# 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_a 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 + v_a ) ( v_p - v_a ) } \over { v_a v_p } \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

\LARGE { \Delta v = ( v_p - v_a ) } \left( \LARGE { \sqrt{ { 2 ( v_p + v_a )^2 } \over { v_p^2 + v_a^2 } } } \LARGE - 1 \right)

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

## Comparison

\mu = 1 , normalized ( actually 3.9860044e14 m2/s3 for the Earth, neglecting J2 oblateness effects)

 r_a/r_p v_p/v_a Hohmann spiral ratio, Hohmann to spiral 1.0000 1.0000 0.0000 0.0000 undef 1.0020 1.0010 0.0010 0.0010 1.0000 1.0201 1.0100 0.0100 0.0100 1.0000 1.2100 1.1000 0.0998 0.1000 0.9977 1.9296 1.3891 0.3790 0.3891 0.9740 6378+250 → 12789 M288 server sky 6.3614 2.5222 1.2724 1.5222 0.8359 6378+250 → 42164 geosynchronous 57.996 7.6155 3.8787 6.6155 0.5863 6378+250 → 384400 Moon ~~\infty ~~\infty 0.4142 1.0000 0.4142 6378+250 → escape

To M288, radius ≈2Re, 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!

SpiralHohmann (last edited 2016-10-22 00:33:11 by KeithLofstrom)