Research 1: M Miller

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
DS
Source plane
DSL Lens plane DL

5. Gravitational Microlensing
– Single Lens
Lens Equation
θ+
β θE
θ- Observer
DS
Source 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)