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.

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

Repeating (2.6) for convenience:

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

(5.1)

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

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

(5.2)

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

(5.3)

where m₀ is the rest mass. Thus the mass rate is:

(5.4)

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

(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.

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

(5.6)

which from (5.3) and (5.4) becomes

(5.7)

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

(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

(5.9)

which when substituted into (5.8) gives

(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.

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

(5.11)

then

(5.12)

consequently with

(5.13 )

Substitution from (5.11) and (5.12) gives

(5.14)

But from (5.7)

(5.15)

Which, when substituted into (5.14) gives after reduction

(5.16)

Also from (5.8) after taking the magnitude

(5.17)

and substitution of this into (5.16) then yields

(5.18)

but from (5.11)

(5.19)

so that this gives in (5.18)

(5.20)

It now only remains to convert the term to proper time as follows; rewriting (5.12) as

and, from (5.19), with

(5.21)

then

(5.22)

and rearrangement of this then gives

(5.23)

so that (5.20) finally becomes

(5.24)

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

To obtain the equation of the orbit from (5.24), it is initially necessary to evaluate the first integral of (5.6). Rearrangement of that equation yields

(5.25)

Integrating (5.25) gives

(5.26a)

which then, after the determination of the constant of integration, k from initial conditions, and, when gives, together with the second part of (5.21):

(5.26b)

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

(5.27)

then

(5.28a)

and

(5.28b)

Insertion of (5.26b), (5.27) and (5.28a & b) into (5.24) then gives the desired result for the equation of the orbit.

(5.29)

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

(5.30)

The equation of the orbit of the smaller particle, from (5.29) and (5.30) then becomes

(5.31)

to solve this equation put

(5.32)

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 (5.32) with respect to j gives

(5.33)

Substitution of (5.32) and (5.33) into (5.31) then yields

(5.34)

This equation can now be solved using standard methods to yield

(5.35)

where the constant of integration has, together with (5.32), been inserted. Rearranging (5.35 ) for dm /dj gives

(5.36)

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

where

(5.37)

and where initial conditions have been chosen such that the constant of integration is zero. Equation (5.37) 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 (5.37), 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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