Maximum likelihood fitting of X-ray power density spectra: Application to high-frequency quasi-periodic oscillations from the neutron star X-ray b

📝 Abstract

High frequency quasi-periodic oscillations (QPOs) from weakly magnetized neutron stars display rapid frequency variability and high coherence with quality factors up to at least 200 at frequencies around 850 Hz. Their parameters have been estimated so far from standard min(chi2) fitting techniques, after combining a large number of Power Density Spectra (PDS), as to have the powers normally distributed. Accounting for the statistical properties of PDS, we apply a maximum likelihood method to derive the QPO parameters in the non Gaussian regime. The method presented is general, easy to implement and can be applied to fitting individual PDS, several PDS simultaneously or their average, and is obviously not specific to the analysis of kHz QPO data. It applies to the analysis of any PDS optimized in frequency resolution and for low frequency variability or PDS containing features whose parameters vary on short timescales, as is the case for kHz QPOs. It is equivalent to the standard chi^2 minimization fitting when the number of PDS fitted is large. The accuracy, reliability and superiority of the method is demonstrated with simulations of synthetic PDS. We show that the maximum likelihood estimates of the QPO parameters are asymptotically unbiased, and have negligible bias when the QPO is reasonably well detected. By contrast, we show that the standard min(chi2) fitting method gives biased parameters with larger uncertainties. The maximum likelihood fitting method is applied to a subset of archival Rossi X-ray Timing Explorer (RXTE) data of the neutron star X-ray binary 4U1608-522. We show that the kHz QPO parameters can be measured on 8 second timescales and that the time evolution of the frequency is consistent with a random walk. This enables us to estimate the intrinsic quality factor of the QPO to be around 260, whereas previous analysis indicated a maximum value around 200 (abridged).

💡 Analysis

High frequency quasi-periodic oscillations (QPOs) from weakly magnetized neutron stars display rapid frequency variability and high coherence with quality factors up to at least 200 at frequencies around 850 Hz. Their parameters have been estimated so far from standard min(chi2) fitting techniques, after combining a large number of Power Density Spectra (PDS), as to have the powers normally distributed. Accounting for the statistical properties of PDS, we apply a maximum likelihood method to derive the QPO parameters in the non Gaussian regime. The method presented is general, easy to implement and can be applied to fitting individual PDS, several PDS simultaneously or their average, and is obviously not specific to the analysis of kHz QPO data. It applies to the analysis of any PDS optimized in frequency resolution and for low frequency variability or PDS containing features whose parameters vary on short timescales, as is the case for kHz QPOs. It is equivalent to the standard chi^2 minimization fitting when the number of PDS fitted is large. The accuracy, reliability and superiority of the method is demonstrated with simulations of synthetic PDS. We show that the maximum likelihood estimates of the QPO parameters are asymptotically unbiased, and have negligible bias when the QPO is reasonably well detected. By contrast, we show that the standard min(chi2) fitting method gives biased parameters with larger uncertainties. The maximum likelihood fitting method is applied to a subset of archival Rossi X-ray Timing Explorer (RXTE) data of the neutron star X-ray binary 4U1608-522. We show that the kHz QPO parameters can be measured on 8 second timescales and that the time evolution of the frequency is consistent with a random walk. This enables us to estimate the intrinsic quality factor of the QPO to be around 260, whereas previous analysis indicated a maximum value around 200 (abridged).

📄 Content

arXiv:1112.0535v1 [astro-ph.HE] 2 Dec 2011 Maximum likelihood fitting of X-ray power density spectra: Application to high-frequency quasi-periodic oscillations from the neutron star X-ray binary 4U1608-522 Didier Barret1 Institut de Recherche en Astrophysique et Plan´etologie & Universit´e de Toulouse (UPS), 9 avenue du Colonel Roche, BP 44346, 31028, Toulouse Cedex 4, France didier.barret@irap.omp.eu and Simon Vaughan X-Ray and Observational Astronomy Group, University of Leicester, Leicester, LE1 7RH, U.K. ABSTRACT High frequency quasi-periodic oscillations (QPOs) from weakly magnetized neutron stars display rapid frequency variability (second timescales) and high coherence with quality factors up to at least 200 at frequencies about 800-850 Hz. Their parameters have been estimated so far from standard min(χ2) fitting techniques, after combining a large number of Power Density Spectra (PDS), as to have the powers normally distributed (the so-called Gaussian regime). Before combining PDS, different methods to minimize the effects of the frequency drift to the estimates of the QPO parameters have been proposed, but none of them relied on fitting the individual PDS. Accounting for the statistical properties of PDS, we apply a maximum likelihood method to derive the QPO parameters in the non Gaussian regime. The method presented is general, easy to implement and can be applied to fitting individual PDS, several PDS simultaneously or their average, and is obviously not specific to the analysis of kHz QPO data. It applies to the analysis of any PDS optimized in frequency resolution and for low frequency variability or PDS containing features whose parameters vary on short timescales, as is the case for kHz QPOs. It is equivalent to the standard χ2 minimization fitting when the number of PDS fitted is large. The accuracy, reliability and superiority of the method is demonstrated with simulations of synthetic PDS, containing Lorentzian QPOs of known parameters. Accounting – 2 – for the broadening of the QPO profile, due to the leakage of power inherent to windowed Fourier transforms, the maximum likelihood estimates of the QPO parameters are asymptotically unbiased, and have negligible bias when the QPO is reasonably well detected. By contrast, we show that the standard min(χ2) fitting method gives biased parameters with larger uncertainties. The maximum likelihood fitting method is applied to a subset of archival Rossi X-ray Timing Explorer (RXTE) data of the neutron star X-ray binary 4U1608- 522, for which, we show that the lower kHz QPO parameters can be measured on timescales as short as 8 seconds. As to demonstrate the potential use of the results of the maximum likelihood method, we show that in the observation analyzed the time evolution of the frequency is consistent with a random walk. We then show that the broadening of the QPO due to the frequency drift scales as √ T, as expected from a random walk (T is the integration time of the PDS). This enables us to estimate the intrinsic quality factor of the QPO to be ∼260, whereas previous analysis indicated a maximum value around 200. Subject headings: accretion, accretion disks, stars: neutron, X-rays: binaries, X-rays: stars 1. Introduction The standard method of weighted least squares (i.e. minimum χ2, hereafter min(χ2)) model fitting is only equivalent to maximum likelihood estimation (MLE) of the model parameters when the data to be fitted are normally (Gaussian) distributed. As is well known, the distribution of M averaged periodogram ordinates (from a stationary, linear stochastic process) follows a χ2 2M distribution with 2M degrees of freedom; as M increases, the χ2 2M tends towards a normal distribution (Groth 1975; Papadakis & Lawrence 1993). In X-ray timing the most commonly used method to reach the so-called Gaussian regime is Bartlett’s method (Bartlett 1948): segment the original time series into M non-overlapping segments, compute a periodogram for each segment, and average over all segments to produce a spectral estimate to be fitted (van der Klis 1989). Typically, one uses M > 50 to produce approximately Gaussian distributed averages (Papadakis & Lawrence 1993). The drawbacks of this method are a loss of frequency resolution by a factor M (reducing the sensitivity for narrow feature detection) and a suppression of the lowest frequencies. There is also a major drawback in the case of the analysis of high frequency quasi-periodic oscillations (QPOs), because their frequency varies rapidly with time, typically on timescales of seconds (Barret et al. 2005b). Since a kHz QPO can be a relatively narrow feature (FWHM ∼2 −4 – 3 – Hz), sub-Hz frequency resolution of the PDS (equivalent to a segment duration of a few seconds) is required to have the QPO profile properly sampled. In order to use min(χ2) fitting one must average a large number of segments, M, which then requires a total integration time exceeding hundreds, even thousands of seconds, depending of the

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