# K2 Optical Emission from OJ 287 and Other $\gamma$-Ray Blazars on Hours-to-
Weeks Timescales from 2014–2018

Ann E. Wehrle Space Science Institute, 4765 Walnut Street, Suite B, Boulder,
CO 80301, USA  Michael Carini Western Kentucky University, Department of
Physics and Astronomy, 1906 College Heights Blvd., Bowling Green, KY 42101,
USA  Paul J. Wiita The College of New Jersey, Department of Physics, 2000
Pennington Rd., Ewing, NJ 08628-0718, USA  Joshua Pepper Lehigh University,
Department of Physics, 413 Lewis Laboratory, 16 Memorial Drive East,
Bethlehem, PA 18015, USA  B. Scott Gaudi The Ohio State University, Department
of Astronomy, 140 W. 18th Ave., Columbus, OH 43210, USA  Richard W. Pogge The
Ohio State University, Department of Astronomy, 140 W. 18th Ave., Columbus, OH
43210, USA  The Ohio State University, Center for Cosmology & AstroParticle
Physics, 191 West Woodruff Ave., Columbus, OH 43210, USA  Keivan G. Stassun
Vanderbilt University, Department of Physics & Astronomy, 6301 Stevenson
Center Ln., Nashville, TN 37235, USA  Steven Villanueva, Jr. NASA Postdoctoral
Program Fellow  The Ohio State University, Department of Astronomy, 140 W.
18th Ave., Columbus, OH 43210, USA  NASA Goddard Space Flight Center,
Exoplanets and Stellar Astrophysics Laboratory (Code 667), Greenbelt, MD
20771, USA

(Version of 29 March 2023. Accepted for publication in the Astrophysical
Journal.)

###### Abstract

We present second observations by K2 of OJ 287 and 7 other $\gamma$-ray AGNs
obtained in 2017–2018, second and third observations of the lobe-dominated,
steep spectrum quasar 3C 207, and observations of 9 additional blazars not
previously observed with K2. The AGN were observed simultaneously with K2 and
the Fermi Large Area Telescope for 51–81 days. Our full sample, observed in
2014–2018, contained 16 BL Lac objects (BL Lacs), 9 Flat Spectrum Radio
Quasars (FSRQs), and 4 other $\gamma$-ray AGNs. Twelve BL Lacs and 7 FSRQs
exhibited fast, jagged light curves while 4 BL Lacs and 2 FSRQs had slow,
smooth light curves. Some objects changed their red-noise character
significantly between repeated K2 observations. The optical characteristics of
OJ 287 derived from the short-cadence K2 light curves changed between
observations made before and after the predicted passage of the suspected
secondary supermassive black hole through the accretion disk of the primary
supermassive black hole. The average slopes of the periodogram power spectral
densities of the BL Lacs’ and FSRQs’ light curves differed significantly, by
$\approx 12$%, with the BL Lac slopes being steeper, and a KS test with a
$p$-value of 0.039 indicates that these samples probably come from different
populations; however, this result is not as strongly supported by PSRESP
analyses. Differences in the origin of the jets from the ergosphere or
accretion disk in these two classes could produce such a disparity, as could
different sizes or locations of emission regions within the jets.

AGN: individual: OJ 287 — Galaxies: active — Galaxies: jets

††software: EVEREST (Luger, R., Agol, R., Kruse, E., Barnes, R., Becker, A.,
Foreman-Mackey, D., & Deming, D. 2016, AJ, 152,100), K2SFF (Vanderburg, A. &
Johnson, J. A. 2014, PASP, 126, 948)

##  1 Introduction

Strong, rapid variability is one of the prime characteristics of the blazar
class of active galactic nuclei. Both the historic sub-categories of blazars,
the BL Lacertae objects (BL Lacs) and the flat spectrum radio quasars (FSRQs),
show these fluctuations in every observable band of the electromagnetic
spectrum. This paper is the last of three to focus on the optical variations
of those blazars measured by the Kepler satellite in its extended K2 mission
that were also bright enough in $\gamma$-rays to have been detected by the
Large Area Telescope on the Fermi Gamma-Ray Space Telescope (Fermi-LAT) in the
first four years of its mission as listed in the Fermi-LAT Second AGN Catalog
or 2LAC (Ackermann et al. 2011). Our earlier observations were reported in
Wehrle et al. (2019), hereafter Paper 1, and Carini et al. (2020), hereafter
Paper 2. These K2 measurements provide unique data, as they are made nearly
continuously over nominally $\sim 80$ day long campaigns at a cadence of 29.4
minutes.

All blazars have long been understood to possess plasma jets with bulk
relativistic flows pointing close to our line of sight (Blandford & Rees 1978;
Urry & Padovani 1995). The strong Doppler boosting of the flux from the
approaching jet means that the its emission dominates over other sources of
radiation in most, if not all, bands. The first main difference between FSRQs
and BL Lacs are that the former are quasars with clear emission lines visible
in their optical and UV spectra while the latter have undetectable or
extremely weak emission lines. Both BL Lacs and FSRQs have non-thermal double-
humped spectral energy distributions (SEDs) (usually plotted as log($\nu
F_{\nu}$) vs. log($\nu$)), but the frequencies at which the intrinsically
weaker BL Lac SEDs peak are normally higher than the corresponding ones for
the more powerful FSRQs (e.g., Fossati et al. 1998). The lower-frequency SED
humps are dominated by synchrotron emission from the ultrarelativistic
electrons in the jet, though quasi-thermal radiation from accretion disks is
detectable in many FSRQs but generally not in BL Lacs (e.g., Ghisellini et al.
2017). The high-energy humps can usually be nicely fitted by leptonic models,
where the same ultrarelativistic electrons upscatter photons to produce
$\gamma$-rays (and sometimes hard X-rays); the seed photons can be the
synchrotron emission itself (synchrotron-self-Compton process) (e.g.,
Ghisellini et al. 1998) but in many cases they apparently come from the
accretion disk, broad line clouds or the dust torus (external Compton process)
(e.g., Dermer et al. 2009; Arsioli & Chang 2018). Alternative, hadronic,
models for the high-energy hump involve contributions from proton-synchrotron
radiation and $\gamma\gamma$ pair production and can sometimes provide better
fits to the SEDs (e.g., Böttcher et al. 2013; Paliya et al. 2018).

In Paper 1 we described and discussed simultaneous Fermi-LAT observations of
nine $\gamma$-ray blazars, including the famous BL Lac, OJ 287, that were made
in 2014–2015. OJ 287 is the best candidate for a blazar with a supermassive
black hole (SMBH) binary at its core (e.g., Sillanpää et al. 1988; Valtonen et
al. 2016). Paper 2 similarly treated an additional ten such blazars, including
the first identified quasar, 3C 273, that were observed in 2015–2017. In Paper
2 we also presented a reanalysis of some of the results from Paper 1, so that
specific comparisons could be made uniformly. In this paper we consider the
final observations of $\gamma$-ray blazars made by K2 in 2017–2018. Included
in the total of 18 sources for which new observations are presented in this
paper are seven sources, including OJ 287, that we had previously observed
(Paper 1) and one source observed three times. These repeated K2 observations
allow us to investigate changes in the characteristics of multi-month-long
light curves and their power spectral densities (PSDs) over spans of a few
years.

We describe the selection of $\gamma$-ray blazars in Section 2. In Section 3,
we describe the K2 observations in Campaigns 14–19, the data reduction, and
the light curve analysis. We present the results of the K2 optical
observations, including the PSDs, in Section 4. Section 5 describes
contemporaneous ground-based observations obtained for three objects which
provided independent confirmation of the K2 brightness corrections. We
describe the contemporaneous Fermi-LAT data and our analysis of them in
Section 6. We discuss our results in Section 7 and we summarize our
conclusions in Section 8.

Table 1: Targets in 2017–2018aaEntries for the five objects omitted from the
final sample are indicated with parentheses.

Name | EPIC ID | Fermi-LAT |  $K_{p}$bbKepler magnitude as tabulated in the EPIC catalog. | $z$ | Class | Campaign | Notes
---|---|---|---|---|---|---|---
|  | Name |  |  |  | Fields |
NVSS J110735+022225 | 201621388 | 4FGL J1107.6+0222 | 18.584 |  $>$1.0735 | BL Lac | 14 | 1
(NVSS J105151+010312 | 248438564 | 4FGL J1051.9+0103 | 18.831 |  $\cdots$ | BL Lac | 14 | 2, 3)
(PMN J1018+0530 | 248586210 | 4FGL J1018.4+0528 | 17.457 | 1.94525 | FSRQ | 14 | 4)
4C +06.41 | 248611911 | see note | 16.888 | 1.27 | FSRQ | 14 | 5
TXS 1013+054 | 251457104 | 4FGL J1016.0+0512 | 18.75 | 1.71272 | FSRQ | 14 |  $\cdots$
(PMN J1059+0225 | 251457105 | see note | 19.0 |  $\cdots$ | unknown | 14 | 6)
3C 207 | 211504760 | 4FGL J0840.8+1317 | 18.264 | 0.6808 | LDRQ | 5, 16, 18 |  $\cdots$
TXS 0836+182 | 211852059 | 4FGL J0839.4+1803 | 17.046 |  $\cdots$ | BL Lac | 16 | 2, 7
NVSS J090226+205045 | 212035517 | 4FGL J0902.4+2051 | 15.741 |  $\cdots$ | unknown | 16 | 2
TXS 0853+211 | 212042111 | 4FGL J0856.8+2056 | 18.345 |  $\cdots$ | BL Lac | 16 | 2
NVSS J090900+231112 | 251376444 | 4FGL J0908.9+2311 | 17.939 |  $>$ 0.432 | BL Lac | 16 | 8
PKS 1335-127 | 212489625 | 4FGL J1337.6$-$1257 | 17.851 | 0.539 | FSRQ | 6, 17 |  $\cdots$
PMN J1318-1235 | 212507036 | 4FGL J1318.7$-$1234 | 18.027 |  $\cdots$ | unknown | 17 |  $\cdots$
PKS 1352-104 | 212595811 | 4FGL J1354.8$-$1041 | 17.399 | 0.332 | FSRQ | 6, 17 |  $\cdots$
RBS 1273 | 212800574 | 4FGL J1329.4$-$0530 | 15.201 | 0.57587 | X-ray QSO | 6, 17 |  $\cdots$
PKS B1329-049 | 229227170 | 4FGL J1332.0$-$0509 | 18.2 | 2.15 | FSRQ | 6, 17 | 9
PKS B1310-041 | 251502828 | 4FGL J1312.8$-$0425 | 18.3 | 0.8249 | FSRQ | 17 |  $\cdots$
RGB J0847+115 | 211394951 | 4FGL J0847.2+1134 | 17.331 | 0.1982 | BL Lac | 5, 18 |  $\cdots$
WB J0905+1358 | 211559047 | 4FGL J0905.6+1358 | 17.213 | 0.2239 | BL Lac | 5, 18 | 10
OJ 287 | 211991001 | 4FGL J0854.8+2006 | 14.819 | 0.3056 | BL Lac | 5, 18 |  $\cdots$
BZB J0816+2051 | 212035840 | 4FGL J0816.9+2050 | 17.486 |  $\cdots$ | BL Lac | 5, 18 |  $\cdots$
(PKS 2335-027 | 246327456 | 4FGL J2338.0$-$0230 | 18.318 | 1.072 | FSRQ | 19 | 11)
(PKS 2244-002 | 251721269 | 4FGL J2247.4$-$0001 | 18.359 | 0.094 | BL Lac | 19 | 11)

Note. — 1\. Two faint objects are 8″ and 12″ from the blazar. The blazar light
curve was separated with a custom aperture. 2\. The SDSS automated photometric
redshift cited in NED is not reliable because quasar spectral template does
not apply to blazars. 3\. The blazar has a stellar object of similar
brightness within 4″ (1 pixel). Light curves could not be separated. 4\.
Target was too faint during K2 observations to have sufficient signal-to-noise
ratio for analysis. 5\. See text for discussion of association. 6\. See text
for explanation of why K2 target was not associated with a 4FGL source. 7\. A
redshift of 0.28 for TXS 0836+182 is an estimate from galaxy morphological
fitting (Abraham et al. 1991). 8\. The blazar has a fainter non-AGN companion
galaxy 3.8″($\sim 1$ pixel) away (see Rosa-González et al. 2017). Both have
the same absorption line redshift. In our custom aperture, the variability
comes from the blazar, not the inactive companion. 9\. Target has alternate
EPIC ID 229228144. 10\. Target has alternate EPIC ID 211559044 which does not
appear in all MAST searches. 11\. Target omitted from final sample due to poor
K2 data quality in Campaign 19 during the last days of the mission.

##  2 Sample Selection

We initially chose 23 targets known to emit $\gamma$-rays (Table 1) by
searching the Fermi-LAT Second AGN Catalog (2LAC, Ackermann et al. 2011) for
the optically-brightest AGN that would be observed in the K2 Campaigns 14–19
fields (programs GO14027, GO16027, GO17010, GO18010, and GO19010, led by
principal investigator A. Wehrle), following the criteria we used in Papers 1
and 2. Although 21 of these targets are indeed blazars, we determined that 2
are not and and we do not have enough information to classify 3 targets as
either FSRQs or BL Lacs. Thirteen blazars overlapped with the infrared and
optically bright samples selected by programs GO14040, GO16040, GO17020,
GO18020, and GO19020 that were led by principal investigator M. Carini. We
give updated $\gamma$-ray names from the Fermi-LAT 8-year Source Catalog
(4FGL, Ajello et al. 2020) and Fermi-LAT 10-year Source Catalog (4FGL-DR2,
Ballet et al. 2020) in Table 1. The $\gamma$-ray blazars in the five fields
had estimated K2 optical magnitudes $K_{p}$ in the 420–900 nm band from 14.818
to 18.831 in the K2 Ecliptic Plane Input Catalog (EPIC; Huber et al. 2016;
Huber & Bryson 2018 and references therein) which was generally drawn from the
USNO-B and SDSS catalogs. OJ 287 is by far the brightest, with $K_{p}=14.818$.
The K2 mission ended a few days into Campaign 19 when the spacecraft could no
longer be accurately pointed. Our two Campaign 19 targets were dropped from
the sample because the data quality was poor during the last few days of the
mission. One target, NVSS J105151+010312 = EPIC 248438564, was dropped because
its light curve could not be separated from that of a nearby star. Another
target, PMN J1018+0530 = EPIC 248586210, was dropped because it was too faint
during the mission. A target, PMN J1059+0225 = EPIC 251457105, that turned out
to be not associated with a 4FGL source was also dropped from the sample. In
our final sample of 18 unique targets observed in 2017–2018, eight are classed
as BL Lacs, six as FSRQs, one as an X-ray QSO, one as a lobe-dominated radio
quasar (LDRQ), and two as blazars of unknown class.

Table 2: Dates of K2 and Fermi-LAT Observations K2 Campaign | Dates | MJD | Duration
---|---|---|---
14 | 2017 Jun 1 – 2017 Aug 19 | 57905-57985 | 81 days
16aaFields 16 and 17 were forward-facing campaigns which enabled simultaneous observations from the ground. | 2017 Dec 7 – 2018 Feb 25 | 58095-58174 | 80 days
17aaFields 16 and 17 were forward-facing campaigns which enabled simultaneous observations from the ground. | 2018 Mar 2 – 2018 May 8 | 58179-58246 | 67 days
18 | 2018 May 13 – 2018 Jul 2 | 58251-58301 | 51 days

##  3  K2 Observations, Data Reduction and Analysis

A journal of the K2 and Fermi-LAT observations is given in Table 2. Campaigns
14, 16, 17 and 18 had no anomalies as described in the K2 Data Release Notes
https://keplergo.github.io/KeplerScienceWebsite/k2-data-release-notes.html.
Campaigns 16 and 17 were “forward-facing” campaigns to enable simultaneous
ground-based data to be obtained. The Campaign 5 and 18 fields, the Campaign 6
and 17 fields, and the Campaign 16 and 18 fields overlapped, hence, the
targets with more than one K2 observation are all from Campaigns 5, 6, 16, 17,
and 18. Campaign 18 was ended early to download its data when it appeared that
the cryogen was running low. Campaign 19 had poor data quality while the
cryogen was running out, and ended after 10 days.

As in our earlier observations, the two longest campaigns’ observations
contained $\sim$3,500 29.4-minute (“30-minute” or “long cadence”) samples,
with the 67 day Campaign 17 containing $\sim$3000 data points, and the 51-day
Campaign 18 containing $\sim$ 2200 data points. We used data corrected by the
EVEREST algorithm Version 2, hereafter “EVEREST” developed by Luger et al.
(2016, 2018) to compensate for the drift of the spacecraft caused by photon
pressure and the motions introduced by the firings of the spacecraft thruster
approximately every 6 or 12 hours (Van Cleve et al. 2016); see Paper 2 for
details. Anomalous data caused by thermal drift during the first couple of
days at the beginnings of the campaigns were discarded. As in Papers 1 and 2,
we were also able to obtain “short cadence” ($\sim 1$ minute intervals) for
the brightest blazar in our sample, here OJ 287.

As we found with the standard Kepler pipeline products for AGN light curves
(Wehrle et al. 2013; Revalski et al. 2014), standard K2 pipeline output for
AGN often removed true astrophysical brightness variations in the process of
removing instrumental effects. As in Papers 1 and 2, we examined the light
curves produced by the standard pipeline processing (SAP, PDCSAP), the K2SFF
processing (Vandenburg & Johnson 2014, and updates online at MAST), and the
EVEREST processing (Luger et al. 2016, 2018). The EVEREST light curves had the
fewest residual instrumental errors coming from thermal drifts in the first
two days of campaigns, isolated low points, and “sawtooth” amplitude
variations at thruster firing (6 and 12 hour) intervals.

We compared the EVEREST light curves of targets that fell on the same modules
to see if they varied simultaneously. We found no evidence of significant
correlated variability. We also checked for evidence of nonlinear amplifier
effects (Kolodziejczak et al. 2010; tabulated in Table 13 of the Kepler
Instrument Handbook, Van Cleve & Caldwell 2016). Two and six targets,
respectively, fell on modules classed as having “strong” and “medium”
(“Moiré”) effects, however, detailed examination of the amplitude variations
showed no visible evidence of “Moiré” distortions. We found no evidence of
(unrelated)“rolling band” effects.

Our long cadence observations of the brightest target, OJ 287, reached a noise
level (standard deviation) after EVEREST processing of 13 ct s-1 ($\sim$
0.09%). The other targets’ long cadence observations had noise levels of 3–13
ct s-1 while the average brightnesses were 244 to 11,728 ct s-1. The noise was
measured during $\sim$ 0.5 day intervals when the light curves exhibited
variations below 1%. The OJ 287 short cadence data were custom processed by us
through the EVEREST software in the same fashion as described in Paper 1.

We examined the K2 “postage stamp” apertures used in processing for each
target to make sure the targets were isolated. EPIC 211394951 was not present
in the EVEREST pipeline processing data products available at MAST; we applied
the EVEREST processing ourselves utilizing a custom aperture. EPIC 251376444
has a close companion only 3.8″away; we used a custom aperture where the
variable signal comes from the AGN, not the companion. The original EVEREST
aperture of EPIC 212035840 had moved the object out of the aperture at the end
of the campaign; hence, it needed custom processing with a carefully chosen
aperture. EPIC 201621388 had two other faint sources in the EVEREST pipeline
aperture, so a custom aperture was designed to exclude contribution from these
sources. EPIC 251502828 has another similar-brightness object about 10″ away,
so we reprocessed it with a custom aperture that excluded the other object.
The final sample in Campaigns 14 –18 contained 18 unique targets with good
quality data, including 3C 207 observed in both Campaigns 16 and 18. In
addition, we found that for 3 of the objects from our previous papers (2 from
Paper 1 (EPIC 211559047, 211394951) and 1 from Paper 2 (EPIC 217154395)) the
default EVEREST aperture was contaminated with other objects. We re-reduced
these sources with custom apertures to eliminate the contribution from these
nearby sources to the blazar’s light curve. The field of EPIC 217154395 was
found to be too crowded to produce a reliable light curve containing only the
AGN flux.

##  4 K2 Optical Results

The K2 EVEREST light curves for our new and reprocessed targets are shown in
the top sub-panels of Figures 1–4. Key features of these light curves are
given in Table 3. During the K2 observations, the greatest variation
(maximum/minimum) detected was the factor of 12.24 flaring we observed in the
FSRQ PKS 1352-104 (EPIC 212595811) during Campaign 6. The smallest variations,
a factor of 1.01 (long cadence) or 1.02 (short cadence), were seen in 3C 273
(EPIC 229151988) and in the X-ray QSO, RBS 1273 (EPIC 212800574), with factors
of 1.02 and 1.01 in Campaigns 6 and 17, respectively. Substantial variations
by factors of 1.07–8.50 were detected in the other objects. All of these
variations are much larger than the respective noise levels (Table 3).

###  4.1 Overall Appearance

We examined the light curves for all the sources in our K2 samples as listed
in Table 3 and shown in Figs. 1 – 4 in this paper and in Figs. 3 – 13 in Paper
1 and Figs. 2 – 5 in Paper 2. We qualitatively characterized the observed
fluctuations into two broad categories: fast, jagged changes for light curves
that contained multiple clear changes of flux trends during the campaigns; and
slow, smooth variations that showed only a few flux modulations over that
period. These characterizations are given in Table 3, in the Max/Min column,
with S denoting smooth light curves and J denoting jagged. Considering each
individual campaign separately (so some sources are double counted and one is
triple counted) we have a total of 39 light curves, of which 29 are jagged and
10 are smooth. There is no significant difference between the blazar types,
with there being 15 jagged light curves for BL Lacs and 10 for FSRQs and 5
smooth light curves for BL Lacs and 2 for FSRQs. The sources in our sample
that are of uncertain blazar type or not true blazars were also split, with 4
jagged and 3 smooth light curves. If we do not count second observations of
blazars independently, there are 12 BL Lacs and 7 FSRQs with jagged light
curves while 4 BL Lacs and 2 FSRQs have smooth light curves. We also checked
for any difference with source redshift; however, both the more common jagged
and the rarer smooth categories included both lower and higher redshift
objects of both BL Lac and FSRQ types, so there is no evidence of a trend with
redshift.

We also considered the ranges of optical fluxes in terms of the ratio of the
maximum to minimum count rates seen during these observations, as given in
Table 3. The mean value of Maximum/Minimum for the 12 long cadence FSRQ light
curves is 2.96 if we include the very large flare seen in PKS 1532$-$104
during Campaign 6 but 2.08 if we exclude that flare from that light curve. The
corresponding mean value for the 20 BL Lac light curves is 1.49, so it appears
that FSRQs are more active by this measure. If we consider the median of the
Maximum/Minimum values the difference is somewhat less: 1.52 for FSRQs
(regardless of the flare’s inclusion) and 1.32 for BL Lacs. The fact that
FSRQs have higher average Doppler boosting factors than BL Lacs (e.g.,
Liodakis et al. 2017) could explain this difference. We also tabulate, in the
ninth column of Table 3, another quantity characterizing the variations, the
“Coefficient of Variation” (defined as (average count rate)/(standard
deviation during full campaign)). This variability measure is more commonly
used at other wavebands and so may be useful for multi-wavelength studies.

###  4.2 Slopes of Optical Power Spectral Densities

The primary analysis technique for the K2 data in this paper, as in Papers 1
and 2, is the computation and examination of PSDs. A key difference in this
paper is that the PSDs were constructed utilizing a periodogram-based method,
rather than a direct application of the discrete Fourier transform (e.g.,
Edelson & Krolik 1988), as in Papers 1 and 2. Our rationale for this choice is
provided in detail in the Appendix. These PSDs are shown in the bottom sub-
panels of Figures 1–4 and their slopes and errors are listed in Table 3. These
PSDs are usually characterized by red noise, with $P(\nu)\propto\nu^{\alpha}$,
with $\alpha<0$, for $\nu<\nu_{b}$; above this “break frequency” it has a
white noise character ($\alpha=0$) which is expected when the signal is
dominated by measurement errors. To minimize the red noise leak (transfer of
variability power from low to high frequencies), we computed the PSDs after
removing any linear trend and then convolved them with a Hamming window
function (Harris 1978), as in Papers 1 and 2. To best estimate the PSD slopes,
we then binned the logarithm of the power in intervals of 0.08 in log $\nu$
and found a linear fit to the portion of the PSD displaying power law behavior
(bottom sub-panels of Figs. 1–4) as defined in Paper 2. This “sweet spot” in
the PSDs, where the abundant, adequately-sampled and low-noise data were well
fitted by a single power law, was generally between log $\nu[Hz]=-5.0$ and
$-6.4$, corresponding to timescales between 1.16 and 36.6 days. Error bars in
the PSD plots represent the rms scatter in the data points in each bin and
where there is only one point per bin, no error bar is shown.

It is useful to consider the effects of flares on the PSDs. We note that there
is only one light curve in which a strong isolated flare was observed: PKS
1352$-$104 during Campaign 6 (Figure 2d). In this case we saw that a single
PSD slope is not a good fit so we removed the last 16.5 days of data and
recomputed the PSD, as shown in Figure 2e. The results with and without the
big flare are both given in Table 3. All of the other source light curves are
typical of blazars, with multiple flares of varying modest strengths observed
during the length of these K2 campaigns. It would be unproductive to cut out
any of these variations in this study of variability properties and we have
not done so. Discontinuities with strong drops in power at certain frequencies
are the dips associated with the weak peaks that originate in a few flares
peaking in the light curve that have similar timescales of a few days. They
are not due to individual strong flares because such flares cause flattening
of the low frequency part of the PSD (compare Figures 2d and 2e). They are not
due to windowing because windowing effects are minimized by using the Hamming
window which produces dips in the light curve at exactly the same frequency
for all light curves of a given length. The K2 campaigns are not all the same
length: the PSDs obtained for each target during a campaign do not all show
dips and peaks at the same frequencies.

In our power spectral response analysis (PSRESP; e.g., Uttley et al. 2002),
the results of which are given in the last columns of Table 3, we test a model
for the entire observed PSD; this model is a power law in the red noise
portion of the PSD with a hard break to the white noise (slope = 0.0) portion.
The location of the hard break to white noise is identified by estimating the
frequency at which the slope of the PSD transitions to 0.0. This “turnover”
frequency varies over a decade or more in frequency from object to object and
is seen to change for the same object between different observations on
different CCD modules. Thus, the slopes from our PSRESP analysis represent the
slope along the entire power law portion of the PSD, from the variable
turnover to white noise down to the lowest sampled frequency. The confidence
factors quoted in Table 3 represent the confidence of the adopted model (power
law + white noise) fit. We note that only a small fraction (17%) of these
confidence factors are $>$ 90%. We discuss the effects of red noise leak and
sampling window on the periodogram-derived PSDs in Appendix A. For consistency
in comparing the PSDs of different sources or of the same source at different
times, as we describe in Paper 2, our analysis was focused on the range of
frequencies where we found the PSD to be well sampled and exhibiting the
lowest noise (i.e., the “sweet spot”; $-6.4<$ log $\nu<-5.0$) using the
periodograms.

For the short cadence data on OJ 287, we anticipated that the PSD, as we found
in OJ 287 (Paper 1) and 3C 273 (Paper 2), would show a plateau induced by
sampling effects between log $\nu=-4.2$ and $-3.8$, corresponding to
timescales between 4.4 and 1.8 hours. A similar plateau in the Campaign 18 OJ
287 PSD did appear, and we corrected for it as we did in Papers 1 and 2,
utilizing the DFOURT and CLEAN methodology (Roberts et al. 1987; Hoegbom
1974).

The slopes of the optical PSDs found by the periodogram method of the 14 new
long cadence blazar light curves over the “sweet spot” red noise frequency
ranges (generally for log ${\nu}$ ranging from $-5.0$ to $-6.4$) vary from
$-1.18\pm 0.38$ to $-2.95\pm 0.35$ (Table 3). The corresponding power-law fits
over the entire range of red-noise found using the PSRESP method are between
$-1.61\pm 0.10$ and $-2.58\pm 0.32.$ We show histograms of both of these
approaches to the PSD slopes in Figure 5 for targets in this paper combined
with those in Paper 1 and Paper 2. The mean value of the 12 FSRQ PSD slopes
determined via the periodogram method is $-1.98\pm 0.42$ and the mean value of
the 20 BL Lac PSD slopes is $-2.22\pm 0.38$. The median values are $-1.93$ and
$-2.27$, respectively. Objects with multiple observations have all measured
slopes included in these histograms. A two-sided KS test yields a KS statistic
of 0.483 and a $p$ value of 0.039. It is thus unlikely the two samples come
from the same distribution and we conclude that there is a probable difference
in the typical PSD slopes between BL Lac objects and FSRQs measured on the
timescales probed by K2 campaigns.

The mean value of the 12 FSRQs PSD slopes determined via the PSRESP method is
$-1.88\pm 0.33$ and the mean value of the 20 BL Lacs is $-2.05\pm 0.32$. The
median values are $-1.86$ and $-2.01$, respectively. Objects with multiple
observations have all measured slopes included in these histograms. A two-
sided KS test yields a KS statistic of 0.35 and a $p$ value of 0.255. We note
that the majority of the slopes determined by the PSRESP method have low
confidence factors (column 14, Table 3), due to the the fact that the PSRESP
method fits the entire PSD, rather than the well defined portion of the PSD we
define as our “sweet spot”. Our PSRESP implementation includes an estimate of
the high frequency turnover to white noise and the fit to the linear part of
the PSD includes all PSD values down to the lowest frequency measured. Thus,
while the probability returned by the two-sided KS test that the two
distributions arise from different parent populations is not as high as that
from the periodogram method, we can understand this because the PSRESP
analysis includes data at low frequencies below the well-sampled “sweet spot”
which tend to flatten the PSD slope.

Figure 1: OJ 287 ( EPIC 211991001) in Campaign 5 and 18. (a) Long cadence
light curve and PSD in Campaign 5; (b) long cadence light curve and PSD in
Campaign 18; (c) short cadence light curve and PSD in Campaign 5; (d) short
cadence light curve and PSD in Campaign 18. The short cadence PSDs for both
campaigns have been fitted twice: the black lines are fitted in the “sweet
spot” between log $\nu=-5.0$ and $-6.4$, and the magenta lines are fitted
between log $\nu=-3.4$ and $-6.4$; the latter PSDs have slopes $-2.65\pm 0.05$
and $-2.26\pm 0.06$, for Campaigns 5 and 18, respectively. Figure 2: Top: 3C
207 (EPIC 211504760), a lobe-dominated radio quasar, was observed in Campaigns
5, 16, and 18. Light curves and PSDs are shown for (a) Campaign 5, (b)
Campaign 16, and (c) Campaign 18. Bottom: PKS 1352$-$104 (EPIC 212595811), the
most variable object in our sample, was observed in Campaigns 6 and 17. For
the observations in Campaign 6, we show the light curves and PSDs: (d) with
the giant flare and (e) without the giant flare; (f) Campaign 17 light curve
and PSD. The Campaign 6 data with the giant flare (d) were fitted in the
“sweet spot” between log $\nu=-5.0$ and $-6.4$ (shown in black), and then
above and below the break in the power law at log $\nu=-5.5$ (shown in magenta
and teal, respectively). Table 3:  Optical Light Curve Characteristics For
Combined Sample

Name | EPIC | Class | CaaCampaign | Average | SDbbStandard deviation during full campaign. | NoiseccStandard deviation during 0.2–0.5 day intervals when source variation was less than $\sim 1\%$. | Max/Mindd(Maximum count rate)/(Minimum count rate). Smooth or jagged character of variations is indicated with S or J. | CVeeCoefficient of Variation, defined as (average count rate)/(standard deviation during full campaign). | SlopeffSlope fitted in the sweet spot between log $\nu=-5.0$ and $-6.4$ unless otherwise noted. | Error | PSRESP | PSRESP | Con- | Notes
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
| ID |  |  | Counts |  |  | Smooth or |  |  |  | Slope | Error | fidenceggThe confidence of the adopted model (power law plus white noise) fit using PSRESP. |
|  |  |  | (ct s-1) | (ct s-1) | (ct s-1) | Jagged |  |  |  |  |  |  % |
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11) | (12) | (13) | (14) | (15)
PKS B1130+008 | 201503438 | BL Lac | 1 | 875 | 17 | 4 | 1.10 J | 51.5 | -2.47 | 0.14 | -1.92 | 0.15 | 66 | $\cdots$
WB J0905+1358 | 211559047 | BL Lac | 5 | 3553 | 410 | 9 | 1.44 S | 8.7 | -1.76 | 0.13 | -2.07 | 0.14 | 49 | 1
3C 207 | 211504760 | LDRQ | 5 | 589 | 31 | 4 | 1.32 J | 19.0 | -1.89 | 0.15 | -2.03 | 0.11 | 95 | $\cdots$
RGB J0847+115 | 211394951 | BL Lac | 5 | 1941 | 225 | 5 | 1.42 J | 8.6 | -2.20 | 0.26 | -2.26 | 0.16 | 55 | 1
OJ 287 | 211991001 | BL Lac | 5 | 15744 | 2417 | 58 | 2.36 J | 6.5 | -2.28 | 0.17 | -1.92 | 0.09 | 59 | 2
OJ 287 | 211991001 | BL Lac | 5 | 16038 | 2432 | 54 | 2.31 J | 6.6 | -2.65 | 0.05 |  $\cdots$ |  $\cdots$ |  $\cdots$ | 3
BZB J0816+2051 | 212035840 | BL Lac | 5 | 852 | 102 | 5 | 1.57 J | 8.4 | -2.69 | 0.26 | -2.69 | 0.16 | 24 | 1, 4
PKS B1329-049 | 229227170 | FSRQ | 6 | 310 | 7 | 3 | 1.14 J | 44.3 | -2.01 | 0.46 | -1.86 | 0.36 | 99 | 5
PKS 1335-127 | 212489625 | FSRQ | 6 | 1233 | 130 | 2 | 1.64 J | 9.5 | -2.35 | 0.21 | -1.50 | 0.10 | 68 | $\cdots$
PKS 1352-104 | 212595811 | FSRQ | 6 | 2465 | 1735 | 4 | 12.24 J | 1.4 | -1.86 | 0.19 | -1.85 | 0.08 | 5 | 6
PKS 1352-104 | 212595811 | FSRQ | 6 | 2465 | 1735 | 4 | 12.24 J | 1.4 | -4.40 | 0.22 |  $\cdots$ |  $\cdots$ |  $\cdots$ | 7
PKS 1352-104 | 212595811 | FSRQ | 6 | 2465 | 1735 | 4 | 12.24 J | 1.4 | -0.83 | 0.09 |  $\cdots$ |  $\cdots$ |  $\cdots$ | 8
PKS 1352-104 | 212595811 | FSRQ | 6 | 2106 | 148 | 7 | 1.71 J | 14.2 | -1.58 | 0.24 | -1.41 | 0.09 | 66 | 9
RBS 1273 | 212800574 | X-ray QSO | 6 | 11738 | 98 | 5 | 1.02 S | 119.8 | -1.38 | 0.14 | -2.05 | 0.12 | 48 | $\cdots$
PKS B1908-201 | 217700467 | BL Lac | 7 | 965 | 10 | 4 | 1.08 J | 96.5 | -1.63 | 0.22 | -1.45 | 0.13 | 88 | $\cdots$
1H 1914-194 | 218129423 | BL Lac | 7 | 15426 | 332 | 20 | 1.11 J | 46.5 | -2.20 | 0.22 | -2.01 | 0.09 | 59 | $\cdots$
PKS B1921-293 | 229228355 | BL Lac | 7 | 2337 | 126 | 13 | 1.29 J | 18.5 | -2.27 | 0.23 | -1.81 | 0.10 | 55 | 10
PKS 0047+023 | 220299433 | BL Lac | 8 | 661 | 125 | 6 | 3.15 J | 5.3 | -1.45 | 0.26 | -1.57 | 0.08 | 11 | $\cdots$
1RXS J120417.0-070959 | 201079736 | BL Lac | 10 | 5965 | 99 | 5 | 1.07 S | 60.3 | -2.30 | 0.14 | -2.13 | 0.14 | 62 | 11
PKS 1216-010 | 201375481 | BL Lac | 10 | 5729 | 791 | 16 | 1.77 J | 7.2 | -2.50 | 0.29 | -2.25 | 0.13 | 41 | 11
3C 273 | 229151988 | FSRQ | 10 | 91737 | 353 | 10 | 1.01 J | 259.9 | -2.19 | 0.19 | -2.66 | 0.16 | 78 | 2, 11
3C 273 | 229151988 | FSRQ | 10 | 91751 | 403 | 52 | 1.02 S | 227.7 | -2.31 | 0.15 |  $\cdots$ |  $\cdots$ |  $\cdots$ | 3
1RXS J121946.0-031419 | 201247917 | BL Lac | 10 | 1554 | 87 | 4 | 1.29 J | 17.9 | -2.43 | 0.23 | -2.17 | 0.11 | 47 | 11
PKS B2320-035 | 246289180 | FSRQ | 12 | 2324 | 383 | 5 | 2.33 J | 6.1 | -2.09 | 0.18 | -1.93 | 0.10 | 23 | $\cdots$
PKS B2335-027 | 246327456 | FSRQ | 12 | 513 | 44 | 3 | 1.43 J | 11.7 | -2.85 | 0.22 | -2.18 | 0.12 | 49 | 4
NVSS J110735+022225 | 201621388 | BL Lac | 14 | 1116 | 26 | 7 | 1.11 S | 42.9 | -2.95 | 0.35 | -2.58 | 0.32 | 88 | 1, 12
4C +06.41 | 248611911 | FSRQ | 14 | 2910 | 109 | 6 | 1.18 J | 26.7 | -2.17 | 0.13 | -1.93 | 0.11 | 99 | 1
TXS 1013+054 | 251457104 | FSRQ | 14 | 244 | 88 | 5 | 8.50 J | 2.8 | -1.88 | 0.17 | -2.02 | 0.11 | 90 | $\cdots$
3C 207 | 211504760 | LDRQ | 16 | 425 | 12 | 5 | 1.25 J | 35.4 | -1.59 | 0.26 | -2.06 | 0.12 | 67 | 13
TXS 0836+182 | 211852059 | BL Lac | 16 | 2066 | 102 | 3 | 1.26 S | 20.3 | -2.38 | 0.17 | -2.11 | 0.10 | 48 | $\cdots$
NVSS J090226+205045 | 212035517 | Unknown | 16 | 4288 | 973 | 13 | 2.48 J | 4.4 | -2.67 | 0.14 | -2.35 | 0.12 | 60 | $\cdots$
TXS 0853+211 | 212042111 | BL Lac | 16 | 642 | 49 | 4 | 1.43 J | 13.1 | -1.76 | 0.18 | -1.90 | 0.11 | 83 | $\cdots$
NVSS J090900+231112 | 251376444 | BL Lac | 16 | 979 | 51 | 6 | 1.24 J | 19.2 | -2.57 | 0.32 | -2.41 | 0.11 | 34 | 1, 4
PKS 1335-127 | 212489625 | FSRQ | 17 | 581 | 60 | 4 | 1.61 J | 9.7 | -1.74 | 0.17 | -1.72 | 0.35 | 88 | $\cdots$
PMN J1318-1235 | 212507036 | unknown | 17 | 468 | 62 | 4 | 1.71 J | 7.5 | -2.55 | 0.21 | -2.01 | 0.12 | 65 | $\cdots$
PKS 1352-104 | 212595811 | FSRQ | 17 | 2509 | 250 | 7 | 1.86 J | 10.0 | -1.18 | 0.38 | -1.62 | 0.07 | 17 | 14
RBS 1273 | 212800574 | X-ray QSO | 17 | 11728 | 24 | 4 | 1.01 S | 488.7 | -4.20 | 0.44 | -3.12 | 0.47 | 92 | 5
PKS B1329-049 | 229227170 | FSRQ | 17 | 570 | 37 | 3 | 1.29 J | 15.4 | -1.93 | 0.23 | -1.92 | 0.12 | 86 | $\cdots$
PKS B1310-041 | 251502828 | FSRQ | 17 | 1686 | 84 | 6 | 1.29 S | 20.1 | -1.73 | 0.10 | -1.86 | 0.12 | 72 | $\cdots$
RGB J0847+115 | 211394951 | BL Lac | 18 | 1234 | 46 | 6 | 1.17 J | 26.8 | -2.02 | 0.17 | -1.82 | 0.12 | 98 | 1
3C 207 | 211504760 | LDRQ | 18 | 506 | 8 | 7 | 1.13 S | 63.3 | -1.57 | 0.22 | -1.38 | 0.18 | 100 | 4
WB J0905+1358 | 211559047 | BL Lac | 18 | 4067 | 305 | 10 | 1.28 S | 13.3 | -2.04 | 0.13 | -1.94 | 0.16 | 82 | 1
OJ 287 | 211991001 | BL Lac | 18 | 13919 | 1029 | 8 | 1.36 J | 13.5 | -1.96 | 0.20 | -1.61 | 0.10 | 60 | 2
OJ 287 | 211991001 | BL Lac | 18 | 14025 | 1267 | 28 | 1.52 J | 11.1 | -2.26 | 0.06 |  $\cdots$ |  $\cdots$ | 60 | 3
BZB J0816+2051 | 212035840 | BL Lac | 18 | 1085 | 79 | 5 | 1.34 J | 13.7 | -2.57 | 0.17 | -2.31 | 0.13 | 89 | 1

Note. — 1. Custom aperture. 2. Long cadence. 3. Short cadence data fitted
between log $\nu=-3.4$ and log $\nu=6.4$. 4. Slope fitted between log
$\nu=-5.2$ and $-6.4$. 5. Slope fitted between log $\nu=-5.6$ and $-6.4$. 6.
Slope fit in the sweet spot of log $\nu=-5.0$ to $-6.4$; however, there is a
break in the power law around log $\nu=-5.5$. 7. Fit above the break at log
$\nu=-5.5$. 8. Fit below the break at log $\nu=-5.5$. 9. Without large flare.
10. PKS B1921-293 may be intermediate between BL Lac and FSRQ because its weak
emission lines vary in equivalent width when the continuum emission changes
(Wills and Wills 1981). 11. Slope fitted between log $\nu=-5.0$ and $-6.2$.
12. Slope fitted between log $\nu=-5.5$ and $-6.4$. 13. Slope fitted between
log $\nu=-5.3$ and $-6.4$. 14. PSRESP used only one power law because no white
noise was present.

Figure 3: Light curves and PSDs for EPIC 212489625 (PKS 1335$-$127) in (a)
Campaign 6 and (b) Campaign 17; for EPIC 212800574 (RBS 1273) in (c) Campaign
6 and (d) Campaign 17.

.

Figure 3, continued: EPIC 229227170 (PKS B1329$-$049) in (e) Campaign 6 and
(f) Campaign 17; EPIC 211394951 (RGB J0847+115) in (g) Campaign 5 and (h)
Campaign 18.  Figure 3, continued: EPIC 211559047 (WB J0905+1358) in (i)
Campaign 5 and (j) Campaign 18; EPIC 212035840 (BZB J0816+2051) in (k)
Campaign 5 and (l) Campaign 18.  Figure 4: Light curves and PSDs for single
campaign observations of: (a) EPIC 201621388 (NVSS J110735+022225); (b) EPIC
211852059 (TXS 0836+182); (c) EPIC 212035517 (NVSS J090226+205045); (d) EPIC
212042111 (TXS 0853+211). Figure 4, continued:  (e) EPIC 212507036 (PMN
J1318$-$1235); (f) EPIC 248611911 (4C +06.41); (g) EPIC 251457104 (TXS
1013+054; (h) EPIC 251376444 (NVSS J090900+231112) Figure 4, continued: (i)
EPIC 251502828 (PKS B1310$-$041)

.

Figure 5: Histograms of PSD slopes for all the long cadence light curves
binned in 0.2 units. Panels(a) and (b) show the BL Lac and FSRQ PSD slopes
determined by periodogram analysis. Panels (c) and (d) show the BL Lac and
FSRQ PSD slopes determined by PSRESP analysis. See Section 4 for details.

##  5 Contemporaneous Ground-Based Observations

Three of the sources in this paper — OJ 287 (EPIC 211991001), NVSS
J090226+205045 (EPIC 212035517) and PKS 1335$-$127 (EPIC 212489625) — have
contemporaneous ground-based observations, allowing us to validate the EVEREST
extraction method and place the observed flux level and variability in the
context of the source’s long term behavior. The ground-based observations were
obtained from 5 different sources: the Robotically Controlled Telescope (RCT)
blazar monitoring program led by M. Carini http://rct.wku.edu/wordpress/, P.
Smith’s $\gamma$-ray blazar optical monitoring program at Steward Observatory
(Smith et al. 2009), http://james.as.arizona.edu/~psmith/Fermi/, the SMARTS
program (Bonning et al. 2012), the DEdicated MONitor of EXotransits and
Transients telescope (DEMONEXT; Villanueva et al. 2018), and the Zwicky
Transient Facility (ZTF; Masci et al. 2018; Bellm et al. 2019). For OJ 287 and
NVSS J090226+205045, the offset between the times series from different
telescopes was calculated by minimizing the KS statistic between overlapping
segments of the time series. For PKS 1335$-$127, a different procedure was
used (see below for details).

###  5.1 OJ 287 (EPIC 211991001)

Figure 6a shows the contemporaneous ground-based and K2 data for OJ 287 during
Campaigns 5 and 18 and also plots observations between these campaigns and for
a while before the first one. Here we utilized ground-based R band data from
the RCT, from P. Smith (Steward Observatory) and the SMARTS program. Figure 6b
displays the K2 observations of OJ 287 from Campaign 18, together with ground-
based data from the RCT and from Steward. The agreement between these various
ground-based measurements and those from K2, as processed by EVEREST, is very
good. During both K2 campaigns, OJ 287 was in a low-to-middle brightness
range.

###  5.2 NVSS J090226+205045 (EPIC 212035517)

Figure 6c displays the light curve composed of contemporaneous ground-based
and Campaign 18 K2 data for 4FGLJ0902.4+2051 (EPIC 212035517). The ground-
based data are R-band observations from DEMONEXT. DEMONEXT is a 0.5 m
PlaneWave CDK20 f/6.8 Corrected Dall-Kirkham Astrograph telescope at Winer
Observatory in Sonoita, Arizona. It has a 2048 $\times$ 2048 pixel Finger
Lakes Instruments Proline CCD3041 camera, with a $30.7\arcmin\times
30.7\arcmin$ field of view and a pixel scale of 0.90″pixel-1. We binned the
DEMONEXT data into 30-minute time bins to match the K2 long cadence length. We
find excellent agreement of the overall shapes of the light curves between the
K2 and DEMONEXT observations, indicating that the EVEREST algorithm again
performed well in correcting the forward-facing and backward-facing K2 data.

###  5.3 PKS 1335$-$127 (EPIC 212489625)

PKS 1335$-$127 (212489625) was observed by K2 in two different epochs:
Campaign 6 in 2015 and Campaign 17 in 2018. The combined K2 and ground-based
light curves are shown in Figure 6d. Our contemporaneous ground-based
observations using the RCT and ZTF were only during Campaign 17. As there were
not enough overlapping observations between the K2 Campaign 17 and ZTF
observations to perform a meaningful KS analysis, we determined the offset by
eye. In order to include the Campaign 6 observations in the same long-term
light curve, we identified 5 objects which were observed on the same K2
channel (module and output combination) as the blazar and had similar
brightnesses as well as little or no long-term variability. We calculated an
average offset between the Campaigns 6 and 17 K2 count rates for these 5
sources, and applied that to the blazar Campaign 6 observations. The K2
observations in both campaigns were obtained when the source was in mid- to
low-brightness states. The ground-based observations show it was substantially
brighter in between the K2 measurements, and much more active in 2018 and
2019. It varied by a factor of about 12 between MJD 58300 and 58675.

Figure 6: Combined ground-based and K2 EVEREST light curves. Ground-based
photometry magnitudes were converted to flux via the standard magnitude-flux
relationship and scaled to match the K2 data. (a) OJ 287 during Campaigns 5
and 18 in 2014 and 2018, respectively. Ground-based R-band photometry is from
the RCT program led by M. Carini, from the Steward Observatory program (Smith
et al. 2009), and from SMARTS (Bonning et al. 2012). (b) Combined ground-based
and K2 EVEREST light curve of OJ 287 during only Campaign 18 in 2018. The
R-band photometry has been scaled to match the first K2 observation. For
reference, 2016 Jan 1 is MJD 57388.0 and 2018 Dec 31 is MJD 58483.0. (c) NVSS
J090226+205045 (EPIC 212035517) during Campaign 18. Ground-based R-band
photometry is from DEMONEXT (Villanueva et al. 2018). (d) PKS 1335$-$127 (EPIC
212489625) during Campaigns 6 and 17. Ground-based R-band photometry is from
the RCT program and from the Zwicky Transient Factory (ZTF; Masci et al. 2018;
Bellm et al. 2019). The R-band photometry has been scaled to match the second
K2 observations on 2018 March 3 – 2018 July 8 (MJD 58180–58307).

##  6 Fermi-LAT Observations

We reduced all-sky survey mode Fermi-LAT data using the method described in
Papers 1 and 2. Pass 8 data (Atwood et al. 2013) were downloaded from the
Fermi Science Support Center, with radius 20$\arcdeg$, for the 67-81 day
timespans concurrent with the K2 campaigns. The downloaded Fermi-LAT datasets
were centered on three of the four $10\arcdeg$-square K2 campaign fields. We
offset the fourth Fermi-LAT dataset’s center by $5\arcdeg$ from the K2
Campaign 18 field’s center which brought the Fermi-LAT field center closer to
our prime target OJ 287. We used the Fermi Science Tools for unbinned
likelihood, instrument response function P8R2_SOURCE_V6, “SOURCE class” events
(parameters “evclass = 128, evtype = 3”) in the energy range 0.1–500 GeV and
set the maximum zenith angle to $90\arcdeg$. We used diffuse models
gll_iem_v07.fits and iso_P8R3_SOURCE_V3_v1.txt. For the likelihood
calculations, we used the spectral parameters of the 4FGL catalog (Ajello et
al. 2020; Ballet et al. 2020) obtained by running the python script
“make4FGLxml.py” with “gll_psc_v27.fit”) as initial values for the bright
sources and power laws for the K2 targets with one exception, within a
30$\arcdeg$ radius. We allowed sources within a 20$\arcdeg$ radius and the
diffuse emission models to vary. At the time we selected targets for
observation with K2, the $\gamma$-ray source 2FGL J1040.7+0614 (later
catalogued as 3FGL J1040.4+0615 but not included in the 4FGL catalog) had not
been identified with an optical counterpart. We picked the nearby radio source
4C+06.41 (RA, DEC $=10h40m31.630s,+6$°$17^{\prime}21.70^{\prime\prime}$) as a
likely counterpart and observed it with K2. Subsequently, Garrappa et al.
(2019) identified two radio sources that could be associated with 3FGL
J1040.4+0615 and discovered $\gamma$-ray emission from both radio sources by
analyzing 9.8 years of Fermi LAT data. The two objects, 4C+06.41 and GB6
J1040+0617, are only 22″apart. We used the spectral parameters derived by
Garrappa et al. (2019) as initial values for the likelihood calculations for
the two objects and used LAT data only when the Sun was farther than
15$\arcdeg$ from them (the LAT data train was shortened by three days at the
end), following recommendations by S. Garrappa (personal communication). We
then applied the fermiPy UpperLimits scripts (Wood et al. 2017,
https://github.com/fermiPy/fermipy/tree/master/fermipy) with the 4FGL catalog
spectral shapes to calculate 95% confidence level upper limits for undetected
targets in the 0.1–500 GeV energy range.

We present the power-law spectral indices, fluxes, and test statistic (TS)
values and flux upper limits in Table 4 for the targets observed by K2 in
Campaigns 14–18. Of these 18 unique K2 targets, we detected 6 above TS = 16
($\gtrsim 4\sigma$). Two twice-observed targets, PKS 1335-127 and OJ 287, were
detected in both earlier and later campaigns. Three twice-observed targets
were detected during earlier campaigns but not in later campaigns. BZB
J0816+2015 was observed in Campaign 5 as well as 18 but was not included in
Paper 1; only upper limits were found for both periods. Searching the online
database for the Fermi All-sky Variability Analysis (Abdollahi et al. 2017,
https://fermi.gsfc.nasa.gov/ssc/data/access/lat/fava_catalog/) showed no
evidence of variability during the K2 observations for any of our targets.

Table 4: Fermi-LAT Results Name | EPIC | Fermi |  $\gamma$-ray |  $\gamma$-ray | Test | 4FGL $\gamma$-ray | 4FGL $\gamma$-ray | 4FGL
---|---|---|---|---|---|---|---|---
| ID | Source | FluxaaFluxes, upper limits, and test statistics were in the energy range 0.1–500 GeV, measured during K2 campaigns in 2017–2018. | Flux ErroraaFluxes, upper limits, and test statistics were in the energy range 0.1–500 GeV, measured during K2 campaigns in 2017–2018. | Stat- | FluxbbData obtained from Fermi-LAT Fourth Source Catalog - Data Release 2 (4FGL-DR2, Ajello et al. 2020 and Ballet et al. 2020), in the energy range 1–100 GeV, observed between 2008 August 4, to 2018 August 2, retrieved from https://fermi.gsfc.nasa.gov/ssc/data/  | Flux ErrorbbData obtained from Fermi-LAT Fourth Source Catalog - Data Release 2 (4FGL-DR2, Ajello et al. 2020 and Ballet et al. 2020), in the energy range 1–100 GeV, observed between 2008 August 4, to 2018 August 2, retrieved from https://fermi.gsfc.nasa.gov/ssc/data/  | Detection
|  | Name | ph s-1 cm-2 | ph s-1 cm-2 | isticaaFluxes, upper limits, and test statistics were in the energy range 0.1–500 GeV, measured during K2 campaigns in 2017–2018. | ph s-1 cm-2 | ph s-1 cm-2 | Signif-
|  |  |  |  |  |  |  | icance $\sigma$
NVSS J110735+022225 | 201621388 | 4FGL J1107.6+0222 |  $<$ 1.97E-9 |  $\cdots$ |  $\cdots$ | 5.34E-10 | 4.87E-11 | 17.159
4C +06.31 | 248611911 | see notecc4C +06.31 (also known as PKS 1038+0641) was tentatively associated with 2FGL J1040.7+0614, but it is not in the 4FGL. Another $\gamma$-ray source, also not in the 4FGL, associated with quasar GB6 J1040+0617, has been found 22’ away from PKS 1038+064 (Garrappa et al. 2019). | 5.76E-08 | 2.64E-10 | 42.1 |  $\cdots$ |  $\cdots$ |  $\cdots$
TXS 1013+054 | 251457104 | 4FGL J1016.0+0512 | 4.41E-08 | 6.30E-10 | 31.0 | 1.20E-09 | 6.57E-11 | 30.8646
3C 207 (in C16) | 211504760 | 4FGL J0840.8+1317 |  $<$ 6.29E-9 |  $\cdots$ |  $\cdots$ | 1.95E-10 | 3.49E-11 | 7.3078
TXS 0836+182 | 211852059 | 4FGL J0839.4+1803 |  $<$ 5.29E-9 |  $\cdots$ |  $\cdots$ | 3.08E-10 | 3.75E-11 | 11.5571
NVSS J090226+205045 | 212035517 | 4FGL J0902.4+2051 | 1.05E-08 | 7.93E-11 | 26.1 | 1.34E-09 | 6.41E-11 | 37.7239
TXS 0853+211 | 212042111 | 4FGL J0856.8+2056 |  $<$ 4.36E-09 |  $\cdots$ |  $\cdots$ | 4.68E-10 | 4.88E-11 | 14.5981
NVSS J090900+231112 | 251376444 | 4FGL J0908.9+2311 | 1.80E-09 | 2.57E-11 | 16.6 | 4.83E-10 | 5.04E-11 | 18.2402
PKS 1335-127 | 212489625 | 4FGL J1337.6-1257 | 1.12E-08 | 3.03E-09 | 18.0 | 1.72E-09 | 8.17E-11 | 37.3988
PMN J1318-1235 | 212507036 | 4FGL J1318.7-1234 |  $<$ 1.54E-8 |  $\cdots$ |  $\cdots$ | 5.52E-10 | 4.88E-11 | 16.794
PKS 1352-104 | 212595811 | 4FGL J1354.8-1041 |  $<$ 5.54E-8 |  $\cdots$ |  $\cdots$ | 8.66E-10 | 6.20E-11 | 23.4487
RBS 1273 | 212800574 | 4FGL J1329.4-0530 |  $<$ 1.63E-8 |  $\cdots$ |  $\cdots$ | 4.55E-10 | 5.97E-11 | 10.8777
PKS B1329-049 | 229227170 | 4FGL J1332.0-0509 |  $<$ 2.15E-9 |  $\cdots$ |  $\cdots$ | 2.65E-09 | 9.33E-11 | 54.4994
PKS B1310-041 | 251502828 | 4FGL J1312.8-0425 |  $<$ 8.13E-8 |  $\cdots$ |  $\cdots$ | 1.52E-09 | 7.41E-11 | 40.2091
RGB J0847+115 | 211394951 | 4FGL J0847.2+1134 |  $<$ 2.58E-9 |  $\cdots$ |  $\cdots$ | 4.20E-10 | 4.38E-11 | 19.037
3C 207 (in C18) | 211504760 | 4FGL J0840.8+1317 |  $<$ 7.13E-9 |  $\cdots$ |  $\cdots$ | 1.95E-10 | 3.49E-11 | 7.3078
WB J0905+1358 | 211559044 | 4FGL J0905.6+1358 |  $<$ 2.59E-8 |  $\cdots$ |  $\cdots$ | 1.03E-09 | 6.44E-11 | 31.9515
OJ 287 | 211991001 | 4FGL J0854.8+2006 | 7.10E-08 | 2.09E-10 | 139.0 | 5.65E-09 | 1.23E-10 | 100.2436
BZB J0816+2051 | 212035840 | 4FGL J0816.9+2050 |  $<$ 4.77E-8 |  $\cdots$ |  $\cdots$ | 3.48E-10 | 4.47E-11 | 13.0976

##  7 Discussion

One of the key advantages of the K2 mission schedule for our investigation was
the fact that 7 of these blazars could be observed during two different
periods (Campaigns 5 and 18 or Campaigns 6 and 17) separated by just about 3.0
years and about 2.7 years, respectively. Additionally, one object (211504760)
was observed in Campaign 16 as well as in 5 and 18, though it is classified as
a lobe-dominated radio quasar instead of a FSRQ or BL Lac. In this paper we
have reduced the data and computed the PSDs for all these observations in a
uniform fashion and thus can fairly investigate whether the nature of the
fluctuations are stable or change over time scales of a few years. Table 3
gives the directly fitted periodogram PSD slopes and their nominal errors in
the ninth and tenth columns and it appears that these values do not change
much between campaigns. To quantify this impression, if we consider that these
PSD slopes are not significant if the difference between those slopes is less
than the sum of their errors, then 3 of the 9 PSDs produced from the long
cadence K2 data have changed significantly: the BL Lac WB J0905+1358 (EPIC
211559047), but just barely; the FSRQ PKS 1335-127 (EPIC 212489625); and the
X-ray QSO RBS 1273 (EPIC 212800574). If we instead compare the difference of
the slopes to twice the sum of errors taken in quadrature, then only EPIC
212489625 and EPIC 212800574 are clearly substantially different. The PSRESP
slopes tend to have smaller nominal errors than the directly estimated slopes
and so a larger number of differences between epochs are seen. Using either of
the above criteria the BL Lacs OJ 287 (EPIC 211991001), RGB J0847+115 (EPIC
211394951) and BZB J0816+2051 (EPIC 212035840) as well as the FSRQ PKS
1352-104 (EPIC 212595811) and the LDRQ 3C 207 (EPIC 211504760 - between the
C16 and C18 observations) as well as EPIC 212800574 have different slopes.
However, EPIC 212800574 is exceptional in the sense that while it among the
brightest of our sample it varied by less than 1% and 2% during these
observations, making it more difficult to produce reliable PSDs. By inspecting
the PSDs during these observations, we see that the slopes primarily changed
because the transition from red to white noise was substantially different in
the two campaigns (log $\nu=-4.6$ and $-5.5$, respectively, in Campaigns 6 and
17) corresponding to greater white noise domination in Campaign 17 than in
Campaign 6. As will be discussed in Section 7.4, this difference is probably
not of astrophysical origin.

New short cadence data were only collected for OJ 287 and those are discussed
below.

In Papers 1 and 2 we reviewed previous studies of AGN using Kepler and K2 and
we refer the reader to them for key results from that earlier work,
particularly that of Smith et al. (2018), who analyzed 21 Kepler light curves
of Type 1 AGN and that of Aranzana et al. (2018) who analyzed 252 K2 AGN light
curves, most of which, however, were not blazars. Here we note that in the few
cases where members of our samples overlapped with theirs the PSD slopes we
found were in good agreement with theirs.

###  7.1 Slopes of Optical Power Spectral Densities

In keeping with the usual approach, we have computed the PSDs directly from
the observations and we have made no attempt to correct the fluxes to jet
rest-frame emission. Doing so would require being able to calculate the
opposing effects both of source redshifts and the jets’ relativistic motions
(e.g., Gopal-Krishna et al. 2003). While redshifts are given in Table 1 for a
substantial majority of the sources, they are not available for a few of them
and are uncertain for a few others. For those with solid $z$ values, the
median redshift for FSRQs is substantially higher, at 1.072, while that for BL
Lacs is 0.329. For those sources at higher redshift, UV emission is observed
in the K2 optical band and any big-blue-bump accretion disk contribution could
dilute emitted flux variations. On the other hand, only for very few of these
targets are even rough estimates of their jet Doppler factors available, so we
cannot examine their effects on an object-by-object basis. The long term study
of radio knot velocities in the MOJAVE program includes 409 jets and a key
conclusion of this work is that the highest apparent jet speeds are seen only
in blazars with low peak synchrotron frequencies (Lister et al. 2019).
Specifically, FSRQs exhibit larger typical Doppler factors, while high-
frequency BL Lacs have the smallest, with the low- and intermediate- frequency
BL Lacs lying in between those extremes (Lister et al. 2019). A larger Doppler
factor means that rest-frame variability appears both stronger and to occur
over a shorter time scale when observed. We performed simulations to
investigate if the Doppler factor and redshift differences between BL Lacs and
FSRQs could explain the difference we saw and found no evidence that they
could (see Appendix).

As described in Paper 1, we and others found slopes steeper than $-2$ for
three FSRQs and a Seyfert 1.5 from Kepler data in its original observing mode
(Wehrle et al. 2013; Revalski et al. 2014). Mushotzky et al. (2011), Kasliwal
et al. (2015), and Smith et al. (2018) also found some significantly steeper
slopes for various Type 1 AGN (including some blazars) observed for multiple
quarters with Kepler. Recently, Goyal (2021) has determined PSDs from
intranight optical light curves of 14 blazars taken from ground-based
telescopes during the 29 monitoring sessions during which they showed
significant variability. These measurements probe sub-hour timescales and she
used a power spectral response method to estimate PSDs for the more variable
19 of those light curves. Those PSD slopes showed a wide range between $-1.4$
and $-4.0$, but with an average of $-2.9\pm 0.3$, somewhat steeper than the
red-noise variability usually found on longer timescales (Goyal 2021) as well
as those found here.

For power spectra that are very steep or show bends or breaks at lower
frequencies it may be necessary to go beyond analyzing light curves in terms
of single PSD slopes as we have done here and has usually been done in similar
studies. The most common way to do so is to consider continuous-time
autoregressive moving average (CARMA) models (e.g., Kelly et al. 2009, 2014;
Simm et al. 2016; Goyal et al. 2018; Moreno et al. 2019). These characterize
light curves with both a perturbation spectrum and an impulse-response
function that allows for the interpretation of variability timescales. The
lowest order CARMA(1,0) model is isomorphic to the Ornstein-Uhlenbeck, or
damped-random-walk, stochastic model (Kelly et al. 2009). Higher order
CARMA($p,q$) models, which connect the light curve and its first $p$ time
derivatives to the noise and its first $q$ time derivatives, can simulate
light curves with multiple breaks in the power spectra slopes and also yield
PSD slopes steeper than the $-2$ limit of a damped-random-walk model.

In Paper 1 we gave an extensive discussion of the literature discussing the
PSDs likely to arise from accretion disks and relativistic jet models; here we
summarize a few of those points and also discuss some more recent papers. Most
physical models for light curve variability, particularly those relativistic
jet-based ones most relevant to blazar fluctuations, have not been extended or
analyzed to produce simulated PSDs. Simulations of propagating jets, including
relativistic turbulence, were able to produce light curves with a range of PSD
slopes between $\sim-1.5$ to $\sim-2.6$ (Pollack et al. 2016). The turbulent
extreme multi-zone model for a relativistic jet with a standing shock can
produce multi-band light curves that look like observed variations (Marscher
2014) but there have been no published PSDs from this very promising approach.
Recently, Zhang & Giannios (2021) have considered the light curves and PSDs
arising from a “striped blazar jet”, which is characterized by alternating
magnetic polarities. Such a jet, with a structure proposed by Giannios &
Uzdensky (2019), would be launched from the immediate vicinity of the SMBH,
essentially via the Blandford & Znajek (1977) mechanism, and be seeded with
reversing magnetic polarities through the magneto-rotational instability in
the accretion disk (e.g., Balbus & Hawley 1991). Their attractive model yields
PSD slopes near $-2$ via magnetic reconnection for short-term variability
($\lesssim$ weeks) but a wider range of PSD slopes ($-3$ to $-1.3$) over
longer terms, and can consistently explain jet acceleration, jet emission on
large scales, as well as typical emission signatures of blazars (Zhang &
Giannios 2021).

Fluctuations in brightness detected by K2 that directly arise from accretion
disks may also be present in many blazars, particularly FSRQs, where disk
radiation can make a significant contribution to the optical emission during
low states. A recent study of a sample of 67 AGN with measured SMBH masses
finds typical optical PSD slopes around $-2.0$ on timescales of days to
several weeks, but a flattening at lower frequencies on $\sim 100$d timescales
that can be associated with the expected thermal timescale for the ultraviolet
emitting radius from accretions disks (Burke et al. 2021). In general, when
PSDs are calculated for simulated light curves from accretion disk
fluctuations, both phenomenological models (e.g., Mangalam & Wiita 1993) and
sophisticated three-dimensional general relativistic simulations (Noble &
Krolik 2009; Cowperthwaite & Reynolds 2014; O’Riordan et al. 2017) tend to
produce PSD slopes in the range $-1.3$ to $-2.1$, at least for sources where
the disk is viewed nearly face-on, as expected for blazars. As shown in Figure
5, while both BL Lacs and FSRQs exhibit a range of PSD slopes, those of BL
Lacs are typically steeper, with a mean of $-2.22$ while those of FSRQs have a
mean of $-1.98$ using the periodogram approach in the sweet spot, and
respectively $-2.05$ and $-1.88$ using the PSRESP method.

We can understand this difference theoretically, as the jet simulations do
tend to produce steeper PSD slopes than do those involving accretion disks,
although both can yield slopes around $-2$. This type of difference between
the blazar sub-types is aligned with the recent result of Xiao et al. (2022)
who make a strong case that the jets of FSRQs are mostly powered by accretion
disks while those of BL Lacs are powered by extracting black hole rotational
energy. Steeper PSD slopes either mean there is relatively more power at long
timescales or, equivalently, there is less power at short timescales. We
expect that BL Lacs, typically seen at lower redshifts, will have had more
time for their SMBHs to grow and thus should have somewhat higher SMBH masses,
as well as typically higher spins, than FSRQs. All relevant timescales,
including those for variations increase with SMBH mass, either linearly for
processes depending on the ergosphere (Blandford-Znajek) or inner accretion
disk, or sub-linearly for thermal accretion disk processes (e.g. Burke et al.
2021). Then the jets that are likely to be powered by rapidly spinning black
holes with higher mass (i.e., BL Lacs) are expected to have relatively more
power at longer timescales and relatively less power at short timescales than
jets arising only from accretion disks around less massive (FSRQ) SMBHs.

The difference in PSD slopes of the optical emission between BL Lacs and FSRQs
we have observed also could be understood in a framework where the key
physical distinction is related to the location from which those photons
emerge within otherwise similar jets. The optical photons from low synchrotron
peaked blazars, such as FSRQs, are at frequencies above the synchrotron peak,
whereas they are below that peak for the majority of BL Lacs, which are high
synchrotron peaked. In standard shock-in-jet models (e.g. Marscher & Gear
1985) the radiating particles are accelerated to the highest energies they can
achieve in a small region in the immediate vicinity of a shock front where
both particle densities and magnetic field strengths are greatest. Photons of
lower frequencies emerge at greater distances behind the shock that involve
larger volumes. Hence, purely from a simple light crossing time argument, we
expect faster variations to be more dominant from smaller regions and so from
photons on the higher frequency side of the peak, i.e., for FSRQs as opposed
to BL Lacs. Such a preponderance of fluctuations on shorter timescales
corresponds to relatively more spectral power at higher frequencies, hence, a
typically shallower PSD slope for FSRQs than for BL Lacs. This distinction
between the amount of variability in BL Lacs and FSRQs can also be explained
in terms of the optical emission from the former coming from below the
synchrotron peak and so arising from relativistic electrons that are cooling
more slowly than those in FSRQs, where the equivalent electrons are nearer the
peak and rapidly losing energy (e.g., Hovatta et al. 2014).

###  7.2 $\gamma$-ray Activity Level and Blazar Classes

During the Campaign 14–18 K2 observations, six of the 18 unique AGN were
detected with Fermi-LAT in the energy band 0.1-500 GeV at significance levels
$\gtrsim 4\sigma$ (TS $>16$). The maximum/minimum optical variability
amplitude during the K2 campaigns was lower for the AGN not detected with the
Fermi-LAT than for the LAT-detected AGN. Specifically, the maximum to minimum
ratios for $\gamma-$ray detected vs. non-detected AGN were 2.73 vs. 1.13 for
the Campaign 14–18 sample. This could indicate that more active states
occurred simultaneously at both bands. Hovatta et al. (2014) also found that
$\gamma-$ray detected FSRQs are more optically variable than the BL Lacs. We
note that Bhatta (2021) has considered the optical variability of 12
$\gamma$-ray blazars over the course of a decade or so by combining data from
the SMARTS and Steward Observatory monitoring programs, supplemented with
AAVSO and Catalina Surveys data. The only source in common with our entire
sample is 3C 273, for which Bhatta found no correlation between the optical
and $\gamma$-ray fluxes. In contrast, Bhatta (2021) found that for the other
11 sources in his sample, these two bands were highly correlated, implying a
co-spatial origin of those emissions.

The slopes of the optical power spectral densities of the $\gamma-$ray blazars
detected and not detected by the contemporaneous Fermi-LAT observations were
very similar, averaging $-2.16$ and $-2.24$, respectively. This result is in
agreement with what we found for the original PSD slopes of targets in Papers
1 and 2 when refitted with “sweet spot” frequency ranges. The ranges of
maximum to minimum ratios are also similar, with the exception of the outlier
X-ray QSO RBS 1273 with slope $-4.20$. The similarity of the optical
variability characteristics argues in favor of the dominant optical emission
during the K2 observations coming from the relativistic jet.

A study by Ryan et al. (2019) of 13 blazar $\gamma$-ray light curves argued
that CARMA(1,0) models do not always adequately describe the variability
properties but that the slightly more complex CARMA(2,1) models apparently may
do so, as they can naturally produce the low-frequency PSD breaks that seemed
to be present in 4 of them. In contrast, a recent analysis of monthly binned
$\gamma$-ray light curves of a much larger sample of 236 bright blazars found
PSD slopes concentrated around $-1$ but ranging between 0 and $-2$ (Burd et
al. 2021). These Fermi-LAT observations were examined in terms of Ornstein-
Uhlenbeck, or CARMA(1,0), models and it was found that the light curves could
all be reasonably be fit by them. A critical result was that these
$\gamma$-ray flux variations thus could be described in terms of a stochastic
model that required only three parameters — mean flux, correlation length and
random amplitude — each of which was rather narrowly peaked around well-
defined values (Burd et al. 2021).

The dissipation of magnetic energy in jets can provide the source of
accelerated particles and radiation from them. For quite some time, shear
driven Kelvin-Helmholtz instabilities have been known to destabilize
propagating relativistic jets (e.g., Hardee & Clarke 1992).
Magnetohydrodynamic (MHD) current-driven kink instabilities arising in
strongly magnetized jets are expected to produce significant fluctuations in
the emission from those jets (e.g., Appl et al. 2000). Recently, several
papers have used simulations to discuss these MHD kink instabilities in
connection with particle acceleration via magnetic reconnection (Acharya et
al. 2021; Bodo et al. 2021; Kadowaki et al. 2021; Medina-Torrejón et al.
2021). For instance, Acharya et al. (2021) have performed a suite of special
relativistic MHD simulations of a portion of a jet column that show the
development of kink instabilities and their impact on the jet density,
pressure, velocity, magnetic fields, and magnetic energy dissipation. The
initial conditions considered various magnetic field energies, bulk Lorentz
factors, and number of axial wavelengths fitting in the simulation box. These
simulated jet zones fairly quickly become very inhomogeneous, with filamentary
structures dominating the plasma densities, magnetic fields, and estimated
photon fluxes. Once Doppler boosting is taken into account, substantial
variations in the observed intensity are produced by these twisting filaments
and give rise to a quasi-periodic structure in the light curve, with a strong
peak in an otherwise quite flat PSD (Acharya et al. 2021).

Another approach to understanding the development of magnetic reconnection in
relativistic jets has been taken by Medina-Torrejón et al. (2021) and Kadowaki
et al. (2021). This group has used a series of three-dimensional special-
relativistic MHD simulations of a portion of a relativistic jet to investigate
how current-driven kink instabilities grow in these conditions. Those
instabilities appear to naturally lead to turbulence that yields zones where
magnetic reconnection appears to arise. That magnetic reconnection can
accelerate low-energy protons to relativistic energies in blazar jets and
might well explain both $\gamma$-ray and neutrino emission (Medina-Torrejón et
al. 2021). Developing this approach even further, they show how the turbulence
in both magnetic and kinetic energies eventually approach 3-D Kolmogorov
spectra (Kadowaki et al. 2021) and yield significant variations in emitted and
observed fluxes. They produce a simulated $\gamma$-ray light curve that looks
generally similar to that of the Fermi-LAT observations of the very bright and
clearly variable high-energy source, Mrk 421 (Kushwaha et al. 2017). They find
a PSD slope of $\approx-2$ for that simulated $\gamma$-ray light curve while
the observations had a modestly flatter ($\approx-1.5$) PSD slope. An
extension of this work to produce optical light curves and PSDs would be
welcome.

However, it is important to note that all of the relativistic jet light curve
simulations of which we are aware have only had the computing power needed to
focus on the emissions from rather limited portions of the jet, whether from
turbulence in the region of a shock (e.g., Marscher 2014; Pollack et al.
2016), or from some form of magnetic reconnection in a portion of the jet
(Zhang & Giannios 2021; Acharya et al. 2021; Kadowaki et al. 2021). So if a
larger portion of the jet can be included in future simulations we would
expect that the fractional amplitudes of flux variations would be reduced as
fluctuations arising from multiple volumes partially cancel out, thereby
reducing any signals of quasi-periodicities in the process and very possibly
altering the overall PSD slopes.

###  7.3 Results on OJ 287

1\. While there are no substantial differences in the PSD slopes from the long
cadence data during Campaign 5 and 18 observations, there are clearly
different slopes when the short cadence data are considered. The long cadence
data of Campaign 5 and 18 respectively have slopes of $-2.28\pm 0.17$ and
$-1.96\pm 0.20$ in the “sweet spot” range from log $\nu=-5.0$ to $-6.4$, while
the short cadence data have slopes of $-2.65\pm 0.05$ and $-2.26\pm 0.06$,
respectively, when we fitted the full allowed frequency range from log
$\nu=-3.4$ to $-6.4$. The substantially larger frequency range spanned by the
short-cadence data allow for the reduced errors. We note that when the short
cadence data PSDs are evaluated only over the same “sweet spot” used for the
long cadence PSDs, the slopes are in agreement. Thus, there seems to have been
a change in the PSD slopes before and after the predicted passage of the
secondary SMBH through the accretion disk of the primary SMBH (e.g., Lehto &
Valtonen 1996; Valtonen et al. 2016). Of course, this modeled event may not be
the cause of the PSD slope change.

2\. OJ 287 was about 13-14% brighter in Campaign 5 than in Campaign 18 three
years later. During the intervening three years, the blazar brightened as
predicted by the $\sim 12$ year periodicity (see Figure 6), (e.g., Valtonen et
al. 2016), then faded. As mentioned earlier, the blazar was in the low-to-
middle range of its historical brightness during both K2 campaigns.

3\. There was no sign of a plateau at 5-day timescales in the Campaign 18 data
as there was in Campaign 5 (see Paper 1 for details). We note that Campaign 5
was 47% longer than Campaign 18, so any long-lived plateau would have been
easier to detect in Campaign 5. The causative agent for the 5-day timescale
phenomenon may have dissipated sometime in the 3-year interval that included
the predicted passage of the secondary SMBH through the primary SMBH’s
accretion disk. A recent paper on the MOMO (Multiwavelength Observations and
Modeling of OJ 287) project using Swift satellite data (Komossa et al. 2021)
provides an up-to-date discussion of variability in optical and X-ray bands;
the introduction of that paper gives an excellent brief review of OJ 287’s
variability in many wavebands. Among the key findings of their work is that a
structure function analysis of the optical–UV data indicates a characteristic
timescale of $\sim 5$ days when the source was at low-levels but apparently
somewhat longer when OJ 287 was undergoing outbursts. Komossa et al. (2021)
note that this $\sim 5$ d timescale agrees with our earlier (Paper 1) finding
of a flattening of OJ 287’s K2 PSD slope around 5.8 days during a relatively
low flux period in 2015.

4\. The minimum observed timescale of astrophysical variation, as seen by
inspecting the short cadence light curve and using the transition from white
noise to red noise at log $\nu=-3.4$ in the PSD, was about 17 minutes, similar
to the Campaign 5 data. The observed brightness variations in Campaign 18,
listed in Table 3 as the ratio of maximum to minimum brightness, were 1.36 and
1.52 for long and short cadence, respectively. Such differences in long
cadence and short cadence range values are clearly due to 30-minute smoothing
in the long cadence data over sharp peaks and valleys sometimes visible in the
short cadence light curves.

5\. As noted above, the maximum to minimum brightness ratios in Campaign 18
were 1.36 and 1.52 for long and short cadences, respectively. The
corresponding values for Campaign 5 were 2.31 and 2.36. Hence, there was more
variation in Campaign 5 than Campaign 18, as can be seen in Figure 6. The
overall “jagged”, extremely variable, character of the light curves remained
the same after three years. This optically violent variable nature is typical
of OJ 287, as ground-based observations have shown over many years (see
Komassa et al. 2021 for a review). The K2 observations showed that the source
was consistently active at these fast timescales, as presaged by observations
during dedicated ground-based campaigns. We note that variations on timescales
of an hour are sometimes, but not always, seen in ground-based observations of
OJ 287 (see, for example, Gupta et al. 2019 and references therein).

6\. OJ 287 was strongly detected by Fermi-LAT during Campaign 18, with flux of
$7.10\times 10^{-8}\pm 2.09\times 10^{-10}$ ph s-1 cm-2 in the energy band
0.1-500 GeV with a test statistic of 139.0. During Campaign 18, it was
brighter in the $\gamma$-ray band than during Campaign 5, when the flux was
$5.29\times 10^{-8}\pm 9.21\times 10^{-9}$ ph s-1 cm-2 with test statistic of
146.9.

###  7.4 Results on 8 Other Objects Observed 2 or 3 Times

In the following descriptions, we use the Fermi-LAT data presented in Papers 1
and 2 for Campaigns 1–12 and in this paper (Table 4) for Campaigns 14–18. We
use the K2 data presented in Table 3 for all campaigns because it has been
processed in a uniform manner that benefitted from improved correction for
instrumental effects. For historical background on each object, please see
Papers 1 and 2.

3C 207 (EPIC 211504760): This lobe-dominated radio quasar at redshift 0.6808
was not detected by Fermi-LAT during any of our three K2 observations. The
Campaigns 5, 16 and 18 upper limits were $<3.12\times 10^{-8},<6.29\times
10^{-9}$, and $<7.13\times 10^{-9}$ ph s-1 cm-2, respectively, in the energy
band 0.1-500 GeV. With K2, we observed substantial slow and “smooth”
variations in the count rates with maximum/minimum values of 1.32, 1.25, and
1.13, respectively. The PSD slopes in Campaigns 5, 16, and 18 were not
substantially different from each other: $-1.89\pm 0.15,-1.59\pm 0.26,$ and
$-1.57\pm 0.22$, respectively. Superluminal motion has been observed with VLBI
(e.g., Lister et al. 2016), but its lack of core-dominance may indicate that
its jet is tilted farther from our line of sight than those of the bona fide
blazars in our sample. A larger tilt may give rise to a smoother light curve.
Alternatively, the smoother light curve may arise from a jet with only one or
two synchrotron-emitting blobs adding to emission from the corona or disk,
unlike the “jagged” light curves that may come from core-jets with many
independently-emitting blobs.

PKS 1335-127 (EPIC 212489625): One of the best-known and widely studied FSRQ
blazars, it has a redshift of 0.539. It was detected with Fermi-LAT in
Campaign 17 with flux $1.12\times 10^{-8}\pm 3.03\times 10^{-9}$ ph s-1 cm-2
in the energy band 0.1-500 GeV with test statistic 18.0. In Campaign 6, it was
detected with flux $4.25\times 10^{-8}\pm 7.49\times 10^{-9}$ ph s-1 cm-2 with
test statistic 33.3. With K2 during both campaigns, we detected “jagged” light
curves with fractal-like variability at all scales. The PSD slopes were
$-2.35\pm 0.21$ and $-1.74\pm 0.17$ in Campaigns 6 and 17, respectively. The
maximum/minimum ratios were 1.64 and 1.61, respectively.

PKS 1352$-$104 (EPIC 212595811): This strong, bright FSRQ at redshift $=0.332$
flared dramatically during our first K2 observations in Campaign 6, and was by
far the most variable object in our sample. The K2 light curve in Campaign 6,
excluding the flare in the last 16.5 days of Campaign 6, had a PSD slope of
$-1.58\pm 0.24$. In Campaign 17, the maximum/minimum was 1.86, with PSD slope
of $-1.18\pm 0.38$. It is the only object for which a big flare caused a
significant change in PSD slopes between campaigns, though the two light
curves have a similar “jagged” appearance. We used the slope ($-1.86\pm 0.10$)
from the full light curve in Campaign 6 in Figure 5. With Fermi-LAT, the
blazar was weakly detected with flux $2.88\times 10^{-8}\pm 6.45\times
10^{-9}$ ph s-1 cm-2 in Campaign 6 and not detected in Campaign 17 with upper
limit of $5.54\times 10^{-8}$ ph s-1 cm-2.

RBS 1273 (EPIC 212800574): This X-ray QSO, also known as 1RXS 132928.0-053132,
has $K_{p}=15.201$ and redshift 0.57587. The light curves showed three or four
big broad bumps in both Campaigns 6 and 17. This X-ray QSO, consequently, has
the steepest PSD slopes in the sample. Comparing the two PSDs, we see that the
white noise extends to much different frequencies in the two K2 observations.
The white noise characteristics are consistent with those of stars on the same
channels and thus probably do not indicate astrophysical changes in the AGN.
RBS 1273 was not detected during either campaign 5 or 17 by Fermi-LAT with
upper limits of $9.88\times 10^{-9}$ ph s-1 cm-2 and $1.63\times 10^{-8}$ ph
s-1 cm-2, respectively.

PKS B1329$-$049 (EPIC 229227170, with alternate EPIC id 229228144): This high
redshift FSRQ ($z=2.15$) with $K_{p}=18.2$ was unexpectedly faint during
Campaigns 6 and 17 with average count rates of 310 and 570 cts s-1 while the
maximum/minimum ratios were 1.14 and 1.29, respectively. There was no
significant difference in the PSD slopes of $-2.01\pm 0.46$ and $-1.93\pm
0.23$. It was not detected by Fermi-LAT in either campaign 6 or 17, with upper
limits of $2.92\times 10^{-8}$ ph s-1 cm-2 and $2.15\times 10^{-9}$ ph s-1
cm-2, respectively.

RGB J0847+115 (EPIC 211394951): This strong X-ray BL Lac object, detected at
TeV energies, is also known as RX J0847.1+1133. It is slightly extended (2.3″)
on SDSS observations with $z$ = 0.1982 based on weak lines. It was highly
variable in Campaign 5 and moderately variable in Campaign 18 (maximum/minimum
ratios of 1.47 and 1.17, respectively). The PSD slopes were similar: $-2.20\pm
0.26$ and $-2.02\pm 0.17$. It was detected by Fermi-LAT in Campaign 5 with
flux $9.38\times 10^{-9}\pm 7.06\times 10^{-9}$ ph s-1 cm-2 but not in
Campaign 18 with upper limit $2.58\times 10^{-9}$ ph s-1 cm-2.

WB J0905+1358 (EPIC 211559044, with alternate EPIC id 211559047): Also known
as MG1 J090534+1358, this BL Lac has slow, smooth light curves with
maximum/minimum ratios of 1.44 and 1.28 in Campaigns 5 and 18. Small
variations on timescales of a few days are superimposed on slow large
amplitude bumps, with resulting PSD slopes of $-1.76\pm 0.13$ and $-2.04\pm
0.13$, respectively. It was detected by Fermi LAT in Campaign 5 with flux
$4.89\times 10^{-9}\pm 3.53\times 10^{-9}$ ph s-1 cm-2 but not in Campaign 18
with upper limit $2.59\times 10^{-8}$ ph s-1 cm-2.

BZB J0816+2051 (EPIC 212035840): This BL Lac, with $K_{p}=17.486$ has no known
redshift; the optical spectrum is featureless, see, for example, SDSS DR16
(http://skyserver.sdss.org/dr16). It had jagged light curves in Campaigns 5
and 18, with maximum/minimum ratios of 1.57 and 1.34. Its respective PSD
slopes were steep $-2.69\pm 0.26$ and $-2.57\pm 0.17$. It was not detected by
Fermi-LAT in Campaign 18 with an upper limit of $4.77\times 10^{-8}$ ph s-1
cm-2. In Campaign 5 the upper limit was $3.67\times 10^{-9}$ ph s-1 cm-2.

###  7.5 Results on Other Individual AGN with Single Observations

We present our results on the other individual AGN with single observations as
follows. Light curves and binned PSDs are shown in Figure 4 (a-i), ordered by
EPIC number.

4C +06.41= PKS 1038+064 (EPIC 248611911): One of two possible counterparts for
the IceCube neutrino event 141209A detected in 2014 (IceCube Collaboration
2017, 2018), this is an FSRQ with $K_{p}=16.888$ at redshift 1.27. Garrappa et
al. (2019) resolved the 2FGL and 3FGL $\gamma$-ray sources near the position
of 4C +06.41 by using 9.8 years of Fermi-LAT data into two sources 22″apart
which are coincident with 4C +06.41 at RA, DEC = 10h41m17.1625s,
+06d10m16.924s, and GB6 J1040+0617 at RA, DEC = 10h40m31.630s, +06d17m21.70s.
Over the course of the 9.8 years of Fermi LAT observations, the two
$\gamma$-ray sources varied in brightness. The multi-wavelength light curves
in Garrappa et al. (2019) showed GB6 1040+0617 had an increase in brightness
in the $\gamma$-ray band and flared in the optical band while 4C+06.41 was not
detected with Fermi-LAT during the arrival of IceCube neutrino event 141209A.
Hence, Garrappa et al. (2019) found that GB6 1040+0617 is more likely than PKS
1038+064 to be the origin of IceCube neutrino event 141209A. Nonetheless, 4C
+06.41 remains a plausible neutrino counterpart. During our K2 observations,
4C +06.41 was at least 7 times brighter than GB6 1040+0617, specifically,
4C+06.41 was strongly detected with Fermi-LAT with flux of $5.76\times
10^{-8}\pm 2.64\times 10^{-10}$ ph s-1 cm-2 with test statistic of 42.1 while
GB6 J1040+0617 had a Fermi-LAT upper limit of $8.42\times 10^{-9}$ ph s-1 cm-2
. Hubble Faint Object Spectrograph observations showed an absorption line
system at $z=1.224$, in addition to the previously identified system at
$z=0.441$ (Bechtold et al. 2002); we do not expect that the absorption line
systems affected the K2 photometry observations. During our observations in
Campaign 14, the maximum/minimum ratio was 1.18 while the PSD slope was
$-2.17\pm 0.13$.

NVSS J110735+022225 (EPIC 201621388): This is a BL Lac object with
$K_{p}=18.584$ with a spectroscopic redshift lower limit of 1.0735 obtained
from MgII absorption line detection by Paiano et al. (2017). The NED-preferred
redshift of 1.08213 was obtained with SDSS DR7 photometry and is not reliable
because there are no well-detected lines; subsequent SDSS data releases’
redshifts have similar problems. The object is in a crowded optical field with
one object about 10″away. During Campaign 14, the K2 light curve had a
maximum/minimum ratio of 1.11 and PSD slope of $-2.95\pm 0.35$. It was not
detected by Fermi-LAT with an upper limit of $<1.97\times 10^{-9}$ ph s-1 cm-2
during Campaign 14.

TXS 1013+054 (EPIC 251457104): This is an FSRQ with $K_{p}=18.75$ and a
reliable SDSS redshift of 1.71272 determined from 6 emission lines. During
Campaign 14, the jagged K2 light curve showed very high variability, with
maximum/minimum of 8.50, and a PSD slope of $-1.88\pm 0.17$. Simultaneously,
it was detected with Fermi-LAT with flux of $4.41\times 10^{-8}\pm 6.3\times
10^{-10}$ ph s-1 cm-2 and test statistic 31.0.

TXS 0836+182 (EPIC 211852059): This is a BL Lac with $K_{p}=17.046$ and
unknown redshift. An estimated redshift of 0.28 was obtained by Abraham et al.
(1991) based on galaxy morphological fitting. The other redshift given in NED
is $1.155507\pm 0.264737$ from an SDSS spectrum which has no reliably detected
emission lines. The K2 maximum/minimum ratio was 1.26, with PSD slope of
$-2.38\pm 0.17$. It was not detected with Fermi-LAT during the K2 observations
with an upper limit of $<5.29\times 10^{-9}$ ph s-1 cm-2.

NVSS J090226+205045 (EPIC 212035517): This object has $K_{p}=15.741$. The
redshift and class are not known. The two redshifts in NED were obtained by
different generations of SDSS analysis; the most recent was $1.52204\pm
1.41678$. It was detected by Fermi-LAT with flux of $1.05\times 10^{-8}\pm
7.93\times 10^{-11}$ ph s-1 cm-2 with test statistic 26.1. The K2 variability
was strong in the jagged light curve, with maximum/minimum ratio 2.48 and PSD
slope of $-2.67\pm 0.14$.

TXS 0853+211 (EPIC 212042111): This BL Lac has $K_{p}=18.345$ and unknown
redshift. The spectrum is nearly featureless: various SDSS data releases
listed redshifts of $0.2762\pm 4.00000$, $1.58702\pm 1.00000$, or classified
the object as an AO star. The K2 light curve is very jagged, with
maximum/minimum ratio of 1.43 and PSD slope of $-1.76\pm 0.18$. It was not
detected by Fermi-LAT during the K2 observations with an upper limit of
$<4.36\times 10^{-9}$ ph s-1 cm-2.

PMN J1318-1235 (EPIC 212507036): This object of unknown class has
$K_{p}=18.027$ and no known redshift. The jagged K2 light curve has
maximum/minimum ratio of 1.71 and a relatively steep PSD slope of $-2.55\pm
0.21$. It was not detected by Fermi-LAT during the K2 observations with an
upper limit of $<1.54\times 10^{-8}$ ph s-1 cm-2.

NVSS J090900+231112 (EPIC 251376444): As mentioned earlier, this BL Lac has a
fainter, non-AGN companion galaxy 3.8″away (Rosa-González et al. 2017) with
the same absorption line redshift. The K2 light curve is jagged, with a
maximum/minimum ratio of 1.24 and steep PSD slope of $-2.57\pm 0.32$. It was
detected by Fermi-LAT with flux of $1.80\times 10^{-9}\pm 2.57\times 10^{-11}$
ph s-1 cm-2 with test statistic of 16.6.

PKS B1310-041 (EPIC 251502828): This blazar has $K_{p}=18.3$. Sowards-Emmerd
et al. (2004) show an optical spectrum obtained with the HET/LRS with derived
redshift of 0.8249 and classed the object as an FSRQ. The K2 light curve is
smooth, dropping precipitously toward the end of the K2 observations, with
maximum/minimum ratio of 1.29 and PSD slope of $-1.73\pm 0.10$. It was not
detected by Fermi-LAT during the K2 observations with an upper limit of
$<8.13\times 10^{-8}$ ph s-1 cm-2.

##  8 Summary

Our main results are as follows:

1\. Using a KS test with a $p$ value of 0.039, we find a statistically
significant difference between the periodogram PSD slopes of the BL Lacs and
FSRQs in our combined sample of Fermi-LAT detected blazars observed by K2,
with the BL Lac slopes steeper; however, this result is not as strongly
supported by PSRESP analysis probably because it includes data at low
frequencies below the well-sampled “sweet spot” which tend to flatten the
slope. We find that neither redshift nor doppler boosting effects can account
for this difference. Differences in the origin of the jets from the ergosphere
or accretion disk in these two classes could produce such a disparity, as
could the different sizes and locations of emission regions within the jets.

2\. We see small differences in PSD slopes of some AGN for which we had long
cadence K2 observations at two or three epochs. The significance of the
differences depends on whether the periodogram or PSRESP slopes are used.

3\. For OJ 287, although the two long cadence periodogram PSD slopes are
consistent, the short cadence PSD slopes show a substantial difference over
frequencies where the astrophysical red-noise dominated over instrumental
white noise, specifically, on timescales shorter than a few hours. This change
may be related to the passage of the secondary SMBH through the accretion disk
of the primary SMBH which the binary black hole model predicted (Valtonen et
al. 2016) would have occurred between those K2 observations.

4\. The K2 light curves appeared either fast and jagged or slow and smooth.
Three-quarters of the blazars had jagged light curves. No class differences
were observed: specifically, 12 BL Lacs and 7 FSRQs had jagged light curves; 4
BL Lacs and 2 FSRQs had smooth light curves.

5\. Sources with Fermi-LAT detections during their K2 observations usually
showed greater optical variability.

Our other results include:

6\. We found no correlation between redshift and PSD slope.

7\. Contemporaneous ground-based observations for a subset of the sources
confirm that the EVEREST Version 2 processing properly reproduces the
astrophysical variability in blazars.

8\. Simulations show that for an input red noise process with a specific
slope, a Fourier-based time series analysis produces a spread of slopes around
the input PSD slope (see Appendix). Hence, PSD slopes for a single observation
of a single source should be cautiously interpreted.

In conclusion, the K2 mission provided an unique window onto the optical
variability properties of blazars, thanks to the sub-hour cadence and
essentially uniform temporal sampling of these observations made over quite
extended durations. Such measurements really cannot be obtained from the
ground. Currently, the Transiting Exoplanet Survey Satellite (TESS) is
observing a larger number of blazars with measurements over shorter total
durations (28 days) than K2 with the first of these results now available
(Weaver et al. 2020; Raiteri et al. 2021). The TESS short cadence observations
can probe the higher frequency regions of the the PSDs of the very brightest
blazars. As these short timescales are where we noted a significant difference
between the two observations of OJ 287, multiple observations of other very
bright blazars by TESS should be illuminating. In the future, the multi-day
cadence monitoring of very large samples of blazars and other AGN will come
from the LSST program on the Vera Rubin Telescope. The LSST observations will
provide superb long duration data sets and produce critical complementary
information on the optical variability discussed here.

We thank the anonymous reviewer for their comments. We thank Josefa Becerra
Gonzalez, Deirdre Horan, and Arti Goyal for helpful discussions. We are
grateful to Rodrigo Luger (University of Washington and Flatiron Institute)
for advice on processing short cadence data through EVEREST software. We thank
Paul Smith (Steward Observatory) for helpful discussions and for making his
data publicly available. We thank Simone Garrappa for advice and discussion
about the 4C+06.41 field. We acknowledge support from NASA K2 Guest
Investigator Program grant NNX16AI61G. This paper includes data collected by
the K2 mission. Funding for the K2 mission is provided by the NASA Science
Mission Directorate. B.S.G. was supported by Thomas Jefferson Chair for
Discovery and Space Exploration at the Ohio State University. The 0.5-m
DEdicated MONitor of EXotransits and Transients telescope (DEMONEXT) is
located at and operated by the Winer Observatory in Arizona and funded by the
Massachusetts Institute of Technology, The Ohio State University, and
Vanderbilt University. The 1.3-m Robotically Controlled Telescope (RCT)
located at Kitt Peak National Observatory is operated by the RCT Consortium.
Data from the Steward Observatory spectropolarimetric monitoring project were
used. This program is supported by Fermi Guest Investigator grants NNX08AW56G,
NNX09AU10G, NNX12AO93G, and NNX15AU81G. The Fermi LAT Collaboration
acknowledges generous on-going support from a number of agencies and
institutes that have supported both the development and the operation of the
LAT as well as scientific data analysis. These include the National
Aeronautics and Space Administration and the Department of Energy in the
United States, the Commissariat a l’Energie Atomique and the Centre National
de la Recherche Scientifique / Institut National de Physique Nucléaire et de
Physique des Particules in France, the Agenzia Spaziale Italiana and the
Istituto Nazionale di Fisica Nucleare in Italy, the Ministry of Education,
Culture, Sports, Science and Technology (MEXT), High Energy Accelerator
Research Organization (KEK) and Japan Aerospace Exploration Agency (JAXA) in
Japan, and the K. A. Wallenberg Foundation, the Swedish Research Council and
the Swedish National Space Board in Sweden. Additional support for science
analysis during the operations phase is gratefully acknowledged from the
Istituto Nazionale di Astrofisica in Italy and the Centre National d’ Études
Spatiales in France. This research has made use of the NASA/IPAC Extragalactic
Database (NED) which is operated by the Jet Propulsion Laboratory, California
Institute of Technology, under contract with the National Aeronautics and
Space Administration. Some of the data presented in this paper were obtained
from the Mikulski Archive for Space Telescopes (MAST). STScI is operated by
the Association of Universities for Research in Astronomy, Inc., under NASA
contract NAS5-26555. Support for MAST for non-HST data is provided by the NASA
Office of Space Science via grant NNX09AF08G and by other grants and
contracts. The specific observations analyzed can be accessed via
https://dx.doi.org/10.17909/ajrj-2a29 (catalog
https://dx.doi.org/10.17909/ajrj-2a29). Some observations were obtained with
the Samuel Oschin 48-inch Telescope at the Palomar Observatory as part of the
Zwicky Transient Facility project. ZTF is supported by the National Science
Foundation under Grant No. AST-1440341 and a collaboration including Caltech,
IPAC, the Weizmann Institute for Science, the Oskar Klein Center at Stockholm
University, the University of Maryland, the University of Washington,
Deutsches Elektronen-Synchrotron and Humboldt University, Los Alamos National
Laboratories, the TANGO Consortium of Taiwan, the University of Wisconsin at
Milwaukee, and Lawrence Berkeley National Laboratories. Operations are
conducted by COO, IPAC, and UW. Kepler, K2, Fermi, RCT, SMARTS, ZTF.

##  9 Appendix: Simulations

###  9.1 Comparison of PSD Analysis Approaches

To better understand the potential limitations of the PSD analysis, we
undertook a series of simulations comparing calculating the PSD from a Fast
Fourier Transform (FFT,) from a straightforward implementation of the discrete
Fourier transform (DFT), and from a periodogram (Scargle 1982) analysis. Light
curves were initially simulated following the method of Timmer & Koenig
(1995), and then PSDs were calculated.

In one of the most frequently cited publications on PSDs of light curves,
Fougere (1985) simulated 100 red noise light curves with a 5 minute duration
and 2.8 second cadence, matching that of ionospheric scintillation data from
the MARIST satellite, then calculated PSDs and determined their slope. He
showed that for end-matched and windowed data with an input PSD slope =
$-2.00$, an average PSD slope of $1.9841\pm 0.0688$ was found when the PSD was
determined via an FFT. No binning of the PSD was done prior to his fitting
process. As described in Section 4, we used a periodogram rather than an FFT
(since the K2 data are not precisely evenly spaced) to determine the PSD, and
we binned the PSD by 0.08 in log $\nu$ space prior to fitting. In order to
test the effects of this binning, we produced 100 red noise light curves and
PSDs with the same duration and cadence as in Fougere (1985), determined the
PSD via both a FFT and a periodogram analysis, and then binned and fit the PSD
slope in the same manner as in this paper. For the PSD produced by the FFT, we
found an average slope of $-1.92\pm 0.04$ and for the PSD produced by the
periodogram analysis, we found an average slope = $-2.00\pm 0.04$. It does not
appear that the use of binning or the use of periodogram rather than the FFT
to produce the PSD has any significant effect on the slope calculation though
the periodogram may be preferable. The resulting histograms of PSD slopes are
shown in Figure 7a,b.

.

Figure 7: Histograms of simulated PSD slopes calculated via (a) an FFT and (b)
a periodogram with the same duration and cadence as Fougere (1985). Use of a
periodogram to determine the PSD and binning of the PSD prior to fitting do
not affect the determination of the PSD slope. Histograms of simulated PSD
slopes calculated via (c) an FFT, (d) a DFT and (e) a periodogram for 100
simulated light curves with the K2 observation cadence and duration, generated
with an input PSD slope of $-2.0$. On average, the periodogram returns the
closest match to the input PSD slope. (f) Histogram of PSD slopes determined
from our periodogram analysis of 1000 light curves simulated with an input PSD
slope of $-2.0$, with each 800-day light curve broken into 10 segments, and
with a K2 template sampling pattern applied.

We noticed that in the simulations described above, an input slope of $-2.0$
returned a range of measured output PSD slopes. In order to understand the
implications of this for the use of the PSD to determine the underlying
process, we simulated 100 light curves with an input slope $=-2.0$, a turnover
to white noise at log($\nu$) = $-5.0$, 30-minute sampling and a duration of 80
days, thus matching the K2 light curve characteristics of the objects in our
sample. We calculated the PSDs using three techniques: FFT, discrete Fourier
transform and a periodogram analysis. We performed the fits to the slopes in
exactly the same manner as for the K2 light curves in this paper. The
resulting histograms of the distribution of fitted PSD slopes are shown in
Figure 7c,d,e. For the FFT case, the average slope = $-2.05\pm 0.23$, for the
DFT case the average slope = $-2.22\pm 0.23$ and, for the periodogram case,
the average slope = $-2.02\pm 0.23$. The periodogram approach returns the
average output slope closest to the input slope of $-2.0$; however, in all
three cases, a very similar range of slopes is returned by the analysis. These
simulations show that while a slope returned via a PSD analysis does not
always precisely reflect the input PSD, the computed slopes will usually be
quite close to the input power-spectrum. We then examined what, if any, effect
the slightly uneven sampling of the K2 observations would have on the
determination of the PSD slope via our periodogram methodology. Examining our
K2 light curves, we found that after excluding data points with poor data
quality (determined using the project-provided quality flags), between 5 and
10% of cadences were removed from the light curves. Using the observation of
OJ 287 as a typical (6.8% cadence loss) sampling template, we simulated 1000
light curves with a duration 10 times that of the K2 observations (800 days)
with an input PSD slope of $-2.0$. Each 800 day light curve was broken into
ten 80-day segments (to mimic the usual K2 observation campaign length) and
the typical K2 sampling pattern described above was applied. The resulting
distribution of output slopes after application of our periodogram method is
shown in Figure 7(f). The result is that for an input slope of $-2.0$, the
mean returned slope was $-1.99\pm 0.21$.

We also investigated the effect that our choice of bin size for the PSD has on
the resulting PSD slope determination. We simulated an 80-day light curve with
30-minute sampling, a slope of $-2.0$ and a turnover to white noise at
log($\nu$)= $-5.0$. We fit the PSD in the same manner as the PSDs in this
paper. We find that using bin sizes between 0.05 and 0.20 in log space does
not significantly affect the slope determination.

###  9.2 Effects of Doppler Boosting and Redshifts

To test if the effects of different Doppler boosting or redshifts could be the
cause of the differences in PSD slopes between the BL Lac objects and FSRQs in
our sample, we simulated 1000 rest frame light curves and applied Doppler
boosting corrections to simulate the rest frame light curve observed as a BL
Lac and as an FSRQ. The rest frame light curves were simulated via the process
described in Timmer & Koenig (1995). We assumed an input PSD slope of $-2.0$
and a turn over to white noise at log ($\nu$)$=-5.0$, which is consistent with
the typical turnover to white noise seen in our sample objects. The Doppler
boosting corrections applied were those discussed in Gopal-Krishna et al.
(2003). We used the mean redshifts of BL Lacs (0.329) and FSRQs (1.05) and we
assumed a spectral index of $-1.0$. We assumed average Doppler boosting
factors of 9.2 for BL Lacs and 15.2 for FSRQs, as given in Liodakis et al.
(2017). We computed the PSDs and their slopes using the same method as for our
sample objects, described in Section 4 and Papers 1 and 2.

We found no evidence that Doppler boosting (or redshift differences, which
partially cancel each other) would create the difference in PSD slopes seen in
our sample. The mean PSD slope of the rest frame light curves boosted to the
above BL Lac parameters was $-2.02\pm 0.25$ and for rest frame light curves
boosted to FSRQ parameters also was $-2.04\pm 0.25$, essentially identical to
the input slope in both cases. Figure 8 shows histograms of the distribution
of PSD slopes. The distributions look almost identical; a two sided KS test
returns $p=0.60$ for the null-hypothesis that these two distributions arise
from the same parent population.

Figure 8: Histograms of the PSD slopes of 1000 simulated red noise AGN light
curves with an input PSD slope of $-2.0$, boosted by Doppler factors of 9.2
for BL Lacs (black) and 15.2 for FSRQs (purple). The average slope and range
of slopes are not affected by Doppler boosting or redshift differences.

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