#format jsmath
= 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.
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== 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'''.

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== 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 $

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== Comparison ==

$ \mu = 1 $ , normalized  ( actually 3.9860044e14 m^2^/s^3^ for the Earth, neglecting J,,2,, oblateness effects)

|| $r_a/r_p$ || $v_p/v_a$ || Hohmann || spiral ||<-2> '''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                ||

[[ attachment:spiral-to-hohmann.ods | libreoffice spreadsheet ]]

To M288, radius ≈2R,,e,,, 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!