I've been talking about v_inf, or v∞, or hyperbolic excess velocity despite not knowing what the heck I've been talking about.
I admit I still don't. But I did find something I can understand: black-box equations.
Erik Francis explains it all: from any orbit, which goes at v_cir velocity, you want into some other orbit... Hohmann, probably; then off-Hohmann. What's the delta-V? And no, you don't get just to subtract one from the other; we're doing astrophysics here. We're not sailing around interstellar space. We're in a solar system, probably a planetary system. We're locked in an orbit.
Where escape-velocity is the velocity just to get off that first orbit (if angled directly away), you're in orbit - and if it's a circular orbit, you are decidedly angled 90° from that. v_inf looks at other angles - like the angle to get to Hohmann, or to 2L4.
Francis says: √(v_inf2 + v_esc2) - v_cir. That assumes you don't have the kick-ass thrust needed for an Oberth manoeuvre.
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