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Differences between revisions 1 and 8 (spanning 7 versions)
⇤ ← Revision 1 as of 2016-09-07 03:38:03 →
Size: 675
Editor: KeithLofstrom
Comment:
← Revision 8 as of 2016-09-07 18:35:19 → ⇥
Size: 3939
Editor: KeithLofstrom
Comment:
Deletions are marked like this. Additions are marked like this.
Line 9: Line 9:
Transfer orbit from $ r_a $to$ r$ _p 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 ) } $

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

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 square of the velocity ratio.
Line 12: Line 38:
== Spiral, continuous thrust === == Spiral, continuous thrust ==
Line 16: Line 42:
$ \Large v = \sqrt{ \mu / r } ~~~~~ r = \mu / v^2 ~~~~~ v = L / r ~~~~~ L = \mu / v ~~~~~ v = \mu / L ~~~~~ r = L^2 / \mu $ $ \large v = \sqrt{ \mu / r } ~~~~~ r = \mu / v^2 ~~~~~ v = L / r ~~~~~ L = \mu / v ~~~~~ v = \mu / L ~~~~~ r = L^2 / \mu $
Line 18: Line 44:
$ \Large d L = r d v = ( L^2 d v / \mu ) d v ~~~~~ d v = ( \mu / L^2 ) d L $ $ \large d L = r ~ d v = ( L^2 d v / \mu ) d v ~~~~~ d v = ( \mu / L^2 ) d L $
Line 22: Line 48:
$ \Large \Delta v = \int_{v_p}^{v_a} ( \mu / L^2 ) d L = \mu / L_p - \mu / L_a = v_p - v $ $ \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 $
|| $v_p$ || $v_a$ || hohmann || spiral || ratio ||
|| 1.0000 || 1.0000 || 0.0000 || 0.0000 || 1.0000 ||
|| 1.0010 || 1.0000 || 0.0010 || 0.0010 || 1.0000 ||
|| 1.0100 || 1.0000 || 0.0100 || 0.0100 || 1.0000 ||
|| 1.1000 || 1.0000 || 0.0998 || 0.1000 || 0.9977 ||
|| 1.3891 || 1.0000 || 0.3790 || 0.3891 || 0.9740 || 6378+250 -> 12789 M288 server sky ||
|| 2.5222 || 1.0000 || 1.2724 || 1.5222 || 0.8359 || 6378+250 -> 42165 geosynchronous ||
|| 7.6155 || 1.0000 || 3.8787 || 6.6155 || 0.5863 || 6378+250 -> 384400 Moon ||
|| 1.0000 || 0.0000 || 0.4142 || 1.0000 || 0.4142 || 6378+250 -> escape ||

[[ attachment:spiral-to-hohmann.ods | 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!

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

Perigee orbit at rp : vp=rp 

Apogee orbit at ra : va=ra 

Transfer orbit from ra to rp.

v0t=21ra+1rp=2rarpra+rp    e=ra+rpra−rp 

vpt=(1+e)v0t=2rara+rp2rarpra+rp=2(ra+rp)rarp 

vat=(1−e)v0t=2rpra+rp2rarpra+rp=2(ra+rp)rarp 

v at perigee: vp=vpt−vp 

v at apogee: va=va−vat 

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 ) }

~~~~~~~ \large = { \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

\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 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

v_p

v_a

hohmann

spiral

ratio

1.0000

1.0000

0.0000

0.0000

1.0000

1.0010

1.0000

0.0010

0.0010

1.0000

1.0100

1.0000

0.0100

0.0100

1.0000

1.1000

1.0000

0.0998

0.1000

0.9977

1.3891

1.0000

0.3790

0.3891

0.9740

6378+250 -> 12789 M288 server sky

2.5222

1.0000

1.2724

1.5222

0.8359

6378+250 -> 42165 geosynchronous

7.6155

1.0000

3.8787

6.6155

0.5863

6378+250 -> 384400 Moon

1.0000

0.0000

0.4142

1.0000

0.4142

6378+250 -> escape

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!

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