APPENDIX A.

Derivation of the Median Value of the Radius Vector Magnitude from a Focal Point in an Elliptic Orbit.

In an elliptic orbit the radius vector magnitude is given by
re = Le
1+ ecosf
(A.1)
The average value of re around the complete orbit is then
áre ñ = Le
2p
 ó
õ 		2p

0 
 1
1+ ecosf
 df
(A.2)
Because the orbit is symmetrical, (A.2) may be written
áre ñ = Le
p
 ó
õ 		p

0 
 1
1+ ecosf
 df
(A.3)
Using the substitution
b = tan f
2
(A.4)
then
cosf = 1- b2
1+ b2
     and    df = 2db
1+ b2
(A.5)
The integration limits then become
when  f = p,      b = tan p/2  = ¥    and    when  f = 0,      b = tan0  = 0
(A.6)
These substitutions convert (A.3) to
áre ñ = Le
p( 1-e )
 ó
õ 		¥

0 
1
æ
è 		1+e
1-e
ö
ø 		  + b2
 db
(A.7)
This is a standard integral that evaluates to
áre ñ = Le
( 1-e 2 )1/2
(A.8)

It is interesting to note that the average value of the radius vector magnitude from a focal point around the complete orbit is equal to the semi-minor axis.


P4 Version 1.3.0
Ó P.G.Bass, April 2008

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