Estimating Orbital Period of Exoplanets in …Inverse Ray Shooting observer lens plane source plane...

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

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

Geometry of the System

O

x′ y′

z′ ≡ z

x

y ≡ y′′

ïï

x′′

z′′

i

á/2− i

á/2− i


Inverse Ray Shooting

observer

lens plane

source plane


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


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


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

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


Simulation 1 (periodogram)


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


Source trajectory



Amplification curve

q = 0.8, a = 0.23, e = 0, i = ï = 0°, P = tE/3


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


Source trajectory



Amplification curve

q = 0.8, a = 0.23, e = 0.5, i = 45°, ï = 0°, P = 2tE


Fit to Real Data

Event OGLE-2011-BLG-1127/MOA-2011-BLG-322


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.


http://dx.doi.org/10.1093/mnras/stt2363

http://arxiv.org/abs/1401.6288

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


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


Dimensionless Quantities

Einstein Radius

RE =

√4GMc2

DdsDd

Ds

Einstein Angle

ÚE =RE

Dd=

√4GMc2

Dds

DsDd

Critical Superficial Mass Density

Îcr =c2Ds

4áGDdDds


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


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

)+ · · ·


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


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


Estimating Orbital Period of Exoplanets in …Inverse Ray Shooting observer lens plane source plane...

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