SECTION
I. A
Simple Point Mass Lens
We begin with a point mass lens, because we
can easily find the images analytically and then demonstrate how to
obtain the same results with gravlens.
Part 1. Analytical Analysis
First, we find the Einstein radius using the
general equation:
The point mass lens equation can be rewritten as:
where 
is the image position and 
is the
source position. Solving this equation yields the two image positions :
Once we have the image positions
we can calculate the magnification of each image. Due to the
symmetry of this case, we can use the following equation to obtain the
magnification of each image.
Now we will solve these equations. Solving for
the Einstein radius yields:
We will assume
that the source position is one half the Einstein radius and determine
the image positions:
Lastly, we find the magnification of each image:
Part 2.
Gravlens Analysis
Now, we can use the gravlens
software to
obtain the same results. Because we will be dealing with a case
of a point mass at the origin, we must set the gridlo1 variable to
a small non-zero value (e.g. gridlo1
= 1.0e-4) to avoid the central
singularity (see §4.1.3 in
the
gravlens
manual). To
do this, simply enter the command:
>
set gridlo1 = .0001
Because our value for the Einstein radius will be 2.02 arc seconds, we
must set the gridhi1
variable to a larger value (e.g. gridhi1
= 5.0) than the default so the outer image is within the grid used in
the software. We do this via the command:
>
set gridhi1 = 5.0
Once that variable is set, you can begin to specify the mass models
using the command:
> startup 1 1
You will then be prompted to specify the mass model and its
parameters. A point mass model
is specified as ptmass.
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 only one which we must specify for our case is the mass
scale or p[1].
The lens will be centered at the origin or
represented by
the Cartesian coordinate position (0, 0) so p[2] and p[3] are set to
zero. For the point mass model, this represents the Einstein
radius of the lensing system. Using the Einstein radius we found
in our analytical calculations, we enter:
ptmass 2.02 0 0 0 0 0 0 0 0 0
Next, you will be prompted to specify which values are allowed to vary
for the galaxy. This is a option that is useful for the lens
modeling part of the software. Since we do not wish to allow any
of the parameters to vary, we simple enter all zeros:
0 0 0 0 0 0 0 0 0 0
You have now defined the model and its parameters.
Note: As an
alternative to performing this interactive run, you can place the mass
model specification in a file and
read it with the command
> startup
<file>
Here the startup file must contain the following model and
galaxy information that would be entered through the prompt:
1 1
ptmass 2.02 0 0 0 0 0 0
0 0 0
0 0 0 0 0 0 0 0 0 0
We can now
find the image positions and corresponding magnifications of the
lensing system using the findimg
command. This command takes two
parameters, u and v, which specify the
Cartesian coordinate position of the source. The relation between
the source position, , used
in the analytical analysis and the Cartesian coordinate position (u, v) is given by the equation:
We already have determined that the
source position,
,
is one-half the Einstein radius of the lens. Therefore, 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 = 1.01
v = 0.00
With this information, we can now use gravlens
to find the images
and their magnifications by entering the command:
> findimg 1.01 0.00
This should return:
findimg results:
1.010000e+000 0.000000e+000 # source
# 2 images:
-1.577168e+000 -1.210594e-016 -5.914103e-001
2.587168e+000 -2.908380e-023 1.591410e+000
The first two values in each row represent the image position, (x, y)
and the third represents the image magnification. The results
produced by the software can be translated to match the analytical
calculations
found in Part I by using the relation between the analytical image
positions, ,
and the (x, y) coordinates for each image
position as done with the
source position given by and (u,
v). The analytical
results may differ from the gravlens calculations a
little due to rounding in our calculations.
Note: As an
alternative to performing this interactive run, you can perform the
entire run by placing all the above commands in a file. To do
this you simply enter the command:
gravlens <file>
at the command prompt. For example, the input file for this run would
contain the following lines:
set gridlo1 = .0001
set gridhi1 = 5.0
startup 1 1
ptmass 2.02 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
findimg 1.01 0.00