Estimating Orbital Period of Exoplanets in …Inverse Ray Shooting observer lens plane source plane...
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Università del Salento and INFN Lecce
Estimating Orbital Period of Exoplanetsin Microlensing Events
Mosè Giordano
19th International Conference on Microlensing
Annapolis, MD
January 20, 2015
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Binary Lens with Orbital Motion
The parameters needed to model microlensing events by binary lens withorbital motion are
• Paczyński curve parameters: t0 u0 tE Ú
• finite source effects: â?• binary lens: s q
• binary lens with orbital motion: a e i ï
In addition, with small mass ratios q there is the close-wide degeneracys←→ s−1
What if we knew the orbital period of the lenses
P = 2á
√a3
G (m1 + m2)= 2á
√a3
Gm1(1 + q)
independently from a fit?
Mosè Giordano (UniSalento and INFN Lecce) Estimating Orbital Period of Exoplanets in Microlensing Events January 20, 2015 2 / 15
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Geometry of the System
O
x′ y′
z′ ≡ z
x
y ≡ y′′
ïï
x′′
z′′
i
á/2− i
á/2− i
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Inverse Ray Shooting
observer
lens plane
source plane
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Inverse Ray Shooting (cont.)
Solve the lens equation “backwards”
Ø = z −N¼i=1
êi (z − zi )‖z − zi‖2
Conditions
• source area subdivided in at least 103 pixels
• each pixel on the source plane matches at least 100 pixels on the lensplane
Pros and cons
3 precise, also on caustics
7 very slow, high number of photons to be “shot”
3 any lens configuration
7 only point-like source
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Witt & Mao Method
Binary-Lens Equation in complex formalism (details?)
Ø = z +ê1
z1 − z+
ê2
z2 − z
Put the lenses on points z1 = −z2 along the real axis (zj = z j )
p5(z) =5¼i=0
cizi = 0
Amplification
Þ(Ø) =N¼i=1
|Þi | =N¼i=1
ái
detJ
∣∣∣∣∣z=zi
Pros and cons3 fast7 only point-like source3 any lens configuration7 doesn’t work near caustics
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Hexadecapole Approximation
ï
â/2
â/2
S0,0
S1,0S1,1
S1,2 S1,3
S2,0S2,1
S2,2S2,3
S3,0
S3,1
S3,2
S3,3
Approximation of the amplification functionwith a Taylor series up to the fourth order
Þfinite(â) =2áF
∞¼n=0
Þ2n
∫ â
0S(w)w2n+1 dw
= Þ0 +Þ2â
2
2
(1− È
5
)+Þ4â
4
3
(1− 11È
35
)+ · · ·
Pros and cons
3 fast (no amplification map required)
3 extended source
3 any lens configuration and any radialluminosity profile of the source
7 far enough from the caustics
Details?Mosè Giordano (UniSalento and INFN Lecce) Estimating Orbital Period of Exoplanets in Microlensing Events January 20, 2015 7 / 15
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Simulation 1
−2−1.5−1−0.5
00.5
11.5
Ù
1
2
3
4
5
Am
plifi
cati
on
−0.001
0
0.001
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
Resi
dual
s
à
−3−2−10123t/tE
−1
−0.5
0
0.5
1
−1
−0.5 0
0.5 1
Lensing star orbit
Companion planet orbit
Source trajectory
Central caustic curve
Best-fitting Paczynski curve
Amplification curve
q = 10−3, a = 0.2, e = 0.5, i = 45°, ï = 0°, P = tE/4
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Simulation 1 (periodogram)
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Simulation 2
−2−1.5−1−0.5
00.5
11.5
Ù
1
1.5
2
2.5
3
3.5
Am
plifi
cati
on
−0.04−0.02
00.020.040.060.08
0.1
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
Resi
dual
s
à
−3−2−10123t/tE
−0.03
−0.02
−0.01
0
0.01
0.02
0.03
−0.0
3
−0.0
2
−0.0
1 0
0.0
1
0.0
2
0.0
3
Lensing star orbit
Companion planet orbit
Source trajectory
Central caustic curve
Best-fitting Paczynski curve
Amplification curve
q = 0.8, a = 0.23, e = 0, i = ï = 0°, P = tE/3
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Simulation 2 (periodogram)
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Simulation 3
−4−3−2−1
01234
Ù
1
1.5
2
2.5
3
Am
plifi
cati
on
−0.04−0.02
00.020.040.060.08
0.1
−4 −2 0 2 4
Resi
dual
s
à
−6−4−20246t/tE
−3
−2
−1
0
1
2
3
−3 −2 −1 0 1 2 3
Lensing star orbit
Companion planet orbit
Source trajectory
Central caustic curve
Best-fitting Paczynski curve
Amplification curve
q = 0.8, a = 0.23, e = 0.5, i = 45°, ï = 0°, P = 2tE
Mosè Giordano (UniSalento and INFN Lecce) Estimating Orbital Period of Exoplanets in Microlensing Events January 20, 2015 12 / 15
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Simulation 3 (periodogram)
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Fit to Real Data
Event OGLE-2011-BLG-1127/MOA-2011-BLG-322
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Conclusions
Orbital period of the lenses should be shorter than the Einstein time ofthe event or we must have a long observational window
We fit the observed amplification curve to a simple Paczyński curve, withfour easily-guessable free parameters, and then perform a periodogramon the residuals: the period so obtained is the period of the binarysystem
We need to remove a very small region around the central peak from theresiduals before performing the periodogram
Periodic feature with the same period far from the peak =⇒ sourceperiodicity (binary system, intrinsic variable, etc. . . )
Reference
A. Nucita, M. Giordano, F. De Paolis, and G. Ingrosso. “Signatures ofrotating binaries in microlensing experiments”. In: Monthly Notices ofthe Royal Astronomical Society 438 (Mar. 2014), pp. 2466–2473. doi:10.1093/mnras/stt2363. arXiv: 1401.6288.
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Lens Equation
O
S
I
optical axis
A
R
à
LDd Dds
Ds
Ù=ÔDs
ÓDds
Ds
Ddà=ÚDs
observer
lensplane
source plane
Ú Ô
Ó
Lens Equation
~Ô = ~Ú − ~ÓDds
Ds⇐⇒ ~Ù = ~à
Ds
Dd− ~ÓDds ⇐⇒ ~y = ~x − ~Ó
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Critic and Caustic Curves
Amplification Matrix
Ji j =�yi�xj
Amplification
Þ =1
detJ
Critic CurvesLocus of the points in the lens plane in which Þ→∞ ⇐⇒ detJ → 0
Caustic CurvesLocus of the points in the source plane in which Þ→∞ ⇐⇒ detJ → 0
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Dimensionless Quantities
Einstein Radius
RE =
√4GMc2
DdsDd
Ds
Einstein Angle
ÚE =RE
Dd=
√4GMc2
Dds
DsDd
Critical Superficial Mass Density
Îcr =c2Ds
4áGDdDds
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Complex Formalism
Introduced by Witt (1990)Complex Coordinates:Source Plane: z = x + iyLens Plane: Ø = à + iÙMass Distribution
Î(z) =N¼j=1
mjÖ2(z − zj )
Lens Equation
Ø = (1−Ü)z +Õz −N¼j=1
êjz − z j
Critic Curves Parametrization
N¼j=1
êj(z − z j )2
= (1−Ü)eiï−Õ
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Hexadecapole Approximation: details
Far from the caustics, amplification can be expanded in Taylor series
Þ(à,Ù) =∞¼n=0
n¼i=0
Þn,i (à − à0)i (Ù− Ù0)n−i
Amplification of an extended source
Þfinite(â;à0,Ù0) =
∫ â
0wS(w)dw
∫ 2á0
Þ(à0 + w cosÚ,Ù0 + w sinÚ)dÚ∫ â
0wS(w)dw
∫ 2á0
dÚ
=2áF
∞¼n=0
Þ2n
∫ â
0S(w)w2n+1 dw
With linear limb-darkening (S(w) = (1− È (1− (3/2)√
1−w2/â2))F /áâ2)
Þfinite(â;à0,Ù0) = Þ0 +Þ2â
2
2
(1− È
5
)+Þ4â
4
3
(1− 11È
35
)+ · · ·
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Hexadecapole Approximation: details (cont.)
Mw,+ =14
3¼j=0
Þ(à0 + w cos(ï+ já/2),Ù0 + w sin(ï+ w sin(ï+ já/2)))−Þ0
≈ 14
3¼j=0
4¼n=0
n¼i=0
Þn,iwn(cos(ï+ já/2))i (sin(ï+ já/2))n−i −Þ0
=(Þ4,0 +Þ4,4)(3 + cos(4ï)) + (Þ4,3 +Þ4,1)sin(4ï) +Þ4,2(1− cos(4ï))
8+Þ2w
2
Mw,× =14
3¼j=0
Þ(à0 + w cos(ï+ (2j + 1)á/4),Ù0 + w sin(ï+ w sin(ï+ (2j + 1)á/4)))
−Þ0
≈(Þ4,0 +Þ4,4)(3− cos(4ï))− (Þ4,3 +Þ4,1)sin(4ï) +Þ4,2(1 + cos(4ï))
8w4
+Þ2w2
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Hexadecapole Approximation: details (cont.)
Recipe:
• determine amplification on the thirteen points
• use these amplifications to calculate Mâ,+, Mâ,×, and Mâ/2,+
• calculate Þ2â2 and Þ4â
4 with relations
Þ2â2 =
16Mâ/2,+ −Mâ,+
3
Þ4â4 =
Mâ,+ + Mâ,×
2−Þ2â
2
• insert Þ2â2, Þ4â
4, and amplification Þ0 of the central monopole insideequation
Þfinite(â;à0,Ù0) = Þ0 +Þ2â
2
2
(1− È
5
)+Þ4â
4
3
(1− 11È
35
)+ · · ·
to get the amplification of a finite source
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