Perturbative Approach to a Non-spherically distorted Gravitational - PowerPoint PPT Presentation

Perturbative Approach to a Non-spherically distorted Gravitational Lens Masumi KASAI kasai@phys.hirosaki-u.ac.jp Hirosaki University 2011.6.6@former Research Inst. Theoretical Physics, a.k.a. “ Rironken ”

非球対称重力レンズへの 摂動的アプローチ 葛西 真寿 弘前大学 大学院理工学研究科 kasai@phys.hirosaki-u.ac.jp 2011.6.6@ 旧理論物理学研究所 a.k.a. “ 理論研 ”

Compact lens model a point mass α i = 4 GM | b | 2 b i Some generalizations... • rotational e ff ect a ... ( ( GM ) 2 ) • higher order e ff ects O ... • binary lens, etc...

Compact lens model a point mass α i = 4 GM | b | 2 b i Some generalizations... • rotational e ff ect a ... ( ( GM ) 2 ) • higher order e ff ects O ... • binary lens, etc...

Compact lens model a point mass α i = 4 GM | b | 2 b i Some generalizations... • rotational e ff ect a ... ( ( GM ) 2 ) • higher order e ff ects O ... • binary lens, etc...

Compact lens model a point mass α i = 4 GM | b | 2 b i Some generalizations... • rotational e ff ect a ... ( ( GM ) 2 ) • higher order e ff ects O ... • binary lens, etc...

E ff ect of non-spherical distortion bending angle b i b j b k b i b j ( ) α i = 4 GM | b | 2 + 8 G 2 Q jk , − Q i j c 2 c 2 | b | 6 | b | 4 mass multipole moments ∫ ρ d 3 x M = ( ) ∫ X i X j − 1 2 δ i j | X | 2 d 3 X ρ Q ij = trace-free quadrupole moment on the lens plane

E ff ect of non-spherical distortion bending angle b i b j b k b i b j ( ) α i = 4 GM | b | 2 + 8 G 2 Q jk , − Q i j c 2 c 2 | b | 6 | b | 4 mass multipole moments ∫ ρ d 3 x M = ( ) ∫ X i X j − 1 2 δ i j | X | 2 d 3 X ρ Q ij = trace-free quadrupole moment on the lens plane

E ff ect of non-spherical distortion bending angle b i b j b k b i b j ( ) α i = 4 GM | b | 2 + 8 G 2 Q jk , − Q i j c 2 c 2 | b | 6 | b | 4 mass multipole moments ∫ ρ d 3 x M = ( ) ∫ X i X j − 1 2 δ i j | X | 2 d 3 X ρ Q ij = trace-free quadrupole moment on the lens plane

Normalization & diagonalization ( e ) 0 Q ij ⇒ 0 − e Lens Equation x 2 + y 2 − e ( x 2 − 3 y 2 ) x x β x = x − ( x 2 + y 2 ) 3 x 2 + y 2 − e (3 x 2 − y 2 ) y y β y = y − ( x 2 + y 2 ) 3 β = ( β x , β y ): source position θ = ( x , y ): image position

Lens Equation x 2 + y 2 − e ( x 2 − 3 y 2 ) x x β x = x − ( x 2 + y 2 ) 3 x 2 + y 2 − e (3 x 2 − y 2 ) y y β y = y − ( x 2 + y 2 ) 3 Higher order simultaneous polynomials How to solve? cf. Asada (2005): exact, analytic approach polar coordinates = ⇒ 10th order eq. 4 or 6 real solutions (on-axis case only)

Lens Equation x 2 + y 2 − e ( x 2 − 3 y 2 ) x x β x = x − ( x 2 + y 2 ) 3 x 2 + y 2 − e (3 x 2 − y 2 ) y y β y = y − ( x 2 + y 2 ) 3 Higher order simultaneous polynomials How to solve? cf. Asada (2005): exact, analytic approach polar coordinates = ⇒ 10th order eq. 4 or 6 real solutions (on-axis case only)

order of magnitude e ∼ 10 − 5 ( M ⊙ ) 3 ( 10 7 km ) 2 ( ) 2 R v ) ( ≪ 1 10 6 km 10 km s − 1 M R E a more practical approach Solve perturbatively with respect to e

order of magnitude e ∼ 10 − 5 ( M ⊙ ) 3 ( 10 7 km ) 2 ( ) 2 R v ) ( ≪ 1 10 6 km 10 km s − 1 M R E a more practical approach Solve perturbatively with respect to e

zeroth-order solutions ( e = 0 ) Lens Equation ( x 2 + y 2 )( β x − x ) + x = 0 , ( x 2 + y 2 )( β y − y ) + y = 0 2 images x = x ± 0 ≡ f ± β x , y = y ± 0 ≡ f ± β y √ 1 + 4 β − 2 f ± ≡ 1 ± √ β 2 x + β 2 , β = y 2 (cf. a trivial solution ( x , y ) = (0 , 0) excluded)

first-order solutions ( 0 < e ≪ 1 ) x − 3 β 2 )( f ± ) 2 − 1 (4 β 2 x ± = x ± 0 + x ± 1 = f ± β x + e ˜ β 2 ( f ± β 2 + 1)( f ± β 2 + 2) β x (3 β 2 − 4 β 2 y )( f ± ) 2 + 1 y ± = y ± 0 + y ± 1 = f ± β y + e ˜ β 2 ( f ± β 2 + 1)( f ± β 2 + 2) β y That’s all folks... ?

first-order solutions ( 0 < e ≪ 1 ) x − 3 β 2 )( f ± ) 2 − 1 (4 β 2 x ± = x ± 0 + x ± 1 = f ± β x + e ˜ β 2 ( f ± β 2 + 1)( f ± β 2 + 2) β x (3 β 2 − 4 β 2 y )( f ± ) 2 + 1 y ± = y ± 0 + y ± 1 = f ± β y + e ˜ β 2 ( f ± β 2 + 1)( f ± β 2 + 2) β y That’s all folks... ?

More solutions? Perturbing 2 zeroth-order solutions ⇓ always get 2 first-oder solutions. However, algebraic structure of the lens equation tells the existence of more than 2 solutions. How to get more solutions perturbatively? Perturbative generation from nothing?

Perturbation around the excluded solution x = 0 + x 1 , ˜ y = 0 + y 1 ˜ Solutions for the “minor” images ( ) y ± = ± √ e x ± = 1 1 + 1 − 1 ˜ 2 e β x , ˜ 2 e 2 e β y Totally 4 images obtained!

e β x β y (1) y num (2) y appr Error | (2) − (1) | 3 . 9 × 10 − 5 0.01 0.0 0.2 1.10087 1.10091 1 . 0 × 19 − 4 -0.89881 -0.89891 3 . 7 × 10 − 6 0.09950 0.099500 6 . 8 × 10 − 5 -0.10157 -0.10150 1 . 8 × 10 − 5 0.01 0.0 0.5 1.27780 1.27782 2 . 0 × 10 − 4 -0.77262 -0.77282 8 . 5 × 10 − 5 0.09809 0.09800 2 . 7 × 10 − 4 -0.10327 -0.10300 7 . 3 × 10 − 5 0.02 0.0 0.5 1.27479 1.27486 8 . 4 × 10 − 4 -0.76402 -0.76486 1 . 8 × 10 − 4 0.13802 0.13784 9 . 5 × 10 − 4 -0.14878 -0.14784 Error ∼ O ( e 2 ), also depends on β

Amplification factor Perturbative solutions ˜ x ( β ) , ˜ y ( β ) work well. What about the amplification factor? can be directly calculated from � � � det ∂ ( x ± , y ± ) ( ) � � A ± = � = A ± 1 + e ∆ ± ( β ) � � � � 0 ∂ ( β x , β y ) � �

Error of A ± e β x β y parity 0.01 0.0 0.2 + 0.45% 2.24% − 0.02 0.0 0.2 1.82% + 9.10% − 0.02 0.2 0.0 1.77% + 8.55% − 0.02 0.5 0.0 + 0.09% 4.14% − Not so good. Error often exceeds O ( e ).

Better accuracy without higher-order calculations � � � det ∂ ( x ± , y ± ) ( ) � � A ± = � = A ± 1 + e ∆ ± ( β ) � � � � 0 ∂ ( β x , β y ) � � Pad´ e approximant ) − 1 ( ± ≡ A ± 1 − e ∆ ± ( β ) A P 0

Better accuracy without higher-order calculations � � � det ∂ ( x ± , y ± ) ( ) � � A ± = � = A ± 1 + e ∆ ± ( β ) � � � � 0 ∂ ( β x , β y ) � � Pad´ e approximant ) − 1 ( ± ≡ A ± 1 − e ∆ ± ( β ) A P 0

Error of A ± Error of A ± e β x β y parity P 0.01 0.0 0.2 + 0.45% 0.04% 2.24% 0.26% − 0.02 0.0 0.2 1.82% 0.15% + 9.10% 1.25% − 0.02 0.2 0.0 1.77% 0.20% + 8.55% 0.66% − 0.02 0.5 0.0 + 0.09% 0.02% 4.14% 0.80% − Pad´ e approximant successfully reduces error, without doing higher-order calculations.

Error of A ± Error of A ± e β x β y parity P 0.01 0.0 0.2 + 0.45% 0.04% 2.24% 0.26% − 0.02 0.0 0.2 1.82% 0.15% + 9.10% 1.25% − 0.02 0.2 0.0 1.77% 0.20% + 8.55% 0.66% − 0.02 0.5 0.0 + 0.09% 0.02% 4.14% 0.80% − Pad´ e approximant successfully reduces error, without doing higher-order calculations.

Changes in image properties image separation of two “major” images   2( β 2 x − β 2   y ) { } ∆ x ≡ x + − x − = ∆ x 0         e  1 + e × O (1)  1 − β 2 ( β 2 + 4) − 1  = ∆ x 0              only slightly changes Amplification di ff erence P ≃ 1 − 2 A di ff ≡ A + P − A − β 2 e can be significantly changed if β 2 ∼ e , even if e ≪ 1

Changes in image properties image separation of two “major” images   2( β 2 x − β 2   y ) { } ∆ x ≡ x + − x − = ∆ x 0         e  1 + e × O (1)  1 − β 2 ( β 2 + 4) − 1  = ∆ x 0              only slightly changes Amplification di ff erence P ≃ 1 − 2 A di ff ≡ A + P − A − β 2 e can be significantly changed if β 2 ∼ e , even if e ≪ 1

“Anomalous” Flux Ratio Non-spherical distorted lens, even if it’s tiny ( e ≪ 1), can cause significant amount of flux anomalies in the lensed images , whereas it only slightly changes the image positions. P ≃ 1 − 2 A di ff ≡ A + P − A − β 2 e “anomaly” = unexpectedly large di ff erence from a point mass case

Summary Perturbative approach to a non-spherically distorted gravitational lens • Perturbatively generated solutions from “nothing” • E ffi cient Pad´ e approximant • “Anomalous” flux ratio by a tiny distortion?