The magnification theorem Olaf Wucknitz wucknitz@astro.uni-bonn.de Three decades of gravitational lenses, JENAM, 21 April 2009
2.5 decades of the magnification theorem General theorems in lensing odd image theorem even image corollary cusp relation flux ratio anomalies lensing is achromatic chromatic microlensing magnification theorem? titlepage introduction summary contents bonus back forward previous next fullscreen 1
Light deflection Naive Newtonian calculation: α = dz dl = 1 c dl c Φ Φ = GM R α Soldner 1801 (Newton) Einstein 1915 (general relativity) α = 2 G c 2 M r α = 4 G c 2 M r titlepage introduction summary contents bonus back forward previous next fullscreen 2
Distortion/magnification source lensed M θ s θ lens equation θ s = θ α(θ) first derivative dθ s = dθ α θ dθ = M 1 dθ magnification / mapping matrix M(θ) = ( 1 α ) 1 θ (area) magnification = amplification µ = ± det M titlepage introduction summary contents bonus back forward previous next fullscreen 3
Magnification amplification observer s view observer Ω lens source Ω s solid angles measure apparent size magnification = Ω Ω s source s view _ Ω s observer lens _ Ω source flux distributed over solid angle amplification = Ω Ω s titlepage introduction summary contents bonus back forward previous next fullscreen 4
Example: point-mass 4 µ + µ 2 0-2 -4-2 -1.5-1 -0.5 0 0.5 1 1.5 2 θ s [ Wambsganss (1998), Liv. Rev. Rel. 1, 12 ] titlepage introduction summary contents bonus back forward previous next fullscreen 5
I ve seen this before... [ Einstein (1936) ] [ see also Refsdal (1964) ] [ Einstein notebooks 1910 1912 ] titlepage introduction summary contents bonus back forward previous next fullscreen 6
Potential, light travel time potential ψ α(θ) = ψ(θ) Poisson equation 2 ψ(θ) =: 2κ(θ) = 2σ = 2 Σ Σ c light-travel time for virtual ray (θ s fixed) Fermat-potential φ(θ) = (θ θ s) 2 Fermat s principle: real images are minima, (e.g. unperturbed image) maxima, or saddle-points of φ t = D dd s c D ds φ(θ) 2 ψ(θ) titlepage introduction summary contents bonus back forward previous next fullscreen 7
Magnification theorem [ Schneider (1984) ] Hessian of Fermat-potential is inverse magnification matrix µ 1 = 1 α θ = 2 φ θ 2 = 1 κ γ x γ y γ y 1 κ + γ x diagonalise: rotate shear, γ = γ x + γ y µ 1 = 1 κ γ 0 0 1 κ + γ minimum: both eigenvalues positive Poisson: convergence κ = σ 0 sum: 2(1 κ) > 0 κ < 1 0 κ < 1 µ 1 = (1 κ) 2 γ 2 0 < µ 1 1 titlepage introduction summary contents bonus back forward previous next fullscreen 8
An apparent paradox source in centre amplif. 1 in all directions integrate over sphere total flux amplification! conservation of energy? solution to energy crisis? lensing cannot create photons titlepage introduction summary contents bonus back forward previous next fullscreen 9
The standard explanation lens distorts geometry area of surface shrinks! have to compare with same mean geometry compare with same mean density in Universe [ Weinberg (1976) ] titlepage introduction summary contents bonus back forward previous next fullscreen 10
Bad excuse, because... equivalent: refraction or Newtonian deflection does not change geometry same formalism same paradox! so far: tangential plane no problem in the plane now: do it on the sphere! titlepage introduction summary contents bonus back forward previous next fullscreen 11
Deflection angle for the sphere calculation for D s α = 2GM c 2 x 0 0 dz [x 2 0+(z z 0 ) 2] 3/2 α = m 2 cot θ 2 m θ 3 2 m/θ (m/2) cot (θ/2) α θ s α M D d θ z x 1 0 0 π/2 θ π [ compare Avni & Shulami (1988) and Jaroszynski & Paczynski (1996) ] titlepage introduction summary contents bonus back forward previous next fullscreen 12
Magnification for the sphere for point-mass µ 1 = m (1 ) 2 = 2.35 10 11 1 α θ = 1 + m 2 m2 θ 4 [ 1 + O ( θ 2)] planar approximation µ 1 = 1 m2 θ 4 10 µ tot µ + µ tot 1 µ + 1 3 2 µ µ 1 1 0-1 -2 [10 11 ] 0.1 1 10 θ s [arcsec] 0 500 1000 1500 2000 2500 θ s [arcsec] -3 titlepage introduction summary contents bonus back forward previous next fullscreen 13
Lensing on the sphere far from optical axis: µ < 1 in this situation: magnification theorem not valid integration over sphere: mean µ is 1 no paradox modified Poisson equation [ ] 2 ψ(θ) =: 2κ(θ) = 2 σ(θ) σ not always κ 0 failure of theorem field lines decay! titlepage introduction summary contents bonus back forward previous next fullscreen 14
Field lines on the sphere FIELD LINES titlepage introduction summary contents bonus back forward previous next fullscreen 15
Back to gravitation and summary short summary flat spacetime with refractive medium (or Newtonian) magnification theorem not valid modified Poisson equation equivalent: gravity with appropriate reference situation constant area of sphere magnification theorem invalid, no paradox advantage: magnification is function of deflection not equivalent: inappropriate reference situation constant affine distance, area not constant focusing theorem valid, apparent paradox planar proof does still not hold for sphere (e.g. mass shell) [ Wucknitz (2008), MNRAS 386, 230 ] titlepage introduction summary contents bonus back forward previous next fullscreen 16
Bonus: From deflection to magnification magnification on the plane derivative of deflection angle constant deflection not relevant magnification on the sphere derivatives still relevant constant deflection too! parallel transport rigid rotation titlepage introduction summary contents bonus back forward previous next fullscreen 17
Exact magnification on the sphere (1) lens equation Θ µ s? = Θ µ a µ (Θ) is not a vector equation (for finite a)! alternative description start at Θ µ move in the direction of a µ along a geodesic for a total distance of a geodesic equation ẍ α + Γ α µν ẋ µ ẋ ν = 0 affine parameter λ from 0 (Θ) to 1 (Θ s ) titlepage introduction summary contents bonus back forward previous next fullscreen 18
Exact magnification on the sphere (2) start x µ (0) = Θ µ ẋ µ (0) = a µ end x µ (1) = Θ µ s slightly displaced start: geodesic deviation D 2 ξ α Dλ 2 = ẋβ ẋ µ ξ ν R α µβν curvature tensor R α µβν := Γ α µν,β Γ α µβ,ν + Γ α ρβγ ρ µν Γ α ρνγ ρ µβ = 1 K 2 ( δ α β g µν δ α ν g µβ ) titlepage introduction summary contents bonus back forward previous next fullscreen 19
Exact magnification on the sphere (3) coordinates and to a µ result for geodesic deviation ξ(1) = M 1 ξ(0) magnification matrix M 1 = a ; 1 a ; a ; sin(a/k) a/k 1 a ; a ; a ; 1 a ; cos(a/k) a ; sin(a/k) a/k titlepage introduction summary contents bonus back forward previous next fullscreen 20
Affine distance, light travel time metric (c = 1) ds 2 = (1 + 2Ψ) dt 2 (1 2Ψ) dx 2 affine distance, light travel time: measured at observer s position light travel time: T = (1 + Ψ 0 ) dx (1 2Ψ) affine distance: L = (1 Ψ 0 ) dx general focusing theorem: µ > 1 for constant affine distance titlepage introduction summary contents bonus back forward previous next fullscreen 21
Lensing by a spherical shell shell of radius (metric) r 0 with mass M σ GM/c2 r 0 for σ 1 limit r 0, M 0 with σ = const no change of global geometry unlensed situation (re-)move sphere or... affine distance Λ = r 1 2σ constant focusing theorem: compare with constant Λ focusing titlepage introduction summary contents bonus back forward previous next fullscreen 22
Contents 1 The magnification theorem 2 Light deflection 3 Distortion/magnification 4 Magnification amplification 5 Example: point-mass 6 I ve seen this before... 7 Potential, light travel time 8 Magnification theorem 9 An apparent paradox 10 The standard explanation 11 Bad excuse, because... 12 Deflection angle for the sphere 13 Magnification for the sphere 14 Lensing on the sphere 15 Field lines on the sphere 16 Back to gravitation and summary 17 Bonus: From deflection to magnification titlepage introduction summary contents bonus back forward previous next fullscreen 23
18 Exact magnification on the sphere (1) 19 Exact magnification on the sphere (2) 20 Exact magnification on the sphere (3) 21 Affine distance, light travel time 22 Lensing by a spherical shell 23 Contents titlepage introduction summary contents bonus back forward previous next fullscreen 24