1. Studying cool planets around distant low-mass stars Planet detection by gravitational microlensing Martin Dominik Royal Society University Research Fellow SUPA, University of St Andrews, School of Physics & Astronomy

2. Deflection of light by gravity (1911) α = 2GM c2ξc2ξ bending angle suggested: measure during Solar eclipse measurable at Solar limb: α = 0. ″ 85

3. Deflection of light by gravity (1915) α = 4GM c2ξc2ξ bending angle confirmed by measurement of stellar positions during Solar eclipse (Eddington 1919) measurable at Solar limb: α = 1. ″ 7

4. Bending of starlight by stars (Gravitational microlensing) SL You are here

5. α = 4GM c2ξc2ξ bending angle I−I− I+I+ ξ η α L S DLDL DSDS α Deriving the gravitational lens equation side view O

6. c2c2 4GM DL DSDL DS DS−DLDS−DL x = (lens equation) ) 1/2 The gravitational lens equation and its solutions ξ D L θ E η D S θ E y = 1 x y = x − angular Einstein radius α = 4GM c2ξc2ξ bending angle I−I− I+I+ ξ η θ E = ( x ± = [y ± (y 2 +4) 1/2 ] 1 2 two images side view

7. y (y 2 +4) 1/2 y 2 +2 1 2 angular Einstein radius with ‘typical’ D S ~ 8.5 kpc and D L ~ 6.5 kpc θ E ~ 600 (M/M ☉ ) 1/2 as μ ) DL DSDL DS DS−DLDS−DL θ E = ( 1/2 4GM c2c2 total magnification A(y) = ∑ x±x± y dx±dx± dydy = ± Image distortion and magnification lens equation relates radial coordinates, polar angle conserved radial distortiontangential distortion x±x± y dx±dx± dydy x ± = [y ± (y 2 +4) 1/2 ] observer’s view

8. Microlensing light curves SL (t-t 0 )/t E μ ~ 15 μ as d -1 t E ~ 40 (M/M ☉ ) 1/2 days tE = θE/μtE = θE/μ tEtE t − t 0 y(t) = [ u 0 2 + ( ) 2 ] 1/2 y (y 2 +4) 1/2 y 2 +2 A(y) = A[y(t)] defined by u 0, t 0, t E

9. Notes about gravitational lensing dated to 1912 on two pages of Einstein’s scratch notebook

11. Solid angle of sky covered by N L Einstein circles NL πθE2NL πθE2 (neglect overlap) u 3/√5 ≈ 1.34 optical depth τ = probability that given source star is inside Einstein circle quantify alignment by defining corresponds to brightening in excess of 34% with mass volume density ρ (D L ) and mass spectrum f(M) f(M) dM = 1 ∫ N → f(M) D L 2 dD L d Ω dM ∫ ρ(DL)ρ(DL) M ) θ E = ( 1/2 c2c2 4GM DL DSDL DS DS−DLDS−DL for ρ = const., x = D L /D S τ = D S 2 ρ x(1-x) dx = D S 2 ρ ∫ 4πG4πG c2c2 2πG2πG 3c23c2 0 I τ ~ 6 × 10 -7 35% disk, 65% bulge Galactic disk ρ = 0.1 (pc) 3 M☉M☉ D S ~ 8 kpc, Microlensing optical depth τ ~ 2 × 10 -6 (Galactic disk) (total)

12. Microlensing event rate Γ = N S = 2 × 10 -5 N S (yr) -1 τ event time-scale t E ~ 20 days π 2 = t E average duration of ‘event’ with A > 1.34 (unit circle: area π, width 2, average length ) π 2 1936: no computers 1965: computers not powerful enough Microlensing surveys are a major data processing venture for 1000 events per year monitor N S ~ 5 × 10 7 stars ~ 85 ongoing events at any time (Note: target observability shows seasonal variation) during t E, source star moves on the sky by θ E event rate

13. First reported microlensing event MACHO LMC#1 Nature 365, 621 (October 1993)

14. Optical Gravitational Lensing Experiment 1.3m Warsaw Telescope, Las Campanas (Chile) 1.8m MOA Telescope, Mt John (New Zealand) τ ~ 10 -6 for microlensing event → ~1000 events alerted per year daily monitor ≳ 100 million stars, Current microlensing surveys (2007)

15. foreground object occults the background object foreground object occults the lensing images of the background object Lensing or eclipse ? Prediction from data prior to first caustic peak Text Lensing regime: D L /D S ≈ 1/2, region broadens with increasing D S Eclipse regime: D S − D L ≫ D L or D S ≫ D L eclipsing planets around observed (source) stars microlensing planets around lens stars eclipsing stellar binaries Consequences: Condition for eclipse: θ ≪ θ E : lensing (brightening) θ ≫ θ E : eclipse (dimming) ) DL DSDL DS DS−DLDS−DL θ E = ( 1/2 4GM c2c2 vs. angular Einstein radius angular radius θ = R L /D L of intervening object

16. Which host stars? Microlensing detects planets around lens stars Host stars are selected by means of chance alignment Their mass distribution is related to stellar mass function Stellar mass probed by microlensing Microlensing prefers detecting planets around red-dwarf stars

17. Multiple point-mass lens y = x − x |x|2|x|2 lens equation (2D) for single point-mass lens in weak-field limit, superposition of deflection terms, but not of light curves θ θEθE x = β θEθE y = two-dimensional position angles β (source) and θ (image) angular Einstein radius ) DL DSDL DS DS−DLDS−DL θ E = ( 1/2 4GM c2c2 total mass M, i-th object with mass fraction m i at x (i), ∑ m i = 1 |x − x (i) | 2 x − x (i) y = x − ∑ m i

18. The binary point-mass lens solving for (x 1,x 2 ) leads to 5th-order complex polynomial angular separation δ = d θ E q = m 2 /m 1 m 2 = 1 − m 1 mass ratio completely characterized by two dimensionless parameters (d,q) ( ) in centre-of-mass system: x (1) = d, 0 x (2) = − d, 0 q 1+q 1 1 q y 1 = x 1 − − q 1+q x 1 − d q 1+q x 1 − d+ x 2 2 ( ) 2 1 1+q x 1 + d 1 1+q x 1 + d + x 2 2 2 y 2 = x 2 − − 1 1+q q x2x2 q x 1 − d+ x 2 2 ( ) 2 1 1+q x 1 + d x2x2 + x 2 2 2 m 1 = q 1+q 1 m 2 = time-scale of planetary deviations ≪ orbital period either 3 or 5 images

19. Magnification and caustics 3 topologies for a binary lens “resonance” at d ~ 1 point lens: critical points at x = 1 (Einstein circle), map to y = 0 (point caustic at position of lens) caustics { y c = y(x c ) | } ∂y ∂x (x c ) = 0 det ( ) A(y) = ∑ | | -1 magnificatio n ∂y ∂x (xj)(xj) det ( ) j A(y) → ∞ ∂y ∂x for det ( ) (x j ) = 0for at least one j

20. Planetary-regime binary-lens caustics and excess magnification q = 10 -2 red: caustics for q ≪ 1: two-scale problem lens star produces 2 images if planet close to one of these, it creates a further split maximal effect if lens star creates image near the planet tidal field of the star creates extended ‘planetary caustic’ there is also a ‘central caustic’ close to the lens star for d ~ 1, both merge into a single caustic d [1] planet-star separation in units of stellar angular Einstein radius D = d [1] − d [1] 1 inside Einstein circle (|D| < 1) for (√5−1)/2 < d [1] < (√5+1)/2 position of planetary caustic (fulfills lens equation) (“lensing zone”) blue shades: dimmer green shades: brighter shading levels: more than 1%, 2%, 5%, 10%

21. Caustics and excess magnification (II) q = 10 -2 q = 10 -3

22. q = 10 -2 q = 10 -3 q = 10 -4 Caustics and excess magnification (III) Detection efficiency light curve determined by 1D cut through 2D magnification map detection efficiency ε (d,q; u 0,t 0,t E ) = P ( detectable signal in event (u 0,t 0,t E ) | planet (d,q) ) for event with (u 0,t 0,t E ), only a fraction of all trajectory angles lead to detectable signal

23. q = 10 -2 q = 10 -3 q = 10 -4 Planetary deviations t E = 20 d wideclose ρ ✶ = θ ✶ / θ E = 0.025 (giant R ✶ ~ 15 R ☉ ) [d]

24. Signal amplitude limited by finite angular source size θ ✶ Planetary signals For point-like sources, both signal duration and probability scale with this factor Linear size of deviation regions scale with q 1/2 (planetary caustic) or q (central caustic) t E = 20 d planetary causticcentral caustic main-sequence star R ✶ ~ 1 R ☉ vs giant R ✶ ~ 15 R ☉