Measuring the Astrometric Signature of Transiting Planets...
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Measuring the Astrometric Signature of Transiting Planets with SIM Scott Gaudi, Ohio State University
Transcript of Measuring the Astrometric Signature of Transiting Planets...
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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%$& 2.5µas
Rp
RJ
'
()
*
+,
2
R*
R"
'
()
*
+,
%1
d
20pc
'
()
*
+,
%1
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Astrometric Wobble and Transit Signal• HD 189733• M*=0.81M, R*=0.79R, d=19.3pc• Mp=1.15MJ, Rp=1.16RJ, a=0.031AU• Astrometric Wobble Amplitude
• Astrometric Transit Amplitude
• Ratio of Amplitudes
• Ratio of Durations
!
!T "#*$
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"
#
$%
&
'(
Period of Orbit
Transit Duration=!a
R*
~ 30
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Parameter Dependence• Transit Duration
• Parallel Amplitude
• Perpendicular Amplitude
Duration = 2!0(1" b
2)
1/2, where !
0#PR
*
2$a
~!T(1" b
2)1/2
~!Tb
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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
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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.
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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
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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
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To Do
• Practical application.– Refined S/N estimates.– Suitable reference stars?– Optimal strategies.– Expected uncertainties.
• Additional systems?• Astrometric signature of eclipsing binaries.
