j} - \frac{3}{4} \left( C_2Q \frac{\partial C_1}{\partial \sigma _j} + C_1Q \frac{\partial C_2}{\partial \sigma _j}+C_1C_2 \frac{\partial Q}{\partial \sigma _j} \right) \cos \psi + \frac{3}{16} C_2Q \right. \\&\times \,\left. \left( Q \frac{\partial C_2}{\partial \sigma _j}{+}C_2 \frac{\partial Q}{\partial \sigma _j} \right) \sin 2\psi {+} \left( B_3+\frac{3}{4} C_1C_2Q \sin \psi {+} \frac{3}{16} C_2^2 Q^2 \cos 2\psi \right) \frac{\partial \psi }{\partial \sigma _j} \right] . \end{aligned} \end{aligned}$$
(77)

Then, for \(j = 1,2\),

$$\begin{aligned} \begin{aligned} \frac{\partial \mathrm {\Omega }^\prime }{\partial \lambda _j} =&-\,c^2 \alpha _3(-2 \alpha _1)^{-\frac{1}{2}} \left( A_3 + \sum _{k=1}^4 kA_{3k} \cos kv \right) \frac{\partial v}{\partial \lambda _j} \\&+\,\frac{\alpha _3 u^{\frac{1}{2}}}{\alpha _2} \left( B_3 + \frac{3}{4} C_1C_2Q \sin \psi + \frac{3}{16} C_2^2 Q^2 \cos 2\psi \right) \frac{\partial \psi }{\partial \lambda _j} \end{aligned} \end{aligned}$$
(78)

and

$$\begin{aligned} \frac{\partial \mathrm {\Omega }^\prime }{\partial \beta _3} = 1 . \end{aligned}$$
(79)

However, in the current investigation, the partial derivatives of three other quantities are also modified. First, the partials of the polar component of angular momentum, \(\alpha _3\), are modified to remove singularities. Next, the partials of \(M_s\) and \(\psi _s\) are modified to convert all the partials to be with respect to the spheroidal Delaunay elements, \(\lambda _j\), instead of the Jacobi constants \(\beta _j\).

Beginning with the partials of \(\alpha _3\), if near the equatorial singularity such that the element set containing Q is used, then the only changes are that Eqs. (46) and (47) must be applied to the existing partials with respect to \(q_j\) from Walden and Watson (1967). Recall that \(q_j\) is defined differently in this paper, where \(q_1 = a\), \(q_2 = e\), and \(q_3 = Q\). To avoid the polar orbit singularity, the partials of \(\alpha _3\) with respect to \(\sigma _j\) must be completely rederived from Eq. (35) and taken with respect to \(\tilde{\sigma }_j\), where \(\tilde{\sigma }_1 = a\), \(\tilde{\sigma }_2 = e\), \(\tilde{\sigma }_3 = \tilde{S}\). The partials are determined as

$$\begin{aligned} \frac{\partial \alpha _3}{\partial \tilde{\sigma }_j} = \mathrm {sgn \,}\alpha _3 \left( \tilde{S} u_\alpha \frac{\partial \alpha _2}{\partial \tilde{\sigma }_j} - \delta _{3j} \alpha _2 u_\alpha + \frac{1}{2} \frac{\alpha _2 \tilde{S}}{u_\alpha } \frac{\partial u_\alpha }{\partial \tilde{\sigma }_j} \right) , \end{aligned}$$
(80)

where

$$\begin{aligned} u_\alpha \equiv \left\{ 1 - \frac{c^2}{a_0 p_0} S - \frac{\left( \frac{2z_\delta }{p_0}\right) ^2 \left( 1-\frac{c^2}{a_0 p_0} S \right) }{\left[ 1 + \frac{c^2}{a_0 p_0} (1-2S) \right] ^2} S \right\} ^{\frac{1}{2}} \end{aligned}$$
(81)

from Eq. (35) and, for \(j = 1,2\),

$$\begin{aligned} \begin{aligned} \frac{\partial u_\alpha }{\partial \tilde{\sigma }_j}&= S \left\{ -c^2 \frac{\partial }{\partial \tilde{\sigma }_j}(a_0p_0)^{-1} + \frac{\left( \frac{2z_\delta }{p_0} \right) ^2 \left[ \frac{2}{p_0} \left( 1-\frac{c^2}{a_0p_0} S \right) \frac{\partial p_0}{\partial \tilde{\sigma }_j} + c^2 S \frac{\partial }{\partial \tilde{\sigma }_j}(a_0p_0)^{-1} \right] }{\left[ 1 + \frac{c^2}{a_0p_0} (1-2S) \right] ^2} \right. \\&\qquad + \left. \frac{2c^2 \left( \frac{2z_\delta }{p_0} \right) ^2 (1-2S) \left( 1-\frac{c^2}{a_0p_0} S \right) }{\left[ 1+\frac{c^2}{a_0p_0} (1-2S) \right] ^3} \frac{\partial }{\partial \tilde{\sigma }_j}(a_0p_0)^{-1} \right\} . \end{aligned} \end{aligned}$$
(82)

For \(j = 3\) (with respect to \(\tilde{S}\)),

$$\begin{aligned} \begin{aligned} \frac{\partial u_\alpha }{\partial \tilde{S}} =&\left\{ -c^2 \left( S \frac{\partial }{\partial S}(a_0p_0)^{-1} + \frac{1}{a_0p_0} \right) \right. \\&+ \left. \frac{\left( \frac{2z_\delta }{p_0} \right) ^2 \left[ \left( 1-\frac{c^2}{a_0p_0} S \right) \left( -1+\frac{2S}{p_0} \frac{\partial p_0}{\partial S} \right) + c^2 S \left( S \frac{\partial }{\partial S}(a_0p_0)^{-1}+\frac{1}{a_0p_0} \right) \right] }{\left[ 1+\frac{c^2}{a_0p_0} (1-2S) \right] ^2} \right. \\&+ \left. \frac{2c^2 \left( \frac{2z_\delta }{p_0} \right) ^2 S \left( 1-\frac{c^2}{a_0p_0} S \right) }{\left[ 1+\frac{c^2}{a_0p_0} (1-2S) \right] ^3} \left[ (1-2S) \frac{\partial }{\partial S}(a_0p_0)^{-1} - \frac{2}{a_0p_0} \right] \right\} \frac{\partial S}{\partial \tilde{S}} . \end{aligned} \end{aligned}$$
(83)

Notice the strong similarities between these partials of \(u_\alpha \) and those of u located in Walden and Watson (1967).

Next, all the partials can be converted to be with respect to the spheroidal Delaunay elements. The partials of Walden and Watson (1967) with respect to \(\beta _j\) can be decomposed as

$$\begin{aligned} \frac{\partial (\cdot )}{\partial \beta _j} = \frac{\partial (\cdot )}{\partial \lambda _j} \frac{\partial \lambda _j}{\partial \beta _j} . \end{aligned}$$
(84)

By inspection, \(\beta _1\) and \(\beta _2\) only appear explicitly in the expressions for \(l_0\) and \(l_0 + g_0\). Therefore, to obtain the desired partial \(\partial (\cdot ) / \partial \lambda _j\) for an arbitrary quantity, one option is to simply not perform the final step of the chain rule in the existing partials. In other words, the partial derivative \(\partial \lambda _j / \partial \beta _j\) should not be computed. This goal is accomplished by simply computing \(\partial M_s / \partial \lambda _j\) and \(\partial \psi _s / \partial \lambda _j\) and otherwise building up the partials in the same way as in Walden and Watson (1967), except that \(\partial \beta _j\) is replaced by \(\partial \lambda _j\) in all the equations. None of the other partial derivatives need to be modified when changing the independent variables to spheroidal Delaunay elements. The specific partials to be modified are as follows. For \(j = 1,2,3\), the new partials of \(M_s\) are determined as

$$\begin{aligned} \frac{\partial M_s}{\partial \lambda _j} = \delta _{1j} \end{aligned}$$
(85)

and those of \(\psi _s\) are given by

$$\begin{aligned} \frac{\partial \psi _s}{\partial l_0} = \frac{\partial \psi _s}{\partial g_0} = 1; \quad \frac{\partial \psi _s}{\partial \beta _3} = 0 . \end{aligned}$$
(86)

The simple partials of \(M_s\) and \(\psi _s\) given in Eqs. (85) and (86) replace the complicated expressions for their partials with respect to the Jacobi constants.

1.4 C.4 Partials of Vinti’s nonsingular transformation to ECI coordinates

When the linear transformation from oblate spheroidal orbital elements to ECI coordinates is desired, the partials of Vinti’s nonsingular transformation must be computed. These partials have not been published previously and are given here.

The partials of position and velocity require the partials of several constants associated with the nonsingular transformation. Beginning with constant quantities, the nonsingular equation for \(\zeta \) is used, given by

$$\begin{aligned} \zeta = \frac{C_1}{2(1-C_2)} , \end{aligned}$$
(87)

where \(C_2\) is \(O(J_2^2)\) and much smaller than unity. The equation for \(\zeta \) given in Eq. (87) is Eq. (154) in Vinti (1966a), but Walden and Watson (1967) used the singular form \(\zeta = P/(1-S)\). The nonsingular partial derivatives of \(\zeta \), for \(j = 1,2,3\), are determined as

$$\begin{aligned} \frac{\partial \zeta }{\partial \sigma _j} = \frac{1}{2(1-C_2)} \left( \frac{\partial C_1}{\partial \sigma _j} + 2\zeta \frac{\partial C_2}{\partial \sigma _j} \right) . \end{aligned}$$
(88)

The remaining constants are unique to the nonsingular transformation. First, note from Vinti (1969) the constants \(H_k\) and \(C_3\) determined as

$$\begin{aligned} H_1&= \sqrt{ \frac{1 + S + (1 - S) \sqrt{ 1 - C_3^2 z_\delta ^2} }{2} } , \end{aligned}$$
(89a)
$$\begin{aligned} H_2&= \frac{1}{2} Q \left[ \sqrt{1 - C_3z_\delta } - \sqrt{1 + C_3z_\delta } \right] , \end{aligned}$$
(89b)
$$\begin{aligned} H_3&= \frac{1}{2} \left[ (1 + P) \sqrt{1 - C_3z_\delta } + (1 - P) \sqrt{1 + C_3z_\delta } \right] , \end{aligned}$$
(89c)

and

$$\begin{aligned} C_3 = \frac{2u}{p_0 (1 - C_2 S)} = \frac{2 \zeta }{z_\delta } . \end{aligned}$$
(90)

Vinti (1969) denoted \(C_3\) as r, but r is avoided here to alleviate confusion between this quantity and the magnitude of a position vector. From Eq. (90), the partials of \(C_3\) are determined as

$$\begin{aligned} \frac{\partial C_3}{\partial \sigma _j} = \frac{2}{p_0^2 (1-C_2S)^2} \left\{ p_0 (1-C_2S) \frac{\partial u}{\partial \sigma _j} - u \left[ (1-C_2S) \frac{\partial p_0}{\partial \sigma _j} - p_0 \left( \delta _{3j}C_2 + S \frac{\partial C_2}{\partial \sigma _j} \right) \right] \right\} \end{aligned}$$
(91)

for \(j = 1,2,3\), or more simply as

$$\begin{aligned} \frac{\partial C_3}{\partial \sigma _j} = \frac{2}{z_\delta } \frac{\partial \zeta }{\partial \sigma _j} . \end{aligned}$$
(92)

For \(j = 1,2,3\), the partials of \(H_k\) are obtained from Eq. (89) as

$$\begin{aligned} \frac{\partial H_1}{\partial \sigma _j}= & {} \frac{1}{4 H_1} \left\{ \delta _{3j} \left[ 1-(1-4\zeta ^2)^{\frac{1}{2}} \right] - \frac{(1-S) 4\zeta }{(1-4\zeta ^2)^{\frac{1}{2}}} \frac{\partial \zeta }{\partial \sigma _j} \right\} , \end{aligned}$$
(93)
$$\begin{aligned} \frac{\partial H_2}{\partial \sigma _j}= & {} \frac{1}{2} \left[ (1-2\zeta )^{\frac{1}{2}} - (1+2\zeta )^{\frac{1}{2}} \right] \frac{\partial Q}{\partial \sigma _j} - \frac{1}{2} Q \left[ (1-2\zeta )^{-\frac{1}{2}} + (1+2\zeta )^{-\frac{1}{2}} \right] \frac{\partial \zeta }{\partial \sigma _j} ,\nonumber \\ \end{aligned}$$
(94)

and

$$\begin{aligned} \frac{\partial H_3}{\partial \sigma _j} = \frac{1}{2} \left[ (1-2\zeta )^{\frac{1}{2}} - (1+2\zeta )^{\frac{1}{2}} \right] \frac{\partial P}{\partial \sigma _j} - \frac{1}{2} \left[ \frac{1+P}{(1-2\zeta )^{\frac{1}{2}}} - \frac{1-P}{(1+2\zeta )^{\frac{1}{2}}} \right] \frac{\partial \zeta }{\partial \sigma _j} . \end{aligned}$$
(95)

To propagate a relative Vinti trajectory in ECI coordinates using an STM, the partials for \(\dot{\mathrm {\Omega }}^\prime \), \(\dot{\psi }\), and \(\dot{v}\) are required in addition to those of position and velocity. From Eq. (55) for \(\dot{\mathrm {\Omega }}^\prime \) in Vinti (1969), for \(j = 1,2,3\),

$$\begin{aligned} \begin{aligned} \frac{\partial \dot{\mathrm {\Omega }}^\prime }{\partial \sigma _j} =&-c^2 \left[ \frac{\partial \alpha _3}{\partial \sigma _j} (-2\alpha _1)^{-\frac{1}{2}} + \alpha _3 (-2\alpha _1)^{-\frac{3}{2}} \frac{\partial \alpha _1}{\partial \sigma _j} \right] \left( A_3 + \sum _{k=1}^4 kA_{3k} \cos kv \right) \dot{v} \\&- \frac{c^2 \alpha _3}{(-2\alpha _1)^{\frac{1}{2}}} \left\{ \left[ \frac{\partial A_3}{\partial \sigma _j} + \sum _{k=1}^4 k\cos kv \frac{\partial A_{3k}}{\partial \sigma _j} - \left( \sum _{k=1}^4 k^2 A_{3k} \sin kv \right) \frac{\partial v}{\partial \sigma _j} \right] \dot{v} \right. \\&+ \left. \left( A_3 + \sum _{k=1}^4 kA_{3k} \cos kv \right) \frac{\partial \dot{v}}{\partial \sigma _j} \right\} + \frac{u^{\frac{1}{2}}}{\alpha _2} \left( \frac{\partial \alpha _3}{\partial \sigma _j}-\frac{\alpha _3}{\alpha _2} \frac{\partial \alpha _2}{\partial \sigma _j}+\frac{\alpha _3}{2u} \frac{\partial u}{\partial \sigma _j} \right) \\&\times \left( B_3+\frac{3}{4} C_1C_2Q \sin \psi +\frac{3}{16} C_2^2 Q^2 \cos 2\psi \right) \dot{\psi } \\&+ \frac{\alpha _3 u^{\frac{1}{2}}}{\alpha _2} \left\{ \left[ \frac{\partial B_3}{\partial \sigma _j} + \frac{3}{4} \left( C_2Q \frac{\partial C_1}{\partial \sigma _j}+C_1Q \frac{\partial C_2}{\partial \sigma _j}+C_1C_2 \frac{\partial Q}{\partial \sigma _j} \right) \sin \psi \right. \right. \\&+ \left. \left. \frac{3}{8} C_2Q \left( Q \frac{\partial C_2}{\partial \sigma _j} + C_2 \frac{\partial Q}{\partial \sigma _j} \right) \cos 2\psi + \left( \frac{3}{4}C_1C_2Q \cos \psi - \frac{3}{8} C_2^2 Q^2 \sin 2\psi \right) \frac{\partial \psi }{\partial \sigma _j} \right] \dot{\psi } \right. \\&+ \left. \left( B_3 + \frac{3}{4} C_1C_2Q \sin \psi + \frac{3}{16} C_2^2 Q^2 \cos 2\psi \right) \frac{\partial \dot{\psi }}{\partial \sigma _j} \right\} . \end{aligned} \end{aligned}$$
(96)

Also, for \(j = 1,2\),

$$\begin{aligned} \begin{aligned} \frac{\partial \dot{\mathrm {\Omega }}^\prime }{\partial \lambda _j} =&- \frac{c^2 \alpha _3}{(-2\alpha _1)^{\frac{1}{2}}} \left[ - \left( \sum _{k=1}^4 k^2 A_{3k} \sin kv \right) \frac{\partial v}{\partial \lambda _j} \dot{v} + \left( A_3 + \sum _{k=1}^4 kA_{3k} \cos kv \right) \frac{\partial \dot{v}}{\partial \lambda _j} \right] \\&+ \frac{\alpha _3 u^{\frac{1}{2}}}{\alpha _2} \left[ \left( \frac{3}{4}C_1C_2Q \cos \psi - \frac{3}{8} C_2^2 Q^2 \sin 2\psi \right) \dot{\psi } \frac{\partial \psi }{\partial \lambda _j} \right. \\&+ \left. \left( B_3 + \frac{3}{4} C_1C_2Q \sin \psi + \frac{3}{16} C_2^2 Q^2 \cos 2\psi \right) \frac{\partial \dot{\psi }}{\partial \lambda _j} \right] . \end{aligned} \end{aligned}$$
(97)

From Eq. (23) for \(\dot{v}\) in Vinti (1969) with \(-2 \alpha _1 = \mu / a_0\), for \(j = 1,2,3\),

$$\begin{aligned} \begin{aligned} \frac{\partial \dot{v}}{\partial \sigma _j} =&\frac{[(-2 \alpha _1)(\rho ^2 - 2b_1\rho + b_2^2)]^{\frac{1}{2}}}{\rho (\rho ^2 + c^2 \eta ^2)} \left[ \delta _{1j} (1-e^2)^{\frac{1}{2}} - \delta _{2j} \frac{ae}{(1-e^2)^{\frac{1}{2}}} \right] + \dot{v} \left[ -\frac{1}{\rho } \frac{\partial \rho }{\partial \sigma _j} \right. \\&+ \left. \frac{1}{2\alpha _1} \frac{\partial \alpha _1}{\partial \sigma _j} - 2 \left( \rho \frac{\partial \rho }{\partial \sigma _j} + c^2 \eta \frac{\partial \eta }{\partial \sigma _j} \right) (\rho ^2 + c^2 \eta ^2)^{-1} \right] \\&- \frac{2a \alpha _1 (1-e^2)^{\frac{1}{2}}}{\rho (\rho ^2 + c^2 \eta ^2) [(-2 \alpha _1)(\rho ^2 - 2b_1\rho + b_2^2)]^{\frac{1}{2}}} \left[ (\rho -b_1) \frac{\partial \rho }{\partial \sigma _j} - \rho \frac{\partial b_1}{\partial \sigma _j} + b_2 \frac{\partial b_2}{\partial \sigma _j} \right] . \end{aligned} \end{aligned}$$
(98)

Also, for \(j = 1,2\),

$$\begin{aligned} \begin{aligned} \frac{\partial \dot{v}}{\partial \lambda _j} =&\dot{v} \left[ -\frac{1}{\rho } \frac{\partial \rho }{\partial \lambda _j} + \frac{1}{2\alpha _1} \frac{\partial \alpha _1}{\partial \lambda _j} - 2 \left( \rho \frac{\partial \rho }{\partial \lambda _j} + c^2 \eta \frac{\partial \eta }{\partial \lambda _j} \right) (\rho ^2 + c^2 \eta ^2)^{-1} \right] \\&- \frac{2a \alpha _1 (1-e^2)^{\frac{1}{2}}}{\rho (\rho ^2 + c^2 \eta ^2) [(-2 \alpha _1)(\rho ^2 - 2b_1\rho + b_2^2)]^{\frac{1}{2}}} (\rho -b_1) \frac{\partial \rho }{\partial \lambda _j} . \end{aligned} \end{aligned}$$
(99)

From Eq. (57) for \(\dot{\psi }\) in Vinti (1969) with \(\alpha _2 = \sqrt{\mu p_0}\), for \(j = 1,2,3\),

$$\begin{aligned} \begin{aligned} \frac{\partial \dot{\psi }}{\partial \sigma _j} =&\left\{ (1 + C_1\eta - C_2\eta ^2)^{\frac{1}{2}} \frac{\partial \alpha _2}{\partial \sigma _j} + \frac{\alpha _2}{2} \left[ \eta \frac{\partial C_1}{\partial \sigma _j} + \left( C_1 - 2C_2\eta \right) \frac{\partial \eta }{\partial \sigma _j} - \eta ^2 \frac{\partial C_2}{\partial \sigma _j} \right] \right. \\&\times \left. (1 + C_1\eta - C_2\eta ^2)^{-\frac{1}{2}} \right\} [u^{\frac{1}{2}} (\rho ^2 + c^2 \eta ^2)]^{-1} - \frac{\alpha _2 (1 + C_1\eta - C_2\eta ^2)^{\frac{1}{2}}}{\rho ^2 + c^2 \eta ^2} \\&\times \left[ \frac{1}{2 u^{\frac{3}{2}}} \frac{\partial u}{\partial \sigma _j} \frac{2}{u^{\frac{1}{2}}(\rho ^2 + c^2 \eta ^2)} \left( \rho \frac{\partial \rho }{\partial \sigma _j} + c^2 \eta \frac{\partial \eta }{\partial \sigma _j} \right) \right] . \end{aligned} \end{aligned}$$
(100)

Also, for \(j = 1,2\),

$$\begin{aligned} \begin{aligned} \frac{\partial \dot{\psi }}{\partial \lambda _j} =&\frac{\alpha _2 \left( C_1 - 2C_2\eta \right) }{2u^{\frac{1}{2}} (\rho ^2 + c^2 \eta ^2) (1 + C_1\eta - C_2\eta ^2)^{\frac{1}{2}}} \frac{\partial \eta }{\partial \lambda _j} \\&- \frac{2 \alpha _2 (1 + C_1\eta - C_2\eta ^2)^{\frac{1}{2}}}{u^{\frac{1}{2}} (\rho ^2 + c^2 \eta ^2)^2} \left( \rho \frac{\partial \rho }{\partial \lambda _j} + c^2 \eta \frac{\partial \eta }{\partial \lambda _j} \right) . \end{aligned} \end{aligned}$$
(101)

The partials of \(\dot{\mathrm {\Omega }}^\prime \), \(\dot{\psi }\), and \(\dot{v}\) with respect to \(\beta _3\) vanish. Now, the new partials of the ECI state can be obtained in terms of the preceding partials and those of Walden and Watson (1967) and Walden (1968). Note that Eqs. (104), (107), (110), and (114) each contain a Kronecker delta, \(\delta _{3j}\), in one of the terms. These small terms are slightly different depending on whether the partials are taken with respect to \(\tilde{S}\) or Q. Each of these four equations expresses the partials with respect to \(\tilde{S}\). To obtain the partials with respect to Q, simply make the following changes:

$$\begin{aligned} \frac{\delta _{3j}}{H_1} \longrightarrow -\frac{\delta _{3j}}{2H_1\tilde{S}} \frac{\partial S}{\partial Q} \end{aligned}$$
(102)

and

$$\begin{aligned} \frac{\partial (\cdot )}{\partial \tilde{\sigma }_j} \longrightarrow \frac{\partial (\cdot )}{\partial q_j} . \end{aligned}$$
(103)

Now, beginning with positions, for \(j = 1,2,3\),

$$\begin{aligned} \begin{aligned} \frac{\partial X}{\partial \tilde{\sigma }_j}&= \frac{X \rho }{\rho ^2+c^2} \frac{\partial \rho }{\partial \tilde{\sigma }_j} + (\rho ^2+c^2)^{\frac{1}{2}} \left\langle \cos \mathrm {\Omega }^\prime \cos \psi \frac{\partial H_1}{\partial \tilde{\sigma }_j} - H_1 \left( \sin \mathrm {\Omega }^\prime \cos \psi \frac{\partial \mathrm {\Omega }^\prime }{\partial \tilde{\sigma }_j} \right. \right. \\&\quad + \left. \left. \cos \mathrm {\Omega }^\prime \sin \psi \frac{\partial \psi }{\partial \tilde{\sigma }_j} \right) + \mathrm {sgn \,}\alpha _3 \left\{ \left( -\frac{\delta _{3j}}{H_1} + \frac{\tilde{S}}{H_1^2} \frac{\partial H_1}{\partial \tilde{\sigma }_j} \right) \sin \mathrm {\Omega }^\prime (H_2+H_3 \sin \psi ) \right. \right. \\&\quad - \left. \left. \frac{\tilde{S}}{H_1} \left[ \cos \mathrm {\Omega }^\prime (H_2+H_3 \sin \psi ) \frac{\partial \mathrm {\Omega }^\prime }{\partial \tilde{\sigma }_j} + \sin \mathrm {\Omega }^\prime \right. \right. \right. \\&\quad \times \left. \left. \left. \left( \frac{\partial H_2}{\partial \tilde{\sigma }_j} + \sin \psi \frac{\partial H_3}{\partial \tilde{\sigma }_j} + H_3 \cos \psi \frac{\partial \psi }{\partial \tilde{\sigma }_j} \right) \right] \right\} \right\rangle . \end{aligned} \end{aligned}$$
(104)

Also, for \(j = 1,2\),

$$\begin{aligned} \begin{aligned} \frac{\partial X}{\partial \lambda _j} =&\frac{X \rho }{\rho ^2+c^2} \frac{\partial \rho }{\partial \lambda _j} + (\rho ^2+c^2)^{\frac{1}{2}} \left\{ - H_1 \left( \sin \mathrm {\Omega }^\prime \cos \psi \frac{\partial \mathrm {\Omega }^\prime }{\partial \lambda _j} + \cos \mathrm {\Omega }^\prime \sin \psi \frac{\partial \psi }{\partial \lambda _j} \right) \right. \\&- \left. \mathrm {sgn \,}\alpha _3 \frac{\tilde{S}}{H_1} \left[ \cos \mathrm {\Omega }^\prime (H_2+H_3 \sin \psi ) \frac{\partial \mathrm {\Omega }^\prime }{\partial \lambda _j} + H_3 \sin \mathrm {\Omega }^\prime \cos \psi \frac{\partial \psi }{\partial \lambda _j} \right] \right\} \end{aligned} \end{aligned}$$
(105)

and

$$\begin{aligned} \begin{aligned} \frac{\partial X}{\partial \beta _3}&= (\rho ^2+c^2)^{\frac{1}{2}} \left[ - H_1 \sin \mathrm {\Omega }^\prime \cos \psi - \mathrm {sgn \,}\alpha _3 \frac{\tilde{S}}{H_1} \cos \mathrm {\Omega }^\prime (H_2+H_3 \sin \psi ) \right] . \end{aligned} \end{aligned}$$
(106)

Next, for \(j = 1,2,3\),

$$\begin{aligned} \begin{aligned} \frac{\partial Y}{\partial \tilde{\sigma }_j} =&\frac{Y \rho }{\rho ^2+c^2} \frac{\partial \rho }{\partial \tilde{\sigma }_j} + (\rho ^2+c^2)^{\frac{1}{2}} \left\langle \sin \mathrm {\Omega }^\prime \cos \psi \frac{\partial H_1}{\partial \tilde{\sigma }_j} + H_1 \left( \cos \mathrm {\Omega }^\prime \cos \psi \frac{\partial \mathrm {\Omega }^\prime }{\partial \tilde{\sigma }_j} \right. \right. \\&- \left. \left. \sin \mathrm {\Omega }^\prime \sin \psi \frac{\partial \psi }{\partial \tilde{\sigma }_j} \right) + \mathrm {sgn \,}\alpha _3 \left\{ \left( \frac{\delta _{3j}}{H_1} - \frac{\tilde{S}}{H_1^2} \frac{\partial H_1}{\partial \tilde{\sigma }_j} \right) \cos \mathrm {\Omega }^\prime (H_2+H_3 \sin \psi ) \right. \right. \\&+ \left. \left. \frac{\tilde{S}}{H_1} \left[ -\sin \mathrm {\Omega }^\prime (H_2+H_3 \sin \psi ) \frac{\partial \mathrm {\Omega }^\prime }{\partial \tilde{\sigma }_j} + \cos \mathrm {\Omega }^\prime \right. \right. \right. \\&\times \left. \left. \left. \left( \frac{\partial H_2}{\partial \tilde{\sigma }_j} + \sin \psi \frac{\partial H_3}{\partial \tilde{\sigma }_j} + H_3 \cos \psi \frac{\partial \psi }{\partial \tilde{\sigma }_j} \right) \right] \right\} \right\rangle . \end{aligned} \end{aligned}$$
(107)

Also, for \(j = 1,2\),

$$\begin{aligned} \begin{aligned} \frac{\partial Y}{\partial \lambda _j} =&\frac{Y \rho }{\rho ^2+c^2} \frac{\partial \rho }{\partial \lambda _j} + (\rho ^2+c^2)^{\frac{1}{2}} \left\{ H_1 \left( \cos \mathrm {\Omega }^\prime \cos \psi \frac{\partial \mathrm {\Omega }^\prime }{\partial \lambda _j} - \sin \mathrm {\Omega }^\prime \sin \psi \frac{\partial \psi }{\partial \lambda _j} \right) \right. \\&+ \left. \mathrm {sgn \,}\alpha _3 \frac{\tilde{S}}{H_1} \left[ -\sin \mathrm {\Omega }^\prime (H_2+H_3 \sin \psi ) \frac{\partial \mathrm {\Omega }^\prime }{\partial \lambda _j} + H_3 \cos \mathrm {\Omega }^\prime \cos \psi \frac{\partial \psi }{\partial \lambda _j} \right] \right\} \end{aligned} \end{aligned}$$
(108)

and

$$\begin{aligned} \begin{aligned} \frac{\partial Y}{\partial \beta _3}&= (\rho ^2+c^2)^{\frac{1}{2}} \left[ H_1 \cos \mathrm {\Omega }^\prime \cos \psi - \mathrm {sgn \,}\alpha _3 \frac{\tilde{S}}{H_1} \sin \mathrm {\Omega }^\prime (H_2+H_3 \sin \psi ) \right] . \end{aligned} \end{aligned}$$
(109)

Considering the velocities, for \(j = 1,2,3\),

$$\begin{aligned} \frac{\partial \dot{X}}{\partial \tilde{\sigma }_j}&= \frac{\rho }{\rho ^2+c^2} \left( \dot{\rho } \frac{\partial X}{\partial \tilde{\sigma }_j} + X \frac{\partial \dot{\rho }}{\partial \tilde{\sigma }_j} \right) + \frac{\dot{\rho } X}{\rho ^2+c^2} \left( 1 - \frac{2\rho ^2}{\rho ^2+c^2} \right) \frac{\partial \rho }{\partial \tilde{\sigma }_j} - \dot{\mathrm {\Omega }}^\prime \frac{\partial Y}{\partial \tilde{\sigma }_j} - Y \frac{\partial \dot{\mathrm {\Omega }}^\prime }{\partial \tilde{\sigma }_j} \nonumber \\&\quad + \frac{\rho G_X}{(\rho ^2+c^2)^{\frac{1}{2}}} \dot{\psi } \frac{\partial \rho }{\partial \tilde{\sigma }_j} + (\rho ^2+c^2)^{\frac{1}{2}} \left\langle -\cos \mathrm {\Omega }^\prime \sin \psi \frac{\partial H_1}{\partial \tilde{\sigma }_j} + H_1 \left( \sin \mathrm {\Omega }^\prime \sin \psi \frac{\partial \mathrm {\Omega }^\prime }{\partial \tilde{\sigma }_j} \right. \right. \nonumber \\&\quad - \left. \left. \cos \mathrm {\Omega }^\prime \cos \psi \frac{\partial \psi }{\partial \tilde{\sigma }_j} \right) + \mathrm {sgn \,}\alpha _3 \left\{ \left( -\frac{\delta _{3j}}{H_1} + \frac{\tilde{S}}{H_1^2} \frac{\partial H_1}{\partial \tilde{\sigma }_j} \right) H_3 \sin \mathrm {\Omega }^\prime \cos \psi \right. \right. \nonumber \\&\quad - \left. \left. \frac{\tilde{S}}{H_1} \left[ \sin \mathrm {\Omega }^\prime \cos \psi \frac{\partial H_3}{\partial \tilde{\sigma }_j} + H_3 \left( \cos \mathrm {\Omega }^\prime \cos \psi \frac{\partial \mathrm {\Omega }^\prime }{\partial \tilde{\sigma }_j} - \sin \mathrm {\Omega }^\prime \sin \psi \frac{\partial \psi }{\partial \tilde{\sigma }_j} \right) \right] \right\} \right\rangle \dot{\psi } \nonumber \\&\quad + (\rho ^2+c^2)^{\frac{1}{2}} G_X \frac{\partial \dot{\psi }}{\partial \tilde{\sigma }_j}, \end{aligned}$$
(110)

where

$$\begin{aligned} G_X \equiv -H_1 \cos \mathrm {\Omega }^\prime \sin \psi - \mathrm {sgn \,}\alpha _3 \frac{\tilde{S}}{H_1} H_3 \sin \mathrm {\Omega }^\prime \cos \psi . \end{aligned}$$
(111)

Also, for \(j = 1,2\),

$$\begin{aligned} \begin{aligned} \frac{\partial \dot{X}}{\partial \lambda _j} =&\frac{\rho }{\rho ^2+c^2} \left( \dot{\rho } \frac{\partial X}{\partial \lambda _j} + X \frac{\partial \dot{\rho }}{\partial \lambda _j} \right) + \frac{\dot{\rho } X}{\rho ^2+c^2} \left( 1 - \frac{2\rho ^2}{\rho ^2+c^2} \right) \frac{\partial \rho }{\partial \lambda _j} - \dot{\mathrm {\Omega }}^\prime \frac{\partial Y}{\partial \lambda _j} - Y \frac{\partial \dot{\mathrm {\Omega }}^\prime }{\partial \lambda _j} \\&+ \frac{\rho G_X}{(\rho ^2+c^2)^{\frac{1}{2}}} \dot{\psi } \frac{\partial \rho }{\partial \lambda _j} + (\rho ^2+c^2)^{\frac{1}{2}} \left[ H_1 \left( \sin \mathrm {\Omega }^\prime \sin \psi \frac{\partial \mathrm {\Omega }^\prime }{\partial \lambda _j} - \cos \mathrm {\Omega }^\prime \cos \psi \frac{\partial \psi }{\partial \lambda _j} \right) \right. \\&- \left. \mathrm {sgn \,}\alpha _3 \frac{\tilde{S}}{H_1} H_3 \left( \cos \mathrm {\Omega }^\prime \cos \psi \frac{\partial \mathrm {\Omega }^\prime }{\partial \lambda _j} - \sin \mathrm {\Omega }^\prime \sin \psi \frac{\partial \psi }{\partial \lambda _j} \right) \right] \dot{\psi } + (\rho ^2+c^2)^{\frac{1}{2}} G_X \frac{\partial \dot{\psi }}{\partial \lambda _j} \end{aligned} \end{aligned}$$
(112)

and

$$\begin{aligned} \begin{aligned} \frac{\partial \dot{X}}{\partial \beta _3} =&\frac{\rho \dot{\rho }}{\rho ^2+c^2} \frac{\partial X}{\partial \beta _3} - \dot{\mathrm {\Omega }}^\prime \frac{\partial Y}{\partial \beta _3} + (\rho ^2+c^2)^{\frac{1}{2}} \left[ H_1 \sin \mathrm {\Omega }^\prime \sin \psi \right. \\&- \left. \mathrm {sgn \,}\alpha _3 \frac{\tilde{S}}{H_1} H_3 \cos \mathrm {\Omega }^\prime \cos \psi \right] \dot{\psi } . \end{aligned} \end{aligned}$$
(113)

Next, for \(j = 1,2,3\),

$$\begin{aligned} \frac{\partial \dot{Y}}{\partial \tilde{\sigma }_j}= & {} \frac{\rho }{\rho ^2+c^2} \left( \dot{\rho } \frac{\partial Y}{\partial \tilde{\sigma }_j} + Y \frac{\partial \dot{\rho }}{\partial \tilde{\sigma }_j} \right) + \frac{\dot{\rho } Y}{\rho ^2+c^2} \left( 1 - \frac{2\rho ^2}{\rho ^2+c^2} \right) \frac{\partial \rho }{\partial \tilde{\sigma }_j} + \dot{\mathrm {\Omega }}^\prime \frac{\partial X}{\partial \tilde{\sigma }_j} + X \frac{\partial \dot{\mathrm {\Omega }}^\prime }{\partial \tilde{\sigma }_j} \nonumber \\&+ \frac{\rho G_Y}{(\rho ^2+c^2)^{\frac{1}{2}}} \dot{\psi } \frac{\partial \rho }{\partial \tilde{\sigma }_j} + (\rho ^2+c^2)^{\frac{1}{2}} \left\langle -\sin \mathrm {\Omega }^\prime \sin \psi \frac{\partial H_1}{\partial \tilde{\sigma }_j} - H_1 \left( \cos \mathrm {\Omega }^\prime \sin \psi \frac{\partial \mathrm {\Omega }^\prime }{\partial \tilde{\sigma }_j} \right. \right. \nonumber \\&+ \left. \left. \sin \mathrm {\Omega }^\prime \cos \psi \frac{\partial \psi }{\partial \tilde{\sigma }_j} \right) + \mathrm {sgn \,}\alpha _3 \left\{ \left( \frac{\delta _{3j}}{H_1} - \frac{\tilde{S}}{H_1^2} \frac{\partial H_1}{\partial \tilde{\sigma }_j} \right) H_3 \cos \mathrm {\Omega }^\prime \cos \psi \right. \right. \nonumber \\&+ \left. \left. \frac{\tilde{S}}{H_1} \left[ \cos \mathrm {\Omega }^\prime \cos \psi \frac{\partial H_3}{\partial \tilde{\sigma }_j} - H_3 \left( \sin \mathrm {\Omega }^\prime \cos \psi \frac{\partial \mathrm {\Omega }^\prime }{\partial \tilde{\sigma }_j} + \cos \mathrm {\Omega }^\prime \sin \psi \frac{\partial \psi }{\partial \tilde{\sigma }_j} \right) \right] \right\} \right\rangle \dot{\psi } \nonumber \\&+ (\rho ^2+c^2)^{\frac{1}{2}} G_Y \frac{\partial \dot{\psi }}{\partial \tilde{\sigma }_j} , \end{aligned}$$
(114)

where

$$\begin{aligned} G_Y \equiv -H_1 \sin \mathrm {\Omega }^\prime \sin \psi + \mathrm {sgn \,}\alpha _3 \frac{\tilde{S}}{H_1} H_3 \cos \mathrm {\Omega }^\prime \cos \psi . \end{aligned}$$
(115)

Also, for \(j = 1,2\),

$$\begin{aligned} \begin{aligned} \frac{\partial \dot{Y}}{\partial \lambda _j} =&\frac{\rho }{\rho ^2+c^2} \left( \dot{\rho } \frac{\partial Y}{\partial \lambda _j} + Y \frac{\partial \dot{\rho }}{\partial \lambda _j} \right) + \frac{\dot{\rho } Y}{\rho ^2+c^2} \left( 1 - \frac{2\rho ^2}{\rho ^2+c^2} \right) \frac{\partial \rho }{\partial \lambda _j} + \dot{\mathrm {\Omega }}^\prime \frac{\partial X}{\partial \lambda _j} + X \frac{\partial \dot{\mathrm {\Omega }}^\prime }{\partial \lambda _j} \\&+ \frac{\rho G_Y}{(\rho ^2+c^2)^{\frac{1}{2}}} \dot{\psi } \frac{\partial \rho }{\partial \lambda _j} + (\rho ^2+c^2)^{\frac{1}{2}} \left[ -H_1 \left( \cos \mathrm {\Omega }^\prime \sin \psi \frac{\partial \mathrm {\Omega }^\prime }{\partial \lambda _j} + \sin \mathrm {\Omega }^\prime \cos \psi \frac{\partial \psi }{\partial \lambda _j} \right) \right. \\&- \left. \mathrm {sgn \,}\alpha _3 \frac{\tilde{S}}{H_1} H_3 \left( \sin \mathrm {\Omega }^\prime \cos \psi \frac{\partial \mathrm {\Omega }^\prime }{\partial \lambda _j} + \cos \mathrm {\Omega }^\prime \sin \psi \frac{\partial \psi }{\partial \lambda _j} \right) \right] \dot{\psi } + (\rho ^2+c^2)^{\frac{1}{2}} G_Y \frac{\partial \dot{\psi }}{\partial \lambda _j} \end{aligned} \end{aligned}$$
(116)

and

(117)

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Biria, A.D., Russell, R.P. A satellite relative motion model including \(J_2\) and \(J_3\) via Vinti’s intermediary. Celest Mech Dyn Astr 130, 23 (2018). https://doi.org/10.1007/s10569-017-9806-4

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