Table 8. Open in new tab Optimal SIE and...

	SIE	SIE	ε − γ	ε − γ	NSIE + γ
\|$\chi ^{2}_{r}$\|	1.5466	12.204	18.165	–	0.7319
f	\|$0.6649_{-0.0094}^{+0.0100}$\|	\|$0.6479_{-0.0097}^{+0.0097}$\|	–	–	\|$0.6634_{-0.0096}^{+0.0110}$\|
ϖ	\|$1.9720_{-0.0029}^{+0.0028}$\|	\|$1.9686_{-0.0024}^{+0.0024}$\|	–	–	\|$2.0111_{-0.0023}^{+0.0023}$\|
ε	–	–	\|$0.6653_{-0.0094}^{+0.0100}$\|	1.5949	–
γ	–	–	\|$0.1047_{-0.0094}^{+0.0100}$\|	-0.0292	\|$0.0138_{-0.00012}^{+0.00012}$\|
ω	–	–	\|$2.7437_{-0.0094}^{+0.0100}$\|	1.1729	\|$2.7107_{-0.0010}^{+0.0010}$\|
θ_S	\|$0.0652_{-0.0021}^{+0.0021}$\|	\|$0.0669_{-0.0035}^{+0.0035}$\|	\|$0.0595_{-0.0094}^{+0.0100}$\|	-0.0166	\|$0.0632_{-0.0011}^{+0.0012}$\|
θ	\|$6.1261_{-0.0146}^{+0.0150}$\|	\|$6.0629_{-0.0408}^{+0.0408}$\|	\|$6.0589_{-0.0094}^{+0.0100}$\|	6.0524	\|$6.1236_{-0.0118}^{+0.0119}$\|
θ_E	\|$0.8974_{-0.0023}^{+0.0023}$\|	\|$0.9024_{-0.0021}^{+0.0021}$\|	\|$0.8812_{-0.0094}^{+0.0100}$\|	0.8865	\|$0.9396_{-0.0019}^{+0.0019}$\|
ρ_c	–	–	–	–	\|$0.0332_{-0.0013}^{+0.0013}$\|
Δx_A	0.0031	0.0163	0.0064	–	0.0028
Δx_B	0.0087	0.0097	0.0082	–	0.0020
Δx_C	0.0082	0.0006	0.0210	–	0.0030
Δx_D	0.0036	0.0268	0.0335	–	0.0027

SIE

ε − γ

NSIE + γ

first order

|$\chi ^{2}_{r}$|

1.5466

12.204

18.165

–

0.7319

|$0.6649_{-0.0094}^{+0.0100}$|

|$0.6479_{-0.0097}^{+0.0097}$|

–

|$0.6634_{-0.0096}^{+0.0110}$|

|$1.9720_{-0.0029}^{+0.0028}$|

|$1.9686_{-0.0024}^{+0.0024}$|

–

|$2.0111_{-0.0023}^{+0.0023}$|

–

|$0.6653_{-0.0094}^{+0.0100}$|

1.5949

–

|$0.1047_{-0.0094}^{+0.0100}$|

-0.0292

|$0.0138_{-0.00012}^{+0.00012}$|

–

|$2.7437_{-0.0094}^{+0.0100}$|

1.1729

|$2.7107_{-0.0010}^{+0.0010}$|

θ_S

|$0.0652_{-0.0021}^{+0.0021}$|

|$0.0669_{-0.0035}^{+0.0035}$|

|$0.0595_{-0.0094}^{+0.0100}$|

-0.0166

|$0.0632_{-0.0011}^{+0.0012}$|

|$6.1261_{-0.0146}^{+0.0150}$|

|$6.0629_{-0.0408}^{+0.0408}$|

|$6.0589_{-0.0094}^{+0.0100}$|

6.0524

|$6.1236_{-0.0118}^{+0.0119}$|

θ_E

|$0.8974_{-0.0023}^{+0.0023}$|

|$0.9024_{-0.0021}^{+0.0021}$|

|$0.8812_{-0.0094}^{+0.0100}$|

0.8865

|$0.9396_{-0.0019}^{+0.0019}$|

ρ_c

–

|$0.0332_{-0.0013}^{+0.0013}$|

Δx_A

0.0031

0.0163

0.0064

–

0.0028

Δx_B

0.0087

0.0097

0.0082

–

0.0020

Δx_C

0.0082

0.0006

0.0210

–

0.0030

Δx_D

0.0036

0.0268

0.0335

–

0.0027

	SIE	SIE	ε − γ	ε − γ	NSIE + γ
\|$\chi ^{2}_{r}$\|	1.5466	12.204	18.165	–	0.7319
f	\|$0.6649_{-0.0094}^{+0.0100}$\|	\|$0.6479_{-0.0097}^{+0.0097}$\|	–	–	\|$0.6634_{-0.0096}^{+0.0110}$\|
ϖ	\|$1.9720_{-0.0029}^{+0.0028}$\|	\|$1.9686_{-0.0024}^{+0.0024}$\|	–	–	\|$2.0111_{-0.0023}^{+0.0023}$\|
ε	–	–	\|$0.6653_{-0.0094}^{+0.0100}$\|	1.5949	–
γ	–	–	\|$0.1047_{-0.0094}^{+0.0100}$\|	-0.0292	\|$0.0138_{-0.00012}^{+0.00012}$\|
ω	–	–	\|$2.7437_{-0.0094}^{+0.0100}$\|	1.1729	\|$2.7107_{-0.0010}^{+0.0010}$\|
θ_S	\|$0.0652_{-0.0021}^{+0.0021}$\|	\|$0.0669_{-0.0035}^{+0.0035}$\|	\|$0.0595_{-0.0094}^{+0.0100}$\|	-0.0166	\|$0.0632_{-0.0011}^{+0.0012}$\|
θ	\|$6.1261_{-0.0146}^{+0.0150}$\|	\|$6.0629_{-0.0408}^{+0.0408}$\|	\|$6.0589_{-0.0094}^{+0.0100}$\|	6.0524	\|$6.1236_{-0.0118}^{+0.0119}$\|
θ_E	\|$0.8974_{-0.0023}^{+0.0023}$\|	\|$0.9024_{-0.0021}^{+0.0021}$\|	\|$0.8812_{-0.0094}^{+0.0100}$\|	0.8865	\|$0.9396_{-0.0019}^{+0.0019}$\|
ρ_c	–	–	–	–	\|$0.0332_{-0.0013}^{+0.0013}$\|
Δx_A	0.0031	0.0163	0.0064	–	0.0028
Δx_B	0.0087	0.0097	0.0082	–	0.0020
Δx_C	0.0082	0.0006	0.0210	–	0.0030
Δx_D	0.0036	0.0268	0.0335	–	0.0027

Table 8.

Open in new tab

Optimal SIE and ε − γ lens parameters for the Q2237+0305 system, derived by the ferret sophisticated genetic algorithm which is a component of the Qubist Global Optimization Toolbox.

	SIE	SIE	ε − γ	ε − γ	NSIE + γ
		first order		first order
\|$\chi ^{2}_{r}$\|	1.5466	12.204	18.165	–	0.7319
f	\|$0.6649_{-0.0094}^{+0.0100}$\|	\|$0.6479_{-0.0097}^{+0.0097}$\|	–	–	\|$0.6634_{-0.0096}^{+0.0110}$\|
ϖ	\|$1.9720_{-0.0029}^{+0.0028}$\|	\|$1.9686_{-0.0024}^{+0.0024}$\|	–	–	\|$2.0111_{-0.0023}^{+0.0023}$\|
ε	–	–	\|$0.6653_{-0.0094}^{+0.0100}$\|	1.5949	–
γ	–	–	\|$0.1047_{-0.0094}^{+0.0100}$\|	-0.0292	\|$0.0138_{-0.00012}^{+0.00012}$\|
ω	–	–	\|$2.7437_{-0.0094}^{+0.0100}$\|	1.1729	\|$2.7107_{-0.0010}^{+0.0010}$\|
θ_S	\|$0.0652_{-0.0021}^{+0.0021}$\|	\|$0.0669_{-0.0035}^{+0.0035}$\|	\|$0.0595_{-0.0094}^{+0.0100}$\|	-0.0166	\|$0.0632_{-0.0011}^{+0.0012}$\|
θ	\|$6.1261_{-0.0146}^{+0.0150}$\|	\|$6.0629_{-0.0408}^{+0.0408}$\|	\|$6.0589_{-0.0094}^{+0.0100}$\|	6.0524	\|$6.1236_{-0.0118}^{+0.0119}$\|
θ_E	\|$0.8974_{-0.0023}^{+0.0023}$\|	\|$0.9024_{-0.0021}^{+0.0021}$\|	\|$0.8812_{-0.0094}^{+0.0100}$\|	0.8865	\|$0.9396_{-0.0019}^{+0.0019}$\|
ρ_c	–	–	–	–	\|$0.0332_{-0.0013}^{+0.0013}$\|
Δx_A	0.0031	0.0163	0.0064	–	0.0028
Δx_B	0.0087	0.0097	0.0082	–	0.0020
Δx_C	0.0082	0.0006	0.0210	–	0.0030
Δx_D	0.0036	0.0268	0.0335	–	0.0027

	SIE	SIE	ε − γ	ε − γ	NSIE + γ
		first order		first order
\|$\chi ^{2}_{r}$\|	1.5466	12.204	18.165	–	0.7319
f	\|$0.6649_{-0.0094}^{+0.0100}$\|	\|$0.6479_{-0.0097}^{+0.0097}$\|	–	–	\|$0.6634_{-0.0096}^{+0.0110}$\|
ϖ	\|$1.9720_{-0.0029}^{+0.0028}$\|	\|$1.9686_{-0.0024}^{+0.0024}$\|	–	–	\|$2.0111_{-0.0023}^{+0.0023}$\|
ε	–	–	\|$0.6653_{-0.0094}^{+0.0100}$\|	1.5949	–
γ	–	–	\|$0.1047_{-0.0094}^{+0.0100}$\|	-0.0292	\|$0.0138_{-0.00012}^{+0.00012}$\|
ω	–	–	\|$2.7437_{-0.0094}^{+0.0100}$\|	1.1729	\|$2.7107_{-0.0010}^{+0.0010}$\|
θ_S	\|$0.0652_{-0.0021}^{+0.0021}$\|	\|$0.0669_{-0.0035}^{+0.0035}$\|	\|$0.0595_{-0.0094}^{+0.0100}$\|	-0.0166	\|$0.0632_{-0.0011}^{+0.0012}$\|
θ	\|$6.1261_{-0.0146}^{+0.0150}$\|	\|$6.0629_{-0.0408}^{+0.0408}$\|	\|$6.0589_{-0.0094}^{+0.0100}$\|	6.0524	\|$6.1236_{-0.0118}^{+0.0119}$\|
θ_E	\|$0.8974_{-0.0023}^{+0.0023}$\|	\|$0.9024_{-0.0021}^{+0.0021}$\|	\|$0.8812_{-0.0094}^{+0.0100}$\|	0.8865	\|$0.9396_{-0.0019}^{+0.0019}$\|
ρ_c	–	–	–	–	\|$0.0332_{-0.0013}^{+0.0013}$\|
Δx_A	0.0031	0.0163	0.0064	–	0.0028
Δx_B	0.0087	0.0097	0.0082	–	0.0020
Δx_C	0.0082	0.0006	0.0210	–	0.0030
Δx_D	0.0036	0.0268	0.0335	–	0.0027

Notes. The parameters ε represents the slope of the power-law mass distribution, γ the magnitude of the external shear, ω the orientation of the external shear and ρ_c the size of the core radius in arcsec. The lower and upper limits correspond to the range of the lens parameters used by the genetic algorithm routine. The distances between the lensed images and the modelled images are represented by Δx.

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