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

(5.31)

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

(5.32)

This is solvable for W giving

(5.33)

Now, if the astronomical situation is such that the additional inequality a m ₀ << 1 is valid, (5.33) finally reduces to

(5.34)

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

(5.35)

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

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

(5.35a)

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

(5.36)

Substitution of this into (5.35a) then yields

(5.37)

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 , 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

(5.38)

which may immediately be further approximated to

(5.39)

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 (5.39) are now substituted into (5.11) to give

(5.40)

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

(5.41)

The total angle of deflection may now be approximated from the two boundary conditions of the orbit where m =0. At these boundaries if , then applying these conditions to (5.41) gives

(5.42)

and the total angle of deflection of the trajectory is then

(5.43)

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.

On to the next Section: Conclusions

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