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DESCRIPTION
Research 1: M MillerTRANSCRIPT
Discovering Extrasolar Planets via Gravitational Microlensing
Michael Miller Denis Sullivan
Overview
Introduction to Gravitational Microlensing Multiple lens systems
Complex representation Analysing data Concluding remarks
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
LensObserver
Source
Image
bM
Gravitational Microlensing – Single Lens
Two approximations: Thin lens approximation Small angle approximation
φ ≈ sin(φ) ≈ tan(φ)
Source plane Lens plane
Observer
DLDSL
DS
θE
Gravitational Microlensing – Single Lens
Source plane Lens plane
Observer
DLDSL
DS
Lens Equation
β θE
θ+
θ-
θ
θ
Gravitational Microlensing – Single Lens
Lens Equation
Gravitational Microlensing – Single Lens
Lens Equation
θE1
θ+z+
wβθ-z-
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
Source plane Lens plane
Observer
Multiple Lenses – Two Lenses
Lens Equation
Star + planet (or binary stars)
y
xw
z
z
z
r2
r1
Positions represented by vectors
Multiple Lenses – Two Lenses
Lens Equation
Star + planet (or binary stars)
Cannot be solved analytically Solved numerically
Inverse-Ray Tracing “Brute force approach”
Semi-Analytical Method
y
xw
z
z
z
r2
r1
Positions represented by vectors
Multiple Lenses – Two Lenses
Lens Equation
iy
xw
z
z
z
r2
r1
Positions represented by complex numbers
Star + planet (or binary stars)
Five roots Five images? 3 or 5 images Numerically solve polynomial
using Jenkins-Traub algorithm Substitute z back into Lens
Equation recalculated w = source position w
z is physical image recalculated w ≠ source position w
z is not physical image
Star + planets
For N lenses No. of roots = N2 + 1 Numerically solve polynomial
using Jenkins-Traub algorithm Substitute z back into Lens
Equation recalculated w = source position w
z is physical image recalculated w ≠ source position w
z is not physical image
y
xw
z
z
z
r2
r1
Multiple Lenses
Lens Equation
z
z
r3
r4
Positions represented by complex numbers
3 lens animation
Multiple Lenses - Three Lenses
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
Analysing Data
MOA OGLE
microFUN
Fit theoretical light curve to data
Light curve parameters Mass ratio(s) Einstein crossing time Source radius Impact parameter Lens position(s)
Lens separation(s) + angle(s) Lens Motion Parallax
Least squares fit Vary parameters to minimise χ2
When χ2 is minimised, values for parameters are parameter values for event
Analysing Data
In units of θE
Depend on Mass
MOAOGLE
microFUNχ2: minimised!
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
Concluding remarks
Acknowledgements VUW Optical Astrophysics Research Group Marsden Fund MOA Collaboration
(Microlensing Observations in Astrophysics)