GRAVITATIONAL LENSING LECTURE 2 Docente: Massimo Meneghetti AA - PowerPoint PPT Presentation

GRAVITATIONAL LENSING LECTURE 2 Docente: Massimo Meneghetti AA 2015-2016

CONTENTS ➤ Lens equation ➤ Lensing potential

From last lecture DEFLECTION OF LIGHT IN GENERAL RELATIVITY How to define the effective diffraction index?

DEFLECTION OF LIGHT IN GENERAL RELATIVITY

SCHWARZSCHILD METRIC ◆ − 1 ✓ 1 − 2 GM ◆ ✓ 1 − 2 GM ds 2 = dR 2 − R 2 (sin 2 θ d φ 2 + d θ 2 ) c 2 dt 2 − Rc 2 Rc 2 x = r sin θ cos φ r 1 + 2 GM y = r sin θ sin φ R = rc 2 r z = r cos φ dl 2 = [ dr 2 + r 2 (sin 2 θ d φ 2 + d θ 2 )] In the weak field limit: ◆ − 1 ✓ 1 − 2 GM 1 + 2 GM ✓ ◆ 1 − 2 GM 1 − 2 GM 1 = ≈ Rc 2 c 2 R Rc 2 c 2 r q 1 + 2 GM c 2 r 1 + 2 GM 1 = ✓ ◆ 1 − 2 GM 1 − GM c 2 r q 1 + 2 GM ≈ c 2 r c 2 r c 2 r ✓ ◆ 1 + 2 GM 1 − GM 1 − 2 GM ≈ ≈ c 2 r c 2 r c 2 r 1 + 2 GM ≈ c 2 r

SCHWARZSCHILD METRIC IN THE WEAK FIELD LIMIT ◆ − 1 ✓ 1 − 2 GM ◆ ✓ 1 − 2 GM ds 2 = dR 2 − R 2 (sin 2 θ d φ 2 + d θ 2 ) c 2 dt 2 − Rc 2 Rc 2 ✓ ◆ ✓ ◆ 1 − 2 GM 1 + 2 GM ds 2 = c 2 dt 2 − dl 2 rc 2 rc 2 Φ = − GM r

DEFLECTION OF LIGHT BY A BLACK HOLE suggested reading: http://arxiv.org/pdf/0911.2187v2.pdf Generic static spherically symmetric metric: u=impact parameter r m =minimum distance between the photon and the BH

DEFLECTION OF LIGHT BY A BLACK HOLE for the Schwarzschild metric 3 GM √ 3 c 2

DEFLECTION ANGLE OF A POINT MASS Z + ∞ ↵ = 2 ~ ˆ ~ r ⊥ Φ dz c 2 −∞ Φ = − GM r

AGAIN ON THE EDDINGTON EXPEDITION ➤ The goal of Eddington expeditions was to measure a shift in the position of the Hyades stars due to solar deflection ➤ What is the exact shift we should expect to measure? ➤ What is the relation between the intrinsic and the apparent positions of the stars?

LENS EQUATION L S

COMOVING DISTANCE suggested reading: http://arxiv.org/pdf/astro-ph/9905116v4.pdf comoving distance (along the line of sight) = distance between two points which remains constant if the two points are moving with the Hubble flow proper distance = distance between the two points measured by rulers at the time they are being observed D c = D pr (1 + z )

ANGULAR DIAMETER DISTANCE A(z) B(z) D M δθ = comoving transversal distance D M δθ O = angular diameter distance = ratio of the physical (proper) transverse size to its angular size

LENS EQUATION L S β = θ − D LS α ˆ D S

LENS EQUATION L S β = θ − D LS α ˆ D S

LENS EQUATION L S β = θ − D LS α ˆ D S

LENS EQUATION L S β = θ − D LS α ˆ D S

LENS EQUATION L S β = θ − D LS α ˆ D S

DEFLECTION BY EXTENDED LENSES Z + ∞ ↵ = 2 ➤ Remaining in the weak field ~ ˆ ~ r ⊥ Φ dz c 2 −∞ limit, one can use the superposition principle ➤ The deflection angle by a system of point masses is the vectorial sum of the deflection angles of the single lenses ➤ This can be easily generalized to the case of a continuum distribution of mass ➤ Assumption: thin screen approximation

DEFLECTION ANGLE OF AN AXIALLY SYMMETRIC LENS

DEFLECTION ANGLE OF AN AXIALLY SYMMETRIC LENS 2 π ξ 0 < ξ ξ

LENS EQUATION ✓ − D LS ˆ � = ~ ~ ↵ ( ~ ✓ ) ~ D S ~ ⌘ ~ ⇠ ~ ~ � = ✓ = D S D L ✓ ) = D LS ˆ ↵ ( ~ ↵ ( ~ ✓ ) ~ ~ D S � = ~ ~ ✓ − ~ ↵

OTHER NOTATIONS ~ ~ ⌘ ✓ ) = D LS ⇠ ˆ ↵ ( ~ ↵ ( ~ ~ � = ~ ~ ~ ✓ ) � = ~ ~ ✓ − ~ ↵ ✓ = D S D S D L θ 0 = ξ 0 = η 0 D L D S ↵ ( ✓ ) x ) = ~ = D L D LS ˆ ↵ ( ~ y = ~ ↵ ( ~ x ) ↵ ( ~ ✓ ) ~ x − ~ ~ ~ ✓ 0 ⇠ 0 D S

LENSING POTENTIAL This formula tells us that the deflection is caused Z + ∞ ↵ = 2 by the projection of the Newtonian gravitational ~ ˆ ~ r ⊥ Φ dz c 2 potential on the lens plane. −∞ We introduce the effective lensing potential

LENSING POTENTIAL This formula tells us that the deflection is caused Z + ∞ ↵ = 2 by the projection of the Newtonian gravitational ~ ˆ ~ r ⊥ Φ dz c 2 potential on the lens plane. −∞ We introduce the effective lensing potential the lensing potential is the projection of the 3D 1 potential

LENSING POTENTIAL This formula tells us that the deflection is caused Z + ∞ ↵ = 2 by the projection of the Newtonian gravitational ~ ˆ ~ r ⊥ Φ dz c 2 potential on the lens plane. −∞ We introduce the effective lensing potential the lensing potential is the projection of the 3D 1 potential 2 the lensing potential scales with distances

OTHER PROPERTIES OF THE LENSING POTENTIAL The deflection angle is the gradient of the lensing potential The laplacian of the lensing potential is twice the convergence

OTHER PROPERTIES OF THE LENSING POTENTIAL The deflection angle is the gradient of the lensing potential The laplacian of the lensing potential is twice the convergence