Astron. Astrophys. 319, 515-524 (1997)

3. Power spectra

The root mean square residuals technique used in this section for the estimation of red noise power density and associated random walk noise strengths is discussed in detail by Cordes (1980) and Deeter (1984). The applications made by Baykal et al., (1993a, b) are the examples of its use in characterizing random walk processes.

For the case of th-order red noise (or r th-time integral of white noise time series) with strength , the mean squared residual for data spanning an interval T is proportional to . The proportionality factor (or normalization coefficient) depends on the degree m of the polynomial removed prior to computing the mean square residual; this factor can be obtained by determining the expected mean square residual for unit strength red noise () over a unit interval (), either by Monte-Carlo methods (Cordes 1980) or by mathematical evaluation (Deeter 1984). The mean square residual, after removing a polynomial of degree m over an interval of length T, is then given by

where is the proportionality factor which has been derived from a unit-strength noise process. Our preliminary analysis showed that the angular velocity residuals can be characterized by first order red noise (or random walk). This noise process is adopted in computing the normalization coefficients. The normalization coefficient is computed by Monte Carlo simulation, sampling the simulated data as the real data set was sampled (Cordes 1980). Because most of the BATSE data are approximately equally spaced at 1 day intervals, the normalization coefficient differs by no more than 15 from the theoretical prediction for equally spaced data (assuming a sampling interval of one day) as given in Table 1 and 2 of Deeter 1984.

As a next step, time scales are sampled at nearly octave spacings for the power density estimators (Deeter 1984). To do this the entire length of the data is taken as the longest time-scale. Then the intervals are halved successively to get down to the shortest practical time scale. Intervals with insufficent data points are discarded. The mean squared residuals for the remaining intervals are calculated and converted into the noise strengths () using Eq. 1. The contributions of the measurement errors have been computed by converting the estimated variances of angular velocities into noise strengths and by employing the identical procedure above.

Estimated noise strengths on each time scale are combined into a single power density estimate (, where f is the frequency, Rice (1954); Boynton (1981)) by averaging, and are represented by a frequency equal to the inverse of the time scale. In order to estimate the power density error bars, the effective number of degrees of freedom, for each estimate is determined by generating its statistical distribution by using Monte Carlo simulations of a first order red noise process, sampled in the same way as our angular velocity data (Blackman & Tukey 1958; Deeter & Boynton 1982). Then the power density error bars are computed for a given effective number of degrees of freedom by computing the power density values corresponding to the and points of the distribution; these are equivalent to confidence level for a Gaussian (Blackman & Tukey 1958). In Fig. 3, we display the resulting power density estimates (or noise strengths) of the angular acceleration fluctuations.

Fig. 3. Power spectrum of angular accelerations (or noise strengths, see the text). The asterisks denote the measurement errors.

In the analysis, we have found the crossover frequency between the errors and noise strengths around days). We have fitted a straight line to the 8 lower octave power density estimates and found which is consistent with white noise in the angular accelerations () with the mean power density (or noise strength) rad² /sec³.

In order to crudely estimate the step size of the fluctuations and the rate R of torque events we make use of the autocorrelation of angular acceleration ) time series. We took the numerical time derivative of the angular velocity () time series at time scales corresponding to the cross over frequency (see Fig. 7b) and represented the autocorrelation function in Fig. 4. Autocorrelation function of white noise time series can be expressed as (Jenkins & Watts 1968),

where is the white noise variance at zero time lag . As seen from Fig. 4, actually the autcorrelation function shows correlation for time lags () up to 8-19 days.

Fig. 4. Autocorrelation function of angular accelerations time series.

The step size of the fluctuation during this time scale can be estimated
. By using the values rad² /sec⁴ and , from the autocorrelation function and the random walk noise strength estimate from the power spectrum we estimated the rate of torque events as

This implies a torque event occurs within days and lasts days. These events appear as a delta function (or unresolved events) if we are studying the long time scale fluctuations and make the power spectra of flat at low frequency (or long time scales days). But at high frequencies (or short time scales days), it is quite natural to see decrease in power density estimates. If we express the white noise time series as a sum of Gaussians , with a small finite variance , then we would expect the power spectrum to have the form
. Therefore at frequencies one should see a decrease in the power spectra at order of . This is qualitatively seen in our low resolution power spectra (see Fig. 3).

© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
helpdesk@link.springer.de