Characterizing the Variable X-Ray and UV–Optical Flux Behavior of Blazars

We normalized our data with respect to the average flux for each data set. Outliers in both UV/optical and X-ray flux were eliminated from our data sets for all parts of our analysis utilizing the interquartile range (IQR). Such outliers were calculated to be observations that are either less than the difference between the median and 1.5× the IQR or greater than the sum of the median and 1.5× the IQR (Dovoedo & Chakraborti 2015):

$$\begin{eqnarray}&&\mathrm{lower}\ \mathrm{outlier}={Q}_{1}-(1.5\times \mathrm{IQR})\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&\mathrm{upper}\ \mathrm{outlier}={Q}_{3}+(1.5\times \mathrm{IQR}).\end{eqnarray} \tag{ 2 }$$

The outliers tended to be very distant from most of the observations in the data sets, and changing the factor of the IQR to values such as 1.4 and 1.6 yielded no difference in the analysis of the resulting data set. These points were eliminated if not found within a cluster of similar flux densities (20% above and below the flux of the observation in question within 10 days) that would indicate an X-ray flare or a duration of unusually high or low emission.

We utilized a time-delay method for each data set to gain a better understanding of flux changes on timescales of varying length. This was accomplished by calculating the time differences between one observation and all other observations within a data set of length n, given by

$$\begin{eqnarray}&&{\rm{\Delta }}t={t}_{n}-{t}_{n-1}.\end{eqnarray} \tag{ 3 }$$

The result from one iteration is an array of time offsets that contains the differences between point n in the data set and all other observations in the set. We repeated this process for each data point until a set of arrays containing the time differences between all observations was created, along with their corresponding fluxes. We sorted this array from the smallest to the greatest time difference to gauge the behavior of the source over Δt days.

Data sets with less than 50 observations were "stacked" into a group, in which the UV/optical and X-ray flux densities have been normalized by their respective average fluxes. Flux differences were determined as a function of the time difference (Δt) for each object, then combined afterwards. The stacking of these data sets results in a source containing 243 data points that is similar to a typical data set, allowing us to have sufficient sampling to compare sources with smaller data sets. The sources and their respective filters/bands are listed as: 1ES 1028 (B), 1RXS J003334.6-192130 (V), 1RXS J022716.6+020154 (B), 1RXS J111706.3+201410 (V), 1RXS J150759.8+041511 (V), 1RXS J153501.1+532042.1+532042 (B), H2356-309 (V), PG 1437+309 (V), PKS 2005-489 (V), and Ton 116 (B).

The sources that are within the stack are fainter than those with more observations, and overall errors (positive and negative) are greater for the stack than for its individual sources. The increased error results from both the effect of propagating errors for more than one source and the fact that the stacked objects had larger than average individual errors in their raw data.

When planning an observations, there is a time difference between choosing a target (based on its brightness) and the date the observation is obtained. For a space mission, this time difference can be weeks to months, so it is important to have a quantitative estimate of the typical flux variation as a function of time separation. One approach is the analysis of an X-ray- or UV/optical-triggered response in flux, and here we use a complementary approach, the structure function (SF), which is a numerical method that allows one to determine the variation between two time differences and its behavior as the time differences become longer. The shape of the SF can help determine the proximity of a source's behavior to certain types of noise, such as red noise, which is observed on QSOs on longer timescales, or white (1/f, flickering) noise, which is likely related to intrinsic disk or accretion processes (Manners et al. 2002).

The structure function provides the mean difference in the flux densities as a function of the separation in the sampling interval (e.g., Simonetti et al. 1985; Hufnagel & Bregman 1992). The measurement is defined as

$$\begin{eqnarray}&&\mathrm{SF}=\displaystyle \frac{1}{N}{\rm{\Sigma }}{[a(t)-a(t+{\rm{\Delta }}t)]}^{2}\end{eqnarray} \tag{ 4 }$$

where Δt represents the sampling interval in which the flux differences are subtracted, a is the flux at a given sampling interval, and N is the number of data points that are within the interval.

We set the standard sampling interval to one day with the time-delay data. Small sampling intervals of 0.1, 0.3, and 0.5 days were added for sources with multiple time-delay values that are below one day. These sources, Mrk 421, 3C 454.3, PG 1553+113, 2MASX J14283260+4240210, 3C 273, and PKS 2155-304, have an overall greater number of observations that are closer together in time. For each Δt interval, we averaged the sum of the squares of all flux differences that are within the interval to yield the structure function.

We utilized the standard deviation in the mean as the error of the structure function for each Δt. The first steps of the calculation are defined as

$$\begin{eqnarray}&&{\rm{\Delta }}a=a(t)-a(t+{\rm{\Delta }}t)\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&{\rm{\Delta }}{a}_{\mathrm{err}}=\sqrt{a(t{)}_{\mathrm{err}}^{2}+a(t+{\rm{\Delta }}t{)}_{\mathrm{err}}^{2}}\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&{\rm{\Delta }}{a}_{\mathrm{err}}^{2}={\rm{\Delta }}{a}^{2}\sqrt{2{\left(\displaystyle \frac{{\rm{\Delta }}{a}_{\mathrm{err}}}{{\rm{\Delta }}{a}^{2}}\right)}^{2}}\end{eqnarray} \tag{ 7 }$$

where Δa is the flux difference between two observations that are within the same sampling interval. This process is repeated for each flux difference within one Δt and then for each sampling interval.

One takes the standard deviation of the individual errors within one Δt, where there are n component values for a Δt. The standard deviation in the mean, or uncertainty for one particular sampling interval, is calculated in the usual way:

$$\begin{eqnarray}&&{\sigma }_{\mathrm{SF},\mathrm{mean}}=\displaystyle \frac{{\sigma }_{{\rm{\Delta }}{a}_{\mathrm{err}}^{2}}}{\sqrt{n-1}}.\end{eqnarray} \tag{ 8 }$$

Fitting the structure function is carried out parametrically in the log(SF)–log(Δt) space, assuming a log-normal distribution. Single-segment linear fitting (power law) is appropriate and utilized for sources with fewer than 100 observations, but most objects require a piecewise dual-segment linear fit. The more complicated fit allows one to determine a turnover point (xi) in which the SF flattens and exhibits behavior that switches from one noise pattern to another (i.e., red noise to white or "flickering" noise). Two variations of the simple linear equation of the form

$$\begin{eqnarray}&&{\rm{log}}({\rm{SF}})=m\,{\rm{log}}({\rm{\Delta }}t)+b\end{eqnarray} \tag{ 9 }$$

are utilized to fit our data for the dual-segment linear fit. Before the intersection point of the two lines (xi), the data are fitted to the first equation:

$$\begin{eqnarray}&&{\rm{log}}({\rm{SF}})={k}_{1}\,{\rm{log}}({\rm{\Delta }}t)+{a}_{1}\end{eqnarray} \tag{ 10 }$$

where y is the SF at a given Δt, k₁ is the slope of the first line and a₁ is the y-intercept. The second, flattened line of the piecewise curve fit with slope k₂ has a domain that begins at the turnover point (xi). The structure function value that corresponds to the Δt of the turnover and xi comprise the coordinates of the intersection point Λ:

$$\begin{eqnarray}&&{\rm{\Lambda }}={k}_{1}\times {xi}+{a}_{1}\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&{\rm{log}}({\rm{SF}})={k}_{2}({\rm{log}}({\rm{\Delta }}t)-{xi})+{\rm{\Lambda }}.\end{eqnarray} \tag{ 12 }$$

The four variables xi, k₁, k₂, and a₁ are returned from the curve fit function as log values. We set bounds for these parameters to ensure that the values returned are within the sampling interval used to fit the structure function. We limited the value of xi to be between −1 and 2.5 (in log-normal space), as the curve fit is applied for Δt that are between these orders of magnitude. The quantities k₁ and k₂ were set to be greater than zero, with k₂ limited to be between zero and k₁, creating a flatter slope.

In the single-segment linear fits, the only parameters that are determined are k₁ and a₁, and one equation (below) is utilized to fit the SF data:

$$\begin{eqnarray}&&{\rm{log}}({\rm{SF}})={k}_{1}{\rm{\Delta }}t+{a}_{1}.\end{eqnarray} \tag{ 13 }$$

The restriction applied is that k₁ must exceed zero (an object does not have negative variable behavior). Both fitting routines produce a two-dimensional array whose diagonals provide the variance of the parameter estimates, with the square root of the array diagonals yielding the standard deviation errors of each of the parameters.

The stack comprises 12 objects, all containing 50 or fewer individual observations in the X-ray, B, and V wave bands. The structure function for the stack is calculated by assessing the SF for each source within the stack separately before overlaying each scatter plot of the SF in the final figure.

The errors of the stack structure function are calculated by the same method as the other objects. The combined stack is utilized to determine the errors because the SFs of the individual objects do not always contain more than one observation in each Δt value with which to calculate σ_SF,mean.

A linear curve fit is applied to the time-delay and flux arrays of the combined stack to determine the variable behavior of the stack as if it were an individual object. A piecewise curve fit of the stack SF yields no turnover point, so the more complicated fit is not necessary.

The SF and curve fits were calculated for the most populated filter (UV/optical and X-ray) for every source (Figure 4, Table 4). Objects with over 50 individual observations provided the most precise curve fits and the smallest parameter uncertainties. The six sources with the most well-defined SF will be the primary subject of discussion, although the figures for SF calculations on smaller data sets are included. The SF curve fits (Figure 7) of 3C 454.3 (W1 and X-ray), Mrk 421 (M2 and X-ray), PG 1553+113 (W1 and X-ray), 2MASX J14283260+4240210 (U and X-ray), 3C 273 (V and X-ray), and PKS 2155-304 (V and X-ray) include small sampling intervals of 0.1, 0.3, and 0.5 days (or a combination of such intervals).

Most of the objects exhibit a piecewise curve fit that increases until it reaches a turnover point, where the slope of the fit decreases. The second slope (k₂) remains close to zero, with the average across all SF calculations at 0.18, and it is most often less than half of the first slope (k₁). The average k₁ value is about 0.61 in the X-ray and 0.57 in the UV/optical with an overall average of 0.59, but this parameter varies between 0.0 and 1.84 across all sources. The curve fits for Mrk 421 (X-ray, Figure 7), 3C 454.3 (W1, Figure 7), PG 1553+113 (X-ray, Figure 7), 2MASX J14283260+4240210 (X-ray, Figure 7), 3C 273 (X-ray, Figure 7), 1RXS J150759.8+041511 (V), S5 0836+71 (X-ray), 1RXS J111706.3+201410 (V and X-ray), 1ES 1028 (X-ray), H2356-309 (X-ray), PG 1437+389 (V and X-ray), Ton 116 (B and X-ray), and 1RXS J153501.1+532042 (B) have k₁ values that are between 0.7 and 1.3 (or within 30% of 1.0), which is consistent with red noise, which has a power-law index of 1.0.

A turnover point in either the UV/optical or X-ray occurs before 100 days in 10 out of the 12 sources that utilize a piecewise curve fit, and it occurs before 10 days in eight of the 12. Objects PKS 2155-304 (X-ray, Figure 7) and the stack (B + V and X-ray, Figure 7) exhibit white noise, or flickering, where the fit is mostly flat (k₁ ∼0), and do not feature a distinct turnover point. Objects 1RXS J022716.6+020154 (X-ray), 1RXS J150759.8+041511 (X-ray), 1RXS J151747.3+652522 (U and X-ray), and PKS 2005-489 (X-ray) also have power-law indices that are indicative of white noise, but the limited number of observations does not indicate a high degree of precision in their power-law indices.

Across all sources (Table 5), one observes that there are significantly smaller SF values for UV/optical data than for X-ray data. X-ray SF calculations for each sampling interval are typically 2–4 times larger than the UV/optical SF calculations for Δt < 300 days. Beyond these Δt, the X-ray SF values become closer to those in the UV/optical bands. A notable exception is object 1RXS J022716.6+020154, whose UV/optical values vary between 0.3× and 4× the X-ray values. Standard deviation error (in the mean) is typically 20%–80% of the average SF for each sampling interval at both the UV/optical and X-ray wavelengths. A complete figure set of the six most documented sources' curve fits (including the stack) is available to view in the online journal (seven images).

Table 5. SF Curve Fit Parameters and Uncertainties

Source Name	Wave band	Δt < 1 day	Δt	xi	±	k1	±	k2	±	a1	±	SF (30 Δt)	$\sqrt{\mathrm{SF}}$ (30 Δt)	SF (100 Δt)	$\sqrt{\mathrm{SF}}$ (100 Δt)
1RXS J003334.6-192130	V	N	175	⋯	⋯	0.014	0.196	⋯	⋯	−2.527	0.347	0.005	0.071	0.006	0.078
1RXS J003334.6-192130	X-ray	N	175	0.845	0.345	1.835	1.815	0.000	0.287	−2.179	1.154	0.242	0.492	0.242	0.492
1RXS J022716.6+020154	B	N	205	⋯	⋯	0.274	0.293	⋯	⋯	−2.658	0.563	0.004	0.059	0.006	0.075
1RXS J022716.6+020154	X-ray	N	205	⋯	⋯	0.000	0.291	⋯	⋯	−0.655	0.556	0.159	0.399	0.159	0.399
S5 0836+71	W2	N	200	1.898	0.946	0.273	0.233	0.041	0.701	−2.526	0.361	0.006	0.078	0.008	0.089
S5 0836+71	X-ray	N	200	1.913	0.270	0.743	0.206	0.010	0.672	−2.373	0.323	0.034	0.185	0.066	0.257
1ES 1028	B	N	236	1.929	1.488	0.615	0.445	0.341	2.184	−3.113	0.733	0.006	0.078	0.013	0.112
1ES 1028	X-ray	N	236	2.053	0.645	0.856	0.412	0.228	4.090	−2.213	0.702	0.112	0.334	0.261	0.511
Mrk 421	M2	Y	500	0.784	0.173	0.664	0.050	0.391	0.010	−1.92	0.030	0.076	0.276	0.114	0.337
Mrk 421	X-ray	Y	500	0.305	0.097	1.037	0.110	0.149	0.012	−0.663	0.059	0.705	0.839	0.816	0.903
1RXS J111706.3+201410	V	N	167	⋯	⋯	1.041	0.181	⋯	⋯	−3.982	0.324	0.007	0.082	0.008	0.091
1RXS J111706.3+201410	X-ray	N	167	⋯	⋯	0.144	0.204	⋯	⋯	−1.096	0.364	0.115	0.339	0.158	0.398
1RXS J122121.7+301041	W2	N	200	1.452	0.832	0.541	0.296	0.299	0.233	−2.50	0.330	0.018	0.134	0.025	0.159
1RXS J122121.7+301041	X-ray	N	200	1.000	0.650	0.454	0.582	0.000	0.152	−1.179	0.413	0.185	0.430	0.204	0.452
3C 273	V	Y	200	0.000	1.994	0.302	0.835	0.032	0.126	−2.631	0.488	0.003	0.051	0.003	0.052
3C 273	X-ray	Y	200	0.699	0.709	0.734	0.516	0.000	0.172	−1.736	0.279	0.056	0.237	0.056	0.237
Ton 116	B	N	154	⋯	⋯	1.29	0.376	⋯	⋯	−4.387	0.676	0.003	0.055	0.018	0.135
Ton 116	X-ray	N	154	⋯	⋯	0.968	0.334	⋯	⋯	−2.797	0.601	0.041	0.202	0.151	0.389
2MASX J14283260+4240210	U	Y	150	0.661	0.621	0.635	0.339	0.0911	0.128	−2.295	0.183	0.016	0.126	0.018	0.133
2MASX J14283260+4240210	X-ray	Y	150	0.716	0.270	0.925	0.214	0.131	0.094	−1.870	0.120	0.078	0.279	0.091	0.302
PG 1437+389	V	N	250	1.204	1.397	1.249	11.823	0.045	0.189	−3.808	8.745	0.003	0.055	0.006	0.078
PG 1437+389	X-ray	N	250	1.041	2.818	0.957	3.994	0.606	0.277	−2.119	3.346	0.138	0.372	0.289	0.538
1RXS J150759.8+041511	V	N	144	⋯	⋯	0.711	0.340	⋯	⋯	−2.853	0.592	0.016	0.126	0.038	0.195
1RXS J150759.8+041511	X-ray	N	144	⋯	⋯	0.000	0.314	⋯	⋯	−1.142	0.547	0.074	0.273	0.074	0.273
1RXS J151747.3+652522	U	N	200	⋯	⋯	0.222	0.132	⋯	⋯	−2.569	0.205	0.006	0.076	0.007	0.086
1RXS J151747.3+652522	X-ray	N	200	⋯	⋯	0.000	1.956	⋯	⋯	−0.762	0.305	0.173	0.416	0.173	0.416
1RXS J153501.1+532042	B	N	155	⋯	⋯	1.188	0.295	⋯	⋯	−4.162	0.515	0.004	0.061	0.019	0.137
1RXS J153501.1+532042	X-ray	N	155	⋯	⋯	0.275	0.370	⋯	⋯	−1.479	0.645	0.083	0.289	0.124	0.353
PG 1553+113	W1	Y	350	1.842	0.097	0.682	0.041	0.270	0.057	−2.358	0.061	0.045	0.211	0.087	0.295
PG 1553+113	X-ray	Y	350	0.888	0.196	0.835	0.105	0.349	0.027	−1.449	0.066	0.314	0.561	0.479	0.692
PKS 2005-489	V	N	206	⋯	⋯	0.380	0.221	⋯	⋯	−3.477	0.353	0.001	0.036	0.003	0.053
PKS 2005-489	X-ray	N	206	⋯	⋯	0.584	0.199	⋯	⋯	−2.353	0.318	0.035	0.187	0.109	0.330
PKS 2155-304	V	Y	250	⋯	⋯	0.454	0.010	⋯	⋯	−2.180	0.195	0.033	0.183	0.056	0.236
PKS 2155-304	X-ray	Y	200	⋯	⋯	0.089	0.102	⋯	⋯	−0.878	0.199	0.186	0.431	0.203	0.450
3C 454.3	W1	Y	150	0.766	0.159	0.792	0.091	0.208	0.037	−1.542	0.051	0.331	0.575	0.384	0.620
3C 454.3	X-ray	Y	150	1.06	0.116	0.663	0.059	0.123	0.041	−1.233	0.044	0.187	0.423	0.210	0.458
H2356-309	V	N	300	⋯	⋯	0.133	0.162	⋯	⋯	−2.505	0.334	0.005	0.070	0.005	0.077
H2356-309	X-ray	N	300	1.866	0.290	0.915	0.314	0.000	0.372	−2.354	0.415	0.100	0.316	0.221	0.471
stack	B, V	N	100	⋯	⋯	0.138	0.104	⋯	⋯	−2.531	0.159	0.971	0.986	0.976	0.988
stack	X-ray	N	200	⋯	⋯	0.240	0.100	⋯	⋯	−1.331	0.154	1.049	1.024	1.049	1.024

Note. Δt notes the sampling interval out to which the curve fit is applied, and the SF and its square root at 30 and 100 days present the source's variability on a timescale of one to three months (a factor to consider when observing variable behavior or mission planning, which takes place on a similar timescale). Fields marked with ⋯ indicate that the curve fit used for that data set is a single fit, rather than a piecewise fit, therefore parameters k₂ and xi are not applicable.

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To conclude, the six most observed sources and the stack were fitted to piecewise linear or single-segment linear curve fits to determine the variability behavior of the source's flux and the turnover point at which the source's SF flattened and switched between noise patterns. Of the six sources and the stack, four have turnover points less than 10 days, and PKS 2155-304 and the stack appear to "flicker" in the X-ray and UV/optical. Of all objects, Mrk 421 (X-ray, Figure 7), Ton 116 (X-ray), and 1RXS J111706 (V) have the closest k₁ values to that of red noise (1.0), and the average value for this parameter is 0.61 in the X-ray and 0.57 in the UV/optical across all 19 sources and the stack. The UV/optical SF values are significantly smaller than those of the X-rays until Δt ∼ 300 days for most sources.

The X-ray fractional flux variation (square root of the SF) can provide insight when mission planning because its value at a specific sampling interval allows one to infer the variation of the source's flux at a given Δt. These variation predictions will be more accurately obtained with a greater number of observations, though the flux of a source may change drastically after selecting it for observation. The structure function and variation of each source at 30 and 100 days are shown in Table 5, which characterizes the difference between a monitoring observation and a potential mission observation, respectively.

The sources with the greatest SF values and variation at 30 days are 1RXS J022716.6+020154, Mrk 421, 1RXS J122121.7+301041, and 1RXS J150759.8+041511 (all X-ray), with variations that are within 30% of a 1.0 normalized flux unit, indicating that the X-ray flux varies, on average, by this amount at 30 days from the initial observing time. These sources are followed by 3C 454.3 (X-ray and W1), the stack, 1RXS J111706.3+201410, and PG 1553+113 (the last three in the X-ray), with variations of 0.5–0.7 normalized flux units.

To mitigate the presence of potentially low flux measurements, we consider that a strategy using an X-ray or UV/optical trigger (instead of all of the data) to observe a source will more likely yield overall greater luminosities and improve mission planning.

The structure function calculations show that, relative to an observation prior to constructing an observing schedule (at time t₀), the object can dim enough that the nominal goal for the signal-to-noise ratio is not achieved. This approach does not make use of the object flux at t₀ (an untriggered strategy), but we find that one can improve the result by committing to an observation only if the flux at t₀ is above some flux level, taken as a fraction of the mean flux (the triggered strategy). This can be a valuable procedure because, in mission planning, one must set up an observing schedule days or weeks in advance, therefore necessitating a reasonably accurate flux forecasting method to determine whether a source is bright enough to observe.

We utilized our minimum-triggering procedure with binned data sets exceeding 50 observations, including the stack that we created. We created threshold fluxes that are specific percentages of the calculated mean. We set these thresholds as 0.9F_avg, 1.0F_avg, and 1.1F_avg to compare with the entire binned data set to take into account the most probable range of fluxes that would be documented in an observing run.

For each of these limits, we took the average of the X-ray fluxes in each bin that met or exceeded the calculated threshold flux, or averaged all of the fluxes within the bin if there were no fluxes exceeding the trigger. The result is an array of calculated averages that correspond to each bin. Higher and lower threshold percentages were also considered, but increased triggering values greatly diminished the percentage of the data set with which one can utilize triggering.

The penalty for using a triggered response is that the trigger criteria will only be satisfied a fraction of the time. Thus, one would monitor several targets with the goal that at least one will lie above the trigger value to fill out an observing program. A trade-off exists between the trigger value and the fraction of observations exceeding the trigger for the X-ray observations of 3C 454.3, where 50% of the observations lie below 0.88F_avg and 75% lie below 1.31F_avg (Figure 5).

Using a trigger as a gateway for observing requires that multiple objects be monitored at the scheduling stage, with the requirement that one object will exceed the trigger a large fraction of the time. As an example, we consider a total sample of 20 targets distributed around the sky, with five that are at optimum sky position for the telescope and are monitored at the moment of making a scheduling decision. We can apply binomial probabilities and calculate that for a trigger excluding 50% of the fluxes (five trials), one or more sources will exceed the trigger 98% of the time. For a stricter trigger that requires the flux to be in the upper 25% of the fluxes, this rate of success drops to 76%, This indicates a scheduling failure about one-quarter of the time, which is probably an unacceptable situation. For this reason, we only consider triggers where the rate of success lies in the 90%–98% range, and this roughly corresponds to flux trigger values of 0.9F_X,avg–1.1F_X,avg, although we examine this below more carefully because the variability behavior is different among targets.

We will compare the resulting, averaged fluxes that are produced by X-ray triggers and those that are produced by UV/optical triggers to determine which method is ideal for triggering an X-ray observation. While it is probable that a brighter X-ray triggering threshold will yield overall brighter X-ray flux averages (increasing the threshold will raise the average, and lower fluxes within the bin will be eliminated), the advantage of utilizing a UV/optical trigger is not as guaranteed.

We rebinned the data into larger intervals because it is often the case that there are few flux differences in one-day intervals. We set a bin size of four days in data sets larger than 20 observations from this time-offset array. This span of time allowed us to adequately simplify our data for threshold. This binning is applied to the time-delay triggering analysis, where a resolution of less than one week is adequate. By binning, there are enough measurements in a bin to apply Gaussian statistics, rather than Poisson statistics, which are optimally used in discrete cases and are more difficult to analyze for this purpose.

6.1.1. Error Propagation of Triggering Procedure

We utilize the first and third quartiles to examine the spread of the data in each bin. The asymmetric vertical error bars comprise two different values. The section of the error bar that is above the binned and averaged flux, ${\rm{\Delta }}{\bar{F}}_{\mathrm{bin},\mathrm{upper}}$ is defined as

$$\begin{eqnarray}&&{\rm{\Delta }}{\bar{F}}_{\mathrm{bin},\mathrm{upper}}={Q}_{3}-{\bar{F}}_{\mathrm{bin}}\end{eqnarray} \tag{ 14 }$$

where Q₃ is the third quartile of the flux measurements in the bin. The lower section of the error bar that is below the binned and averaged flux, ${\rm{\Delta }}{\bar{F}}_{\mathrm{bin},\mathrm{lower}}$, is defined as

$$\begin{eqnarray}&&{\rm{\Delta }}{\bar{F}}_{\mathrm{bin},\mathrm{lower}}={\bar{F}}_{\mathrm{bin}}-{Q}_{1}\end{eqnarray} \tag{ 15 }$$

where Q₁ is the first quartile of the flux measurements in the bin. This method is utilized in our investigation to measure the spread of fluxes in each bin instead of the standard deviation in the mean because the former procedure shows us the different flux ranges above and below the calculated average and does not assume that the underlying distribution of flux measurements within a bin is Gaussian.

We present results with triggers that are 0.9F_X,avg, 1.0F_X,avg, and 1.1F_X,avg in each data set for X-ray and UV/optical triggering. Section 6.4 examines and compares such trends resulting from each of the X-ray and UV/optical triggering techniques, followed by an analysis of the results as factors for selecting observing durations.

The binned and averaged flux of each source remains close to the mean calculated flux with no trigger, rarely deviating more than 20% above or below this calculated mean. Untriggered fluxes tend to vary more from their previous value with increasing Δt until a point is reached (usually occurring before 50 days) at which the flux variation at a given time remains the same or becomes difficult to predict (flickers). This behavior occurs in nearly all objects exceeding 50 individual observations, which is illustrated in our structure function analysis.

The advantages of X-ray triggering vary widely by source. As a whole, the 1.1F_X,avg trigger yields the greatest increase in the resulting X-ray flux (compared to no trigger). Selecting increasingly higher trigger fluxes from 0.9F_X,avg to 1.1F_X,avg produces consistently higher resultant fluxes when observing an object while it is bright. However, an object exceeds a trigger for only a fraction of the time, so there are fewer trigger opportunities for another higher flux. A triggering value is in the range 0.9F_X,avg–1.1F_X,avg, because it will lead to successful triggers around 35%–52% of the time while avoiding poor observational outcomes in the absence of triggering. Table 6 illustrates the percentage of observations exceeding these values (and their median counterparts) for the six objects with the best data (Figure 5).

The advantage to raising a trigger's percentage of the average flux is more evident in well-documented sources (over 100 data points). 3C 454.3 (Figure 2) exhibits similar X-ray triggering behavior to Mrk 421 (Figure 2) and PG 1553+113 (Figure 2). Across all filters, setting a trigger that is 0.9F_X,avg yields a brightness advantage that is fairly consistent from 0 to 30 days when compared to fluxes produced by no trigger, though the exact percentages vary among the three sources. This advantage continues up to 100 days.

Sources 2MASX J14283260+4240210 (Figure 2) and PKS 2155-304 (Figure 2) each exhibit similar brightness advantages with respect to each trigger, though their individual percentage advantages over the untriggered data are different. In the former object, raising the X-ray observing trigger from 0.9F_X,avg–1.0F_X,avg and 1.0F_X,avg–1.1F_X,avg both produce an average 5% increase in resulting flux, with the 1.0F_X,avg–1.1F_X,avgg trigger increase generating a slightly higher (7%–9%) brightness advantage from 0 to 24 days. PKS 2155-304 is a highly variable source that appears to flicker with time, but each trigger increase typically yields a varying 0%–10% flux advantage.

Table 7 describes the flux brightness advantages produced by each X-ray trigger in comparison to untriggered X-ray flux (averaged into four-day bins) and summarizes the results in Figure 6. The complete figure set (seven images) can be viewed in the online journal.

Overall, increasing the trigger percentage of the average X-ray flux produces advantages that are emphasized in well-documented sources and less evident in those that flicker or have fewer observations. We set triggering values of 0.9F_X,avg, 1.0F_X,avg, and 1.1F_X,avg to produce higher resulting fluxes without sacrificing too many observing opportunities; these threshold values produce successful triggers that occur 35%–52% of the time. The highest X-ray flux advantages appear in Mrk 421 (Figure 4), 3C 454.3, PG 1553+113, and PKS 2155-304 (Figure 6), though the latter features irregular flux behavior from 0 to 300 days. Mrk 421 appears to "sustain" the advantage produced by 0.9F_X,avg, 1.0F_X,avg, and 1.1F_X,avg triggers for the longest period of time, followed by other bright, variable sources such as 3C 454.3 and PG 1553+113 (Table 8).

A brighter UV/optical trigger does not yield as great an advantage between thresholds as X-ray-triggered results, although it appears as if a brighter UV/optical trigger yields overall higher flux averages than a less bright UV/optical trigger (Tables 9 and 10). Mirroring the X-ray threshold triggering procedure, we also assume that the observing period will occur within 60 days of the last known monitoring observation, but larger Δt of up to 100 days are also illustrated.

These advantages vary extensively by source. In general, a 1.0F_O,avg trigger yields X-ray fluxes that are typically 5%–10% greater than those resulting from a 0.9F_O,avg trigger, though some data sets have too few data points or possess flux advantages that are not discernible from the results of the previous trigger. The largest increases (76% and 81%) are observed in the V band of source 1RXS J122121.7+301041 at roughly 50 and 100 days, respectively. Increasing the trigger to 1.1F_O,avg outputs binned averages that are modestly brighter than those triggered by 1.0F_O,avg and to roughly the same magnitude as the previous trigger increase, if not slightly lower at 3%–10% from 0 to 100 days.

Unlike the X-ray-triggered results, some increases in UV/optical triggering flux yield lower flux measurements, which indicates that the UV/optical and X-ray fluxes do not always positively correlate. Mrk 421, 3C 454.3, PG 1553+113, 2MASX J14283260+4240210, 3C 273, PKS 2155-304, and the stack feature at least one decrease in resulting flux that exceeds 20% any time from 0 to 100 days for either triggering increase. Though increasing the UV/optical observing threshold value generally yields a higher X-ray flux with each increase, it comes at the cost of fewer data points. Returning to the six most well-documented sources and the stack, Figure 3 illustrates the average X-ray flux resulting from each UV/optical trigger and the advantages of setting this trigger at a higher percentage of the average UV/optical flux. The left panels illustrate the binned and averaged flux for the first 100–500 Δt. The green vertical lines represent the error of each bin for the 1.0F_O,avg triggered data. The right panels compare the binned and averaged fluxes produced by each triggering value and the untriggered flux. The advantage is calculated by dividing the flux yielded by the trigger by the untriggered flux value for the same Δt. The complete figure set (seven images) is available in the online journal.

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View figure set (7 panels)

Tarball of figure set images

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View figure set (7 panels)

Tarball of figure set images

To conclude, we find that a brighter UV/optical trigger produces overall higher X-ray flux averages across the most documented objects. However, this is not the case in all objects, and some, including Mrk 421 and 3C 454.3, sometimes display an anticorrelations between their X-ray and UV/optical fluxes. Also, triggers set by UV/optical data are less successful than those set by using X-ray monitoring observations Much like the results of X-ray triggering, the UV/optical-triggered findings indicate that the advantages of raising the trigger are not as evident for objects that flicker or contain fewer observations, especially as increasing the trigger threshold comes at the cost of fewer data points.

We observed trends in the data sets that confirmed an overall advantage to X-ray triggering over UV/optical triggering upon the creation of our bins and application of the threshold method. Sources that exhibit a UV/optical triggering advantage over an X-ray trigger are defined as yielding a higher average flux when this UV/optical trigger was used, in comparison to an X-ray trigger of the same percentage of the average X-ray flux.

We found that out of the 51 data sets, five illustrated a higher optically triggered flux than X-ray-triggered averages (for over half of the time period of 0–100 days), seven exhibited a temporary optically triggered advantage, two had too few observations to apply binning and triggering techniques, and the rest (37 data sets) did not yield a higher optically triggered flux for each 10% increase in the triggering percentage (Table 11).

3C 454.3 (B, M2, U, W1, W2) is the object that possesses this advantage more than 50% of the time from 0 to 100 days, indicating that there is a good chance that the X-ray flux will remain above average for at least 20 days using UV/optical triggers. It is worth noting that this source is particularly bright in X-rays and has been well documented for over 10 yr, with data sets ranging from 294 to 356 individual observations.

Objects with more observations are more likely to have a correlation between X-ray and UV/optical fluxes, positive or negative. Our data suggest that accurate measurements are more likely to lead to strong correlations; large uncertainties in the observations of smaller data sets yielded correlations that were either poor or indeterminate. Although an increase in a UV/optical observing threshold is more likely to be paired with an increase in overall X-ray flux, the result is less bright than an X-ray-triggered flux in nearly all cases.

Some objects do not appear to fit into either group and exhibit optically triggered advantages (for 1.0F_O,avg or 1.1F_O,avg triggers) sporadically or for short periods of time for under half of the timescale. These sources are 1RXS J122121.7+301041 (W1), 2MASX J14283260+4240210 (B, U, W1, W2), 3C 273 (M2, U), and PKS 2155-304 (V). In these instances, the optically triggered flux tends to yield an inconsistent, modest advantage (on average 5%–10%) and features one or two large advantages of up to 25%, usually with the 1.1F_O,avg UV/optical trigger between 40 and 100 days.

There are three ways an object is observed by Swift: objects scheduled in programs without trigger conditions, calibration sources (sources that are regularly observed because of their brightness), and observations that are triggered by bright outbursts, often targets of opportunity (TOOs). For the first category, there is no bias by brightness; this is also true for the calibration sources. However, a brightness trigger or TOO can provide a biased view of the variability and skew the triggering results, but this information is not retained for such observation. The most extreme outbursts that cause TOOs will produce outliers, which we exclude.

Most sources observed frequently were not TOO objects, according to information provided to us by the Science Operation Center. However, in objects such as Mrk 421, 3C 454.3, PKS 2005-489, and 1RXS J122121.7+301041, up to a third of their observations appear to occur at irregular intervals ranging from one to six or more months, with some exposures occurring more than once a day or for several consecutive days, indicating that they were observed as a TOO for a fraction of their exposures or were not observed due to their brightness.

Our investigation has a bearing on mission planning for the scientific missions of Arcus, Athena, or Lynx. A goal is to avoid observing targets when they are significantly fainter than normal, which would compromise the signal-to-noise ratio of the observation. The time between scheduling an observation (based on its brightness at the time) and its execution can be months, so we investigated flux variation over time periods of 30 and 100 days.

We utilized the square root of the structure function to quantify an object's variability at a given sampling interval (Δt). The most variable sources at 30 days (in X-rays) are 1RXS J022716.6+020154 and Mrk 421 (as measured by variabilities exceeding 100% of the previous flux value), followed by 1RXS J122121.7+301041 and 1RXS J150759.8+041511 (with variabilities exceeding 0.8 normalized flux values). In the UV/optical wavelengths, 3C 454.3 is the most variable object at 30 days and has a variability of about 0.51 normalized flux units. The objects with the next highest variabilities have values that are approximately half of the variability of 3C 454.3 for the same sampling interval, indicating that the UV/optical variability of 3C 454.3 is unusually high compared to other AGNs at 30 days, though these variabilities are more similar across objects in the X-ray band.

The X-ray variability of the targets at 100 days is on average 30% greater than their variability at 30 days (190% greater for UV/optical), and the X-ray flux still appears to vary to a greater extent than the UV/optical. The objects with the highest X-ray and UV/optical variabilities at 100 days (PG 1437+389, Mrk 421, PKS 2155-304, 1RXS J150759.8+041511, and 3C 273) also exhibit the greatest ranges in flux, especially in the X-ray band, making their flux more difficult to predict. The stack behaves like a typical source at 30 and 100 days, with X-ray and UV/optical variabilities comparable to those of other objects. For both sampling intervals, the X-ray variability is four to five times larger than the UV/optical variability, and the variability at 100 days is 50% and 26% greater than the variability at 30 days for the X-ray and UV/optical bands, respectively.

One wants to avoid observing an object for a period of time in which its flux is lower than its triggering threshold. We investigate the length of time for which the flux of the top six objects remains below the 1.0F_X,avg triggering threshold in Figure 10.

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We utilized X-ray and UV/optical triggering to improve the likelihood of observing an object at an adequately high flux (generally meeting or exceeding the range 0.9F_X,avg–1.1F_X,avg). Objects with over 100 observations exhibit a more evident brightness advantage when the trigger's percentage of the average flux is increased. Compared to no trigger, fluxes from these objects that are triggered by a value of 0.9F_X,avg are consistently brighter from 0 to 30 days and remain at an advantage from 30 to 100 days, though this ranges from approximately 5% up to 80% for the six most documented objects. Increasing the triggering value to 1.0F_X,avg and 1.1F_X,avg leads to fluxes that are usually around 5%–10% brighter than the flux yielded by the 0.9F_X,avg trigger, but this comes at the cost of fewer successful triggers.

Generally, it would not be ideal to optically trigger results because they yield lower X-ray fluxes than those that are triggered by X-rays. Out of 19 objects and the stack, one (3C 454.3) suggests that optically triggered X-ray fluxes are brighter than those triggered by X-rays for at least half of the timescale from 0 to 100 days (discussed below); four sources (2MASX J14283260+4240210, 3C 273, PKS 2155-304, and 1RXS J122121.7+301041) illustrate a temporary or short-lived (25–50 days) advantage to UV/optical triggering, and the rest of the objects (79%, including the stack and its individual objects) do not suggest an advantage. On average, the 0.9F_O,avg trigger produces fluxes that are 11.7% less luminous than those triggered by 0.9F_X,avg, and these disadvantages of UV/optical triggering for 1.0F_O,avg and 1.1F_O,avg are 13.0% and 14.7%, respectively.

3C 454.3 is the only object that illustrates strong advantages of UV/optical triggering. However, as outlined in Section 7.1, there exists the possibility that selective observation (TOO) yields fluxes that are higher than the typical day-to-day behavior of the source, skewing our triggering results higher (even in the absence of outliers). Thus, to eliminate the effect of prompted observations on our triggering analysis, one would need to utilize a data set composed of entirely regularly monitored observations.

Some objects (2MASX J14283260+4240210, 3C 273 in the B, M2, W1, W2 bands, PKS 2005-489, and S8 0836+71 in the M2 band) present optically triggered fluxes for a significant time frame (over 30 out of 100 days) that are up to 20% less bright than the X-ray-triggered flux of the same triggering threshold, but still produce brighter flux than utilizing no trigger at all. As there is not a large difference between the triggered fluxes, and UV/optical triggering can be realized on the ground, it may be a viable option to optically trigger X-ray observations to mitigate costs or save time for these targets.

Our 19 sources are primarily blazars, which would be the main targets for extragalactic X-ray absorption line studies. Our understanding of variability behavior will be expanded by extending our X-ray and UV/optical variability analysis and triggering procedure to other classes of AGNs. Geometric factors such as the inclination of the quasar's accretion disk, the direction of the jet, and the size or the mass of the central black hole may change the patterns and correlations seen between the X-ray and UV/optical flux.

One can also trigger observations with other wavelengths of light if it is more cost-effective or easily realized, though UV/optical triggering is usually preferred when observing on the ground. Current databases of AGNs typically have a reasonably large quantity of observations at such wavelengths for well-documented objects, so new, monitored data may be required for a more thorough analysis.