GRAVITATIONAL LENSING LECTURE 4 Docente: Massimo Meneghetti AA - - PowerPoint PPT Presentation

GRAVITATIONAL LENSING

LECTURE 4

Docente: Massimo Meneghetti AA 2015-2016

CONTENTS

➤ Distortion and magnification (continuation) ➤ Second order lensing: flexion ➤ Time delays

EXAMPLE: FIRST ORDER DISTORTION OF A CIRCULAR SOURCE

β2

1 + β2 2 = β2

✓ β1 β2 ◆ = ✓ 1 − κ − γ 1 − κ + γ ◆ ✓ θ1 θ2 ◆

In the reference frame where A is diagonal:

β1 = (1 − κ − γ)θ1 β2 = (1 − κ + γ)θ2

β2 = β2

1 + β2 2 = (1 − κ − γ)2θ2 1 + (1 − κ + γ)2θ2 2

This is the equation of an ellipse with semi-axes:

a = β 1 − κ − γ b = β 1 − κ + γ

EXAMPLE: FIRST ORDER DISTORTION OF A CIRCULAR SOURCE

convergence: responsible for isotropic expansion or contraction shear: responsible for anisotropic distortion Ellipticity:

e = a − b a + b = γ 1 − κ = g

EXAMPLE: FIRST ORDER DISTORTION OF A CIRCULAR SOURCE

What is the orientation of the ellipse? Let’s find the eigenvectors corresponding to the eigenvalue Eλt = N(A − λtI) = = N ✓ γ − γ1 −γ2 −γ2 γ + γ1 ◆

λt

with v1 = γ2 γ − γ1 v2 Result: After some math: ~ v = (v1, v2) = |v|(cos 0, sin 0)

cos φ0 = ± cos φ

φ0 = φ φ0 = φ + π

SHEAR DISTORTIONS

γ1 γ2

MAGNIFICATION

Kneib & Natarajan (2012)

MAGNIFICATION

dS

Kneib & Natarajan (2012)

MAGNIFICATION

dS dI

Kneib & Natarajan (2012)

MAGNIFICATION

dS dI

Kneib & Natarajan (2012)

CONSERVATION OF SURFACE BRIGHTNESS

Iν = dE dtdAdΩdν

The source surface brightness is In phase space, the radiation emitted is characterized by the density

f(~ x, ~ p, t) = dN d3xd3p

dN = dE hν = dE cp

d3x = cdtdA

d3~ p = p2dpdΩ

In absence of photon creations or absorptions, f is conserved (Liouville theorem)

f(~ x, ~ p, t) = dN d3xd3p = dE hcp3dAdtd⌫dΩ = Iν hcp3

Since GL does not involve creation or absorption of photons, neither it changes the photon momenta (achromatic!), surface brightness is conserved!

MAGNIFICATION

Kneib & Natarajan (2012)

Lensing changes the amount of photons (flux) we receive from the source by changing the solid angle the source subtends

Fν = Z

I

Iν(~ ✓)d2✓ = Z

S

IS

ν [~

(~ ✓)]µd2

MAGNIFICATION

dS

Kneib & Natarajan (2012)

Lensing changes the amount of photons (flux) we receive from the source by changing the solid angle the source subtends

Fν = Z

I

Iν(~ ✓)d2✓ = Z

S

IS

ν [~

(~ ✓)]µd2

MAGNIFICATION

dS dI

Kneib & Natarajan (2012)

Lensing changes the amount of photons (flux) we receive from the source by changing the solid angle the source subtends

Fν = Z

I

Iν(~ ✓)d2✓ = Z

S

IS

ν [~

(~ ✓)]µd2

MAGNIFICATION

dS dI

Kneib & Natarajan (2012)

Lensing changes the amount of photons (flux) we receive from the source by changing the solid angle the source subtends

Fν = Z

I

Iν(~ ✓)d2✓ = Z

S

IS

ν [~

(~ ✓)]µd2

CRITICAL LINES AND CAUSTICS

Both convergence and shear are functions of position on the lens plane:

 = (~ ✓) = (~ ✓)

The determinant of the lensing Jacobian is

det A = (1 − κ − γ)(1 − κ + γ) = µ−1

The critical lines are the lines where the eigenvalues of the Jacobian are zero: Along these lines the magnification diverges! Via the lens equations, they are mapped into the caustics… tangential critical line radial critical line

SECOND ORDER LENS EQUATION

Aij

SECOND ORDER LENS EQUATION

Aij ∂Aij ∂θk = Dijk

SECOND ORDER LENS EQUATION

Aij ∂Aij ∂θk = Dijk

SECOND ORDER LENS EQUATION

Aij ∂Aij ∂θk = Dijk