Gravitational Lens Models

Here are some results from gravitational lens models.

SIE Lens Models

Here is a table of lens model parameters for singular isothermal ellipsoid (SIE) lens models. Each lens is described by the best-fit critical radius (b), lens position (xl, yl), ellipticity (e=1-b/a), and major axis position angle, in addition to the chi-squared fit of the model. The error bars are standard errors using a chi-squared renormalized to equal Ndof at the minimum. The coordinate center refers to the component from which the lens position is measured.

These models have been used to predict time delays.
 
Lens Center b (") xl (") yl (") e=1-b/a PA (°) Chi2/Ndof Notes
Q0142-100  1.149±0.004 1.764±0.003 -0.574±0.003 0.27±0.02 84±2 0/0 (1)
B0218+357 A 0.1581 -0.17 0.045 0.01092 -5.96 62.92/7 (1) SIE' 
MG0414+0534  B 1.14±0.03  0.53±0.09  -1.34±0.05  0.38±0.09  79.1±1.2  111/6   
B0712+472  A 0.69±0.01  -0.77±0.04  0.08±0.05  0.25±0.11  50.5±0.9  23.9/6   
MG0751+2716    0.41      0.34  64     
SBS0909+523
A
0.5±0.1
0.4±0.1
0.0±0.1
0.3±0.4
56±15
0/0
(1)
BRI0952-0115  0.513±0.002 -0.396±0.003 -0.506±0.003 0.183±0.008 66±1 0/0  (1) 
Q0957+561    3.09      0.64  69 ± 1     
LBQS1009-0252  A 0.757±0.006 -0.537±0.003  -1.097±0.003 0.08±0.04 -58±5 0/0 (1) 
Q1017-207=J03.13  A 0.438±0.002 -0.66±0.02 0.02±0.02 0.22±0.03 -85±11 0/0 (1) 
B 1030+071
A
0.587±0.004
0.84±0.01
-1.09±0.01
0.46±0.06
-38±4
0/0
(1)
B 1030+071 G'
 
0.289±0.002
0.35±0.02
-1.23±0.94
=0
=0
 
extra comp.
HE1104-1805  A 1.451±0.005 0.974±0.004 -0.510±0.004 0.341±0.007 22.3±0.2 0/0 (1) 
PG1115+080  C 1.08±0.03  -0.32±0.04  -1.45±0.06  0.48±0.07  66.7±0.7  483/6   
MG1131+045    0.92      0.33  -26     
1208+1011
A
0.239±0.001
0.105±0.002
-0.381±0.005
 =0
=0 
 0/0
(1), SIS
HST12531-2914  G 0.54±0.04  -0.03±0.04  0.02±0.05  0.38±0.17  19.0±3.7  35.2/6   
H1413+117  A 0.58±0.02  -0.18±0.02  0.58±0.03  0.48±0.07  21.6±1.1  149/4   
HST14176+5226  G 1.34±0.09  0.01±0.09  0.18±0.08  0.52±0.11  48.6±2.5  111/6   
B1422+231  B 0.65±0.02  -0.75±0.03  -0.67±0.02  0.63±0.03  -53.2±1.0  124/ 6   
MG1549+3047    1.15      0.07  -48     
B1608+656  B 1.07±0.04  0.96±0.13  0.88±0.04  0.35±0.16  68.6±1.6  790/6   
MG1654+1346    0.98      0.27  -81     
PKS 1830-211
A
0.48±0.03
-0.50±0.08
-0.45±0.08
0.2±0.2
-20±22
0/0
(1)
B1933+503  4 0.45±0.01  -0.41±0.01  0.25±0.01  0.60±0.03  -46.2±0.6  3.1/4   
2237+0305  A 0.85±0.01  0.09±0.01  0.93±0.01  0.31±0.02  66.6±0.3  15.8/6   

Notes: (1) From Lehar etal 2000, astro-ph/9909072.
 


SIE+shear Lens Models

Here is a table of lens model parameters for SIS+shear lens models. Each lens is described by the best-fit critical radius (b), lens position (xl, yl), shear, and major axis position angle, in addition to the chi-squared fit of the model.  The error bars are standard errors using a chi-squared renormalized to equal Ndof at the minimum.  For underconstrained models with Ndof<0, the parameter ranges giving acceptable chi-squared are given as (min__max).  Parameters held fixed are shown as (=value).  The coordinate center refers to the component from which the lens position is measured.

These models have been used to predict time delays.
 
Lens Center b (") xl (") yl (") e=1-b/a PA (°) shear PA (°) Chi2/Ndof Notes
Q0142-100   A 1.195__1.215  1.764±0.003  -0.574±0.003  0__0.27  =61  0.069__0.089  81__118 0/-1  (1) 
MG0414+0534  B 1.18±0.03  0.47±0.06  -1.31±0.04  =0  =0  0.10±0.03  78.0±1.3  117/6   
B0712+472  A 0.69±0.02  -0.74±0.03  0.05±0.04  =0 =0 0.05±0.04  50.1±1.1  27.9/6   
MG0751+2716    0.40      =0 =0  0.09  64     
BRI0952-0115  0.475__0.527  -0.396±0.003  -0.506±0.003  0_0.38  =59  0.016__0.1  66__144 0/-1  (1) 
LBQS1009-0252  A 0.778±0.016     =0 =0  0.018 (+0.024,-0.014) 49 (+55,-15)  0/0  
Q1017-207=J03.13  A 0.467±0.008     =0  =0  0.108±0.018 75.8±6.0 0/0  
HE1104-1805  A 1.39__1.40 0.974±0.004  -0.510±0.004  0__0.27 =63  0.125__0.142 2__22 0/-1 (1) 
PG1115+080  C 1.14±0.01  -0.33±0.02  -1.36±0.03  =0  =0  0.12±0.02  65.4±0.6  251/6   
MG1131+045  D 0.92±0.01  -0.02±0.02  0.02±0.02  =0  =0  0.11±0.01  -27.2±2.5     
HST12531-2914  G 0.55±0.03  -0.03±0.03  0.03±0.03  =0  =0  0.15±0.05  18.7±2.8  19.5/6   
H1413+117  A 0.61±0.01  -0.17±0.02  0.55±0.02  =0  =0  0.11±0.02  21.7±1.1  142/4   
HST14176+5226  G 1.42±0.06  0.03±0.08  0.20±0.07  =0  =0  0.15±0.04  48.9±2.4  95.7/6   
B1422+231  B 0.77±0.01  -0.72±0.02  -0.65±0.01  =0  =0  0.26±0.01  -54.3±0.5  40.3/6   
B1608+656  B 1.10±0.03  0.88±0.08  0.89±0.03  =0  =0  0.06±0.04  68.9±1.7  837/6   
MG1654+1346  QSO 0.98±0.01  2.26±0.02  -2.08±0.01  =0  =0  0.08±0.01  -79.4±1.0     
B1933+503  4 0.50±0.01  -0.43±0.01  0.30±0.01  =0  =0  0.15±0.02  -45.6±0.9  9.9/4   
2237+0305  A 0.87±0.01  0.10±0.01  0.93±0.01  =0  =0  0.07±0.01  66.8±0.3  20.4/6   

Notes: (1) From Lehar etal 2000, astro-ph/9909072.


M/L Lens Models

 
Lens
Center
b (")
xl (")
yl (")
Re (")
e=1-b/a
PA (°)
Chi2/Ndof
Notes
Q 0142-100
A
3.27
1.767
-0.552
=0.59
=0.25
=61
75/2
(1)
BRI 0952-0115
A
3.63
-0.406
-0.483
=0.14
=0.47
=59
109/2
(1)
B 1030+071
A
1.91
0.83
-1.09
=0.48
=0.22
=28
0.7/2
(1)
B 1030+071 G'
 
0.94
0.35
-1.23
=0.11
=0
=0
  extra comp.
HE 1104-1805
A
5.80
1.004
-0.356
=0.73
=0.23
=63
2069/2
(1)

Notes: (1) From Lehar etal 2000, astro-ph/9909072.
 
 
 
 


M/L+shear Lens Models

 
Lens
Center
b (")
xl (")
yl (")
Re (")
e=1-b/a
PA (°)
shear
PA (°)
Chi2/Ndof
Notes
Q 0142-100
A
3.12±0.07
1.764±0.003
-0.574±0.003
=0.59
=0.25
=61
0.07±0.02
37±6
0/0
(1)
BRI 0952-0115
A
3.49±0.04
-0.396±0.003
-0.506±0.003
=0.14
=0.47
=59
0.065±0.008
-69±3
0/0
(1)
B 1030+071
A
2.0±0.3
0.84±0.01
-1.09±0.01
=0.48
=0.22
=28
0.1±0.1
Any
0/0
(1)
B 1030+071 G'
 
0.99±0.1
0.35±0.02
-1.23±0.02
=0.11
=0
=0
 
 
 
extra comp.
HE 1104-1805
A
4.65±0.03
0.974±0.004
-0.510±0.004
=0.73
=0.23
=63
0.211±0.004
17.6±0.2
0/0
(1)

Notes: (1) From Lehar etal 2000, astro-ph/9909072.
 
 
 
 


Predicted Time Delays (SIE)

Here is a table of predicted time delays based on singular isothermal ellipse (SIE) lens models. For each lens, the leading image component is given, followed by the time delays in units of the time scale

t0 = (DlDs/Dls)(1+zl)/c .

The value for t0 (in h-1 days) is given if the lens redshift is known. An omega = 1 cosmology is assumed.
 
Lens Lead Time Delays dt/t0 t0 (h-1 days)  Notes
Q0142-100  B= 1.634      48.59   
MG0414+0534  A1= 0.126  A2= 0.129  C= 0.586     
B0712+472  A= 0.018  B= 0.018  D= 0.134  48.87   
BRI0952-0115  B= 0.143         
PG1115+080  A1= 0.482  A2= 0.487  B= 0.733  31.22   
HST12531-2914  B= 0.007  C= 0.043  D= 0.067     
H1413+117  B= 0.016  A= 0.067  D= 0.122     
HST14176+5226  B= 0.141  C= 0.428  D= 0.769  84.60   
B1422+231  A= 0.035  B= 0.047  D= 0.557  30.66   
B1608+656  A= 0.094  C= 0.115  D= 0.385  94.20   
B1933+503  4= 0.117  3= 0.132  6= 0.160  99.03  (1) 
Q2237+0305  A= 0.028  D= 0.071  C= 0.164  3.34   

Notes: (1) For B 1933+503, t0 was computed assuming a source redshift zs=2. It would be smaller by 17.8% (24.7%) for zs=3 (4). Note that only images 1, 3, 4, and 6 (the quadruple images of the flat-spectrum source) were used for the fits; the multiple images of the steep-spectrum source were neglected.


Predicted Time Delays (SIS+shear)

Here is a table of predicted time delays based on SIS+shear lens models. For each lens, the leading image component is given, followed by the time delays in units of the time scale

t0 = (DlDs/Dls)(1+zl)/c .

The value for t0 (in h-1 days) is given if the lens redshift is known. An omega = 1 cosmology is assumed.
 
Lens Lead Time Delays dt/t0 t0 (h-1 days)  Notes
Q0142-100  B= 1.683      48.59   
MG0414+0534  A1= 0.060  A2= 0.062  C= 0.379     
B0712+472  A= 0.008  B= 0.008  D= 0.069  48.87   
BRI0952-0115  B= 0.147         
PG1115+080  A1= 0.235  A2= 0.238  B= 0.410  31.22   
HST12531-2914  B= 0.004  C= 0.035  D= 0.062     
H1413+117  B= 0.008  A= 0.032  D= 0.062     
HST14176+5226  B= 0.082  C= 0.251  D= 0.486  84.60   
B1422+231  A= 0.010  B= 0.013  D= 0.538  30.66   
B1608+656  A= 0.041  C= 0.050  D= 0.179  94.20   
B1933+503  4= 0.053  3= 0.061  6= 0.077  99.03  (1) 
Q2237+0305  A= 0.014  D= 0.036  C= 0.093  3.34   

Notes:

(1) For B 1933+503, t0 was computed assuming a source redshift zs=2. It would be smaller by 17.8% (24.7%) for zs=3 (4). Note that only images 1, 3, 4, and 6 (the quadruple images of the flat-spectrum source) were used for the fits; the multiple images of the steep-spectrum source were neglected. ###End of File