GRAVITATIONAL LENSING LECTURE 11 Docente: Massimo Meneghetti AA - - PowerPoint PPT Presentation

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GRAVITATIONAL LENSING LECTURE 11 Docente: Massimo Meneghetti AA - - PowerPoint PPT Presentation

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GRAVITATIONAL LENSING LECTURE 11 Docente: Massimo Meneghetti AA 2015-2016 TODAYS LECTURE Lensing by multiple point masses Binary lenses COMPLEX LENS EQUATION N For a system of N-lenses we obtained: m i X z s = z z z

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GRAVITATIONAL LENSING

LECTURE 11

Docente: Massimo Meneghetti AA 2015-2016

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TODAY’S LECTURE

➤ Lensing by multiple point masses ➤ Binary lenses

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COMPLEX LENS EQUATION

➤ For a system of N-lenses we obtained: ➤ Taking the conjugate: ➤ We obtain z* and substitute it back

into the original equation, which results in a (N2+1)th order complex polynomial equation

➤ This equation can be solved only

numerically, even in the case of a binary lens

zs = z −

N

X

i=1

mi z∗ − z∗

i

z∗

s = z∗ − N

X

i=1

mi z − zi

p(z) =

N

X

i=0

cizi

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COMPLEX LENS EQUATION

➤ Note that the solutions are not necessarily solutions of the

lens equations (spurious solutions)

➤ One has to check if the solutions are solutions of the lens

equation

➤ Rhie 2001,2003: maximum number of images is 5(N-1) for

N>2

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MAGNIFICATION

➤ In the complex form, the magnification can still be derived

from the lensing Jacobian:

det A = ✓∂zs ∂z ◆2 − ∂zs ∂z∗ ✓ ∂zs ∂z∗ ◆∗ = 1 − ∂zs ∂z∗ ✓ ∂zs ∂z∗ ◆∗ ∂zs ∂z∗ =

N

X

i=1

mi (z∗ − z∗

i )2

det A = 1 −

  • N

X

i=1

mi (z∗ − z∗

i )2

  • 2
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CRITICAL LINES AND CAUSTICS

➤ Therefore the critical lines form where

  • N

X

i=1

mi (z∗ − z∗

i )2

  • 2

= 1

➤ Thus, to find the critical points we solve

N

X

i=1

mi (z∗ − z∗

i )2 = eiφ

φ ∈ [0, 2π]

➤ Again, this can be turned into a complex polynomial of order

2N: for N lenses, there are 2N critical lines and caustics. The solutions can be found numerically.

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CRITICAL LINES AND CAUSTICS

Witt, 1990, A&A, 236, 311 critical lines and caustics originated by 400 stars

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BINARY LENSES

➤ determinant of the Jacobian: ➤ condition for critical points: ➤ resulting fourth grade polynomial: ➤ Lens equation:

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BINARY LENSES: TWO LENSES WITH THE SAME MASS (Q=1) AND VARYING DISTANCE

critical lines caustics

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BINARY LENSES: TWO LENSES WITH THE SAME MASS (Q=1) AND VARYING DISTANCE

critical lines caustics

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BINARY LENSES: TWO LENSES WITH THE SAME MASS (Q=1) AND VARYING DISTANCE

critical lines caustics

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BINARY LENSES: TWO LENSES WITH THE SAME MASS (Q=1) AND VARYING DISTANCE

critical lines caustics

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BINARY LENSES: TWO LENSES WITH THE SAME MASS (Q=1) AND VARYING DISTANCE

critical lines caustics

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BINARY LENSES: TWO LENSES WITH THE SAME MASS (Q=1) AND VARYING DISTANCE

critical lines caustics

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BINARY LENSES: TWO LENSES WITH THE SAME MASS (Q=1) AND VARYING DISTANCE

critical lines caustics

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BINARY LENSES: TWO LENSES WITH THE SAME MASS (Q=1) AND VARYING DISTANCE

critical lines caustics

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BINARY LENSES: TWO LENSES WITH THE SAME MASS (Q=1) AND VARYING DISTANCE

critical lines caustics

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BINARY LENSES: TWO LENSES WITH THE SAME MASS (Q=1) AND VARYING DISTANCE

critical lines caustics

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BINARY LENSES: TWO LENSES WITH THE SAME MASS (Q=1) AND VARYING DISTANCE

critical lines caustics

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BINARY LENSES: TWO LENSES WITH THE SAME MASS (Q=1) AND VARYING DISTANCE

critical lines caustics

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BINARY LENSES: TWO LENSES WITH THE SAME MASS (Q=1) AND VARYING DISTANCE

critical lines caustics

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BINARY LENSES: TWO LENSES WITH THE VARYING MASS AND FIXED DISTANCE

critical lines caustics

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BINARY LENSES: TWO LENSES WITH THE VARYING MASS AND FIXED DISTANCE

critical lines caustics

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BINARY LENSES: TWO LENSES WITH THE VARYING MASS AND FIXED DISTANCE

critical lines caustics

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BINARY LENSES: TWO LENSES WITH THE VARYING MASS AND FIXED DISTANCE

critical lines caustics

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BINARY LENSES: TWO LENSES WITH THE VARYING MASS AND FIXED DISTANCE

critical lines caustics

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BINARY LENSES: TWO LENSES WITH THE VARYING MASS AND FIXED DISTANCE

critical lines caustics

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BINARY LENSES: TWO LENSES WITH THE VARYING MASS AND FIXED DISTANCE

critical lines caustics

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BINARY LENSES: TWO LENSES WITH THE VARYING MASS AND FIXED DISTANCE

critical lines caustics

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BINARY LENSES: TWO LENSES WITH THE VARYING MASS AND FIXED DISTANCE

critical lines caustics

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BINARY LENSES: TWO LENSES WITH THE VARYING MASS AND FIXED DISTANCE

critical lines caustics

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BINARY LENSES: TWO LENSES WITH THE VARYING MASS AND FIXED DISTANCE

critical lines caustics

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BINARY LENSES: TWO LENSES WITH THE VARYING MASS AND FIXED DISTANCE

critical lines caustics

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BINARY LENSES: TWO LENSES WITH THE VARYING MASS AND FIXED DISTANCE

critical lines caustics

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BINARY LENSES: TWO LENSES WITH THE VARYING MASS AND FIXED DISTANCE

critical lines caustics

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BINARY LENSES: TWO LENSES WITH THE VARYING MASS AND FIXED DISTANCE

critical lines caustics

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BINARY LENSES: TWO LENSES WITH THE VARYING MASS AND FIXED DISTANCE

critical lines caustics

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BINARY LENSES: TWO LENSES WITH THE VARYING MASS AND FIXED DISTANCE

critical lines caustics

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BINARY LENSES: TWO LENSES WITH THE VARYING MASS AND FIXED DISTANCE

critical lines caustics

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BINARY LENSES: TWO LENSES WITH THE VARYING MASS AND FIXED DISTANCE

critical lines caustics

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BINARY LENSES: TWO LENSES WITH THE VARYING MASS AND FIXED DISTANCE

critical lines caustics

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BINARY LENSES: TWO LENSES WITH THE VARYING MASS AND FIXED DISTANCE

critical lines caustics

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BINARY LENSES: TWO LENSES WITH THE VARYING MASS AND FIXED DISTANCE

critical lines caustics

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BINARY LENSES: TWO LENSES WITH THE VARYING MASS AND FIXED DISTANCE

critical lines caustics

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BINARY LENSES: TWO LENSES WITH THE VARYING MASS AND FIXED DISTANCE

critical lines caustics

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BINARY LENSES: TOPOLOGY CLASSIFICATION

wide

separate 4- cusp caustics

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BINARY LENSES: TOPOLOGY CLASSIFICATION

intermediate

single 6-cusp caustic

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BINARY LENSES: TOPOLOGY CLASSIFICATION

close

single 4-cusp caustic two triangular caustics

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TRANSITIONS

Touching critical lines

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MULTIPLE IMAGES

➤ Lens equation: ➤ complex polynomial:

p5(z) =

5

X

i=0

cizi

➤ 3 or 5 images

Witt & Mao, 1995, ApJ, 447, L105

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MULTIPLE IMAGES

Mollerach & Roulet, “Gravitational Lensing and Microlensing”

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IMAGE MAGNIFICATION

➤ magnification at the

image position:

➤ total magnification: ➤ of course, the magnification varies as a function of z…