1. Chapter 5 – Microlensing PHY6795O – Chapitres Choisis en Astrophysique Naines Brunes et Exoplanètes

2. Contents 5.1 Introduction 5.2 Description 5.3 Caustic and critical curves 5.4 Other light curve effects 5.5 Microlens parallax and lens mass 5.6 Astrometric microlensing 5.7 Other configurations 5.8 Microlensing observations in practice 5.9 Exoplanet results 5.10 Summary of limitations and strengths 5.11 Future developments 5. MicrolensingPHY6795O – Naines brunes et Exoplanètes2

3. 5.1 Introduction  Principle  Presence of matter distorts space-time. Light path is deflected as a result. Gravity acts a lens.  Light from background source can be amplified by a foreground (lens) source.  34 planets detected by this technique so far (as of February 2015) 5. MicrolensingPHY6795O – Naines brunes et Exoplanètes3

4. 5.1 Introduction (2)  Strong lensing  Effects discernable at an individual object level.  Macrolensing : multiple resolved images, arcs, distorted/amplified sources.  Microlensing : discrete multiple images are unresolved. Relevant for exoplanet detection. Requires extremely precise alignment of observer, source and lens to within the angular Einstein radius, or ~1 mas. Primary lens mass is of order 1 M . Intensity variation of primary lens time scale: several weeks Secondary lens effect due to planet: several hours. Very low probability of event. Requires 100s millions sources to be monitored simultaneously. 5. MicrolensingPHY6795O – Naines brunes et Exoplanètes4

5. 5.1 Introduction (3)  Weak lensing  Effects discernable only in a statistical sense. Applies to a large ensemble of source (galaxies). 5. MicrolensingPHY6795O – Naines brunes et Exoplanètes5

6. Contents 5.1 Introduction 5.2 Description 5.3 Caustic and critical curves 5.4 Other light curve effects 5.5 Microlens parallax and lens mass 5.6 Astrometric microlensing 5.7 Other configurations 5.8 Microlensing observations in practice 5.9 Exoplanet results 5.10 Summary of limitations and strengths 5.11 Future developments 5. MicrolensingPHY6795O – Naines brunes et Exoplanètes6

7. 5.2 Description (1)  Light bending  Let a lensing mass M L with impact parameter b. The deflection angle α GR is on condition that, where R S is the Schwarzchild radius We have 5. MicrolensingPHY6795O – Naines brunes et Exoplanètes7 (5.1) (5.2)

8. 5.2 Description (2)  Angle between source and lens  Two solutions to this quadratic equation. 5. MicrolensingPHY6795O – Naines brunes et Exoplanètes8 (5.3) (5.4)

9. 5.2 Description (3)  Einstein radius (angular and linear) Then equation 5.4 can be written with two solutions The angular separation between the two images is 5. MicrolensingPHY6795O – Naines brunes et Exoplanètes9 (5.5) (5.6) (5.7) (5.8) The two sources are separated by if. Rotationally symmetric configuration  Einstein ring.

10. 5.2 Description (4) 5. MicrolensingPHY6795O – Naines brunes et Exoplanètes10

11. 5.2 Description (5)  The Einstein radius in relevant numerical quantities For a typical lens mass of ~ 1 M  half way to the Galactic center (~8 kpc) where most of background sources are located, the Einstein radius is ~ 1 mas. In linear scale, the Einstein is ~ 4 AU, coincidently similar to the orbital radius of planets in the solar system. This is particularly fortuitous for probing exoplanets. 5. MicrolensingPHY6795O – Naines brunes et Exoplanètes11 (5.10) (5.11)

12. 5.2 Description (6) Magnification Equation 5.7 can be written in unit of the Einstein radius Equation 5.8 becomes 5. MicrolensingPHY6795O – Naines brunes et Exoplanètes12 (5.7) While brightening occurs through light deflection, surface brightness is constant. The magnification is the ratio of the image/source area. For a very small source, we can consider a thin source annulus with angle Δ ϕ. For a point lens, this thin annulus will be mapped into two annuli, one inside the Einstein radius and one outside. (5.13) Images of a thin annulus from by a point lens on the plane of the sky. The dashed line is the Einstein ring. is the angle subtended by the thin annulus.

13. 5.2 Description (7) Magnification The area of the source annulus is given by the product of the radial width and the tangential length. Similarly, each image area is, and the magnification is given by Then, we have The total magnification is for and for. For a perfect alignment, A diverge to infinity but it is limited in practice since the source has a finite size. The highest magnification reported is ~ 3000 (Dong et al. 2006). 5. MicrolensingPHY6795O – Naines brunes et Exoplanètes13 (5.12) (5.14) (5.15)

14. 5.2 Description (8) Einstein crossing time The total magnification varies as a function of time due to the relative transverse motion between source, lens and observer. For a given relative transverse velocity between source and lens,, a typical scale for a lensing event is given by the Einstein radius crossing time One can also express the Einstein crossing time with the (unknown) lens- source relative proper motion For a source in the Galactic bulge at ~ 8 kpc, a lend mass of ~ 1 M  half way to the source, the Einstein time scale for the microlensing event is ~35 days. 5. MicrolensingPHY6795O – Naines brunes et Exoplanètes14 (5.18) (5.19)

15. 5.2 Description (9) Einstein crossing time 5. MicrolensingPHY6795O – Naines brunes et Exoplanètes15 (5.20)

16. Contents 5.1 Introduction 5.2 Description 5.3 Caustic and critical curves 5.4 Other light curve effects 5.5 Microlens parallax and lens mass 5.6 Astrometric microlensing 5.7 Other configurations 5.8 Microlensing observations in practice 5.9 Exoplanet results 5.10 Summary of limitations and strengths 5.11 Future developments 5. MicrolensingPHY6795O – Naines brunes et Exoplanètes16

17. 5.3 Caustics and critical curves (1)  Critical curves are regions in the lens plane where the magnification is infinite. Corresponding regions in the source plane as mapped by the lens equation are termed caustics.  For a single point lens, the caustic is the single point behind and the critical curves (positions of the images of these caustics) is the Einstein ring.  High-magnification microlensing events occur when the source comes near to a caustic.  For a binary lens, a star and an orbiting planet, the caustics and shapes can be formulated in terms of the planet/star ratio q = M p /M ★, the angular star-planet separation d in units of θ E and α, the angle of the source trajectory relative to te binary axis.  The planet induces a secondary peak in the smooth light curve. For Earth-mass planets, the time scale of the secondary microlensing event is 3-5 hours. 5. MicrolensingPHY6795O – Naines brunes et Exoplanètes17

18. 5.3 Caustics and critical curves (2) Position of the planetary caustics  Close to primary mass lens since planet mass is much smaller.  Planet has large effect when its position is near one of the images.  Let x c be the position of the caustics and d the star-planet separation in the lens plane, then the lens equation is written  Inverting this equation yields  This equation is invariant in 1/d, i.e. two star-planet separation ( d, - 1/d ) yields the same caustic position. 5. MicrolensingPHY6795O – Naines brunes et Exoplanètes18 (5.21)

19. 5.3 Caustics and critical curves (3) Light curve classification 5. MicrolensingPHY6795O – Naines brunes et Exoplanètes19 Source trajectory through magnification region of the planet. Source trajectory through magnification regions of both primary and planet. Source trajectory very close to the primary. R E refers to the primary.

20. Contents 5.1 Introduction 5.2 Description 5.3 Caustic and critical curves 5.4 Other light curve effects 5.5 Microlens parallax and lens mass 5.6 Astrometric microlensing 5.7 Other configurations 5.8 Microlensing observations in practice 5.9 Exoplanet results 5.10 Summary of limitations and strengths 5.11 Future developments 5. MicrolensingPHY6795O – Naines brunes et Exoplanètes20

21. 5.4 Other light curve effects (1)  Six microlensing parameters to fit: Other effects  Finite source size θ ★. Add a 7th parameter: 5. MicrolensingPHY6795O – Naines brunes et Exoplanètes21

22. 5.4 Other light curve effects (2)  Limb darkening and star spots  Orbital motion of a binary lens  Three cases: rotating binary lens, rotating binary source and a rotating oberver (Earth orbiting the Sun)  Most dramatic effects for a rotating binary lens.  First claimed planetary microlensing event MACHO-97-BLG-41 was revised with an improved fit involving a binary with a period of 1.5 yr.  Orbital motion of a star-planet lens  Effect seen in the data for the outer planet in the first multiple planetary microlensing event, OGLE-2006-BLG-109L (Gaudi et al. 2008) 5. MicrolensingPHY6795O – Naines brunes et Exoplanètes22

23. 5.4 Other light curve effects (3)  Blending  Due to a physical companion of the source star, lens itself or superposition of another (non-lensing) object along line-of-sight. Separation between light from source and lens is possible once angular separation has increased by a few mas, several years after the lens event. Difference in color between the source and the lens will result in a small displacement of the centroid with wavelength. Yields an estimate of the host star spectral type and, with assumption on underlying stellar population, D L, hence a complete solution to lens equation.  Relative source-lense transverse motion  Cannot be determined from light curve but possible by measuring small change in the elongation of the image,.e.g. with stable PSF from HST. Yield the relative angular proper motion μ LS, hence the Einstein radius from equation 5.19. 5. MicrolensingPHY6795O – Naines brunes et Exoplanètes23

24. Contents 5.1 Introduction 5.2 Description 5.3 Caustic and critical curves 5.4 Other light curve effects 5.5 Microlens parallax and lens mass 5.6 Astrometric microlensing 5.7 Other configurations 5.8 Microlensing observations in practice 5.9 Exoplanet results 5.10 Summary of limitations and strengths 5.11 Future developments 5. MicrolensingPHY6795O – Naines brunes et Exoplanètes24

25. 5.5 Microlens parallax and lens mass (1)  Fitting of the microlensing light curve yields 6 orn7 parameters including q, the planet/host star mass ratio. Determination of the planet mass thus requires an estimate of the host star mass. This can done through the microlens parallax framework.  Le us consider the geometry of the light bending for an impact parameter b = R E 5. MicrolensingPHY6795O – Naines brunes et Exoplanètes25

26. 5.5 Microlens parallax and lens mass (2)  We have the following relations:,, with where ω rel is the relative parallax by analogy with the usual trigonometric parallax ω =AU/ d ( ω in radian and d in AU) 5. MicrolensingPHY6795O – Naines brunes et Exoplanètes26 (5.24) (5.25) (5.26)

27. 5.5 Microlens parallax and lens mass (3)  The microlens parallax ω E expresses the size of the Earth orbit relative to the Einstein radius of the microlensing event projected onto the observer plane  Equations 5.24-5.27 can be rearranged to give  If both θ E and ω E can be determined, then the lens (host star) can be established from equation 5.28. The planet mass follows from the mass ratio parameter q. 5. MicrolensingPHY6795O – Naines brunes et Exoplanètes27 (5.28) (5.27) (5.28) (5.29) (5.30)

28. 5.5 Microlens parallax and lens mass (4)  How does one constrain θ E ?  From finite source size effects. Light curve fitting yields: If the source star is visible and unblended, its angular radius θ ★ can be estimated from the angular size-color relation based on its magnitude and color (Yoo et al. 2004).  Measurements of the relative lens-source proper motion through multi-color imaging secured several months/years after the microlens event.  How does one constrain ω E ?  From observations of the microlensing light curve over an extented observer baseline either from Combined observations from the ground and space Effects of the non-linear motion of the Earth’s orbit; can be measured only for (relatively rare) high-magnification events. 5. MicrolensingPHY6795O – Naines brunes et Exoplanètes28

29. 5.5 Microlens parallax and lens mass (5) Example of a microlens parallax measurement 5. MicrolensingPHY6795O – Naines brunes et Exoplanètes29

30. Contents 5.1 Introduction 5.2 Description 5.3 Caustic and critical curves 5.4 Other light curve effects 5.5 Microlens parallax and lens mass 5.6 Astrometric microlensing 5.7 Other configurations 5.8 Microlensing observations in practice 5.9 Exoplanet results 5.10 Summary of limitations and strengths 5.11 Future developments 5. MicrolensingPHY6795O – Naines brunes et Exoplanètes30

31. 5.6 Astrometric microlensing  In addition to the photometric magnification, microlensed images also leads to a small motion of their photocenter, typically by a fraction of a mas. The astrometric of the centroid is, from Eqn 5.15,  The maximum deflection angle is with u=√2 For a typical bulge lens, with θ E ~300 μ as, the shift is ~0.1 mas.  Astrometric detection through high-proper motion stars  The idea is to follow a nearby high-proper motion star (e.g. Bernard’s star; ~10’’/yr) over several years and identify, through μ as astrometry (e.g. with GAIA), potential microlensing events. 5. MicrolensingPHY6795O – Naines brunes et Exoplanètes31 (5.32) (5.33)

32. Contents 5.1 Introduction 5.2 Description 5.3 Caustic and critical curves 5.4 Other light curve effects 5.5 Microlens parallax and lens mass 5.6 Astrometric microlensing 5.7 Other configurations 5.8 Microlensing observations in practice 5.9 Exoplanet results 5.10 Summary of limitations and strengths 5.11 Future developments 5. MicrolensingPHY6795O – Naines brunes et Exoplanètes32

33. 5.7 Other configurations Other source/lens configurations that might be observed in the future.  Planet orbiting the source star.  Provides information on atmospheric composition, satellites and ring. Planets could be detected as they cross the caustics of the foreground lens. Theoretically possible to detect H 2 O and CH 4 of a close-in Jupiter (Spiegel et al. 2005).  Satellite orbiting a planet  Feasible under favourable conditions.  Planet orbiting a binary system  Free-floating planets  Microlensed transiting planets  Planetasimal disks 5. MicrolensingPHY6795O – Naines brunes et Exoplanètes33

34. Contents 5.1 Introduction 5.2 Description 5.3 Caustic and critical curves 5.4 Other light curve effects 5.5 Microlens parallax and lens mass 5.6 Astrometric microlensing 5.7 Other configurations 5.8 Microlensing observations in practice 5.9 Exoplanet results 5.10 Summary of limitations and strengths 5.11 Future developments 5. MicrolensingPHY6795O – Naines brunes et Exoplanètes34

35. 5.8 Microlensing in practice  Large-scale observing programmes focus on Baades’s Window in the central Galactic bulge.  Pros: high stellar density to maximize number of microlensing events. Region of relatively low extinction.  Cons: crowding and blending.  Observing strategy: Two-step mode 1.Wide-angle survey detects the early stages of a microlensing event using relatively coarse temporal sampling. 2.Once deviation is significant, an alert is issued and an array of smaller, follow-up narrow-angle telescopes distributed in Earth longitudes follow the events with high-precision photometry and a much denser time coverage.  Two major teams  MOA (Microlensing Observations in Astrophysics); NZ/Japan Phase I: 0.6m telescope; Phase II: 2m telescope. 20 sq degrees.  OGLE (Optical Gravitational Lens Experiment) 1.3m telescope in Chile  Plus several teams for follow-up observations 5. MicrolensingPHY6795O – Naines brunes et Exoplanètes35

36. Contents 5.1 Introduction 5.2 Description 5.3 Caustic and critical curves 5.4 Other light curve effects 5.5 Microlens parallax and lens mass 5.6 Astrometric microlensing 5.7 Other configurations 5.8 Microlensing observations in practice 5.9 Exoplanet results 5.10 Summary of limitations and strengths 5.11 Future developments 5. MicrolensingPHY6795O – Naines brunes et Exoplanètes36

37. 5.9 Exoplanet results (1)  Naming convention  After the team who first reported the event (ex: OGLE-2003-BLG- 235)  The lens and source can be specified with the suffixes ‘L’ and ‘S’. Additioncal capital or lower case letters designate companions of stellar or planetary mass respectively OGLE-2006-BLG-109LA, …Lb, …Lc designate specifically the star, and the two known planets of this multiple system.  First detection: OGLE-2003-BLG-235 (Bond et al. 2004)  Original etsimates: M p ~1.5 M J, D L =5.2 kpc refined to M p ~2.6 M J, D L =5.8 kpc, after follow-up HST observations to constrain spectral type of the host (K) star.  First low-mass microlensing event (Beaulieu et al. 2006)  M p ~5.5 M E, D L =6.6 kpc, M host. 5. MicrolensingPHY6795O – Naines brunes et Exoplanètes37

38. 5.9 Exoplanet results (2) First multiple system: OGLE-2006-BLG-109 (Gaudi etal. 2008) 5. MicrolensingPHY6795O – Naines brunes et Exoplanètes38 M P =0.7 and 0.27 M J

39. 5.9 Exoplanet results (3)  Statistical results (Gould et al. 2010)  First constraints on giant/ice giant frequency beyond the snow line.  Based on only 6 detections with mean mass ratio q =5x10 -4  Consistent with a flat distribution in log a (semi-major axis)  Planet frequency 8x that of Doppler surveys at 25x the orbital separation.  Constraint on super-Earth population from Cassan et al. (2012). 5. MicrolensingPHY6795O – Naines brunes et Exoplanètes39

40. Contents 5.1 Introduction 5.2 Description 5.3 Caustic and critical curves 5.4 Other light curve effects 5.5 Microlens parallax and lens mass 5.6 Astrometric microlensing 5.7 Other configurations 5.8 Microlensing observations in practice 5.9 Exoplanet results 5.10 Summary of limitations and strengths 5.11 Future developments 5. MicrolensingPHY6795O – Naines brunes et Exoplanètes40

41. 5.10 Summary of limitations and strengths  Limitations  Low-probability (10 -8 ), non-repeatable events.  Planetary events are short-lived (Jupiter ~1 day; Earth: a few hours) Dense temporal sampling required.  Planet typically at several kpc. Distance difficult to estimate without additional constraints (microlens parallax, knowledge of host star spectral type)  Planet parameters (mass, orbital radius) scale with (possibly uncertain) properties of the host star.  Strengths  Given high frequency monitoring and high photometric accuracy (e.g. in space), the method is sensitive of Earth-mass planets over wide separations.  To first oder, magnification amplitude is independant of the planet mass  Detection of multiple systems possible through well-sampled high magnification events.  Technique largely unbiased in terms of host star properties. Planets and their host star should therefore be found in proportion of their actual frequency in the Galactic disk 5. MicrolensingPHY6795O – Naines brunes et Exoplanètes41

42. Contents 5.1 Introduction 5.2 Description 5.3 Caustic and critical curves 5.4 Other light curve effects 5.5 Microlens parallax and lens mass 5.6 Astrometric microlensing 5.7 Other configurations 5.8 Microlensing observations in practice 5.9 Exoplanet results 5.10 Summary of limitations and strengths 5.11 Future developments 5. MicrolensingPHY6795O – Naines brunes et Exoplanètes42

43. 5.11 Future developments (1)  Ground-based  A network of 2m telescopes across the Earth to monitor several square degrees every 10 mins.  6000 events per yr. 100-fold increase in the number of events proved.  Better sensitivity to wide-separaion and free-floating planets  Sspace-based  Advantages of space environmeny: less crowding/blending, exquisite photometric sensitivity.  Sensitivity to Earth mass planets.  Euclid (ESA) Dark energy mission, 1.2m telescope Launch date: 2020 Will include a 3-12 month microlensing program (Beaulieu et al. 2010).  WFIRST (NASA) Dark energy mission, 2.4m telescope Launch date: 2024 Microlensing survey planned. Should detect Mars-like planets. 5. MicrolensingPHY6795O – Naines brunes et Exoplanètes43

44. 5.11 Future developments- Euclid (2) 5. MicrolensingPHY6795O – Naines brunes et Exoplanètes44 (Credit: Jean-Philippe Beaulieu

45. 5.11 Future developments- Euclid (3) 5. MicrolensingPHY6795O – Naines brunes et Exoplanètes45 Euclid will detect Earth mass planets (Credit: Jean-Philippe Beaulieu

46. 5.11 Future developments- Euclid (4) 5. MicrolensingPHY6795O – Naines brunes et Exoplanètes46 Euclid will detect free-floating planets (Credit: Jean-Philippe Beaulieu