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Gravitational Lensing

Stephen Mullens

Simon Notley

�

Graviitational Lensing Stephen Mullens and Simon Notley, 2005

Abstract

Presented here is a detailed consideration of the modelling of gravitational lensing

using a ray-tracing computer simulation. The model discussed deals exclusively

with forward-engineering techniques – constructing the visible image from the

source and the lens. The mechanics of constructing the model are discussed at

length, including an analysis of the effect on image quality of different computational

parameters. Numerous gravitational lensing phenomena are reproduced using

computer simulation, including the Einstein Ring, arcs and lesser distortions due to

weak lensing. Astronomical images of real galaxies are also processed. The results

of these simulation are shown to be similar to observed gravitational lenses. A brief

discussion of the Einstein Cross is undertaken, although the model discussed here

fails to reproduced observed images in this case.

�


History 4

Theory of Gravitational Lensing 4

Strong Lensing 5

Weak Lensing 5

Microlensing 5

Computational Methods: Simple Model 6

Initial Assumptions 6

The Basic Model 7

Monte Carlo Methods 8

Building the Lensed Image 10

The Einstein Ring 1�

Off-axis and Elliptical Sources 15

Computational Methods: Advanced Model 17

Non point-like gravitational lenses 17

Calculating the mass felt by each ray 18

Lens Pixelation Problems �0

Interpolation Schemes �1

Central defects in lensed images ��

Lensed Images 23

Elliptical Lenses �5

Simulation of Real Life Lensing Observations �6

Conclusions 29

References, Bibliography & Sources 30

�


History

The first written occurrence of the idea of lights path being bent by massive objects came in 1804 from

a German geodesist, mathematician and astronomer, Johann Soldner. A rough translations of the title of

his article was “On the Deflection of a Light Ray from its Straight Motion due to the Attraction of a World

Body which it Passes Closely’”. This publication pre-empts modern ideas of curved space time by over

a hundred years (see Wambsganss 1998)

In 1916 Einstein published his General Theory of Relativity which outlined the quantitative affect of massive

objects in the fabric of space-time. The first confirmation of his theory came from explaining Mercury’s

perihelion shift, the second came during the solar eclipse of 1919. Arthur Eddington and his team

measured the apparent shift in the position of a star viewed close to the sun’s corona and the shift

matched the result predicted by Einstein to approximately �0%, this has since been re-measured and

is correct to within 0.0�%. The solar eclipse was essential for this observation because it blocked out a

significant amount of light, hence making faint stars visible.

In recent years the gravitational lensing phenomena has been used to infer the existence of black holes

and other massive dark matter objects such as MACHOs (Massive compact halo objects). This is

achieved by observing a stellar object over time and watching to see if its intensity changes, if a good

sized lensing object moves in between the source and observer then the star will wink in and out.

Theory of Gravitational Lensing

A gravitational lens can be described as a natural telescope because they are capable of magnifying and

distorting light from distant objects. A gravitational lens occurs because massive objects bend and distort

space-time around them, light rays passing near follow the straightest possible worldline (path of least

action) which causes them to change direction to an external observer. Gravitational lensing is analogous

to normal refraction of light by a glass lens, as the light crosses the lens surface, the refractive index

changes and its direction of travel is changed. As shown by Carroll and Ostlie (1996), the deviation in the

angle in radians of a ray of light as it passes a massive object is

Where r is the closest distance the ray passes from the centre of mass, M is the enclosed mass and

all other symbols have their normal definitions. From this formula it is possible to analytically solve simple

lensing situations with point mass lenses.

r D D

GMrcDD

GMc

ol ls

ls

ol

= = −( )=

=⎛

⎝⎜⎞

⎠⎟⎛

θ φ θ

φ

θ

4

4

2

2⎝⎝⎜⎞⎠⎟ +( )

− = − = −( )

=±

1D D

r r D r D

r r

ls ol

g ol g ls

g

θ φ θ

θ gg

ol

g g g

ol ls

ls ol

C

D

r x y

CD D GM

D D c

2

2 2

2

2

16

−

= +

=+( )

(xx yDD

i j

xyscaleimscale

g gls

ol

, ) ( )( , )

( / )

= − +

=

1

2 ±± + +⎛

⎝⎜⎞

⎠⎟⎛⎝⎜

⎞⎠

( / )imscale DDD

GMcls

ls

ol

2 4 1 422 ⎟⎟

+⎛

⎝⎜⎞

⎠⎟1

DD

ls

ol

Eqn. 1

5


Strong Lensing

This type of lensing occurs when there is a particularly massive lens with a source close to it. There are

three significant phenomena resulting from strong lensing (Cohn 2000)

1) Multiple images - light from a source is distorted and travels over more than one path resulting in

multiple images of the source forming. Well known multiple images include Einstein Crosses where a �

or 5 multiple images are visible in a cross shape, the lens geometry needed to create this is complex and

not well understood.

2) Einstein Ring - If all objects are in alignment it is possible for an Einstein ring to form whereby a single

object appears as a ring of light on the sky. Einstein rings are often distorted and not perfect rings

because lenses rarely have perfect spherical symmetry and perfect alignment

�) Arcs - this is where a partial Einstein ring forms, this may be due to off axis situations, strange shaped

lenses or other objects interfering. Arcs are the most common phenomena as they can be produced in

many situations.

Weak Lensing

Weak lensing occurs when the lensing is not strong enough to produce multiple images or arcs, but is

capable of distorting the apparent shape of the source, ie stretched and magnified. A possible example

of this effect could be a spherical source appearing to be elliptical on the sky; in reality it is not possible to

use this information to infer the shape of the lens because you never really know the shape of the source.

A method to get a good idea of the distribution of mass in the lens is to find a lensing situation where the

distribution of light coming from the lens can only really infer one type of source which nearly always has

the same shape. If you then know the source and what it looks like after lensing, computer reconstruction

can be used to estimate the mass distribution in the lens.

Microlensing

In this case the lensing is not strong enough to produce multiple images, instead the additional light bent

towards the observer increases the brightness of the apparent source. This can be helpful in viewing

sources which are too far away (faint) to be directly visible.

6


Computational Method

Initial Assumptions

In the first instance, only point-like lensing objects were considered, although care was taken to adopt a

method that could be simply extended to cover more complex cases. The method used is based on the

ray-tracing model of light propagation and encompasses the following characteristics.

Firstly, rays of light are considered to travel in straight lines, only being deflected by the lens at a single

discontinuity in the plane of the lens and parallel to the observer. The magnitude of that deflection is a

function of the distance from the lens at the point of deflection.

Secondly, the reciprocity of ray paths means that the path of a ray is independent of the direction of

propagation along it. For this reason, we consider paths from the observer to the source. The gain in

efficiency from this can be seen by consideration of a source object emitting light isotropically. Each ray

has a vanishingly small probability of reaching the observer. By considering rays ‘fired’ from the observer,

efficiency can be increased as time is not wasted following rays that can never be observed.

Clearly, it makes no sense to fire beams in all directions from the observer. Instead we limit them to a small

angular range in the direction of the source, thus maximising efficiency. This also has physical significance

because the observer, typically a telescope, can only take in light from a very limited range of angles. The

range of angles used for this is calculated by consideration of the limiting case when a ray fired from the

telescope will only just strike the edge of the source galaxy.

The final in-built assumption is that, when an angle is being used, small angle approximations can be used

to simplify trigonometric functions. The angles of interest in gravitational lensing are of the order of a few

arc seconds or 10-5 radians.

7


The Basic Model

The structure of the model devised for this investigation is illustrated in Figure 1. The source is treated as

a two-dimensional image in the source plane. From a computational point of view, this can be either an

imported image or an artificially generated one. The gravitational lensing effect is encompassed entirely in

the deflection angle phi as described by Equation 1.

The dimension xyscale is derived from the relative positions of lens, source and observer as well as the

mass of the lens by the relation

This is the parameter that controls the distribution of rays fired from the observer. Rays are chosen that

pass through a square of side length xyscale positioned in the lens plane. The defining property of this

square is that a ray xyscale/� from the central axis will fall imscale/� from the axis at the source plane.

This assumes an axially symmetrical system; observer, lens and source all aligned. The reason for this

The layout of the model to be used. Rays travel in straight lines between planes and all deflection

takes place in the lens plane. The incidence of a ray or projected ray on the lens, source and im-

age plane is described by the co-ordinates within those planes (xL,yL), (xS,yS) and (xI,yI) respectively.

‘Imscale’ is the physical size of the source image, xyscale is defined below.

Lens Plane Image / Source PlaneObserver

Dol

Dls

imsc

ale

xysc

ale

xI

xL x

S

xI, xS and xL have corresponding y co-ordinates within the relevant plane. When using a point source as the lensing object, it is placed in the Lens Plane at (xg,yg).

r D D

GMrcDD

GMc

ol ls

ls

ol

= = −( )=

=⎛

⎝⎜⎞

⎠⎟⎛

θ φ θ

φ

θ

4

4

2

2⎝⎝⎜⎞⎠⎟ +( )

− = − = −( )

=±

1D D

r r D r D

r r

ls ol

g ol g ls

g

θ φ θ

θ gg

ol

g g g

ol ls

ls ol

C

D

r x y

CD D GM

D D c

2

2 2

2

2

16

−

= +

=+( )

(xx yDD

i j

xyscaleimscale

g gls

ol

, ) ( )( , )

( / )

= − +

=

1

2 ±± + +⎛

⎝⎜⎞

⎠⎟⎛⎝⎜

⎞⎠

( / )imscale DDD

GMcls

ls

ol

2 4 1 422 ⎟⎟

+⎛

⎝⎜⎞

⎠⎟1

DD

ls

ol

Fig. 1

Eqn. 2

8


choice of parameter is that it ensures most of the rays fired will fall upon the image. Note that there is still

a possibility of rays passing very close to the lens, such that r is very small and the deflection angle very

large. These rays will mostly fall outside the source image.

Monte Carlo Methods

A Monte Carlo process is one which involves using random numbers to quickly find an approximate

answer to a problem. In this instance, rays will be fired at random angles within the established parameters,

an image of what the observer sees will eventually be built up. The level of detail in the image will be

dependent on the number of these random rays used. This may seem an unnecessary sacrifice of

quality for reduced run-time at this stage, but it was realised that these methods would be vital for later

parts of the investigation and as such they were used from the start.

9


Start

Input source image and imscale,

the mass of the lense,Dol, Dls, xg and yg.

Calculate maximum value of xL that results in a ray from O falling within the source image.

Define xyscale as twice this value

Randomly generate two vectors of length S, scale the values to have

range ±xyscale/2. Call these xL and yL

Choose number of samples

Calculate corresponding (xS,yS) and (xI,yI) and store in vectors

Take the nth vector pair created above, record source intensity

at (xS,yS)

Create new image array of size (1+Dls/Dol)*xyscale. This willbecome the lensed image

n=1

Set element of lensed image array at (xI,yI) equal to recorded

intensity

n<S?

n=n+1

YES

NO

Output lensed image

Add recorded intensityto element of lensed image

array at (xI,yI)

Mapping Addition

End

The operation of the computational model used to map the incidence of the rays on

the three planes and build an image by one of two methods – addition or mapping

Fig. 2

10


Building the lensed image

Figure 2 illustrates the general structure of the program used to generate the image seen by the observer.

In compiling this image, we considered two possible methods of evaluating the relative light intensity at

each point. We named these methods intensity mapping and intensity addition. The two techniques

utilise the same method to trace the path of the beam through the lens, but differ when it comes to

interpreting this to create the final image.

As shown by the flowchart in Figure 2, the program operates on simple premise. It fires a large number of

random beams from the observer, towards the lens, constrained by the parameter xyscale as explained

earlier. Simple geometry is then applied to calculate the point at which the deflected ray strikes the source

plane and the point in the source plane from where it appears to have come. Note that the latter set

of points are said to comprise the image plane. With these sets of co-ordinates stored in arrays, the

lensed image can be built-up one of two ways. Intensity mapping simply involves mapping the intensity

in the source plane to its corresponding image plane co-ordinate. This is computationally efficient and

quickly creates a clear, noiseless image. However, it does not have the physical realism of the additive

approach.

Consider a lens such that the light from a large but dull star was concentrated into a very small area by

the deflection. One would expect this small area to appear brighter to an observer than the original as

the same amount of light is now coming from a smaller area. The mapping approach does not allow for

this sort of intensity magnification or indeed for any similar effect where more than one pixel of the source

contributes to the brightness of a single pixel in the lensed image. The additive method avoids this by

adding the intensity of the corresponding source area to the lensed image each time a ray is fired. This

makes intensity changes possible at the expense of taking much longer to create a noise-free image.

This reduction in efficiency is a product of the Monte Carlo technique. In the mapping model, since

beams falling on the same pixel over write, the criteria for a perfect image is that each pixel is struck by

at least one beam. In the additive model, two beams striking one pixel will add together. Clearly some

pixels will end up brighter or darker simply due to random coincidence rather than the nature of the lens.

In order to minimise this, we need considerably more beams to be fired. The image will tend towards

being ‘perfect’ as the number of beams fired tends to infinity.

For a 512x512 pixel image, we can estimate the number of samples required for both methods statistically.

As stated above, the mapping method requires each pixel to be hit once. Imagine firing one ray at a

time, each one hitting a random pixel (we neglect the small number of rays not hitting any pixel here) and

somehow marking that pixel. The probability of the first ray hitting an unmarked pixel is clearly 1. The

probability of the second doing the same is �6�1��/�6�1�� and so on. Using these probabilities to

estimate a the number of rays needed to hit an unmarked pixel each time (from 1 for the first pixel through

11


to 262144 for the final one) we sum all these numbers to get an estimate of the total number of rays

required. For the 512x512 image, this is ~3x106.

Figure 3 shows the results of using increasing numbers of rays. Note that the image is smooth in the

places that have been imaged, but that it has ‘dead’ white pixels where insufficient rays have been fired.

The additive approach is somewhat more complicated. Here, we need enough rays that the whole of

the image is sampled equally. Clearly this will mean many more rays than before when each pixel had to

be hit only once. The method used to estimate the number of rays required here involves looking at the

distribution of rays over the 51�x51� image. A simple computer program was used to generate random

numbers in the range 1-�6�1��. The frequency of each of these random numbers was recorded. The

aim for an even distribution is to have equal frequencies to within a certain tolerance. The tolerance in this

case is one grey-level in the final image. Using a 64-level image, this means that the frequencies should

be equal to within 1/6� of their value. The statistical condition applied to represent this is that �s=(1/6�),

where s is the standard deviation of the frequency values after scaling such that the mean frequency is

one.

Source passing a point−like stationary lensing object

x − arcseconds

y −

arcs

econ

ds

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1


x − arcseconds

y −

arcs

econ

ds

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1


x − arcseconds

y −

arcs

econ

ds

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1


x − arcseconds

y −

arcs

econ

ds

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Images of a galaxy constructed using the mapping method with 5x104, 5x105, 1x106 and

5x106 rays. As predicted, the image is totally smooth in the final image as all pixels are filled.

Fig. 3

1�


Figure 4 shows the reduction on standard deviation as the number of samples is increased. It is clear

that a large number of samples will be required for smooth images. To meet our criteria for totally smooth

images, the number of rays needed would be in excess of 1010. This would take far too long to run,

simply calculating the standard deviation for this case would take around 1� hours on a �Ghz machine.

Some compromise will have to be reached between performance and run time.

Level of Random Error as a Function of Number of Rays for 512x512 Image

0

5

10

15

20

25

30

35

1.00E+06 1.00E+07 1.00E+08 1.00E+09 1.00E+10

Number of Rays

Sta

nd

ard

Devia

tio

n in

Gre

y L

evels

(fo

r 6

-bit g

reyscale

)

The standard deviation of the grey-levels clearly falls as the number of rays is increased.

However the decrease is extremely slow as the error drops to around 1 grey-level.


x − arcseconds

y −

arcs

econ

ds

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1


x − arcseconds

y −

arcs

econ

ds

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1Source passing a point−like stationary lensing object

x − arcseconds

y −

arcs

econ

ds

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1


x − arcseconds

y −

arcs

econ

ds

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

The additive method applied using 5x106, 1x107, 5x107 and 1x108 rays. The final picture took

some time to create and appears very smooth although it doesn’t fit the criteria specified earlier.

Fig. 4

Fig. 5

1�


The effect of increasing ray numbers in the additive method is shown in Figure 5. Notice how the images

become grainy rather than simply having dead pixels as before. It was decided that the additive methods

was fast enough to be useable in the investigation. However, creating images with more than 1010 rays

was simply not time-efficient, so a compromise was reached and most of the images used in this report

were created using the additive method and 5x107 rays.

The Einstein Ring

For the case where the observer, lens and source are perfectly in line, light rays emitted from the source

are bent in a manner which produces a ring shape from the observers perspective.

Figure 6 shows the path of a ray which travels from the source and is bent just enough by the lens to

strike the observer. If the path of the ray is traced back from the perspective of the observer then it will

appear to come from the top point on the circle. The system illustrated is clearly cylindrically symmetric

about the central axis. This means that the point produced by the single ray can be rotated around the

axis to produce a ring. The ring in this case will have an angular radius of q radians on the sky. This effect

is known as an Einstein Ring, The rays contributing to the ring satisfy the condition

where r is the distance from the origin in the lens plane. This can then be rearranged to find q and the

formula for the angular deflection substituted in.

This formula shows that as the mass of the lens rises, it is possible to produce much larger rings, however

even with massive objects such as black holes, the radius is only of the order of 10-5 radians.

Dls Dol

ObserverSource

φ

θLens Plane

r D D

GMrcDD

GMc

ol ls

ls

ol

= = −( )=

=⎛

⎝⎜⎞

⎠⎟⎛

θ φ θ

φ

θ

4

4

2

2⎝⎝⎜⎞⎠⎟ +( )

− = − = −( )

=±

1D D

r r D r D

r r

ls ol

g ol g ls

g

θ φ θ

θ gg

ol

g g g

ol ls

ls ol

C

D

r x y

CD D GM

D D c

2

2 2

2

2

16

−

= +

=+( )

(xx yDD

i j

xyscaleimscale

g gls

ol

, ) ( )( , )

( / )

= − +

=

1

2 ±± + +⎛

⎝⎜⎞

⎠⎟⎛⎝⎜

⎞⎠

( / )imscale DDD

GMcls

ls

ol

2 4 1 422 ⎟⎟

+⎛

⎝⎜⎞

⎠⎟1

DD

ls

ol

r D D

GMrcDD

GMc

ol ls

ls

ol

= = −( )=

=⎛

⎝⎜⎞

⎠⎟⎛

θ φ θ

φ

θ

4

4

2

2⎝⎝⎜⎞⎠⎟ +( )

− = − = −( )

=±

1D D

r r D r D

r r

ls ol

g ol g ls

g

θ φ θ

θ gg

ol

g g g

ol ls

ls ol

C

D

r x y

CD D GM

D D c

2

2 2

2

2

16

−

= +

=+( )

(xx yDD

i j

xyscaleimscale

g gls

ol

, ) ( )( , )

( / )

= − +

=

1

2 ±± + +⎛

⎝⎜⎞

⎠⎟⎛⎝⎜

⎞⎠

( / )imscale DDD

GMcls

ls

ol

2 4 1 422 ⎟⎟

+⎛

⎝⎜⎞

⎠⎟1

DD

ls

ol

The path of a ray from source to observer and the projection of these rays onto the image plane

Fig. 6

Eqn. 3

Eqn. 4

1�


The first test of the computer model was whether it could reproduce this phenomenon if given a small,

spherical source galaxy and a point like lensing object, both aligned with the observer. Figure 7 shows the

results. The ring calculated above is shown in red and the ring produced by ray addition perfectly aligned

with it, the soft edges being a consequence of the slightly diffuse source.

It is no great surprise that the program and the calculation on the previous page agree so well, as they

are derived from the same ‘ray-tracing’ model. A better test would be whether the program can produce

similar effects to those observed by astronomers. Nearly all observed rings exhibit two bright spots at

opposite sides of the ring, connected by some kind of arc, as seen in Figure 8. Could the program

reproduce these features?

An Einstein Ring (PKS 1830-211) observed using the Australia Telescope. Notice

the dimension agree well with the model, but the shape is more complex.


x − arcseconds

y −

arcs

econ

ds

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

An Einstein Ring formed with Dol=Dls=1x1025m, M=5x1042kg and a simple spherical

source of ~ 0.5 arcsecs diameter. The analytically calculated ring is shown in red

Fig. 7

Fig. 8

15


Off-axis objects and elliptical sources

In order to introduce some asymmetry to the lensed image, it makes sense to break the cylindrical

symmetry of the system. The program used for the point-like lens allows the source to be moved to a

position (xg,y

g) within the lens plane. The distance to the lens at each point is modified accordingly and

thus the image produced is altered. It is not immediately apparent what effect this will have and it is not

easy to solve this case analytically due to the lack of symmetry.

Figure 9 shows promising results with clear evidence of the two bright spots observed in real-life images.

In fact the clear presence of these two intense areas allows the formulation and solution of a second

geometrical problem, similar to that of the Einstein ring. It is apparent from the results illustrated in Figure

9 that the axis with the image plane along which the bright spots develop is defined by the direction of

movement of the lens. For example, moving the lens along the horizontal axis results in the bright areas

forming on this axis, one either side of the origin. Given this additional insight it is possible to calculate

analytically the locations of the bright spots. Working within the plane containing the spots – defined as

the plane containing lens, source and observer where these are all point-like – the problem reduces to

that solved earlier for the Einstein ring with a slight modification. The condition which needs to be satisfied

is

where

This reduces to a quadratic equation in q, the solutions representing the positions of the two bright

areas:

r D D

GMrcDD

GMc

ol ls

ls

ol

= = −( )=

=⎛

⎝⎜⎞

⎠⎟⎛

θ φ θ

φ

θ

4

4

2

2⎝⎝⎜⎞⎠⎟ +( )

− = − = −( )

=±

1D D

r r D r D

r r

ls ol

g ol g ls

g

θ φ θ

θ gg

ol

g g g

ol ls

ls ol

C

D

r x y

CD D GM

D D c

2

2 2

2

2

16

−

= +

=+( )

(xx yDD

i j

xyscaleimscale

g gls

ol

, ) ( )( , )

( / )

= − +

=

1

2 ±± + +⎛

⎝⎜⎞

⎠⎟⎛⎝⎜

⎞⎠

( / )imscale DDD

GMcls

ls

ol

2 4 1 422 ⎟⎟

+⎛

⎝⎜⎞

⎠⎟1

DD

ls

ol

r D D

GMrcDD

GMc

ol ls

ls

ol

= = −( )=

=⎛

⎝⎜⎞

⎠⎟⎛

θ φ θ

φ

θ

4

4

2

2⎝⎝⎜⎞⎠⎟ +( )

− = − = −( )

=±

1D D

r r D r D

r r

ls ol

g ol g ls

g

θ φ θ

θ gg

ol

g g g

ol ls

ls ol

C

D

r x y

CD D GM

D D c

2

2 2

2

2

16

−

= +

=+( )

(xx yDD

i j

xyscaleimscale

g gls

ol

, ) ( )( , )

( / )

= − +

=

1

2 ±± + +⎛

⎝⎜⎞

⎠⎟⎛⎝⎜

⎞⎠

( / )imscale DDD

GMcls

ls

ol

2 4 1 422 ⎟⎟

+⎛

⎝⎜⎞

⎠⎟1

DD

ls

ol

r D D

GMrcDD

GMc

ol ls

ls

ol

= = −( )=

=⎛

⎝⎜⎞

⎠⎟⎛

θ φ θ

φ

θ

4

4

2

2⎝⎝⎜⎞⎠⎟ +( )

− = − = −( )

=±

1D D

r r D r D

r r

ls ol

g ol g ls

g

θ φ θ

θ gg

ol

g g g

ol ls

ls ol

C

D

r x y

CD D GM

D D c

2

2 2

2

2

16

−

= +

=+( )

(xx yDD

i j

xyscaleimscale

g gls

ol

, ) ( )( , )

( / )

= − +

=

1

2 ±± + +⎛

⎝⎜⎞

⎠⎟⎛⎝⎜

⎞⎠

( / )imscale DDD

GMcls

ls

ol

2 4 1 422 ⎟⎟

+⎛

⎝⎜⎞

⎠⎟1

DD

ls

ol

r D D

GMrcDD

GMc

ol ls

ls

ol

= = −( )=

=⎛

⎝⎜⎞

⎠⎟⎛

θ φ θ

φ

θ

4

4

2

2⎝⎝⎜⎞⎠⎟ +( )

− = − = −( )

=±

1D D

r r D r D

r r

ls ol

g ol g ls

g

θ φ θ

θ gg

ol

g g g

ol ls

ls ol

C

D

r x y

CD D GM

D D c

2

2 2

2

2

16

−

= +

=+( )

(xx yDD

i j

xyscaleimscale

g gls

ol

, ) ( )( , )

( / )

= − +

=

1

2 ±± + +⎛

⎝⎜⎞

⎠⎟⎛⎝⎜

⎞⎠

( / )imscale DDD

GMcls

ls

ol

2 4 1 422 ⎟⎟

+⎛

⎝⎜⎞

⎠⎟1

DD

ls

ol


x − arcseconds

y −

arcs

econ

ds

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1


x − arcseconds

y −

arcs

econ

ds

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1


x − arcseconds

y −

arcs

econ

ds

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

Three lensed images formed with a point-like lens positioned at (1,0), (1,1) and (1,3) in the lens

plane - units are 1020m. Note the bright areas on opposing sides of the origin in each image.

Fig. 9

Eqn. 5

Eqn. 6

Eqn. 7a & 7b

16


To plot these, the components xg and yg are used to split r into x and y components. It is interesting to

note that moving the lens galaxy is largely equivalent to moving the source as the system is defined by the

relative position of these element. In fact, the same derivation can be used if the substitution

is made, where (i,j) are co-ordinates of the centre of the source. This is a more convenient formulation for

computational purposes, so the remaining images in this section were generated by moving the source

whilst leaving the lens at the origin.

Figure 10 Shows a series of images depicting a source passing behind a gravitational lens. Note that

the program produces results in very good agreement with the result is derived above. An extra element

of asymmetry has been introduced in this series of pictures by using an elliptical galaxy as the image.

The effect of this is largely as would be expected, it adds an element of ellipticity to observed shapes.

Following the sequence of Figure 10, when the source is far from the lens, it is relatively unaffected, save

for some minor distortions. As it gets closer, it forms arcs and then a ring. Some of the images at this

point are reminiscent of the real case shown in Figure 8.

Although these images are broadly encouraging there are some drawbacks of the point-like lens model.

In an infinitely small lens, the deflection angle will tend to infinity as rays pass close to the lens. This means

r D D

GMrcDD

GMc

ol ls

ls

ol

= = −( )=

=⎛

⎝⎜⎞

⎠⎟⎛

θ φ θ

φ

θ

4

4

2

2⎝⎝⎜⎞⎠⎟ +( )

− = − = −( )

=±

1D D

r r D r D

r r

ls ol

g ol g ls

g

θ φ θ

θ gg

ol

g g g

ol ls

ls ol

C

D

r x y

CD D GM

D D c

2

2 2

2

2

16

−

= +

=+( )

(xx yDD

i j

xyscaleimscale

g gls

ol

, ) ( )( , )

( / )

= − +

=

1

2 ±± + +⎛

⎝⎜⎞

⎠⎟⎛⎝⎜

⎞⎠

( / )imscale DDD

GMcls

ls

ol

2 4 1 422 ⎟⎟

+⎛

⎝⎜⎞

⎠⎟1

DD

ls

ol


x − arcseconds

y −

arcs

econ

ds

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1


x − arcseconds

y −

arcs

econ

ds

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1


x − arcseconds

y −

arcs

econ

ds

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1


x − arcseconds

y −

arcs

econ

ds

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1


x − arcseconds

y −

arcs

econ

ds

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1


x − arcseconds

y −

arcs

econ

ds

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

An elliptical source passing behind a point-like lens mass positioned centrally. The theoretical

Einstein ring and the calculated locations of the bright spots are shown in red. The source used

originally had its major axis running horizontally, it has been twisted due to weak lensing effects.

Eqn. 8

Fig. 10

17


that, however far from the lens the source gets on the sky, there will still be at least one path that a ray

could follow from source to lens to observer. This is what causes the persistent small image close to the

origin in Figure 10. This is an intrinsic weakness of the point-lens model. To advance beyond this stage,

a more sophisticated model allowing for lens objects with physical size must be developed.

Computational Methods: Advanced Model

Non point-like gravitational lenses

So far all the lenses which have been looked at are perfect point like lenses which exist as a single point

on a 2D plane. For each ray the distance between the point lens and is computed and deflections

calculated.

For a lens which is distributed over a 2D plane perpendicular (to the vector source-observer), each ray

feels mass from all over the distribution with the resultant mass in some unknown direction. In real life a

lens would be a three dimensional object with a continuous mass distribution, the lens would also have

absorption and emission characteristics. For the sake of this investigation the lens will be simplified with

a few assumptions and computational simplifications and further complexities added where physically

justifiable and computationally cheap.

Lens mass will be distributed into a pixel structure matrix, that is to say that the lens will consist of a matrix

of pixels, with each pixel having a weight which corresponds to a specific mass. The total mass of the

lens will be normalised to a fix number for each experiment to guarantee consistency.

The lens will not emit any radiation and will be totally transparent to all incoming radiation.

Whether the ray is inside or outside the lensing mass, the position of the centre of mass will always be

the same. The implication of this is that at the start of the program, the centre of mass of the lens can be

worked out and the light rays will always be bent in this direction, but because each ray feels a different

mass, they are bent to a different degree.

18


Calculating the mass felt by each ray

For simplicity lens distributions will be built in MatLab to ensure maximum control over all the factors such

as position. The lens will be a square n by n array of pixels which will be normalised once created.

Now that there is a mass distribution of pixels, the goal is to work out the net mass felt by each pixel in

that array. For example in the middle of a spherically symmetric distribution of mass, a ray would feel no

force as all the mass vectors cancel, but outside of the distribution the ray would feel a point mass at the

centre of mass of the distribution.

The first step should be to find the centre of mass of the lens distribution by vector summing over all

of the elements. This can be done by giving each pixel an additional weight based upon its x and y

displacement from the (1,1) point. Considering x and y separately, multiply each pixel mass by the vertex

displacement then add them all together, this can be algebraically:

Now that the centre of mass of the lens is known (Cx and Cy) this can be set as xL and yL, the position of

a point lens. The next step is to work out how much net mass is felt by a ray at each point in the mass

array. The expectation is that all rays are lensed towards the same point but can feel drastically different

masses.

For each point on the mass matrix with coordinates (x,y), it is necessary to work out the mass felt. With

d being the distance between the coordinate of interested and a point on the array, the vector mass in

both the x and y dimensions can be calculated using:

Cf i j i

f i j

Cf i j j

f i j

xji

ji

yji

ji

=( )

( )

=( )

( )

∑∑∑∑∑∑∑∑

,

,

,

,

Total Mass== f i jji

,( )∑∑

d i x j y

i xd

Mj yd

ji

xji

= −( ) + −( )( ) −( )

=( ) −( )

=

∑∑

∑∑

2 2

M =f i,j

f i,j

M

y

tot MM Mx y2 2+

Eqn. 9a-c

Eqn. 10a-d

19


The vector masses in terms of x and y can then be combined together as the sum of two squares to

find the total mass felt at the point (x,y). When programming this code it may be necessary to restrict the

summations to prevent cases where d is zero, hence causing an undefined result, this occurs when i=x

and j=y.

It is then necessary to carry out this operation on every point on the lens mass distribution array. At this

point it is clear that this is a very computationally demanding process and further experiments will be

required to find a suitably high resolution to give detail, while keeping the program running quickly. To

check this produces accurate results, quick examination of elements in the mass felt matrix for physical

results is useful. At a large distance from the distribution, the mass felt should be the normalised total

mass, whereas in the middle of the distribution the mass felt should tend to zero.

Figure 11 shows some examples of possible lensing masses and their associated mass felt distributions.

The black in the Lens mass distributions indicates the presence of mass, the darker the pixel, the more

mass in the area. The black areas in the Mass felt images indicates the ray feels the total mass of the

lens, as the black becomes more grey it feels less mass until it is white where it feels no mass.

From inspection of figure 11 it is clear that the non point lens method is successful, or at least producing

the expected trend.

Mass Felt

arcseconds

arcs

econ

ds

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1Lens Mass Distribution

arcseconds

arcs

econ

ds

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

Mass Felt

arcseconds

arcs

econ

ds

−1 −0.5 0 0.5 1−1

−0.5

0

0.5


arcseconds

arcs

econ

ds

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

Mass Felt

arcseconds

arcs

econ

ds

−1 −0.5 0 0.5 1−1

−0.5

0

0.5


arcseconds

arcs

econ

ds

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

Mass Felt

arcseconds

arcs

econ

ds

−1 −0.5 0 0.5 1−1

−0.5

0

0.5


arcseconds

arcs

econ

ds

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

Mass Felt

arcseconds

arcs

econ

ds

−1 −0.5 0 0.5 1−1

−0.5

0

0.5


arcseconds

arcs

econ

ds

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

Mass Felt

arcseconds

arcs

econ

ds

−1 −0.5 0 0.5 1−1

−0.5

0

0.5


arcseconds

arcs

econ

ds

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

Lens mass distributions converted to mass felt distributions using equations 9 and 10

Fig. 11

�0


Lens Pixelation Problems

A significant problem with using a matrix of mass elements instead of a continuous mass distribution for the

lens is that there will be only a few specific quantised mass states in the matrix. Rays which intersect the

lens plane at similar positions will likely feel the same mass until the difference is significant enough to push

the ray into a different pixel and then there will be a sudden change of mass. These discontinuities will

produce visual defects in the lensed image, to minimise these defects the obvious solution is to increase

the resolution of the lens mass matrix so that there are more discrete quantised states, unfortunately as

the matrix increases, the computational requirements exponentially scale up.

Figure 12 shows the image defect which is produced when using a 16 pixel by 16 pixel lens, where the

limits on the on the x and y axis represent approximately they maximum range of rays which the telescope

can detect. In this manner no rays will exist further out than this image shows. Conversely figure 13

shows that the effect can be diminished by increasing the resolution of the lens to 1�8 by 1�8 pixels over

the same angular size

Some structure is still clearly visible in the 128x128 lensed image but it much less significant. Further

increases in lens resolution are possible and this reduces the size of the square edge enhancement

patterns.

Lensed Image

arcseconds

arcs

econ

ds

0.1 0.2 0.3 0.4

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Galaxy

arcseconds

arcs

econ

ds

−0.15 −0.1 −0.05 0 0.05 0.1 0.15

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Lens Mass

arcseconds

arcs

econ

ds

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

Lensed Image

arcseconds

arcs

econ

ds

0 0.2 0.4 0.6

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Galaxy

arcseconds

arcs

econ

ds

−0.15 −0.1 −0.05 0 0.05 0.1 0.15

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Lens Mass

arcsecondsar

csec

onds

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

A 16x16 resolution lens which produces significant patterns in the lensed image due to rounding

numbers - zoomed image

A 128x128 resolution lens showing significantly reduced pixel edge enhancements - zoomed image

Fig. 12

Fig. 13

�1


Interpolation Schemes

Another method to reduce the pixelation rounding effect is to introduce an interpolation scheme which

estimates the mass felt more precisely by considering where the ray falls into a pixel. Unfortunately this

process can be very costly as it would require approximately at least another 10 operations for each ray,

where the number of rays used in this experiment are of the order of 50 million.

For a linear first order interpolation, the mass felt by the ray is calculated by examining the four nearest

array vertices in the mass felt array and weighing them based on the proximity of the ray to each specific

vertex.

Where the function f(x,y) denotes the mass felt by a particular ray according to first order interpolation, and

f(xi,yj) denotes one of the surrounding pixels. The mass felt can by found by:

This operation requires 10 additions/subtractions and 4 multiplications to find an approximate value for

the mass felt for each ray. Higher order interpolations schemes such as bicubic (order 3) can be used to

further improve the estimate of the mass felt at each point but these require exponentially more operations

to be carried out. Using these interpolation algorithms the lensed images can be improved with a similar

effect that can be achieved by raising the resolution of the lensed image.

f x y f x y

f x y f x y x

, ,

, ,

( ) = ( )+ ( ) − ( )( )

0 0

1 0 0 0 Δ

+ ( ) − ( )( )+ ( ) + ( ) −

f x y f x y y

f x y f x y f0 1 0 0

1 1 0 0

, ,

, ,

Δ

xx y f x y x y

x x xy y y

0 1 1 0

0

0

, ,( ) − ( )( )

= −= −

Δ Δ

ΔΔ

(0,0) (1,0)

(0,1) (1,1)

(x,y)

Diagram to show how each surrounding vertex influences the mass felt by the ray at position (x,y).

Fig. 14

Eqn. 11a-c

��


Central defects in lensed images

In the model being used it is assumed that the lens is totally transparent. At the centre of a circularly

symmetric lens, the mass felt by a ray is approximately zero and this will produce a slightly distorted image

of the source at the centre of any lensed image. As before the issue of pixelation of the lens will show its

hand here.

Figures 15 and 16, show the diminished angular size of the central region anomaly, as the resolution rises

the central region fixes itself. Physically these rays of light would have definitely been absorbed by the

lensing object and not reach the observer, so it is good that they can be diminished computationally.

A further improvement to the program could be applied whereby rays are actually absorbed by the lens,

and the probability of absorption is related to the mass felt at that point. If a ray felt ~100% of the lensing

mass then it is safe to say it wont be absorbed but alternatively if it only feels �% of the mass it will be in

the central part of the lens and definitely absorbed. Modification of the code to allow this further increase

the computational work needed and does help remove the central defects.

Lensed Image

arcseconds

arcs

econ

ds

0.005 0.01 0.015 0.02 0.025 0.03

0

0.005

0.01

0.015

0.02

Galaxy

arcseconds

arcs

econ

ds

−0.15 −0.1 −0.05 0 0.05 0.1 0.15

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Lens Mass

arcseconds

arcs

econ

ds

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

Lensed Image

arcseconds

arcs

econ

ds

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.02

0.04

0.06

0.08

0.1

0.12

Galaxy

arcseconds

arcs

econ

ds

−0.15 −0.1 −0.05 0 0.05 0.1 0.15

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Lens Mass

arcseconds

arcs

econ

ds

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

A 32x32 resolution lens with zoomed in section on the lensed image showing the central low mass

defect.

A 128x128 resolution lens with zoomed in section on the lensed image, note the axis labels which

show how small the defect is in this case.

Fig. 15

Fig. 16

��


Lensed Images

The first type of non point like lens to be considered is a spherically symmetric lens with a mass distribution

which falls away from a constant value at the centre by 1/radius�.

By changing the size of the lens it is possible to alter the Einstein ring phenomena when compared to total

alignment of a point lens. As the lens expands from 1 pixel (approximate point lens) to a large lens with

the specified profile in figure 17, the Einstein ring collapses inwards. The constant narrow outer line is the

theoretical Einstein ring for a point lens.

Radius

Len

s D

ensi

ty

r2

1

Lensed Image

arcseconds

arcs

econ

ds

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

Galaxy

arcseconds

arcs

econ

ds

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1Lens Mass

arcseconds

arcs

econ

ds

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

Lensed Image

arcseconds

arcs

econ

ds

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

Galaxy

arcseconds

arcs

econ

ds

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1Lens Mass

arcseconds

arcs

econ

ds

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

Lensed Image

arcseconds

arcs

econ

ds

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

Galaxy

arcseconds

arcs

econ

ds

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1Lens Mass

arcseconds

arcs

econ

ds

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

Lens density profile for possible lens type

Lensed images produced by a perfectly aligned spherical source and spherical lens

Fig. 17

Fig. 18

��


Figure 18 shows how the Einstein rings collapse in on itself as the mass distribution of the lens changes,

note the total mass of the lens is constant over each image. The images in question were created by

simulating fifty million rays of light passing over the area of the sky in a solid angle of approximately two

arcseconds squared, and using a 1�8x1�8 lens matrix.

The collapse of the Einstein ring in this case is due to the rays feeling a smaller mass than for a point lens

situation. In the point situation the rays which are detected are those at ~0.5 arc seconds radius, for

the non point cases rays which pass through the lens at this point will possibly be absorbed, but more

importantly will feel a smaller mass, this in turn means that the radius of the ring must shrink to compensate

for this loss of mass. As the lens continues to expand in size the rays will continue to feel less and less

mass, up to the point where they will enter the constant density region where they will mainly be absorbed

and never seen by the observer.

�5


Elliptical Lenses

Figure 19 shows what happens as an elliptical lens is moved off centre to produce arcs. In the first image

the lens is exactly central and it is slowly moved off centre in steps of equal x and y simultaneously. This

is not the only possible situation which can produce arcs.

In the first stage of figure 19 there is diminished intensity of radiation at either side of the ring, this is due

to the Lens mass absorbing some of the light which would have formed the ring. The ring is also slightly

squashed in on the sides do to a smaller mass being felt in these areas.

The next stage is to produce lensing images which match observations as closely as possible, the areas

to focus on will be the three major phenomena, arcs, rings and crosses. By replicating observed images

it should be possible to infer the possible lens mass distribution which produced them in the first place.

Typically the source objects being used will be spherical but other shapes will be tested as well because

a lot of objects in space are elliptical.

Lensed Image

arcseconds

arcs

econ

ds

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

Galaxy

arcseconds

arcs

econ

ds

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1Lens Mass

arcseconds

arcs

econ

ds

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

Lensed Image

arcseconds

arcs

econ

ds

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

Galaxy

arcseconds

arcs

econ

ds

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1Lens Mass

arcsecondsar

csec

onds

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

Lensed Image

arcseconds

arcs

econ

ds

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

Galaxy

arcseconds

arcs

econ

ds

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1Lens Mass

arcseconds

arcs

econ

ds

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

Production of basic arcs and rings using an elliptical lens

Fig. 19

�6


Simulation of Real Life Lensing Observations

Lensed Image

arcseconds

arcs

econ

ds

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

Galaxy

arcseconds

arcs

econ

ds

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1Lens Mass

arcseconds

arcs

econ

ds

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

Figure 20 - Giant arcs

A = Cl2244-02 - A giant arcs viewed via the Hubble

Space Telescope

B = Simulated arc using a circular source galaxy and a

slightly off centre elliptical lens.

A

B

Lensed Image

arcseconds

arcs

econ

ds

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

Galaxy

arcseconds

arcs

econ

ds

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1Lens Mass

arcseconds

arcs

econ

ds

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

Figure 21 - Distorted Einstein Ring

A = MG 1654+1346. Point like quasar firing two radio

jets in opposite directions, one is lensed from Earths frame

of reference, the other is not.

B = Simulated distorted ring shape. Note it is very difficult

to establish which lens mass distribution would produce

the required result.

A

B

�7


Figure 22 - Einstein Cross

A = G2237 + 0305. Einstein Cross

B = Einstein Cross produced via different computer

modelling approach

A

B

Figures 20 and 21 show how basic phenomena can be reproduced using the code created in this

project. In the case of the Einstein ring, the mass distribution was fudged by a trial and error approach

to try to replicate the observed image; many other mass distributions could also have worked. Minor

changes in the lens mass distribution can have a profound effect on the lensing patterns produced and

this made it very difficult to produce an Einstein cross. The Einstein cross shown in Figure 22B was

taken from Newbury (1997) and shows that it is possible to produce an Einstein cross using the basic

assumptions in this report. The key difficultly is finding the correct lens distribution, Newbury (1997) states

it to be an elliptical structure, but does not mention how the mass is actually distributed in this structure.

�8


Lensing real images of Galaxies

Lensed Image

arcseconds

arcs

econ

ds

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

Galaxy

arcseconds

arcs

econ

ds

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1Lens Mass

arcseconds

arcs

econ

ds

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

Figure 23 - Lensing of an elliptical galaxy

A= Source galaxy

B= Galaxy seen after being lensed by a point source with total

alignment of the system

A

B

Lensed Image

arcseconds

arcs

econ

ds

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

Galaxy

arcseconds

arcs

econ

ds

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1Lens Mass

arcseconds

arcs

econ

ds

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

Figure 24 - Lensing of a spiral galaxy

A= Source galaxy

B= Galaxy seen after being lensed by a point source with

total alignment of the system

A

B

Figures 23 and 24 show what happens if pictures of galaxies are lensed using a point lens at the origin.

The fuzziness in figure 23A has the effect of making the Einstein Ring in 23B fuzzy in specific areas. Figure

24A shows a spiral galaxy with a selection of bright objects around it, the spirals in figure 24A produce

small arcs around the Einstein ring in the lensed image of figure 24B, the bright objects occur as dots

around the ring in this image.

�9


Conclusion

This investigation has looked into how gravitational lensing can be simulated using Monte Carlo ray

tracing techniques. The computational model was able to reproduce some basic gravitational lensing

phenomena, but producing more complex situations such as the Einstein Cross proved difficult because

they require a very complicated lensing mass. Even the example shown in the previous section of a

program that successfully predicts a cross, doesn’t give the four circular images seen in the G��7 +

0�05 formation.

Some of the subtleties of real life lensing behaviour might be better modelled with a three dimensional lensing

mass rather than a two dimensional projection. A three dimensional lensing mass would allow a source

galaxy to be repeatedly lensed and hence re-allow previously forbidden rays to strike the observer.

The backwards mapping ray tracing method which was used is possibly an overly simplistic approach

to the problem of gravitational lensing, this may have further complicated producing the Einstein cross

phenomena. This method assumes that the probability of a ray striking any location on the telescope’s

CCD is constant over the whole array, this is unlikely to be exactly the case because the lens architecture

may make some locations preferable. It would make more sense to assume isotropic emission from the

source, rather than isotropic rays reaching the observer. However, this would be very difficult to model to

implement with any reasonable degree of computational efficiency.

Despite these limitations and the failure to produce a convincing Einstein cross, the investigation has

shown that many simple lensing effects can be easily modelled with a fairly straightforward ray-tracing

model. It performed particularly well in dealing with point-like and simple diffuse lenses, producing results

similar to those observed by astronomers, and for this reason it can be considered a broad success.

�0


References

Carroll & Ostilie,1996, An Introduction to Modern Astrophysics, Page 1�05-1�11, Addison Wesley

Longman

Cohn J, �000. Gravitational Lensing, http://astron.berkeley.edu/~jcohn/lens.html

Newbury P, 1997, http://www.iam.ubc.ca/~newbury/lenses/lenses.html

Wambsganss J, 1998, Gravitational Lensing in Astronomy, http://www.livingreviews.org/lrr-1998-1�,

Max-Planck-Gesellschaft.

Bibliography

Chae, Turnshek, Khersonsky, 1998,The American Astronomical Society. A Realistic Grid Of Models

For The Gravitationally Lensed Einstein Cross (Q2237]0305) And Its Relation To Observational Constraints,

The Astrophysical Journal, �95 : 609 E 616

Ratnatunga et al,1995, New ‘‘Einstein Cross’’ Gravitational Lens Candidates In Hubble Space Telescope

Wfpc2 Survey Images, The Astrophysical Journal, �5� : L 5–L 8.

Sources

All figures used in this report were created by the authors, with the exception of those listed below

Fig. 8 http://www.atnf.csiro.au/people/jlovell/

Fig. 20A http://serweb.oamp.fr/kneib/ast_now96/

Fig. 21A www.mira.org/fts0/s_system/161/text/txt005z.htm

Fig. 22A http://www.laughtergenealogy.com/bin/space/einstein-cross.jpg

Fig. 22B http://www.iam.ubc.ca/~newbury/lenses/mfk/src/mfk128.gif

Fig. 23A http://en.wikipedia.org/wiki/NGC_��1�

Fig. 24A http://www.hero.ac.uk/resources/R_2_Galactic_ghost_300.jpg

gravitational lensing · graviitational lensing stephen mullens and simon notley, 2005 history 4...

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