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

On to the Next Section:- Appendix B

Back to the Introduction to this Paper:- Magnetic Dipole Couplin

Back to the Home Page for this Site:- Home