5  Planar Orbital Motion Kinematics In D0.

In this Section, the equation of an orbital motion is first derived and then solved as a second illustration of the manner in which the concept of Existence Velocity, may be applied to relativistic problems of this nature in Pseudo - Euclidean Space - Time.

5.1  Derivation of the General Curvi - Linear Equation of Motion in a Plane in D0.

Repeating (2.7) for convenience:
V = v + jc  æ
è
1 v2

c2
ö
ø
1/2

 
If the energy mass is m then existence momentum, for purely planar motion, is:

M = m  é
ê
ë
×
r
 
n + wrt + jc  æ
ç
è
1
×
r
2
 

c2
 -  w2r2

c2
ö
÷
ø
1/2

 
ù
ú
û
(5.1)

where for mathematical convenience spatial polar axes have been chosen and where, with reference to some stationary origin,

r is the radial distance of the orbit.
×
r
 
  = dr/dt is the radial velocity of the orbit.


w = dj/dt is the angular rate of the orbit.

n and t are the radial and radial normal unit vectors.

Differentiating (5.1) with respect to time yields the force equation thus:

F dM

dt
 =  é
ë
m æ
è
××
r
 
- w2r ö
ø
+
×
m
 
×
r
 
ù
û
n + é
ë
m æ
è
2w 
×
r
 
+
×
w
 
r ö
ø
+
×
m
 
wr ù
û
t
(5.2)


+ j é
ê
ê
ë
×
m
 
c æ
ç
è
1 -
×
r
2
 

c2
 -  w2r2

c2
ö
÷
ø
1/2

 
-
m  æ
è
×
r
 
××
r
 
+ w
×
w
 
 r2 + w2r
×
r
 
ö
ø

c æ
è
1
×
r
2

c2
 -  w2r2

c2
ö
ø
1/2

 
ù
ú
ú
û

If F is purely spatial, then the temporal part of (5.2 ) is zero and m can be determined by simple integration to be:
m
m
0

æ
è
1
×
r
2

c2
 -  w2r2

c2
ö
ø
1/2

 
(5.3)

where mo is the rest mass. Thus the mass rate is:

×
m
 
=
mo æ
è
×
r
 
××
r
 
+ w
×
w
 
r2 + w2r 
×
r
 
ö
ø

c2 æ
è
1
×
r
2

c2
 -  w2r2

c2
ö
ø
3/2

 
(5.4)

Substitution of (5.3) and (5.4) into (5.2) then yields after reduction, (F is now purely spatial),

F
mo

æ
è
1
×
r
2

c2
 -  w2r2

c2
ö
ø
3/2

 
é
ê
ê
ê
ê
ê
ë
ì
í
î
æ
è
××
r
 
 - w2r ö
ø
æ
è
1 w2r2

c2
ö
ø
 +  æ
è
2w
×
r
 
 + 
×
w
 
r ö
ø
wr
×
r
 

c2
ü
ý
þ
n
  +  ì
í
î
æ
è
2w
×
r
 
 + 
×
w
 
r ö
ø
æ
è
1
×
r
2

c2
ö
ø
 +  æ
è
××
r
 
- w2r ö
ø
wr
×
r
 

c2
ü
ý
þ
t
ù
ú
ú
ú
ú
ú
û
(5.5)

This is the most general form of the force equation for spatially accelerated curvi - linear motion in a plane in D0 and which clearly possesses a distinct symmetry.

5.2  The Case of a Purely Radial Force

If F is purely radial (constant angular momentum ) , then in (5.2), in addition to the temporal component, the radial normal component will also be zero. Thus

m  æ
è
2w 
×
r
 
+
×
w
 
 r ö
ø
= -
×
m
 
w r
(5.6)

which from (5.3) and (5.4) becomes

2w 
×
r
 
+
×
w
 
 r = -
w 
×
r
 
æ
è
××
r
 
- w2r ö
ø

c2 æ
è
1
×
r
2

c2
ö
ø
(5.7)

which can also be obtained from the radial normal component of (5.5). Substitution of (5.7) into (5.5) gives

F =
mo æ
è
××
r
 
- w2r ö
ø

æ
è
1
×
r
2

c2
 -  w2r2

c2
ö
ø
1/2

 
æ
è
1
×
r
2

c2
ö
ø
n
(5.8)

This is the equation of motion of a point mass in a plane in D0 subjected to an arbitrary spatial radial force. Note that substitution of (5.7) into (5.4) yields after reduction

×
m
 

mo
×
r
 
æ
è
××
r
 
w2r ö
ø

c2 æ
è
1
×
r
2

c2
 -  w2 r2

c2
ö
ø
1/2

 
æ
è
1
×
r
2

c2
ö
ø
(5.9)

which when substituted into (5.8) gives

F =
×
m
 
c2

×
r
n
(5.10)

This is clearly seen to be identical to the rectilinear case and provides further confirmation that the mass rate effect only exists along coincident elements of the force and velocity vectors.

5.3  Conversion of the Equation of Motion to Proper Time

To determine the equation of the orbit it is first necessary to convert the equation of motion to the proper time of the point mass.

Conversion of (5.8) to the proper time of the point mass, is achieved as follows. With

dtp

dt
 =  æ
ç
è
1
×
r
2
 

c2
 -  w2r2

c2
ö
÷
ø
1/2

 
(5.11)

then

dr

dtp
 = 
×
r
 
dt

dtp
 = 
×
r

æ
è
1
×
r
2

c2
 -  w2r2

c2
ö
ø
1/2

 
(5.12)

consequently with

d2r

dtp2
 = d æ
è
dr

dtp
ö
ø
/dtp  =  é
ë
d æ
è
dr

dtp
ö
ø
/dt ù
û
dt

dtp
(5.13)

Substitution from (5.11) and (5.12) gives

d2r

dtp2
 = 
××
r
 
  æ
è
1 w2r2

c2
ö
ø
 + 
w
×
w
 
 r2 
×
r
 

c2
 + 
w2r
×
r
 
2
 

c2

æ
ç
è
1
×
r
2
 

c2
  -  w2r2

c2
ö
÷
ø
(5.14)

But from (5.7)

w
×
w
 
r2 
×
r
 

c2
 = -
2w2r 
×
r
 
2
 

c2
 - 
w2r2 
×
r
 
2
 
æ
è
××
r
 
- w2r ö
ø

c4 æ
è
1
×
r
2

c2
ö
ø
(5.15)

Which, when substituted into (5.14) gives after reduction

d2r

dtp2
 = 
××
r
 
w2r
×
r
2

c2

æ
ç
è
1
×
r
2
 

c2
ö
÷
ø
æ
ç
è
1
×
r
2
 

c2
 -  w2r2

c2
ö
÷
ø
(5.16)

Also from (5.8) after taking the magnitude

××
r
 
 = w2r æ
ç
è
1
×
r
2
 

c2
 -  w2r2

c2
ö
÷
ø
1/2

 
æ
ç
è
1
×
r
2
 

c2
ö
÷
ø
F

m0
(5.17)

and substitution of this into (5.16) then yields

d2r

dtp2
 = 
F

m0 æ
è
1
×
r
2

c2
 -  w2r2

c2
ö
ø
1/2

 
 + 
w2r

æ
è
1
×
r
2

c2
- w2r2

c2
ö
ø
(5.18)

but from (5.11)

w =  dj

dt
 =  æ
è
dj

dtp
ö
ø
æ
è
dtp

dt
ö
ø
 = wp æ
ç
è
1
×
r
2
 

c2
 -  w2r2

c2
ö
÷
ø
1/2

 
(5.19)

so that this gives in (5.18)

d2r

dtp2
 = 
F

m0 æ
è
1
×
r
2

c2
 -  w2r2

c2
ö
ø
1/2

 
 + wp2 r
(5.20)


It now only remains to convert the term   æ
è
1
×
r
 
2
 
/c2 - w2r2/c2 ö
ø
1/2
 
  to proper time as follows; rewriting (5.12) as


×
r

p 
 = 
×
r

æ
è
1
×
r
2

c2
 -  w2r2

c2
ö
ø
1/2

 
(5.21)

and, from (5.19), with

wp r
wr

æ
è
1
×
r
2

c2
 -  w2r2

c2
ö
ø
1/2

 
(5.22)
then
×
r
 
2
p 
+ wp2 r2
×
r
 
2
 
+ w2r2

æ
è
1
×
r
2

c2
 -  w2r2

c2
ö
ø
(5.23)

and rearrangement of this then gives

æ
ç
è
1
×
r
2
 

c2
 -  w2r2

c2
ö
÷
ø
1/2

 
 =  æ
ç
è
1 + 
×
r
 
2
p 

c2
 +  wp2 r2

c2
ö
÷
ø
-1/2

 
(5.24)

so that (5.20) finally becomes

d2r

dtp2
 = wp2 r æ
ç
è
1 + 
×
r
 
2
p 

c2
 +  wp2 r2

c2
ö
÷
ø
1/2

 
F

m0
(5.25)

and for a purely spatial radial force, is the equation of planar motion in D0 expressed as a function of the proper time.

5.4  Derivation of the Equation of the Orbit

To obtain the equation of the orbit from (5.25), it is initially necessary to evaluate the first integral of (5.6). Rearrangement of that equation yields

×
m

m
 - 2
×
r

r
 - 
×
w

w
= 0
(5.26)

Integrating (5.26) gives


m0 wr2

æ
è
1
×
r
2

c2
 -  w2r2

c2
ö
ø
1/2

 
 = k
(5.27)


which then, after the determination of the constant of integration, k from initial conditions, r = r0 and w = w0 , when 
×
r
 
= 0   gives, together with (5.22):


wp r2
w0 r02

æ
è
1 w02 r02

c2
ö
ø
1/2

 
(5.28)

and in line with convention this constant is designated, h. The equation of the orbit may now be obtained in the usual way thus. Putting

r 1

m
(5.29)

then

dr

dtp
 = - wp

m2
dm

dj
 = -h dm

dj
(5.30)

and

d2r

dtp2
-( hwp ) æ
è
d2m

dj2
ö
ø
-( h2m2 ) æ
è
d2m

dj2
ö
ø
(5.31)

Insertion of (5.28), (5.29) and (5.30) and (5.31) into (5.25) then gives the desired result for the equation of the orbit.

d2m

dj2
 + m = - F

m0 h2m2
é
ë
1 +  ì
í
î
m2 +  æ
è
dm

dj
ö
ø
2

 
ü
ý
þ
h2

c2
ù
û
1/2

 
(5.32)

5.5  Solution of the Equation of the Orbit for Two Oppositely Charged Particles In a Vacuum.

Assuming conditions are such that the only effect between the two particles is an electrostatic one and that their relative size is such that the smaller has negligible effect upon the larger, and ignoring any spin effects, then the force of attraction between them may be expressed as

F = -F0 m2
(5.33)

The equation of the orbit of the smaller particle, from (5.32) and (5.33) then becomes

d2m

dj2
 + m =  F0

m0 h2
é
ë
1 +  ì
í
î
m2 +  æ
è
dm

dj
ö
ø
2

 
ü
ý
þ
h2

c2
ù
û
1/2

 
(5.34)

to solve this equation put

m2 +  æ
è
dm

dj
ö
ø
2

 
 = Q2
(5.35)

this being the inverse of the perpendicular distance from a focal point of the orbit to a tangent at any point on the spatial trajectory. Differentiating (5.35) with respect to j gives

d2m

dj2
 + m = Q dQ

dm
(5.36)

Substitution of (5.35) and (5.36) into (5.34) then yields

Q dQ

dm
 =  æ
è
1 +  h2Q2

c2
ö
ø
1/2

 
F

m0 h2
(5.37)

This equation can now be solved using standard methods to yield

( m - m0 ) F0

m0 c2
 =  é
ë
1 +  ì
í
î
m2 +  æ
è
dm

dj
ö
ø
2

 
ü
ý
þ
h2

c2
ù
û
1/2

 
 +  æ
è
1 +  h2m02

c2
ö
ø
1/2

 
(5.38)

where the constant of integration has, together with (5.35), been inserted. Rearranging (5.38) for dm/dj gives


æ
è
dm

dj
ö
ø
2

 
=
- æ
è
1 -  F02

m02 c2h2
ö
ø
m2
 + 2 æ
è
F0

m0 h2
ö
ø
ì
í
î
æ
è
1 +  h2m02

c2
ö
ø
1/2

 
 -  F0 m0

m0 c2
ü
ý
þ
m
(5.39)
+ m02 æ
è
1 +  F02

m02 c2h2
ö
ø
 -  æ
è
F0 m0

m0 h2
ö
ø
æ
è
1 +  h2m02

c2
ö
ø
1/2

 

This equation is also a standard type that can be solved using conventional methods to yield, after some reduction

m é
ê
ë
( F0 /m0 h2 ){ ( 1 + h2m02 /c2 )1/2 - F0 m0 /m0 c2 }

( 1 - F02 /m02 c2h2 )
ù
ú
û
é
ê
ë
1 +  { m0 h2m0 /F0 - ( 1 + h2m02 /c2 )1/2 }cosF

(( 1 + h2m02 /c2 )1/2 - F0 m0 /m0 c2 )
ù
ú
û
(5.40)

where

F = j æ
è
1 F02

m02 c2h2
ö
ø
1/2

 
(5.41)

and where initial conditions have been chosen such that the constant of integration is zero. Equation (5.40) describes the spatial trajectory of the smaller particle about the larger and clearly, as in the literature [5], is seen to be a rotating conic section. From the second part of (5.40), this rotation is seen to be a function of the finite Spatial Terminal Velocity, c, within D0, and also, that the precession angle is a constant one being, unlike the gravitational case, independent of the term m.


R1 Version 2.3.3
Ó P.G.Bass, February 2008
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