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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

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SLIDE 1

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”

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SLIDE 2

非球対称重力レンズへの 摂動的アプローチ

葛西 真寿

kasai@phys.hirosaki-u.ac.jp

弘前大学 大学院理工学研究科

2011.6.6@旧理論物理学研究所 a.k.a. “理論研”

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SLIDE 3

Compact lens model a point mass αi = 4GM |b|2 bi Some generalizations...

  • rotational effect a...
  • higher order effects O

( (GM)2) ...

  • binary lens, etc...
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SLIDE 4

Compact lens model a point mass αi = 4GM |b|2 bi Some generalizations...

  • rotational effect a...
  • higher order effects O

( (GM)2) ...

  • binary lens, etc...
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SLIDE 5

Compact lens model a point mass αi = 4GM |b|2 bi Some generalizations...

  • rotational effect a...
  • higher order effects O

( (GM)2) ...

  • binary lens, etc...
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SLIDE 6

Compact lens model a point mass αi = 4GM |b|2 bi Some generalizations...

  • rotational effect a...
  • higher order effects O

( (GM)2) ...

  • binary lens, etc...
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SLIDE 7

Effect of non-spherical distortion bending angle

αi = 4GM c2 bi |b|2 + 8G c2 ( 2Q jk bjbkbi |b|6 − Qi j bj |b|4 ) , mass multipole moments

M = ∫ ρ d3x Qij = ∫ ρ ( XiXj − 1 2δi j|X|2 ) d3X trace-free quadrupole moment on the lens plane

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SLIDE 8

Effect of non-spherical distortion bending angle

αi = 4GM c2 bi |b|2 + 8G c2 ( 2Q jk bjbkbi |b|6 − Qi j bj |b|4 ) , mass multipole moments

M = ∫ ρ d3x Qij = ∫ ρ ( XiXj − 1 2δi j|X|2 ) d3X trace-free quadrupole moment on the lens plane

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SLIDE 9

Effect of non-spherical distortion bending angle

αi = 4GM c2 bi |b|2 + 8G c2 ( 2Q jk bjbkbi |b|6 − Qi j bj |b|4 ) , mass multipole moments

M = ∫ ρ d3x Qij = ∫ ρ ( XiXj − 1 2δi j|X|2 ) d3X trace-free quadrupole moment on the lens plane

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SLIDE 10

Normalization & diagonalization

Qij ⇒ ( e −e )

Lens Equation

βx = x − x x2 + y2 − e(x2 − 3y2) x (x2 + y2)3 βy = y − y x2 + y2 − e(3x2 − y2) y (x2 + y2)3

β = (βx, βy): source position θ = (x, y): image position

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SLIDE 11

Lens Equation

βx = x − x x2 + y2 − e(x2 − 3y2) x (x2 + y2)3 βy = y − y x2 + y2 − e(3x2 − y2) y (x2 + y2)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)

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SLIDE 12

Lens Equation

βx = x − x x2 + y2 − e(x2 − 3y2) x (x2 + y2)3 βy = y − y x2 + y2 − e(3x2 − y2) y (x2 + y2)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)

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SLIDE 13
  • rder of magnitude

e ∼ 10−5 ( M⊙ M ) ( R 106 km )3 (107 km RE )2 ( v 10 km s−1 )2 ≪ 1

a more practical approach

Solve perturbatively with respect to e

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SLIDE 14
  • rder of magnitude

e ∼ 10−5 ( M⊙ M ) ( R 106 km )3 (107 km RE )2 ( v 10 km s−1 )2 ≪ 1

a more practical approach

Solve perturbatively with respect to e

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SLIDE 15

zeroth-order solutions (e = 0)

Lens Equation (x2 + y2)(βx − x) + x = 0, (x2 + y2)(βy − y) + y = 0

2 images

x = x±

0 ≡ f ±βx,

y = y±

0 ≡ f ±βy f ± ≡ 1 ± √ 1 + 4β−2 2 , β = √ β2

x + β2 y

(cf. a trivial solution (x, y) = (0, 0) excluded)

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SLIDE 16

first-order solutions (0 < e ≪ 1)

˜ x± = x±

0 + x± 1 = f ±βx + e

(4β2

x − 3β2)( f ±)2 − 1

β2( f ±β2 + 1)(f ±β2 + 2)βx ˜ y± = y±

0 + y± 1 = f ±βy + e

(3β2 − 4β2

y)( f ±)2 + 1

β2(f ±β2 + 1)(f ±β2 + 2)βy

That’s all folks... ?

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SLIDE 17

first-order solutions (0 < e ≪ 1)

˜ x± = x±

0 + x± 1 = f ±βx + e

(4β2

x − 3β2)( f ±)2 − 1

β2( f ±β2 + 1)(f ±β2 + 2)βx ˜ y± = y±

0 + y± 1 = f ±βy + e

(3β2 − 4β2

y)( f ±)2 + 1

β2(f ±β2 + 1)(f ±β2 + 2)βy

That’s all folks... ?

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SLIDE 18

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?

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SLIDE 19

Perturbation around the excluded solution

˜ x = 0 + x1, ˜ y = 0 + y1 Solutions for the “minor” images ˜ x± = 1 2e βx, ˜ y± = ± √e ( 1 + 1 2e ) − 1 2e βy

Totally 4 images obtained!

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SLIDE 20

e βx βy (1) ynum (2) yappr Error |(2) − (1)| 0.01 0.0 0.2 1.10087 1.10091 3.9 × 10−5

  • 0.89881
  • 0.89891

1.0 × 19−4 0.09950 0.099500 3.7 × 10−6

  • 0.10157
  • 0.10150

6.8 × 10−5 0.01 0.0 0.5 1.27780 1.27782 1.8 × 10−5

  • 0.77262
  • 0.77282

2.0 × 10−4 0.09809 0.09800 8.5 × 10−5

  • 0.10327
  • 0.10300

2.7 × 10−4 0.02 0.0 0.5 1.27479 1.27486 7.3 × 10−5

  • 0.76402
  • 0.76486

8.4 × 10−4 0.13802 0.13784 1.8 × 10−4

  • 0.14878
  • 0.14784

9.5 × 10−4

Error ∼ O(e2), also depends on β

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SLIDE 21

Amplification factor

Perturbative solutions ˜ x(β), ˜ y(β) work well.

What about the amplification factor? can be directly calculated from A± =

  • det ∂(x±, y±)

∂(βx, βy)

  • = A±

( 1 + e ∆±(β) )

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SLIDE 22

e βx βy parity Error of A± 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).

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SLIDE 23

Better accuracy without higher-order calculations A± =

  • det ∂(x±, y±)

∂(βx, βy)

  • = A±

( 1 + e ∆±(β) )

Pad´ e approximant AP

± ≡ A±

( 1 − e ∆±(β) )−1

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SLIDE 24

Better accuracy without higher-order calculations A± =

  • det ∂(x±, y±)

∂(βx, βy)

  • = A±

( 1 + e ∆±(β) )

Pad´ e approximant

AP

± ≡ A±

( 1 − e ∆±(β) )−1

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SLIDE 25

e βx βy parity Error of A± Error ofA±

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.

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SLIDE 26

e βx βy parity Error of A± Error ofA±

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.

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SLIDE 27

Changes in image properties

image separation of two “major” images

∆x ≡ x+ − x− = ∆x0       1 −        2(β2

x − β2 y)

β2(β2 + 4) − 1        e        = ∆x0 { 1 + e × O(1) }

  • nly slightly changes

Amplification difference Adiff ≡ A+

P − A− P ≃ 1 − 2

β2 e can be significantly changed if β2 ∼ e, even if e ≪ 1

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SLIDE 28

Changes in image properties

image separation of two “major” images

∆x ≡ x+ − x− = ∆x0       1 −        2(β2

x − β2 y)

β2(β2 + 4) − 1        e        = ∆x0 { 1 + e × O(1) }

  • nly slightly changes

Amplification difference Adiff ≡ A+

P − A− P ≃ 1 − 2

β2 e can be significantly changed if β2 ∼ e, even if e ≪ 1

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SLIDE 29

“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. Adiff ≡ A+

P − A− P ≃ 1 − 2

β2 e

“anomaly” = unexpectedly large difference from a point mass case

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SLIDE 30

Summary

Perturbative approach to a non-spherically distorted gravitational lens

  • Perturbatively generated solutions from “nothing”
  • Efficient Pad´

e approximant

  • “Anomalous” flux ratio by a tiny distortion?