The problem of rendezvous, the meeting of spacecraft in orbit, is an important aspect of mission planning. We imagine a situation where a chaser craft, initially traveling on the same circular orbit as its target and separated from it by a known distance, must select an initial thrust vector that will allow it to meet the target (interception) followed by a second thrust vector that will allow it to match velocities with the target (rendezvous). The analysis presented here provides solutions to this problem in simple algebraic forms while offering many rich challenges that support intuition-building exercises for students across a range of skill levels. An html-javascript orbit calculator is made available with this manuscript as a supporting visual aid and can be used to test the analysis and explore the consequences of different orbital intercept solutions.

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Due to the complicated dependence of the orbital period on α and δ, through ϕ and ϵ, it is possible for thrust vectors with small backward components to increase the orbital period of the elliptical trajectory. This can be proved by evaluating the orbital period of Eq. (7) at α = π / 2, which results in ϕ = π / 2 for all values of δ, and therefore T c / T t = 1 / ( 1 ϵ 2 ) > 1. It follows that this ratio of orbital periods will continue to be greater than unity for some window of thrust vector angles greater than π / 2.
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This can be proven by considering the derivative of the RHS of Eq. (18) with respect to α. Noting that the expression for the orbital period of the chaser depends on ϕ only through powers of cos ( α ) after replacing the sin 2 ( α ) term with 1 cos 2 ( α ) in the expression for ϵ 2, it follows that this derivative is proportional to sin ( α ). Therefore, α = 0 ° and α = 180 ° provide solutions with the lowest sensitivity to small deviations in α. Further analysis of this derivative shows that these are the only two possibilities.
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See supplementary material at https://www.scitation.org/doi/suppl/10.1119/10.0003489 for the html-javascript orbit calculator.

Supplementary Material

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