Differences between revisions 5 and 10 (spanning 5 versions)

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 v=(vpt−vat)−(vp−va)=2ra+rprpra−rarp−(vp−va)=21va2+1vp2vavp−vavp−(vp−va) =2va2vp2vp2+va2vpvavp2−va2−(vp−va)=2va2vp2vp2+va2vpva(vp+va)(vp−va)−(vp−va)=vp2+va22(vp+va)2(vp−va)−(vp−va)

Total Thrust Hohmann 2 impulse

v=(vp−va)

vp2+va22(vp+va)2−1

The factor in large parentheses ranges from approximately 1.0 if vpva to 2−104142 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=rv.

v=r r=v2 v=Lr L=v v=L r=L2

dL=r dv=(L2dv)dv dv=(L2)dL

Integrate:

v=vpva(L2)dL=Lp−La=vp−va

Comparison

vp	va	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!

-  ⇤ ← Revision 5 as of 2016-09-07 18:31:40 → 
  Size: 3919
  Editor: KeithLofstrom
  Comment:
+   ← Revision 10 as of 2016-09-07 18:38:26 → ⇥
  Size: 3922
  Editor: KeithLofstrom
  Comment:
-Deletions are marked like this.
+Additions are marked like this.
 Line 17:
-$ \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_{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 } } $
 Line 19:
-$ \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 } } $
+$ \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 } } $
 Line 25:
-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 ) } $
+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 ) } } $
-Line 27:
+Line 28:
-$ ~~~~~~~ \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:
+----
=== Total Thrust Hohmann 2 impulse ===