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)