eory of Microlensing and Planetary Microlensing:

Basic Concepts

Shude Mao National Astronomical Observatories of China

& University of Manchester

2011 Sagan Exoplanet Summer Workshop, 25/07/2011

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Outline

•  What is microlensing? •  Lens equation, images, and magnifications

•  Binary lens equations •  Critical curves, caustics and planetary

microlensing

•  Summary •  Suggested readings

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What is Galactic microlensing?

Light curve is symmetric, achromatic & non-repeating (Paczynski 1986) Image credit: NASA/ESA

Image separation is too small to resolve, we observe magnification effects.

Microlensing can be used to •  Discover extrasolar planets (~15) •  Study MW structure, stellar mass black holes, … 3

Deflection angle: impulse approximation

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!!!""mV#mV

=F#$tmV

=(GMm /" 2 ) (2" /V)

mV=2GMV 2"

In general relativity, !!!"=2GMc2"

!!

ξ m, V -ξ-ξ

x2

M

Lens equation

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!!"+ Dds"

!!"=Ds

Dd#!", "

!!"=4GM

c2#

!!"+"!"=#!", "!"=4GM

c2Dd#

DdsDs

=#E2

#

!!"

Ds+DdsDs

!!!"=!!"

Dd÷Ds

!E2 =

4GM

c2DdsDdDs

Single point lens equation

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!!"+"!"=#!", "

!"=4GM

c2Dd#

DdsDs

#!"

#=#E2

# #!"

#

!!"=#!"!#E2

# #!"

#

Setting the angular Einstein radius to unity, we have

! =" !1"

Lens images

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Lens equation: ! =" ! 1"

If ! = 0, Solution: ! = ±1

observer Source

lens

Side view Plane of Sky

Einstein Ring

Lens images

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Lens equation: ! =" ! 1"

If ! " 0, Solutions: !± =" ± " 2 + 4

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Lens images

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Positive Parity Negative Parity

+-

Lens mapping

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•  Lens equation is a 2D mapping between the source plane to the image plane

– The mapping may not be unique (multiple images)

•  Gravitational lensing conserves surface brightness – Magnification is just the ratio of the image area

to the source area

– In general,

µ = det !2!!"

!!!" 2

"

#$$

%

&'' = J

(1, J=det !2!!"

!!!" 2

"

#$

%

&'

!!"!"!"

!!"="!"!#!"("!")

Single lens magnification

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•  Magnification

•  |μ+|-|μ_|=1 (μ+>0, μ->0)

•  μtotal=|μ+|+|μ_|=(β2+2)/(β(β2+4)1/2)

µ =!!" d!"!" d"

= !"d!d"

critical curves and caustics: point lens

•  Caustics: point source positions with ∞ magnifications

•  eir images form critical curves

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• Angular scale: !E =4GM

c2DdsDdDs

!

"#

$

%&

1/2

~ 0.5 mas

• Einstein radius rE = Dd!E = 2.2 AU M0.3 M•

!

"#

$

%&

1/2D

2 kpc!

"#

$

%&

1/2

, D = DdDdsDs

• Timescale: tE =rEVt= 21 day M

0.3 M•

!

"#

$

%&

1/2D

2 kpc!

"#

$

%&

1/2Vt

200 km s'1

!

"#

$

%&'1

Order of magnitude

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For microlensing in the Milky Way, distances ~ few kpc

Degeneracy! Can be partially or completely remove for exotic events!

Single lens equation in complex

Normalised lens equation: !!"=!!"!1!!!"

!

In two dimensions, we write !!"= (xs, ys ), !

!"=(x, y)

xs = x ! xx2 + y2

,

ys = y ! yx2 + y2

, " i

In complex notation

zs = z ! zzz= z ! 1

z, zs = xs + ys i, z = x + y i

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Binary lens equations Single lens equation:

zs = z ! zzz= z ! 1

z, (m=1, lens at origin)

This can be easily generalised to binary lenses:

zs = z ! m1z ! z1

!m2

z ! z2, m1 +m2 =1

Taking the conjugate:

zs = z ! m1z ! z1

!m2

z ! z2.

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•  We obtain zbar and substitute it back into the original equation, which results in a 5th order complex polynomial equation.

•  This can be easily solved to find 3 or 5 image positions.

Binary lens magnification

Magnification: µ = J !1

J = det "(xs, ys )"(x, y)

#

$%

&

'( = det "(zs, zs )

"(z, z)#

$%

&

'(

= 1! m1

(z ! z1 )2 +m2

(z ! z2 )2

2

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•  Critical curves (J=0, μ=∞) and caustics can then be derived.

Planetary caustics q=0.01, a=1.5

Far Near resonant

Distance decreases (Erdl & Schneider 1993)

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q=0.01, a=1 q=0.01, a=0.8

Principles of binary and exoplanet lensing

(Mao & Paczynski 1991)

q=0.001

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q=0.1

Properties of planetary microlensing

•  Rate and deviation duration scale roughly ~ (mass ratio)1/2 – Planetary deviation lasts for days for 1MJ planets, but

hours for 1M⊕ planet – Deviation amplitude can be high even for 1 M⊕ planet

•  Microlensing has sensitivity to –  low-mass planets between ~0.6-1.6 Einstein radii

(resonance zone) –  free-floating planets (seen as single events lasting hours

to days) – multiple planets

•  Complementary to other methods

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Summary

•  Gravitational microlensing is “simple”, based on GR.

•  We have derived –  the lens equations, images, and magnifications – binary lens equation in complex notations – equations of critical curves and caustics.

•  Basic principles of extrasolar microlensing •  With upgraded/new ground-based (& space)

experiments, a particularly exciting decade is ahead!

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Suggested reading •  General reviews on galactic microlensing:

–  Mao, S., 2008, “Introduction to microlensing”, in Proceedings of the Manchester Microlensing Conference, Edited by E. Kerins, S. Mao, N. Rattenbury, L. Wyzykowski (with exercises!) online at http://pos.sissa.it//archive/conferences/054/002/GMC8_002.pdf

–  is lecture is a simplified version of the ones I gave at a workshop in Italy http://www.jb.man.ac.uk/~smao/lecture1.pdf, http://www.jb.man.ac.uk/~smao/lecture2.pdf, http://www.jb.man.ac.uk/~smao/lecture3.pdf

–  Gould, A., 2008, “Recent Developments in Gravitational Microlensing”, presented at "e Variable Universe: A Celebration of Bohdan Paczynski", 29 Sept 2007 online at http://uk.arxiv.org/abs/0803.4324

–  Paczyński, B., 1996, “Gravitational Microlensing in the Local Group”, Annual Review of Astronomy and Astrophysics, Vol. 34, p419 online at http://arjournals.annualreviews.org/doi/abs/10.1146/annurev.astro.34.1.419

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