5.0 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 - |
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 - |
c2
|
- |
w2r2
c2
|
ö
÷
ø
|
1/2
|
- |
|
m |
æ
è
|
×
r
|
|
××
r
|
+ w |
×
w
|
r2 + w2r |
×
r
|
ö
ø
|
|
|
c |
æ
è
|
1 - |
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 = |
m0
|
|
æ
è
|
1 - |
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 - |
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 - |
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
|
|
c2
|
ü
ý
þ
|
n |
|
|
+ |
ì
í
î
|
æ
è
|
2w |
×
r
|
+ |
×
w
|
r |
ö
ø
|
|
æ
è
|
1 - |
c2
|
ö
ø
|
+ |
æ
è
|
××
r
|
- w2r |
ö
ø
|
|
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
which can also be obtained from the radial normal component of (5.5).
Substitution of (5.7) into (5.5) gives
|
F = |
|
|
æ
è
|
1 - |
c2
|
- |
w2r2
c2
|
ö
ø
|
1/2
|
|
æ
è
|
1 - |
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
|
= |
|
c2 |
æ
è
|
1 - |
c2
|
- |
w2 r2
c2
|
ö
ø
|
1/2
|
|
æ
è
|
1 - |
c2
|
ö
ø
|
|
|
|
| (5.9) |
which when substituted into (5.8) gives
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 - |
c2
|
- |
w2r2
c2
|
ö
÷
ø
|
1/2
|
|
| (5.11) |
then
|
|
dr
dtp
|
= |
×
r
|
|
dt
dtp
|
= |
|
|
æ
è
|
1 - |
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
|
+ |
c2
|
+ |
c2
|
|
|
|
æ
ç
è
|
1 - |
c2
|
- |
w2r2
c2
|
ö
÷
ø
|
|
|
|
| (5.14) |
But from (5.7)
|
|
c2
|
= - |
c2
|
- |
|
w2r2 |
×
r
|
2
|
|
æ
è
|
××
r
|
- w2r |
ö
ø
|
|
|
| (5.15) |
Which, when substituted into (5.14) gives after reduction
|
|
d2r
dtp2
|
= |
|
|
æ
ç
è
|
1 - |
c2
|
ö
÷
ø
|
|
æ
ç
è
|
1 - |
c2
|
- |
w2r2
c2
|
ö
÷
ø
|
|
|
|
| (5.16) |
Also from (5.8) after taking the magnitude
|
|
××
r
|
= w2r + |
æ
ç
è
|
1 - |
c2
|
- |
w2r2
c2
|
ö
÷
ø
|
1/2
|
|
æ
ç
è
|
1 - |
c2
|
ö
÷
ø
|
|
F
m0
|
|
| (5.17) |
and substitution of this into (5.16) then yields
|
|
d2r
dtp2
|
= |
F
|
m0 |
æ
è
|
1 - |
c2
|
- |
w2r2
c2
|
ö
ø
|
1/2
|
|
+ |
w2r
|
|
| (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 - |
c2
|
- |
w2r2
c2
|
ö
÷
ø
|
1/2
|
|
| (5.19) |
so that this gives in (5.18)
|
|
d2r
dtp2
|
= |
F
|
m0 |
æ
è
|
1 - |
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
|
= |
|
|
æ
è
|
1 - |
c2
|
- |
w2r2
c2
|
ö
ø
|
1/2
|
|
|
| (5.21) |
and, from (5.19), with
|
wp r = |
wr
|
|
æ
è
|
1 - |
c2
|
- |
w2r2
c2
|
ö
ø
|
1/2
|
|
|
| (5.22) |
then
and rearrangement of this then gives
|
|
æ
ç
è
|
1 - |
c2
|
- |
w2r2
c2
|
ö
÷
ø
|
1/2
|
= |
æ
ç
è
|
1 + |
c2
|
+ |
wp2 r2
c2
|
ö
÷
ø
|
-1/2
|
|
| (5.24) |
so that (5.20) finally becomes
|
|
d2r
dtp2
|
= wp2 r + |
æ
ç
è
|
1 + |
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
Integrating (5.26) gives
|
|
m0 wr2
|
|
æ
è
|
1 - |
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):
|
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
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
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
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
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
|
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
|
- 2 |
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
|