Determination of the redshift of an invisible lens

(1)

Determination

of

the

redshift of

an

invisible

lens

Christophe

Jean

and Jean

Surdej

Institut

d'Astrophysique, Université de Liège, Belgium

1. Introduction

2. Fitting

3. Monte Carlo

Simulations

(2)

Observer

Introduction

Deflector

z _r,Ra, Quasar

Flux received bv the observer from component

i

at the wavelength À:

F^,i

: F|

An"-Ext(À" Rv)Lt

with {

the flux emitted by the quasar;

Ai

the magnification of component i; \

trxt(À/ _{, Rv)}the extinction (À'

: , ,

_,

Rv

is a parameter);

'

I

*

zt,

and Li

the path length of component

i

through the deflector. We assume no contamination, no intrinsic colour variations between

the components, no micro-iensing effects and

that

the extinction is

taking place

in

a single lensing plane.

Colour difference between component

i

and a reference component is

then: (lVlsr,t

_-

fuIb,n)

_-

(Mn,ref

_-

M^r,,.f)

:

)5

ft

LilBxr(Àî,

Ârr)

-

Ext (À'r, Rv))

with

Mx,

_-

M^,

:

_-2.5f

"*

(+)

_{\- ^2,/}

and

Lt

:

Li

_-

Lr.r.

(3)

Fitting

(Mxr,,i

_-

Msr,n)

_-

(M>,r,ref

_-

M^r,r.f)

:

Experimental

colour

frfr

Lillxt(À'r,

Rr)

_-

Ext (^!r,

Rr)l

For the

fitted

colour:

o

_{Ext(Ài ,}Rv)

_-

Ext (À'r,

Rv)

is computed for different contiguous

values of

Rv

and z1;

.

L,; is

fitted

by u routine for each selected

(Rv,rt)

point in order

to

minimize:

(trxperimental colour

_-

Fitted colour)2

x'-D

_()r,

- E-x perrmenl al colour

Ax'

Z

S

map is thus obtained:

21 0.01 0 6

Fitted

colour

r.05

_RV

(4)

Monte Carlo

Simulations

Study of the errors affecting the determination of the lens redshift from a set of 100 simulations. Gaussian noise was added

to

the

simulated magnitudes of the 4 selected components.

5 filters:

UBVRI

0.1 5 : t: i--tr norse.

noise:

0.01

0.02

0.03

0.04

0.05 ----x---mag

mag

I /ll ,r -ry-,.x, . -l 'i!rrr! Y:Ufl-N! , _{rç*'^liiâé?:r".}*]!.t:,' //-' . t':!!!1,' , t EY.,-** ç1;,;--çi"-o 7-?*^=I: :Ï:::

t'l'1

lotrul.-t la-xxx-ri 0.1 b

0.05

0

0 0.5 1 1.5 2 2.5

3 z1

(5)

Gaussian noise. 0.01

mao

0.1 b 0.01 0.001

0.5

1.5 Z1

2.5 Gaussian noise: 0.01

mag

0.05

0.04

0.03

0.02

0.01 0

1 1.5 2

2.5

z1 XJ ^xxxx)<xxv ,.x X

XX--UBVR|

_

IJHKL

k \ t- ' "-- z-r. X \, X., ^X-X X-X _X-X x*-**"**l *:-!--++

5 filters:

UBVRI

7 filters: UBVRIJH

9 filters:

UBVRIJHKL

I i T \/,*\ ',,,x i *r\ ,ï\

0.5

(6)

Conclusions

Monte Carlo simulations indicate

that

the redshift of an invisible

lens can be retrieved

with

this method provided

that

we have

accurate photometric observations of the macro-lensed images.

This method is particularly sensitive to the redshift of the deflector

and thus offers an original way of determining the redshift of an

invisible lens.

High quality photometric data (o

₌

0.01 mag) are badly needed.