Worked examples — Launch window — phasing with target orbit

This page drills the phase-angle machinery from the parent note across every case it can throw at you. Before any numbers, we lay out the full grid of scenarios so you can see there are no gaps — then we hit each cell.

We reuse only two equations from the parent, so let us restate them in plain words first.

Two symbols need a picture before we compute. Look at the figure: the chaser (teal) sits at radius $r_1$, the target (plum) at $r_2$. The phase angle $\varphi$ is the plum-to-teal angle measured at Earth's centre at the launch instant. Positive $\varphi$ = target is ahead (further along the direction of motion); negative $\varphi$ = target is behind.

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The scenario matrix

Every phasing problem is one (or a blend) of these cells. Each row is a distinct behaviour the formula produces — different sign, different limit, or a different real-world flavour.

Cell	What makes it distinct	Sign of $\varphi$	Covered by
A. Outward, big jump ($r_2\gg r_1$)	target very slow, moves little during long transfer	large +	Ex 1 (LEO→GEO)
B. Outward, small jump ($r_2\gtrsim r_1$)	target nearly as fast as arc it must cover	small +	Ex 2
C. Equal radii ($r_2=r_1$)	degenerate: no transfer needed, $T_{\text{syn}}\to \infty$	$\varphi =0$	Ex 3
D. Inward transfer ($r_2<r_1$)	target is faster, covers more than $\pi$	− (trails)	Ex 4
E. Extreme inward ($r_2\ll r_1$)	target laps the arrival point, $\varphi$ very negative	large −	Ex 5
F. Synodic — near-equal orbits	tiny $	n_1-n_2	$ → huge window gap
G. Real-world word problem	Mars departure phasing, planetary $\mu$	+	Ex 7
H. Exam twist — negative-to-positive wrap	how to report $\varphi$ as a physical lead in $[0,2\pi )$	wrap	Ex 8

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The worked examples

Throughout, ${\mu }_{\oplus }=3.986\times 10^5{\text{km}}^3/{\text{s}}^2$ unless a different body is named.

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Recall Self-test the matrix

Which cell gives $\varphi =0$? ::: Cell C — equal radii ($r_1=r_2$), no transfer arc needed. Outward transfer sign of $\varphi$? ::: Positive — target must lead. Inward transfer sign of $\varphi$? ::: Negative — target must trail (it is faster and sweeps more than $\pi$). Why is the Earth→Mars phase angle independent of ${\mu }_{\odot }$? ::: Because $\mu$ cancels in the ratio; $\varphi$ is pure geometry of radii. How do you report a $\varphi$ below $-2\pi$? ::: Reduce modulo $2\pi$ into $[0,2\pi )$ to get the physical launch lead. Why is ISS $T_{\text{syn}}\approx 17$ days? ::: Chaser and target radii nearly equal → tiny $|n_1-n_2|$ → very slow relative drift.