Spiral vs Hohmann

Relative merits of a 2 impulse Hohmann versus a continuous thrust spiral

Simple analyses, does not directly account for mass change from depletion of propellant, residual atmospheric drag, or Earth's shadow blocking sunlight to presumably solar-powered electric thrusters.

Hohmann, 2 impulse

Perigee orbit at r_p : \large v_p = \LARGE { \sqrt{ \mu \over r_p } }

Apogee orbit at r_a : \large v_a = \LARGE { \sqrt{ \mu \over r_a } }

Transfer orbit from r_a to r_p .

\large v_{0t} = \LARGE { \sqrt{ { \mu \over 2 } \left( { 1 \over r_a } + { 1 \over r_p } \right) } = \sqrt{ { \mu \over 2 } \left( { r_a + r_p } \over { r_a r_p } \right) } } ~ ~ ~ ~ \large e = \LARGE { { r_a - r_p } \over { r_a + r_p } }

\large v_{pt} = ( 1 + e ) v_{0t} = \LARGE { \left( { 2 r_a } \over { r_a + r_p } \right) \sqrt{ { \mu \over 2 } \left( { r_a + r_p } \over { r_a r_p } \right) } } \large = \LARGE \sqrt{ { { 2 \mu } \over { ( r_a + r_p ) } } { r_a \over r_p } }

\large v_{at} = ( 1 - e ) v_{0t} = \LARGE { \left( { 2 r_p } \over { r_a + r_p } \right) \sqrt{ { \mu \over 2 } \left( { r_a + r_p } \over { r_a r_p } \right) } }\large = \LARGE \sqrt{ { { 2 \mu } \over { ( r_a + r_p ) } } { r_p \over r_a } }

\Delta v at perigee: \large { \Delta v_p = v_{pt} - v_p }

\Delta v at apogee: \large { \Delta v_a = v_a - v_{at} }

Total \large { \Delta v = ( v_{pt} - v_{at} ) - ( v_p - v_a ) } = \Large { { \sqrt{ { 2 \mu } \over { r_a + r_p } } } \Large { \left( \sqrt{ r_a \over r_p } -\sqrt{ r_p \over r_a } \right) } \large - ( v_p - v_a ) } = { \sqrt{ { { \Large 2 } \over { \LARGE { { 1 \over v_a^2 } + { 1 \over v_p^2 } } } } } { \LARGE \left( { v_p \over v_a } - { v_a \over v_p } \right) } { \large - ( v_p - v_a ) } } \Large = \LARGE { \sqrt{ { { 2 v_a^2 v_p^2 } \over { v_p^2 + v_a^2 } } } \left( { v_p^2 - v_a^2 } \over { v_p v_a } \right) { \large - ( v_p - v_a ) } } \Large = \LARGE { \sqrt{ { { 2 v_a^2 v_p^2 } \over { v_p^2 + v_a^2 } } } \left( { ( v_p + v_a ) ( v_p - v_a ) } \over { v_p v_a } \right) { \large - ( v_p - v_a ) } } \large = \LARGE { { \sqrt{ { 2 ( v_p + v_a )^2 } \over { v_p^2 + v_a^2 } } } \large { ( v_p - v_a ) - ( v_p - v_a ) } }

Total Thrust Hohmann 2 impulse

The factor in large parentheses ranges from approximately 1.0 if v_p \approx v_a to \sqrt{2}-1 \approx 0.4142 if the velocity ratio is very large or small; escape velocity. The radius ratio is the inverse square of the velocity ratio.

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 = { \LARGE \int_{v_p}^{v_a} } ( \mu / L^2 ) d L = \mu / L_p - \mu / L_a = v_p - v_a

Comparison

\mu = 1, normalized

libreoffice spreadsheet

To M288, radius 2R, a spiral orbit is only 2.6% extra deltaV from LEO. For GEO, only 20%. If a high Isp ion engine is cheap and available, use it!