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
The \(F(\rho )\) quartic is also degenerate in the equatorial Vinti problem.
Code updates will be available from this website: http://russell.ae.utexas.edu/index_files/vinti.html.
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.
D’Errico, J., “HPF - A Big Decimal Class,” https://www.mathworks.com/matlabcentral/fileexchange/36534-hpf-a-big-decimal-class.
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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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
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
Then, the elements of the matrix inverse can be expressed as
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
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
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
After much algebra, the expression for \(j_{23}\) reduces to
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
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
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:
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
where
For \(j = 1,2,3\), the partials of \(L_m\) can be expressed as
Then, for \(m > 0\), define
With this definition, it can be shown that the recursive relationship
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 \(j = 1,2,3\),
and
where \(P_m^\prime (x)\) is the derivative of the Legendre polynomial with respect to the argument and
A recursive option for computing the derivative of \(P_{m+1}^\prime (x)\) is given by
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
for even n and
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
and, for \(j = 1,2\),
where
Next, the partials of the first-order term, for \(j = 1,2,3\), are computed as
and, for \(j = 1,2\),
where
Finally, the partials of the second-order term, for \(j = 1,2,3\), are determined as
and, for \(j = 1,2\),
where
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 \(j = 1,2,3\),
Then, for \(j = 1,2\),
and
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
where
from Eq. (35) and, for \(j = 1,2\),
For \(j = 3\) (with respect to \(\tilde{S}\)),
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
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
and those of \(\psi _s\) are given by
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
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
The remaining constants are unique to the nonsingular transformation. First, note from Vinti (1969) the constants \(H_k\) and \(C_3\) determined as
and
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
for \(j = 1,2,3\), or more simply as
For \(j = 1,2,3\), the partials of \(H_k\) are obtained from Eq. (89) as
and
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 \(j = 1,2,3\),
Also, for \(j = 1,2\),
From Eq. (23) for \(\dot{v}\) in Vinti (1969) with \(-2 \alpha _1 = \mu / a_0\), for \(j = 1,2,3\),
Also, for \(j = 1,2\),
From Eq. (57) for \(\dot{\psi }\) in Vinti (1969) with \(\alpha _2 = \sqrt{\mu p_0}\), for \(j = 1,2,3\),
Also, for \(j = 1,2\),
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:
and
Now, beginning with positions, for \(j = 1,2,3\),
Also, for \(j = 1,2\),
and
Next, for \(j = 1,2,3\),
Also, for \(j = 1,2\),
and
Considering the velocities, for \(j = 1,2,3\),
where
Also, for \(j = 1,2\),
and
Next, for \(j = 1,2,3\),
where
Also, for \(j = 1,2\),
and
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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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DOI: https://doi.org/10.1007/s10569-017-9806-4