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

f » æ
è
1 - 6am0

1 + e
ö
ø
-1/2

 
ì
í
î
f - W am0 esin( f - W )

( 1 + e - 6am0 )
ü
ý
þ
(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

f » æ
è
1 - 6am0

1 + e
ö
ø
-1/2

 
ì
í
î
f - W + am0 e

( 1 + e - 6am0 )
( Wcosf + sinf ) ü
ý
þ
(5.32)

This is solvable for W giving

W »
ì
í
î
1 æ
è
1 6am0

1+e
ö
ø
1/2

 
ü
ý
þ
f am0 esinf

( 1 + e - 6am0 )

æ
è
1 am0 ecosf

( 1 + e - 6am0 )
ö
ø
(5.33)

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

W » 3am0

1 + e
f + am0 esinf

1 + e
(5.34)

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

W » 6pam0

1 + e
(5.35)

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

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

W » 3am0 f
(5.36)

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

m0 » ac2

h2
(5.37)

Substitution of this into (5.36) then yields

W » 3a2c2

h2
f
(5.38)

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

W » 3a2m02

1 + am0
f am0 sinf

1 + am0
(5.39)

which may immediately be further approximated to

W » am0 sinf
(5.40)

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

m » m0

1 + am0
{ am0 + cos( f - am0 sinf ) }
(5.41)

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

m » m0 ( 2am0 + cosf )
(5.42)


The total angle of deflection may now be approximated from the two boundary conditions of the orbit where m = 0. At these boundaries if f = ±  æ
è
p

2
 + d ö
ø
, then applying these conditions to (5.42) gives



d » 2am0
(5.43)

and the total angle of deflection of the trajectory is then

2d » 4am0
(5.44)

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.



G1 Version 2.2.4
Ó P.G.Bass, November 2009
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