A Bayesian approach to COROT A Bayesian approach to COROT light curves analysislight curves analysis

Francisco Jablonski, Felipe Madsen and Walter Gonzalez

Instituto Nacional de Pesquisas EspaciaisSão José dos Campos, SP

AbstractAbstract

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.

IntroductionIntroduction

• 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 << 1 AU)• The energy injected in their magnetospheres may be orders

of magnitude larger than in Jupiter, since MPLANET > MJup and BPLANET > Bjup

• Estimates of the emitted power for extra-solar planets in Bastian et al. (2000), Zarka et al. (2001) e Farrel et al. (2003)

The Radio-Optical connectionThe 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

Detection of impulsive eventsDetection 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 > 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

The Bayesian approach (1)The Bayesian approach (1)

dDLP

DLPD

)|()(

)|()()|(

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:

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).

(1)

The Bayesian approach (2)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 .

Implementation of the MCMCImplementation of the MCMC

1. Start chain at t=0 with state 0

2. Generate a tentative ', with proposal transition q('|t ) †. Evaluate

3. Generate an uniform random number U=[0,1]

4. If U (t, ') make t+1 = ' (that is, accept the transition)IF U > (t, ') make t+1 = t.

5. Increment t6. Goto step 2

)()|(

)'()'|(,1min)',(

tt

tPDL

PDL(2)

† To simplify, a symmetric q('|t ) was chosen (the Metropolis-Hastings algorithm)

Computational detailsComputational details

The likelihood in Eqs. (1) e (2) may be written as:

N

i

ii yyDL

12

2

2

ˆexp

2

1)|(

where yi, i=1…N, 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:

iy

N

iii

N

iiti yyyyr

1

2'

1

2

2ˆˆ

2

1ln

The models we examined are:

00

00

1ˆ

ˆ

iiay

iiay

i

i

11ˆ

,ˆ

00

000

Tiiiay

Tiiiiay

i

i

(single step with amplitude )

(box of amplitude and duration T)

ResultsResults

0 100 200 300 4000.0

0.2

0.4

0.6

0.8

1.0

Relação S/R = 0.7

I0 = 381

Freq

üênc

ia

Posição do impulso

0.5 1.0 1.5 2.0-40

-30

-20

-10

0

10

20

30

40

I

Relação S/R

0 100 200 300 400 5000.5

1.0

1.5

S/R=0.7 I0 = 381

Inte

nsid

ade

Índice

Figure 1 - Simulated light-curve in which astep occurs at index I0=381. The signal/noise ratio for the event is 0.7

Figure 2 - The distribution of 68500 samples of p(I), that is, the probability that a stepoccurs at a given location in the light-curve of Fig. 1. It is the distribution ( |D) marginalized with respect to the amplitude.

Figure 3 - The difference between the modeof p(I) in Fig. 2 and the injected value I0, asa function of the signal/noise ratio of the event. For each value of signal/noise wegenerated 100 distinct curves with randomlocations of I0, so that an average and standard deviation of I could be calculated.

DiscussionDiscussion

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/N~0.5 (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.

BibliographyBibliography

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Bastian, T., Dulk, G.A., and Leblanc, Y. 2000, ApJ, 545, 1058Defaÿ, C., Deleuil, M., and Barge, P. 2001, A&A, 365, 330 Farrel, W.M., Desch, M.D., Lazio, T.J., Bastian, T. and Zarka, P. 2003, ASP Conf.

Ser. 294: Scientific Frontiers in Research on Extrasolar Planets, ed. D. Deming and S. Seager (San Francisco: ASP), 151

Gilks, W.R., Richardson, S. & Spiegelhalter, D.J. 1995, Markov Chain Monte Carlo in Practice, Chapman & Hall/CRC

MacKay, D.J.C. 2003, Information Theory, Inference, and Learning Algorithms, www.inference.phy.cam.ac.uk/mackay

Zarka, P. Treumann, R.A., Ryabov, B.P. and Ryabov, V.B. 2001, Ap&SS, 277, 293