eory of microlensing and planetary microlensingnexsci.caltech.edu › workshop › 2011 › talks...
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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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