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Fig 2 Symmetry
Requirement of Lensing Demonstrated
From Fig 2 the total
deflection of the light ray, and thus the total angle of gravitational deflection
representing the accumulative effect of gravitation on the light ray is give
as
In this example the
gravitating mass M
is chosen to be positioned at the midpoint on the line of length DS
joining the observer and
the source. The astronomical distance DL
is the distance from the observer to the lens and DSLis
the distance from the lens to the source. Also again, it is important to note that this case is only a
simplified case, by setting DL
= DSL, presented in most academic textbooks. There is no
requirement at all that the lens be positioned exactly at the midpoint for an
observation of a theoretical Einstein ring. Of course, this again would assume
the validity of the light bending rule of General Relativity. This will become
much more clearer in the next section that addresses the axis of symmetry
of the lensed light ray for corresponding near- and far observers. It is
readily seen that the axis of symmetry for a given light ray is perpendicular to the line
joining the source and the observer only for the special case where the lens is
positioned at the midpoint.
In this example, the
radius R of the
analytical Gaussian sphere enclosing the gravitating mass M
is related to the angle h
under which an astronomical object of that size might
appear to an observer and is given by
where R
and DL
are expressed in meters and the angle h
is expressed in radians.
The impact parameter for
all the light rays that would be responsible for producing an image of an
Einstein ring would simply be R.
This is the nearest point of approach of the light rays to the center of the
gravitating mass M.
Since we are dealing with
small angles, from Fig 2, the deflection of the light ray due to the gravitational effect on
approach to the gravitating mass is just simply
From symmetry we have
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