Spectral analysis (Section 4.4).
T
charTperiodvalues are evaluated in the frequency domain as the primary peak of the Power Spectral Density function (PSD), which is computed from the Autocorrelation Function (ACF). The ACF also enables alternative definitions ofT
charto be used, and these are reviewed in Section 4.4. In addition, spectral purity is indicated in the time domain by the extent of the first negative ACF overshoot, while in the frequency domain it is indicated by the absence of secondary PSD peaks.Tperiodis determined from the PSD, with scintle counting and the data folding plot providing cross-checks. This is because a scintle counting exercise does not usually consider the entire time series, while in a data folding exercise there is often some minor uncertainty between the frequencies defined by the plot of amplitude versus folding frequency and the plot of residuals versus folding frequency (see Section 4.3).
One disadvantage of defining
T
charTperiodis that the quasi-random nature of scintillationmeans the variability in a time series is also only quasi-periodic. Scintles observed over an observing period of, say, 10-15 days are not expected to occur at precisely regular intervals or with the same morphology. However, the scintle intervals, amplitudes and widths are consistent enough that
T
charTperiodis a meaningful definition, while the negative overshootof the ACF, and the strength of any secondary PSD peaks, provide measures of the spectral purity of the signal.
Defining
T
charTperiodalso leans towards a determination that interstellar scintillation is theprincipal cause of the flux density variability. Flux density variations intrinsic to the source might be quasi-periodic in nature, but the variability time scales of the sources studied by this research, PKS B1622-253 and PKS B1519-273, display annual cycles, which proves interstellar scintillation to be the cause of the variability in these sources (see Chapter 2). Also, interpreting flux density variability on time scales of days as interstellar scintillation, rather than due to variability of the source, reduces the implied minimum brightness temperature, which in the source-intrinsic interpretation can be many orders of magnitude above the inverse Compton limit.
Ceduna data contain systematic fast (diurnal) small amplitude flux density variations. These fluctuations are removed by the data processing procedure discussed in Section 4.5, but the variability in unprocessed PKS B1144-379 data from (southern source) observing period 4 is clear enough to illustrate the various approaches to determining Tperiod, described below.
4.2
Scintle Counting
Scintle countingrelies on being able to visually identify peaks and troughs in a flux density
time series. This is possible in the case of Ceduna data, and hence this method is used by this research to provide an empirical estimate of Tperiod. Figure 4.1 shows a scintle counting
exercise applied to PKS B1144-379 data for observing period 4, giving an estimate of Tperiod
6 days / 2.5 scintles = 2.4 days.
Figure 4.1 Scintle counting for PKS B1144-379 data for days 160-170 in 2003.
Scintle counting provides a cross-check on Tperiodvalues determined by spectral analysis, and
the fact that the flux density variability can be recognised by eye using scintle counting provides confidence in these Tperiodvalues.
Scintle peaks and troughs may not be obvious in time series with noisy, low amplitude variability. The data processing methodology presented later in this chapter converts the data to a zero-mean time series, and counting the number of zero-crossing events can help to separate true peaks and troughs from spurious rises and dips.
Daily mean flux density plots are routinely produced for all Ceduna data (see Chapter 6), primarily for comparison to calibrator sources. However, these plots sometimes assist Tperiod
estimates when the Tperiodvalues are on the order of days. Figure 4.2 shows the daily mean
flux densities (intensities) of the raw PKS B1144-379 data in Figure 4.1. The plot clearly supports the conclusion that Tperiodis just over 2 days.
Figure 4.2 Scintle counting using daily mean intensities of PKS B1144-379 (circles) and PKS B1934-638 (dots) from observing period 4.
Flux density change rate. Alternative empirical approaches to examining flux density time
series variability are based on estimating the rate of change of flux density, I. Variations on
this theme include:
I d dt Tchar ln Burbidge et al. (1974) 1 max max dt dI I Tchar McAdam (1978)where dIis the rise or fall in flux density over time dt. Definitions of the variability time
scale in terms of flux density change rates are useful when the data are limited and do not fully capture a scintle. This is the case, for example, with observations of blazar variability using facilities such as the Australia Telescope Compact Array, which are often limited by competing demands for telescope time to sessions of 12 hours or so.
Bignall (2003) notes that the flux density change method places a limit on the size of the emitting region, assuming intrinsic variability, but can also overestimate
T
charif a significantfraction of the source’s emission is non-variable. Blazar emissions are highly variable, and at least some of their radio variability is certainly intrinsic to the source, but that portion of a blazar’s emissions associated with a large emission region are non-variable. Fortunately, the Ceduna monitoring program gathered data on a daily basis over many months, avoiding the need to use flux density change methods because scintle periods can be observed directly.
4.3
Data Folding
The assumption of quasi- periodicity in Ceduna blazar data is a fairly good one, although inspection of the flux density time series shows that scintles do not occur at strictly regular intervals, nor with regular morphologies. Fortunately, quasi-periodicity is sufficient for a periodicity search method to yield a credible estimate of the characteristic variability time scale,
T
charTperiod.Periodicity searches obviously work best for time series that are at least several times Tperiod
in length. They can be made either in the frequency domain by computing a periodogram to estimate the power spectral density; or in the time domain, through data folding. The two approaches are equivalent if the periodogram is appropriately defined (Scargle, 1982). The
Matlabsignal processing toolbox provides several periodograms, which were found to
produce nearly identical results to a data folding procedure written by the author. It was decided to use the data folding method to estimate Tperiodin the flux density time
series, although a periodogram could have done the job as well. The data folding method is to fold the data at a trial period, Ttrial(for example, folding a time series at a trial period of 2
days means the time series is divided into 2 day long sections which are then superimposed), subtracting the average value of each data fold so the final data set has a zero mean. A sine wave of the form
t T A t x trial 2 sin ) (
is fitted to the folded data, with the wave amplitude, A, and the phase, , as free parameters.
This was done by applying a non-linear least squares optimisation technique, using the
Matlabfunction lsqnonlin. The residuals are the differences between the folded data and
the best-fit sine wave. This exercise is repeated across a range of trial frequencies, and
Tperiodcorresponds to the frequency at which the minimum sum-squared residual occurs. The
data folding method usually produces a Tperiodcross-check value that agrees with the Tperiod
Occasionally more than one minimum appears in the plot of residuals as a function of data folding frequency. The two possible reasons for this are that either a second variability time scale is present, or the multiple minima are due to harmonics. If the frequency spacing suggests the minima are due to harmonics, two additional data folding plots can assist in identifying the correct characteristic variability frequency, fperiod= 1/ Tperiod. First, a plot of
the variation with frequency of the best-fit wave amplitude, A. Second, a plot of residuals
verses data folding frequency, with the data binned into hourly phase bins.
Figure 4.3 shows this method applied to the PKS B1144-379 data from observing period 4 (i.e. the data shown in Figure 4.1). Tperiodcorresponds to the frequency at which there are
minima in the two residual plots, and a maximum in the amplitude plot. Inspection of these plots in Figure 4.4 makes it clear that 0.42 days-1is the actual variability frequency, and that
the feature at 0.21 days-1is only a sub-harmonic of this frequency.
Figure 4.3 Data folding analysis of the PKS B1144-379 data for observing period 4.
The solid vertical line indicates the variability frequency of 0.42 days-1, corresponding to a peak in the amplitude of the fitted sinusoid, and troughs in both the residual and the 1 h phase bin residual plots.
Figure 4.4 shows the data folding and phase bin plots at the frequency of 0.42 days-1. This
corresponds to a variability period value of Tperiod2.38 days, an estimate that agrees well
Figure 4.4 Data folding and phase bin plots at the variability frequency of 0.42 days-1.
In Figure 4.3, the low residual (sum square error) values that characterise the data folding plot at high frequencies are also associated with near-zero amplitude values. This is simply a consequence of folding a time series on time scales much shorter than its actual variability. A data folding exercise should not be applied to a time series which is so long that the value of Tperiodchanges significantly. Figure 4.5 presents the results of a data folding exercise
carried out on a PKS B1519-273 time series that is 87 days long. Chapters 6 and 7 show that the variability time scale, Tperiod, of this source follows an annual cycle, and the value of
Tperiodchanges significantly
over 87 days, which smears out the data folding features.
Figure 4.5 Data folding of an overly long time series, here 87 days of 2003 PKS B1519-273 data.
Finally, a folding plot is clumpy if Tperiodis close to an integral number of days. Figure 4.6
shows this in the case of PKS B1519-273 data from northern source group observing period 14, which has a period of 3 days, producing four clumps of data in a data folding plot given that the observing period is 10 days long.
Figure 4.6 A clumpy data folding plot, from examination of PKS B1519-273 data for days 58 to 67 in 2004, with Tperiod3 days.