Figure 7.8 shows the TPeriodvalues for both PKS B1622-253 and PKS B1519-273, including
error bars. The time axis spans the 365 days of a normal year, and data gathering started at about days 150 and 100 of 2003 respectively for the two sources. Cycle one refers to data gathered within 12 months of the start date. Cycle two refers to data gathered thereafter.
Figure 7.8 TPeriodvalues for PKS B1622-253 (top) and PKS B1519-273 (bottom),
and superimposed annual cycles for baseline parameters (see text).
Figure 7.8 shows that the variation of TPeriodvalues of both sources agree fairly well with the
annual cycles expected if scintillation is due to interstellar scattering, as described by the annual cycle model presented in Chapter 2. The model is defined by five parameters:
so Characteristic scintle spatial scale. This is the coherence scale, so(km). The
characteristic scintillation time scale is so V , where Vis the velocity with
which the scintillation pattern moves transversely across the line of sight.
R, Anisotropy. The spatial pattern of scintles is assumed to form elliptical contours
with axial ratio R(this was given the symbol in Chapter 2), and the major axes
of the ellipses are inclined at angle to the direction of the scintillation velocity.
V, V ISM – LSR velocity. The ISM scattering material generally has a velocity offset
with respect to the velocity of the Local Standard of Rest, and {V V} are the
The baseline annual cycles shown in Figure 7.8 correspond to reasonable values of these parameters, namely: {R V V so} = {1, 0 rad, 0 km/s, 0 km/s, qx 106km }, where q = 9
for PKS B1622-253 and q = 5 for PKS B1519-273. Figure 7.9 shows the effect on the
annual cycles of small changes to these baseline values in the case of PKS B1622-253.
Figure 7.9 Sensitivity of PKS B1622-253 Tperiodannual cycles to changes in the
baseline parameter values (black lines). PKS B1519-273 is similar. The arrows show the location of annual cycle peaks and troughs.
PKS B1622-253 and PKS B1519-273 lie along similar lines of sight from the Earth, and have similar TPeriodannual cycles, and sensitivity tests on the PKS B1519-273 annual cycle
have similar results to those shown in Figure 7.9. Slow downs (i.e. increases in the TPeriod
values) near days 50 and 250 are robust features of both annual cycles, the slow down near day 250 being the more pronounced of the two.
Characteristic scintle spatial scale. Changes in sohave a scaling effect: increasing soslows
down the variability time scale, since so V . The annual cycle is accentuated, with a
greater difference between the TPeriodvalues in the slow down and speed up times of year,
but it does not affect the location of the annual cycle’s peaks and troughs.
Anisotropy. The convention used herein to describe anisotropy is that R1, with = 0
denoting the case in which the major axis of the elongated scintle is aligned with the scintillation velocity. The value of affects the annual cycle as follows (see Section 2.6):
When = 0, the majoraxis of the elliptical scintles is aligned with the scintillation
velocity. The TPeriodannual cycle is flatter than the R= 1 case, as the Earth encounters
scintle peaks and troughs more slowly during speed-up times of year, and more quickly during the slow-down times. The locations of theTPeriodslow down peaks near days 50
and 250 are shifted to slightly later and earlier days respectively. The locations of the
TPeriodspeed-up troughs near days 150 and 340 are not much affected.
When = 90, the minoraxis of the elliptical scintles is aligned with the scintillation
velocity. The TPeriodannual cycle is accentuated compared to the R= 1 case, as the Earth
encounters peaks and troughs faster during speed-up times of the year, and more slowly during the slow-down times. The locations of theTPeriodslow down peaks near days 50
and 250 are shifted to slightly earlier and later days respectively. Again, the location of the speed-up troughs near days 150 and 340 are not much affected.
ISM – LSR velocity offset. For these two sources, a {V V} velocity offset primarily
influences the TPeriodannual cycles during the slow-down times of year, but hardly affects
the speed-up times of year.
A positive Vincreases TPeriodat the slow down times of year, accentuating the annual
cycle, while a negative Vreduces TPeriodat the slow down times of year, flattening the
annual cycle. The locations of the slow-down times of year are essentially unchanged.
Changing V has little effect on the annual cycle for these two sources. A positive V
causes the location of the slow-down times of year near days 50 and 250 to shift to slightly later and earlier days respectively, while a negative Vcauses the reverse effect. Best fit annual cycles
Figure 7.10 shows the results of multi-parameter optimisation of the annual cycle model fits to theTPerioddata for PKS B1622-253 and PKS B1519-273, with all five model parameters
free to change. The black solid lines are the optimum annual cycle fits. These lie within the error bars of most data points, the error bars being ~95% (i.e. 2-sigma) confidence limits (Section 6.4.2).
Table 7.1 gives the best fit annual cycle parameter values. For both sources the optimum annual cycle is for highly anisotropic scintles and large LSR velocity offsets, particularly V.
This is unsurprising: significant velocity offsets are expected for scintillation with relatively long variability time scales (days rather than hours), which is associated with more distant scattering screens that typically lie hundreds of parsecs from Earth, and thus are in motion with respect to the LSR.
Anisotropic scintles are associated with other blazar observations, notably PKS 0405-385 (Rickett et al., 2002), and J1819+3845 (Macquart & de Bruyn 2006). Many authors have argued that, for scintillation due to scattering by the ISM, electron density fluctuations are expected to be elongated along the local direction of the magnetic field (e.g. Goldreich & Sridhar 1995; Chandran & Backer 2002). Alternatively, anisotropicity may be due to an anisotropic source structure, instead of the scattering medium, a possibility that is explored later in this chapter.
Figure 7.10 Annual cycle model fits to Tperioddata for PKS B1622-253 (top) and PKS
B1519-273 (bottom). Blue circle data points are first year, red cross data points are second year. All-parameter best fits shown as solid black lines. Best fits for R=5 shown as dash blue lines.
PKS B1622-253 R (rad) V(km/s) V(km/s) so(10 6 km) Optimum 18.0 4.0 0.17 0.02 -37.5 3.0 8.4 0.5 9.2 0.9 R= 5 5.0 0.35 -17.1 7.9 7.9 {R} {VV}so 1.9 1.5 -0.4 3.7 9.9 PKS B1519-273 R (rad) V(km/s) V(km/s) so(10 6 km) Optimum 15.1 3.0 0.07 0.02 -52.0 7.0 12.1 1.0 7.1 0.4 R= 5 5.0 0.185 -29.0 8.3 6.1 {R} {VV}so 1.3 1.9 -4.1 3.6 5.7 Table 7.1 Annual cycle model best-fit values for a) All parameters; b) R constrained
Anisotropy – velocity offset interplay
The blue dashed lines in Figure 7.10 show the best fit annual cycles when the anisotropy ratio is constrained to R5, with the best-fit values in Table 7.1 being for R= 5. Table 7.1
also gives the results of selected optimisation exercises: optimisation of {R} is made with
the other parameters fixed at the baseline values associated with Figure 7.8, and similarly for optimisation of {VV}, and optimisation of so.
These exercises show the interplay between scintle anisotropy and the LSR velocity offset, particularly V. The best fit all-parameter annual cycles to the TPerioddata are for significant
anisotropy and velocity offsets which, as noted, is not surprising. However, the best-fits for the individual parameter optimisations require the anisotropy to be small if the LSR velocity offsets are constrained to be zero; and require the best-fit velocity offsets to be small if the scintles are constrained to be isotropic. This behaviour is also evident in the R= 5 best fit
annual cycles, which have significantly lower LSR velocity offsets than the optimum fits.
Goodness of Fit and error estimates
The chi-squared statistic, 2, is the sum-squared error of the residuals, weighted by the error
bars of the data points, with greater weight given to the model fit through data points with small error bars. The extent to which an annual cycle model fits the data can be assessed using the reduced chi-square statistic, 2
Red
(e.g. Press et al., 1986):
2 2 2 2 1 0 N i i i x f m N 2 2 Red where Nis the number of data points, iis the error bar associated with data point xi, and fi
is the best fit annual cycle model value corresponding to this data point. The annual cycle model has N – mdegrees of freedom, where m= 5 parameters. The Red2 value estimates
the ratio of the variance of a model fit, to the variance of the data being modelled. A model is good if 2 1
Red
. If 2 1
Red
, the error bars are probably too large. If 2 1
Red
, either the error bars have been underestimated or the data are not well described by the model. Evaluating these statistics gives the results in Table 7.2.
PKS B1622-253 PKS B1519-273
N 21 14
2 33.89 7.45
2/ (N-m) 2.12 0.83
Table 7.2 Chi-squared and Reduced Chi-Squared statistics.
which clearly show that the annual cycle models for the two sources both fit the data well; and that the empirically determined error estimates (Chapter 6) are appropriate and neither too large nor too small. A gauge of the significance of the detection of an annual cycle in these sources is the comparison of the 2and 2
Red
values in Table 7.2 with the values associated with “fitting” the data by a linear model in which TPeriod= constant. Choosing the
constant to be the mean TPeriodvalues of 4.68 days and 2.26 days for PKS B1622-253 and
PKS B1519-273 respectively gives the following goodness-of-fit values: PKS B1622-253 2= 334 2 Red = 20.9 PKS B1519-273 2= 90 2 Red = 10.0.
Figures 7.11 and 7.12 show the variation of 2 2
2 minin the neighbourhood of theoptimum model parameters. Constant 2boundaries enable confidence levels (i.e. error
bars) to be estimated for model best fit parameters. A change in a parameter with respect to its best-fit value causes a change in chi-square with respect to its minimum value, 2= 2–
(2)
min, and the confidence level associated with 2can be determined either analytically
or by Monte Carlo simulations. In general it depends on the number of degrees of freedom (DOFs), and Press et al. (1986) tabulate commonly needed results.
If each parameter is considered separately, the one DOF values of 2are appropriate, and 2= 1.00 is the 1-sigma (68%) confidence level, while 2= 2.71 is the 90% confidence
level. If two parameters are considered jointly, the two DOF values of 2are appropriate,
and 2= 2.3 is the 1-sigma (68%) confidence level.
Consider, for example, the top sub-plot of Figure 7.11, which shows 2= 1 and 2.3
contours for the variation of the velocity offset components, {V V}. The 1-sigma error
bar for V considered separately is found by projecting the 2= 1 contour limits onto the Vaxis, giving an error bar of 2.0 km/s, and a similar projection of the 2= 1 contour
onto the Vaxis gives an error bar of 0.3 km/s. The 2= 2.3 ellipsegives the 1-sigma
confidence regionif the {V V} two parameters are considered jointly (Press et al., 1986).
However, it is not appropriate to set the 1-sigma error bars for the annual cycle model as the parameter changes corresponding to the one DOF values of 2. To see this, consider the
best fit annual cycles under the R= 5 constraint (dash lines) in Figure 7.10. These are quite
similar to the unconstrained best fit annual cycles (solid black lines), and their reduced chi- square values of 2
Red
= 2.28 and 3.18 for PKS 1519-273 and PKS 1622-253 respectively indicate that the R= 5 constrained models describe the data fairly well, although not as well
Figure 7.11 Variation of 2= 2– (2)min with model parameters in the vicinity of
their best-fit annual cycle values, for PKS 1622-253. In each graph, the non-variable model parameters are fixed at their best-fit values.
2So0.6 x 106 2V4 km/s 2V0.6 km/s
2 0.01 rad
Figure 7.12 Variation of 2= 2– (2)min with model parameters in the vicinity of
their best-fit annual cycle values, for PKS 1519-273. In each graph, the non-variable model parameters are fixed at their best-fit values.
2R 3.7
2 0.015 rad 2V9 km/s
2V1
This leads to the conclusion that the annual cycle models are not particularly sensitive to the interplay between scintle anisotropy and the LSR velocity offset, and the 1-sigma error bars corresponding to the one DOF value of 2= 1.00 are too small because this joint variation
needs to be considered. Table 7.1 therefore gives 1-sigma (68%) error bars corresponding to parameter changes that result in the two DOF value of 2= 2.30, including the spatial scale
parameter, so. These error bars were estimated from 2= 2.30 contours in {R V} plots
and {R V} plots, and are similar to those obtained from the 2= 2.30 contour projections
in Figures 7.11 and 7.12.