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.

5.2.1  Equation of the Spatial Trajectory - The Basic Curve.

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

Substitution of (4.7) then gives the required relationship as

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

where

L is the semi latus rectum

e is the eccentricity of the basic curve

f is the angle of the focal point radius vector to the major axis

W is the angle of precession of an apse of the trajectory.

Differentiating (5.11) with respect to f yields

but from (5.11)

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

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

where b is some constant.

Inserting (5.15) into (5.14) and expanding gives

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

so that in (5.16) this gives after reduction

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

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

Inserting (5.20) and (5.4) for h into (5.19) then gives, after solving for w₀²

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

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

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

Solving (5.21) for e gives

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. w₀ =cm₀, (5.24) reduces to

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

There appears to be a second root under this criteria, i.e. at e=0. However, if the terminal value of w₀ is entered into (5.22), it reduces to 3am₀ = 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.

5.2.2  Precession of the Perihelion of the Basic Curve.

Putting

reduces (5.27) to the following elliptic integral

the solution of which is

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).