The equation of a planetary orbit can be derived in either the axes of D₁ or those of D₀. In the former it is generally known as Einstein’s Equation of Planetary Motion. The latter is derived in [2] from the metric of the space–time of the General Theory. The former is derived here from foregoing results as a first demonstration that gravitational motion in D₁ is identical to that in the space-time of the General Theory. A solution of the orbit is also obtained which when approximated for the appropriate astronomical situations shows that it also satisfactorily represents gravitational motion in the Solar System. As such this solution thereby provides the proof for Statement (iii) in Section 4 above. The equation of the orbit in the axes of D₀ is derived in Appendix B for comparison with that in [2].

Note – In this Section reference is made to results obtained in Appendix C.

To derive the equation of planetary motion in the axes of D₁, consider first (3.6). This equation can be integrated immediately to give

(5.1)

Inserting the usual initial condition determines the constant of integration as

(5.2

so that in (5.1)

(5.3)

Substituting for m from (3.5) then gives

(5.4)

In line with convention, this constant is now designated as h. However, from (C7) it can also be written as

(5.5)

The equation of the orbit can now be derived in the usual manner as follows. Put

(5.6)

and from (5.5) and (5.6) compute the second order differential term in (C9) as

(5.7)

Substitution of this, together with (5.5) and (5.6) into (C9) then yields the desired equation of the orbit thus

(5.8)

In the literature the equation of the trajectory of a planetary orbit has been obtained from an approximate solution of the orbital equation for two particular cases. Firstly, for closed orbits, it has been shown to approximate a precessing ellipse, viz. [2] pp 199, which constructs an approximate solution of (B14), (the equation of the orbit expressed in the axes of D₀), for a truly elliptical orbit. A similar result is obtained in [3], pp247 Example 102, where an approximate solution of (5.8) is obtained. This solution however, applies only to a circular orbit. Secondly, an approximate solution of (B14) has been obtained for an open orbit in the extreme form of a light ray passing close to the geometrical radius of a gravitational source, viz. [2] pp 202, and has thus been shown to be a precessing hyperbola.

In this Section, an exact solution is obtained for the planetary orbit in D₁, (5.8), by assuming it to be a precessing conic section and using, essentially, the method of Frobenús. This solution is then reduced to the above approximate forms via a process of logical simplification.

Note – In this Section reference is made to results obtained in Appendices B and C.

The equation of the basic curve of the trajectory will be derived from the first integral of (5.8) which is á priori obtained from (B8) by computing ds /dt _p from the first order term in (C2), then substitution of (5.5), (5.6) and (B16) to give

(5.9)

Substitution of (4.7) then gives the required relationship as

(5.10)

the first integral of (5.8).

If the integral of (5.10) is assumed to be a precessing conic section then it will take the form

(5.11)

where

Differentiating (5.11) with respect to f yields

(5.12)

but from (5.11)

(5.13)

Thus (5.12) and (5.13) give, after expansion

(5.14)

Comparing (5.14) with (5.10), to obtain the term in m³ with the correct coefficient, it is necessary to put

(5.15)

where b is some constant.

Inserting (5.15) into (5.14) and expanding gives

(5.16)

Again, comparing (5.16) with (5.10), for the coefficient of m² to be –1, it is necessary to put

(5.17)

so that in (5.16) this gives after reduction

(5.18)

Now, equating the coefficient of m in (5.18) with that in (5.10) then gives

(5.19)

L can be determined from initial conditions applied to (5.11) to be

(5.20)

Inserting (5.20) and (5.4) for h into (5.19) then gives, after solving for

(5.21)

Now, for the basic orbit to be circular, i.e. e=0, (5.21) reduces to

(5.22)

For values of w₀ below this, the basic orbit will be degenerative, and for values above elliptical.

For the basic orbit to be parabolic, i.e. e=1, (5.21) reduces to

(5.23)

For values of w₀ below this the basic orbit will be elliptical and for values above hyperbolic.

Solving (5.21) for e gives

(5.24)

which can also be obtained by equating the constant terms in (5.10) and (5.18).

Now, for the case in which w₀ is equal or very close to its terminal value in D₁, its upper boundary, i.e. w ₀ =cm ₀, (5.24) reduces to

(5.25)

and it is noted for future reference that if 4am₀ <<1, (5.25) may be approximated by

(5.26)

There appears to be a second root under this criteria, i.e. at e=0. However, if the terminal value of w ₀ is entered into (5.22), it reduces to 3a m ₀ = 1, which contradicts the inequality. This second root may therefore be discounted for astronomical situations.

Thus, the basic curve of the trajectory is now seen to be given by the combination of the three equations; (5.11), (5.20) and (5.24). The full solution to (5.10) is completed by the determination of the function W .

Determination of W is effected by substituting (5.11), (5.17) and (5.20) into (5.15) to give

(5.27)

Putting

(5.28)

reduces (5.27) to the following elliptic integral

(5.29)

the solution of which is

(5.30)

the constant of integration being zero.

This relationship, together with (5.28) then permits the determination of W for any value of f , for any condition. Thus (5.30), together with (5.11), (5.20) and (5.24), albeit somewhat cumbersome, constitutes the exact solution of (5.10).

On to the next Section: Planetary Orbits, (Part 2)

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