Spiral vs Hohmann
Relative merits of a 2 impulse Hohmann versus a continuous thrust spiral
Simple analyses, does not account for depletion of propellant.
Hohmann, 2 impulse
Transfer orbit from r_a to r _p
== Spiral, continuous thrust ===
Thrust adds specific angular momentum L = r v .
\Large v = \sqrt{ \mu / r } ~~~~~ r = \mu / v^2 ~~~~~ v = L / r ~~~~~ L = \mu / v ~~~~~ v = \mu / L ~~~~~ r = L^2 / \mu
\Large d L = r d v = ( L^2 d v / \mu ) d v ~~~~~ d v = ( \mu / L^2 ) d L
Integrate:
\Large \Delta v = \int_{v_p}^{v_a} ( \mu / L^2 ) d L = \mu / L_p - \mu / L_a = v_p - v