A power-spectrum autocorrelation technique to detect global ...

Astron. Nachr. / AN , No. , 1 – 4 () / DOI

A power-spectrum autocorrelation technique to detect global asteroseismic parameters G.A. Verner⋆ and I.W. Roxburgh Astronomy Unit, Queen Mary, University of London, Mile End Road, London, E1 4NS, UK. The dates of receipt and acceptance should be inserted later stars: oscillations – methods: data analysis

This article describes a moving-windowed autocorrelation technique which, when applied to an asteroseismic Fourier power spectrum, can be used to automatically detect the frequency of maximum p-mode power, large and small separations, mean p-mode linewidth, and constrain the stellar inclination angle and rotational splitting. The technique is illustrated using data from the CoRoT and Kepler space telescopes and tested using artificial data. c WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction 2

2 Pipeline The first stage of the analysis pipeline is to account for the background variation in each power spectrum. A simple background estimate was found to be sufficient to remove the smooth variation in power without attenuating the signal from the p-mode region. Therefore, the background was obtained from the power spectrum by applying a movingmedian filter with a width of 100µHz to obtain an estimate of the mean power, followed by a moving-mean filter with the same width to smooth the background estimate. The power spectrum is then scaled by this background to give ⋆ Corresponding author: e-mail: [email protected]

−1

Power (ppm µHz )

10

0

10

2

Regions of p-mode power can be characterised by the selfrepeating comb-like structure in a Fourier power spectrum. Using a moving-windowed autocorrelation technique applied to the acoustic power spectrum it is possible to automatically detect the frequency of maximum power, large and small separations, mean mode linewidth and, with sufficient signal-to-noise, constrain the inclination angle and rotational splitting. With the rapidly increasing amount of asteroseismic data becoming available from the CoRoT and Kepler space telescopes, there is a need for accurate and flexible analysis pipelines which can automatically obtain global asteroseismic parameters. A number of automated pipelines have recently been described by groups preparing to analyse data from the Kepler mission (Hekker et al. 2010; Huber et al. 2009; Mathur et al. 2010). The autocorrelation technique described here has been applied to high and low signal-to-noise data from the CoRoT spacecraft and tested successfully with artificial data to show that it provides a powerful method to detect global p-mode parameters and is particularly useful for data with low signal-to-noise.

−2

10

−4

10

1000

2000

3000 4000 Frequency (µHz)

5000

6000


5000

6000

15

Power / Background

arXiv:1104.0631v1 [astro-ph.SR] 4 Apr 2011

Key words

10

5

0

1000

2000

Fig. 1 Background fit and removal for HD49933. Top panel shows the background estimate (green) which has been removed in the scaled power-to-background spectrum below. a relative power-to-background spectrum. This is illustrated in Fig. 1 for the case of the CoRoT F2V star HD49933 (Appourchaux et al. 2008; Benomar et al. 2009). Once the background has been removed, the windowed power-spectrum autocorrelation function (PSAC) is generated from the relative power-to-background spectrum (PBS). In each window of width 500µHz (incremented in steps of 1µHz), the unbiased PSAC is calculated from (PBS − 1). Regions with no autocorrelation signal (i.e. no p-mode detection) return a PSAC of approximately zero. Those win-

c WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

2

G.A. Verner & I.W. Roxburgh: Power spectrum autocorrelation techniques to detect asteroseismic parameters

Freq offset (µHz)

150 100 50 1000

2000


5000

6000

100 50

7000

1000

2000


5000

6000

150 100 50 2000


5000

6000

200 150 100

0.2 0 2000


1000

2000

−3

0.4

1000

50 0

7000 Mean collapsed ACF

1000

5000

6000

7000

6

x 10


5000

6000

100 50 1000

2000


5000

6000

7000

1000

2000


5000

6000

7000

2 0 1000

2000


5000

6000

7000

1000

2000


5000

6000

7000

200 150 100 50 0

7000

4

−2

150

250 Freq offset (µHz)

200

200

0

7000

250 Freq offset (µHz)

Freq offset (µHz)

150

0

250

Mean collapsed ACF

250

200

Mean collapsed ACF

Freq offset (µHz)

200

Freq offset (µHz)

250

250

0.15 0.1 0.05 0 −0.05

Normalised ACF

dows containing p-mode signal show a positive and negative correlation as the ridges alternate into and out of phase. By summing the absolute deviations of this PSAC map, the collapsed signal can be fitted to a Gaussian function to give the frequency of maximum power (νmax ) and the width of the p-mode region. This is illustrated in Fig. 2 in the case of HD49933, HD181906 – a low signal-to-noise F8 star observed by CoRoT (Garc´ıa et al. 2009) and KIC6603624 – a G-type star in the first public release from Kepler (Chaplin et al. 2010).

Normalised ACF

Fig. 2 PSAC maps and νmax fits for HD49933 (left), HD181906 (middle) and KIC6603624 (right). Top panels show the windowed PSAC as a function of central frequency, centre panels show the absolute values of the PSAC map, bottom panels show the mean absolute PSAC at each central frequency (black) and a Gaussian fit to the peak (red).

1

0.5

0 0

20

40

60 Frequency (µHz)

80

100

120

1

0.5

0

Normalised ACF

0 20 40 60 80 100 120 When the central p-mode region (νmax ) has been idenFrequency (µHz) tified, an initial estimate of the large separation can be ob1 tained from the narrow-windowed autocorrelation of the time series (TSAC) (Roxburgh 2009a). Further information can 0.5 then be obtained from a fit to the PSAC in in a single win0 dow containing the central region of maximum power. The 0 20 40 60 80 100 120 Frequency (µHz) amount of information available from the PSAC depends on the properties of the modes, and particularly the mode linewidths in relation to the small separations. The PSAC Fig. 3 Artificial normalised PSAC functions showing the is shown in Fig. 3 for three artificial scenarios in different d separation (green dashed line), d separation (blue 01 02 linewidth regimes. dashed line) and ∆ν separation (red dashed line). The arThe PSAC can be fitted to the limit-spectrum PSAC ob- tificial spectra were generated with linewidths of 3.0, 1.0 tained from a sum of Lorentzian peaks parametrised by the and 0.5µHz (from top to bottom). The rotational splitting large separation (∆ν), small separations (d01 , d02 , d13 ), av- is non-negligible only in the bottom panel, where it is set erage mode linewidth, rotational splitting, inclination angle at 0.5µHz. The large separation was set at 100µHz in each and height ratios. Some of these parameters can be fixed, or case. modes of degree ℓ = 3 can be omitted, based on the signalto-noise ratio and a priori information. The PSAC fits to five spectra are shown in Fig. 4. The fit to solar data (from the 3 Testing VIRGO/SPM instrument on the SoHO spacecraft) identifies the ℓ = 3 peak and illustrates the narrow solar line profiles. The pipeline was tested with 177 artificial Kepler-like time The CoRoT F stars have large mode linewidth which limits series each of 30-days duration (Mathur et al. 2009). The the extraction of small frequency spacing. The Kepler G- distributions of fitted parameters about their input values type stars have narrow, solar-like linewidths and clear small are shown in Fig. 5. The determination of νmax returned a frequency separations. In the case of KIC3656476, the cen- value in 83% of cases. The cases where no value was found tral peak has a triple-peaked appearance due to the close corresponded to artificial data with amplitudes of less than 3ppm. The accuracy of the returned value of νmax shows similarity between the values of d01 and d02 .


www.an-journal.org

Astron. Nachr. / AN ()

3

VIRGO/SPM [60d] 400

200

0

(µHz)

max

100

max

0.2 20

40

60

80

100

120

140

160

0

max

HD49933 [137d]

−100

ν

1

Mean absolute deviation of fitted ν

200

0.4

−300

0.6

−400 2

0.4 0

20

40

60

80

100

4

6

8 10 Amax (ppm)

12

40

60

80

100

ACF

∆ν [TSAC fitted] − ∆ν (µHz)

20

KIC6603624 [33d]

5

0

−5

−10

−15 2

0.8 0.6

4

6

8 10 Amax (ppm)

12

40 20 0 2

40

60

80

100

KIC3656476 [33d] 0.8 0.6 0.4

5

0

−5

−10

20

40

60 Frequency (µHz)

80

100 −15 2

4

6

A

max

8 10 (ppm)

12

Mean absolute deviation of fitted d01 (µHz)

(µHz) 01

[PSAC fitted] − d

01

d

2 0 −2 −4 −6

www.an-journal.org

1

1.5 2 2.5 Mean linewidth (µHz)

3

Mean absolute deviation of fitted d02 (µHz)

(µHz) 02

[PSAC fitted] − d

02

6

8 10 Amax (ppm)

12

14

4

6

8 10 (ppm)

12

14

1


3

3.5

1


3

3.5

4 3 2 1

7 6 5 4 3 2 1

A

5

4

3

2

1

6

6

d

4

5

0 0.5

3.5

8

4 2 0 −2 −4 −6 −8 0.5

14

6

4

−8 0.5

12

max

6

a dependence on the amplitude of the modes. For amplitudes up to 4ppm, the mean absolute deviation from the true value was ∼ 100µHz, but this reduces to ∼ 40µHz for amplitudes of 8ppm and less than 10µHz at 10ppm. The large spacing determination from fitting the PSAC shows an improvement in the accuracy of the value from that obtained from the TSAC. Again, there is a dependence on amplitude, but for the lowest amplitude data, the mean absolute deviation in fitted ∆ν decreases from ∼ 4µHz in the TSAC case to ∼ 2µHz in the PSAC case. For the small frequency spacings (d01 and d02 ) the accuracy is more strongly dependent on the mean ‘observed’ mode linewidth than the maximum mode amplitude. The ‘observed’ mode linewidth is a combination of the linewidths of individual multipole components (due to damping and excitation) and rotational splitting. The d01 spacing was found to be more readily obtained and more accurate than d02 . In cases where the mean linewidth of the modes was less than 1µHz, the mean absolute deviation in fitted d01 values was less than 0.4µHz, compared with 1.2µHz for d02 .

8 10 Amax (ppm)

6

0 2

14

8

Fig. 4 Fits to the PSAC function centred on νmax for five spectra - Sun (VIRGO/SPM), HD49933, HD181906, KIC6603624 and KIC3656476 (from top to bottom). Red lines show the fit to the data.

6

8

10

120 ∆ν [PSAC fitted] − ∆ν (µHz)

20

4

7

0 2

15

1 ACF

60

14

0.4

0

80

Mean absolute deviation of ∆ν [PSAC fitted] (µHz)

ACF

1

0

100

8

10

0

120

15

1

0.9

140

14

HD181906 [157d]

0.95

160

Mean absolute deviation of ∆ν [TSAC fitted] (µHz)

ACF

−200

0.8

180

(µHz)

300

0.6

[fitted] − ν

ACF

0.8

1


3

3.5

5

4

3

2

1

0 0.5

Fig. 5 Results of fits to artificial data. Left panels show the scatter of the fitted values from their real values, right panels show the histogram of the absolute deviations from the true value for each parameter with an error bar indicating the standard deviation in each bin. Results are shown for the fits to (from top to bottom): νmax , ∆ν (from TSAC fit), ∆ν (from PSAC fit), d01 and d02 small separations.


4

G.A. Verner & I.W. Roxburgh: Power spectrum autocorrelation techniques to detect asteroseismic parameters

4 Conclusions The power-spectrum autocorrelation provides a useful way to reliably detect global asteroseismic parameters in an automated way. For the best candidates, a full power-spectrum maximum-likelihood fit obtaining individual mode parameters is preferable, but in cases where this is not possible, or is complicated by low signal-to-noise, obtaining an accurate estimate of the large and small separations is desirable. For cases where full power-spectrum fitting is possible, it is still necessary to generate initial guess parameters for the fitting model, which can be estimated more easily if the average global parameters are known. Using the PSAC technique, it is found that the d01 small separation is more easily determined than the more common d02 small separation due to the influence of the d01 spacing creating a distinctive double or triple peak in the PSAC obtained at νmax . It has been shown that the d01 spacing is sensitive to the structure of the inner core and convective envelope (Roxburgh 2009b), it is therefore reassuring that this parameter may be detectable in cases where it is not possible to obtain a robust estimate of the d02 spacing. Acknowledgements. The authors thank the UK Science and Technology Facilities Council (STFC) which supported this work.

References Appourchaux, T., Michel, E., Auvergne, M., et al.: 2008, A&A, 488, 705 Benomar, O., Baudin, F., Campante, T. L., et al.: 2009, A&A, 507, 13 Chaplin, W. J., Appourchaux, T., Elsworth, Y., et al.: 2010, ApJL, arXiv:0912.3367 Garc´ıa, R. A., R´egulo, C., Samadi, R., et al.: 2009, A&A, 506, 41 Hekker, S., Broomhall, A.-M., Chaplin, W. J., et al.: 2010, MNRAS, 402, 2049 Huber, D., Stello, D., Bedding, T. R., Chaplin, W. J., Arentoft, T., Quirion, P.-O., Kjeldsen, H.: 2009, CoAst, 160, 74 Mathur, S., Garc´ıa, R. A., Regulo, C., Creevey, O. L., Ballot, J., Salabert, D.: 2010, A&A, arXiv:1003.4749 Roxburgh, I.: 2009a, A&A, 506, 435 Roxburgh, I.: 2009b, A&A, 493, 185


www.an-journal.org