SECTION
II. A
Singular Isomethermal Sphere Lens With No Shear
This section provides the step by step
analytical solution for the image positions and magnifications in the
case of an SIS lens. It then uses these calculations for
comparison to demonstrate how to use gravlens to obtain such a
solution.
Part 1. Analytical Analysis
For the case of a SIS lens, the deflection
angle, 
, can be
considered to be equivalent to the Einstein radius, 
, of the
lens. The deflection angle is defined by:
where 
is the velocity dispersion of the lens system. In general, this
equation represents a more realistic situation since one usually
doesn't have information regarding the mass of the lens but the
velocity dispersion can be approximated. Also, it should be noted
that
for a given velocity disperion will
remain constant. For this example we will assume the velocity
dispersion is:
Next we solve for the image positions for a source that lies
one-half of the Einstein radius (or equivalently the deflection angle)
away from the lens. For symmetrical case of a SIS lens without
any shear, the source
position is defined as:
Note that no case is given for when 
is zero. For this model, an Einstein ring is produced when the
source
is at the origin (i.e., directly behind the lensing galaxy).
This equation can be used to solve for the image positions of the lens
system.
The magnification for each image is given by the equation:
Now we will solve these equations. Solving for the deflection
angle yields:
The source position is:
Solving
for the image position, , in
each case gives us:
and
For these two images, the magnifications are:
and
Part 2.
Gravlens Analysis
Now, we can use the gravlens
software to
obtain the same
results. From now on we will use an input file for the entire run
rather than doing
an interactive run. For this case we will be using the alpha model
since we
are dealing with an SIS lens. There are other lens models which
can be
used for a SIS lens as well. The definition and relation between
the
code and
model parameters can be found in Table 3.2 of the manual. Of the
ten
parameters (generally described in §3.2 of the manual) the two
which we must specify for our case are the mass scale (p[1]), b', and the power law index
(p[10]), .
Please note that this
is related to the power law index (see the definition for the alpha
model in the Table 3.2 of the manual) and is not related to
the deflection angle (i.e. ).
The mass scale (p[1]) represents the
Einstein
radius of
the
lensing
system. The lens
will
be at the origin or represented by the Cartesian coordinate position
(0, 0) and we will assume no shear so all other parameters will be set
to zero.
You have now defined the model and it's parameters.
We can now
find
the image positions and corresponding magnifications of the lensing
system using the findimg
command. In our analytical analysis
we
found the images for when the source position, 
, is one-half the
Einstein radius of the
lens. For the sake of simplicity, we will assume that
source lies on the y axis (v = 0). Thus, Cartesian
coordinates that define the source position are given by:
u = 0.35
v = 0.00
Thus, the input file for this run must
contain the following information:
startup 1 1
alpha 0.70 0 0 0 0 0 0 0 0 1
0 0 0 0 0 0 0 0 0 0
findimg 0.35 0.00
Now to use
gravlens to perform this
run, at the prompt we enter the command:
> gravlens <file>
This should return the following:
findimg results:
3.500000e-001 0.000000e+000 # source
# 3 images:
-1.334179e-004 -8.243382e-012 2.423417e-007
-3.497999e-001 -8.149017e-015 -1.000000e+000
1.049933e+000 5.790215e-015 3.000000e+000
The results contain a third cental "ghost" image which can be
ignored. It will lie very close to the origin and will have an
extremely low magnification. This "image" is produced due to part
of the code which attempts to smooth out the transition in finding the
images.