An Application of HilbertHuang Transform on the NonStationary Astronomical Time Series: The Superorbital Modulation of SMC X1
 Author: Hu ChinPing, Chou Yi, Wu MingChya, Yang TingChang, Su YiHao
 Organization: Hu ChinPing; Chou Yi; Wu MingChya; Yang TingChang; Su YiHao
 Publish: Journal of Astronomy and Space Sciences Volume 30, Issue2, p79~82, 15 June 2013

ABSTRACT
We present the HilbertHuang transform (HHT) analysis on the quasiperiodic modulation of SMC X1. SMC X1, consisting of a neutron star and a massive companion, exhibits superorbital modulation with a period varying between ~40 d and ~65 d. We applied the HHT on the light curve observed by the AllSky Monitor onboard
Rossi Xray Timing Explorer (RXTE ) to obtain the instantaneous frequency of the superorbital modulation of SMC X1. The resultant Hilbert spectrum is consistent with the dynamic power spectrum while it shows more detailed information in both the time and frequency domains. According to the instantaneous frequency, we found a correlation between the superorbital period and the modulation amplitude. Combining the spectral observation made by the Proportional Counter Array onboardRXTE and the superorbital phase derived in the HHT, we performed a superorbital phaseresolved spectral analysis of SMC X1. An analysis of the spectral parameters versus the orbital phase for different superorbital states revealed that the diversity ofn_{H} has an orbital dependence. Furthermore, we obtained the variation in the eclipse profiles by folding the All Sky Monitor light curve with orbital period for different superorbital states. A dip feature, similar to the preeclipse dip of Her X1, can be observed only in the superorbital ascending and descending states, while the width is anticorrelated with the Xray flux.

KEYWORD
Xray binaries , superorbital modulations , HilbertHuang transform , SMC X1

1. INTRODUCTION
The highmass Xray binary SMC X1 consists of a 1.06 M_{⊙} neutron star (van der Meer et al. 2007) and a 17.2 M_{⊙} supergiant (Reynolds et al. 1993). The orbital period of this system is ~3.89 d obtained from the Xray eclipse. Gruber & Rothschild (1984) found that the SMC X1 exhibits an Xray periodicity with a rough time scale of 60 d. This periodicity was then confirmed by Wojdowski et al. (1998) and the cycle lengths were reported to vary with time. The behavior of the period variation was investigated after several timefrequency analysis techniques, like the dynamic power spectrum (Clarkson et al. 2003) and the slide LombScargle periodogram (Trowbridge et al. 2007), were applied on the AllSky Monitor (ASM) light curve. We present our analysis by using of a recently developed timefrequency analysis technique, the HilbertHuang transform (HHT), to explore the timing properties of the superorbital modulation. Furthermore, based on the phase defined in the HHT, the superorbital phaseresolved analysis on both the spectral behaviors and the variation of orbital profiles are also presented.
2. OBSERVATIONS
The dwell data, collected by the ASM onboard
Rossi Xray Timing Explorer (RXTE ) with ~ 90 minute timing resolution, were used to explore the detailed variation of orbital profile of SMC X1. The data were further binned into oneday lightcurve to investigate the variation of superorbital period. Furthermore, the data collected by the Proportional Counter Array (PCA) onboard
RXTE were also included to obtain the variation of spectral behavior in the different superorbital states.3. THE HILBERTHUANG TRANSFORM
The HHT, which is proposed by Huang et al. (1998), is designed to analyze the nonlinear and nonstationary time series. The normalized Hilbert transform was used to obtain the instantaneous frequency of the superorbital modulation. However, the Hilbert transform can only be applied on the intrinsic mode functions (IMFs) to get meaningful instantaneous frequency. We used the ensemble empirical mode decomposition, proposed by Wu & Huang (2009), to decompose the ASM light curve into several IMFs.
4. TIMEFREQUENCY ANALYSIS
After decomposing the light curve into several IMFs, we applied the normalized Hilbert transform on them to obtain the Hilbert spectrum. Fig. 1 shows the Hilbert spectrum that contains information of frequency and amplitude variation in a color map, which is stacking with the dynamic power spectrum in contour for comparison. We noticed that the Hilbert spectrum is basically consistent with the dynamic power spectrum, but the resolutions are better in both the time and frequency domains. Since the HHT provided us a welldefined phase of the superorbital modulation, we folded the light curve according to the superorbital phase to obtain the asymmetric superorbital profile, which is shown in Fig. 2. Furthermore, we obtained a significant correlation
between the modulation period and the amplitude shown in Fig. 3. This correlation is similar to that between the mainon flux and the superorbital period of Her X1 (Still & Boyd 2004). Both of the examples are inconsistent with the prediction of radiationinduced warp disk (Wijers & Pringle 1999).
5. SUPERORBITAL PHASERESOLVED ANALYSIS
We further studied the superorbital phaseresolved spectroscopic properties of SMC X1 by using all the PCA observations during 1996 and 2004. The standard2 mode spectrum can be fitted with a cutoff power law continuum and a Gaussian emission line centered at 6.4 keV. The
variation of the uneclipsed hydrogen column density (
n_{H} ) versus the superorbital phase is shown in the middle panel of Fig. 4. We noticed that then_{H} values show great diversity in the ascending state but remain stable in the descending state. By further checking the orbital distribution ofn_{H} in the different superorbital states, we obtained that the diversity might be caused by the dip feature occurred in orbital phase ~ 0.7.The dip feature could be further examined by the folded light curve. Since the appearance of dip may have superorbital dependence, a twodimensional folded light curve is a good way to investigate the dip properties. We perform this analysis method on the dwell light curve collected by the ASM. We first folded the Xray photons in the data window of superorbital phase 0.00.05 according to the orbital ephemeris proposed by Wojdowski et al. (1998)
and normalized the uneclipsed count rate to 1. Then, we move the window 0.01 superorbital cycles forward to obtain the next orbital profile. This process was repeated until the end of the data set. The combined folded light curve, drawn in a threedimensional color map, is shown in Fig. 5. We can easily obtain the dip feature in the dynamic folded light curve, especially that occurred in superorbital descending state. From this figure, we also noticed that the dip width has an anticorrelation with the Xray count rate. This dip feature could be associated with the preeclipse dip of Her X1 (Moon & Eikenberry 2001).
6. CONCLUSIONS
This research demonstrated a timefrequency analysis of the nonstationary superorbital modulation of SMC X1 by the HHT. The Hilbert spectrum shows great detail in both the time and frequency domains. Furthermore, we found a correlation between the superorbital period and amplitude, which is inconsistent with the outer warp model. Based on the phase derived in the HHT, we studied the phaseresolved spectra and the variation of orbital profile. The variation in both the
n_{H} and the orbital profile indicate that the system contains a preeclipse dip. The width variation of the dip can be interpreted by the warp and tilted disk model.

[Fig. 1.] The Hilbert spectrum and the dynamic power spectrum of SMC X1.

[Fig. 2.] Folded light curve of superorbital modulation of SMC X1.

[Fig. 3.] The correlation between the superorbital modulation period and the amplitude. The rank correlation coefficient is r = 0.46 with a null hypothesis probability value of 3.2 × 106.

[Fig. 4.] The unabsorbed flux (top panel), nH (middle panel) and powerlaw index (bottom panel) verse the superorbital phase. Black, blue, red, and green symbols represent the spectral parameters in the low, ascending, high, and descending states, respectively.

[Fig. 5.] Twodimensional dynamic folded light curve of SMC X1.