Measuring theAstrometric Signatureof Transiting Planets

with SIMScott Gaudi, Ohio State University

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Order of Magnitude

• Centroid:

• Uniform surface brightness, full transit:

• Amplitude of the centroid shift:

!"x =#*$x I( $x , $y )d $x d $y%%I( $x , $y )d $x d $y%%

, !"y =#*$y I( $x , $y )d $x d $y%%I( $x , $y )d $x d $y%%

!"x=#*

$

1%$x, !"

y=#*

$

1%$y, where $ =

Rp

R*

&

'(

)

*+

2

is the transit depth

!

!T "#*$

1%$& 2.5µas

Rp

RJ

'

()

*

+,

2

R*

R"

'

()

*

+,

%1

d

20pc

'

()

*

+,

%1

Astrometric Wobble and Transit Signal• HD 189733• M*=0.81M, R*=0.79R, d=19.3pc• Mp=1.15MJ, Rp=1.16RJ, a=0.031AU• Astrometric Wobble Amplitude

• Astrometric Transit Amplitude

• Ratio of Amplitudes

• Ratio of Durations

!

!T "#*$

1%$& 2.5µas

Rp

RJ

'

()

*

+,

2

R*

R"

'

()

*

+,

%1

d

20pc

'

()

*

+,

%1

!

!W "a

d

M p

M* +Mp# 1.9µas

a

0.04AU

$

%&

'

()Mp

MJ

$

%&

'

()M*

M"

$

%&

'

()

*1

d

20pc

$

%&

'

()

*1

!

!T

!W

" 1 Rp

RJ

#

$%

&

'(

2

R*

R"

#

$%

&

'(

)1

a

0.04AU

#

$%

&

'(

)1M

p

MJ

#

$%

&

'(

)1

M*

M"

#

$%

&

'(

Period of Orbit

Transit Duration=!a

R*

~ 30

Parameter Dependence• Transit Duration

• Parallel Amplitude

• Perpendicular Amplitude

Duration = 2!0(1" b

2)

1/2, where !

0#PR

*

2$a

~!T(1" b

2)1/2

~!Tb

Observable Parameters

• Photometric

• Astrometric

b, r !Rp

R*

, t0, "

0

b, r !Rp

R*

, t0, "

0, #

*, $

!

φ = angle of projected planet orbit normal w.r.t. sky coordinates

What is it good for?

• Measure M* and R*.• Measure Mp and Rp.• Confirm or constrain polarized reflected

light measurements.• Determine the inclination of the planet orbit

w.r.t. more distant companions.• Remove degeneracies is multiple transiting

planets systems.

Signal-to-Noise Ratio and Uncertainties

• Signal-to-Noise Ratio– Duration decreases with b– Signal amplitude increases with b

• Parameter Uncertainties– Assume photometric parameters

are known– Fisher matrix

S

N=Q

2

3(1! b2 )1/2 (1+ 2b2 )

"

#$%

&'

1/2

, Q ( ()*0)1/2 +T

,

!

"#* /#* ~ "$ ~ (S /N)%1

Known Systems• Estimated S/N per transit

pair– 10% Out-of-transit data– Two transits with

orthogonal baselines– Four V=10 ref. Stars– Three chop cycles– 30s integrations

• Also S/N for 30 total hrs– Three systems with S/N>3

• GJ436b, S/N~7.3• HD189733b, S/N~16.9• HD209458b, S/N~3.9

– Three with 1.5

To Do

• Practical application.– Refined S/N estimates.– Suitable reference stars?– Optimal strategies.– Expected uncertainties.

• Additional systems?• Astrometric signature of eclipsing binaries.