A Bayesian approach to COROT light curves analysis

1
A Bayesian approach to COROT light curves analysis

• Francisco Jablonski, Felipe Madsen and Walter
  Gonzalez
• Instituto Nacional de Pesquisas Espaciais
• São José dos Campos, SP

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Abstract
Space missions like COROT will produce a large
number of detections of planetary transits. Among
the newly detected planetary systems one expects
to find those in which radio emission is
significantly correlated with events of ejection
of matter in the parent star. The planetary radio
emission is exponentially related with the
velocity and power of the wind from the parent
star. In this context, to detect impulsive events
with relative amplitude of the order of 10-3 -
10-4 in the light curve of the parent star is
important for prompt triggering of follow-up
observations in radiofrequencies. In this work,
we investigate the use of a Bayesian approach for
the detection of impulsive events.
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Introduction

• Jupiter presents non-thermal emission in the kHz
  to GHz bands
• Below 40 MHz ? cyclotron emission
• Higher frequencies ? synchrotron emission
• Average power at the GWatt level
• Many ESP closer to parent star than Jupiter (d ltlt
  1 AU)
• The energy injected in their magnetospheres may
  be orders of magnitude larger than in Jupiter,
  since MPLANET gt MJup and BPLANET gt Bjup
• Estimates of the emitted power for extra-solar
  planets in Bastian et al. (2000), Zarka et al.
  (2001) e Farrel et al. (2003)

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The Radio-Optical connection

• In Jupiter, the radio emission increases orders
  of magnitude after events of coronal mass
  ejection (CME) in the Sun
• CME produce global irradiance variations 10-4
  (rms) in the optical
• COROT photometry can detect CME
• Real-time monitoring with COROT would allow early
  warnings to radio-observations to search for
  radio emission after impulsive events in the
  parent star
• Many targets/events could be observed
• Alternatively, off-line analysis of COROT data
  obtained simultaneously with radio data ok
• Fewer targets/events

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Detection of impulsive events

• Bayesian approach like in Aigrain Favata (2002)
  and Defaÿ et al. (2001)
• Poissonic nature of photon noise
• Two-rates, (?1,?2), model for quiescence and
  flares
• ?2 gt ?1 is an important "a priori" information
• More elaborate models (with duration and shape of
  flare included) could be implemented easily
• Selection of best model is natural in the
  Bayesian context

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The Bayesian approach (1)
The bayesian approach allows us to examine the
parameters of a model (here represented by a
vector ?) as if they were statistical variables
of the same nature as the data (represented by a
vector D). The connection between the two
entities is possible via the Bayes Theorem
(1)
Here L(D?) is the likelihood of the data given
the parameters ? P(?) is the distribution of
probabilities representing our "a priori"
knowledge of the model, and ?(?D) represents the
"a posteriori" distribution of probabilities of
the parameters. We are interested, in general, in
the expected value, or in some measure of the
(marginalized) width of ?(? D).
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The Bayesian approach (2)
The term in the denominator of equation (1) is a
normalization factor that does not change the
shape of ?(?D). In practice, it can be ignored
in the calculations. Equation (1) can be
expressed analytically only in very particular
cases. When the number of parameters in ? is
large, one can find the expected value or the
width of ?(?D) only by numerical methods. Grid
methods, however, are exponentially inefficient
with the growth of the number of parameters
(MacKay 2003). The best method to efficiently
examine ?(? D) is the Markov Chain Monte Carlo
Method (Gilks, Richardson Spiegelhalter
1996). Given a set of parameters ?, a
characteristic feature of the Markov chain is
that a state in the space of ? depends only on
the immediately previous state of ?.
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Implementation of the MCMC

• Start chain at t0 with state ?0
• Generate a tentative ?', with proposal transition
  q(?'?t ) . Evaluate
• Generate an uniform random number U0,1
• If U ? ?(?t, ?') make ?t1 ?' (that is,
  accept the transition)IF U gt ?(?t, ?') make ?t1
  ?t.
• Increment t
• Goto step 2

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Computational details
where yi, i1N, is the light-curve, and the
model for each i. For computational reasons it
is better to express the likelihood ratio of Eq.
(2) in logarithmic form. In the case of uniform
priors we have
The models we examined are
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Results
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Discussion
Figure 3 shows that the procedure based on
sampling of the "a posteriori" distribution
?(?D) for a model in which the impulsive event
is a step is quite good. One can see that there
are no trends in the recovered instants even for
S/N0.5 (in this case it is not possible to see
the events "by eye" anymore). The uncertainty in
the localization of the events at low S/N has
obvious importance in the context of generating
alerts for subsequent observations in
radiofrequencies. The next step to use this
method for the detection of impulsive events in
more realistic conditions is to introduce a
spectrum of intrinsic fluctuations in the
simulated light-curves (e.g., from the spectrum
of the fluctuations of solar irradiance observed
by the VIRGO experiment in SoHO). This would
modify substantially the detection thresholds. In
this case, it is important to separate the
contributions to the global likelihood coming
from Poisson noise and from intrinsic
fluctuations associated to the signal
itself. Another development of interest would be
an implementation of the method suitable for use
in real-time.
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Bibliography

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• Gilks, W.R., Richardson, S. Spiegelhalter, D.J.
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