1. Discovering Extrasolar Planets via Gravitational Microlensing Michael Miller Denis Sullivan
2. Overview Introduction to Gravitational Microlensing Multiple lens systems Complex representation Analysing data Concluding remarks
3. What is Gravitational Microlensing? Bending of light in a weak gravitational field Gravitational field from a star or planet The path of the light bends by a small angle as it passes the star or planet Observer “sees” image of star slightly shifted from source Image Source b M Observer Lens
4. Gravitational Microlensing– Single Lens Two approximations: Thin lens approximation Small angle approximation φ ≈ sin(φ) ≈ tan(φ) θE Observer DSSource plane DSL Lens plane DL
6. Gravitational Microlensing– Single Lens Lens Equation
7. Gravitational Microlensing– Single Lens Lens Equation θ+ z+ β w θ z-- θE 1
8. Gravitational Microlensing– Single Lens Intrinsic brightness of the source, does not change Intensity per unit area in each image is the same as the source Magnification, M, is the ratio of observed light, to amount of light if there was no lensing
9. Multiple Lenses– Two Lenses Star + planet (or binary stars) Lens Equation y z r2 w z r1 z x Observer Source plane Lens plane Positions represented by vectors
10. Multiple Lenses– Two Lenses Star + planet (or binary stars) Lens Equation y Cannot be solved analytically Solved numerically z Inverse-Ray Tracing r2 w z “Brute force approach” r1 z x Semi-Analytical Method Positions represented by vectors
11. Multiple Lenses– Two Lenses Star + planet (or binary stars) Lens Equation Five roots Five images? iy 3 or 5 images Numerically solve polynomial z r2 using Jenkins-Traub algorithm z w Substitute z back into Lens r1 z x Equation recalculated w = source position w z is physical image recalculated w ≠ source position w Positions represented by complex numbe z is not physical image
12. Multiple Lenses Star + planets Lens Equation For N lenses y No. of roots = N2 + 1 z Numerically solve polynomial z r4 r2 using Jenkins-Traub algorithm w z Substitute z back into Lens r1 x Equation r3 recalculated w = source position w z z z is physical image recalculated w ≠ source position w Positions represented by complex numbe z is not physical image
13. Multiple Lenses- Three Lenses 3 lens animation
14. Analysing Data Separation between images is ~milliarcseconds Cannot be resolved! Magnification can be measured! Microlensing events recorded by measuring apparent brightness over time (light curve) Fit together data from different collaborations MOA OGLE Fit theoretical light curve to data microFUN
15. Analysing Data Light curve parameters Mass ratio(s) Einstein crossing time Source radius Depend on Mass Impact parameter OGLE In units of θE MOA Lens position(s) Lens separation(s) + angle(s) Lens Motion Parallax χ2: minimised! microFUN Least squares fit Vary parameters to minimise χ2 When χ2 is minimised, values for parameters are parameter values for event
16. Concluding remarks Exact values for θE and total mass cannot be determined directly from microlensing light curve Advantages: Not dependent on light from host star Free-floating planets Not limited by distance from Earth Gives snap-shot of planetary system in short observing time Disadvantage: Alignments of two stars are rare Follow-up (repeated) measurements difficult
17. Acknowledgements VUW Optical Astrophysics Research Group Marsden Fund MOA Collaboration (Microlensing Observations in Astrophysics)
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