5.2.3  Comparison with Existing Approximate Solutions.

To compare the above results with the approximate solutions of the General Theory mentioned earlier, it is only necessary to simplify (5.30) for known astronomical conditions.

Firstly, terms in m₀² may be considered negligible in comparison with unity, (5.30) then becomes

in which (5.28) has been re-inserted.

For astronomical conditions, W will be very small compared to f so that (5.31) may be further simplified to

This is solvable for W giving

Now, if the astronomical situation is such that the additional inequality am₀ << 1 is valid, (5.33) finally reduces to

Consider first, an orbit with an elliptical basic curve. When f = 2p, (5.34) becomes

in agreement with the solution of [2], pp199, Eq(58.43).

For a circular orbit, e = 0 which gives in (5.34)

However, m₀ may be approximated for a circular orbit from (5.19) and (5.20) to be

Substitution of this into (5.36) then yields

which agrees with the approximate solution of [3], pp 247, Example 102, and represents a maximum value of W.

For the special case in which w₀ = cm₀ , its upper boundary, and the geometric radius of the gravitational source is much greater than its gravitational radius, then (5.26) may be used for the eccentricity of the basic trajectory. Thus inserting (5.26) into (5.34) gives

which may immediately be further approximated to

This represents the rotation of the perihelion of the hyperbolic curve and constitutes a minimum value of W. To determine the total angle of deflection of the trajectory, (5.20), (5.26) and (5.40) are now substituted into (5.11) to give

expanding the cosine term and taking the usual approximation yields after reduction

and the total angle of deflection of the trajectory is then

in agreement with the approximate solution in [2], pp202, Eq[59.18], for the "bending of light rays" in close proximity to a gravitational source.

The results of this Section provide the final proof for Statement (iii) in Section 4.

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