#  A QPO in Mkn 421 from Archival $\it{RXTE}$ Data

Evan Smith, Lani Oramas, Eric Perlman  Aerospace, Physics and Space Sciences
Department, Florida Institute of Technology, 150 West University Blvd.,
Melbourne, FL. 32901, USA

###### Abstract

We report a 325($-7,+8$) day quasi-periodic oscillation (QPO) in the X-ray
emission of the blazar Mkn 421, based on data obtained with the
$\it{Rossi~{}X}$-$\it{ray~{}Timing~{}Explorer~{}(RXTE)}$ satellite. The QPO is
seen prominently in the ASM data (at least 15 cycles), due to the fact that it
has had near-continuous sampling for more than a decade. The PCA data, where
the sampling is not uniform and shows many large gaps, provide supporting
evidence at lower significance. QPOs are an important observable in accretion
disks, can be modulated by various orbital timescales, and may be generated by
a number of mechanisms. They have been studied extensively in X-ray binaries,
and should be present in active galactic nuclei (AGN) if they are governed by
a common set of physical principles. In jetted sources, QPOs can probe jet-
disk interactions or helical oscillations. This QPO previously has been
claimed intermittently in X-ray, radio and gamma-ray data, but the continuous,
15-year extent (1996-2011) of the ASM observations (in which Mkn 421 is the
brightest AGN observed) provides a unique window. The QPO appears present for
nearly the entire extent of the ASM observations. We explore various physical
origins and modulating mechanisms, particularly interpretations of the QPO as
a result of disk-jet interactions, either due to an accretion disk limit
cycle, jet instabilities or helical motions. Limit-cycle related oscillations
would not interact with either Keplerian or Lense-Thirring modulated
oscillations, however those associated with jet instabilities or helical
motions in the jet would likely be modulated by Lense-Thirring precession.

###### Subject headings:

quasi-periodic oscillations, active galactic nuclei, $\it{RXTE}$, QPOs, AGN

##  1\. Introduction

Quasi-Periodic Oscillations (QPOs) in accretion disks have been thoroughly
studied in both neutron star (NS) and black hole X-ray binaries (BHXRB). High-
frequency (HF) QPOs in galactic sources are thought to probe the motion of
matter under strong gravity near the surface of the NS or event horizon of the
BH. Models of QPOs include Keplerian orbital motion of matter in the disk,
central compact object spin, general relativistic effects such as Lense-
Thirring precession, or beat frequencies between two of the previous
mechanisms. The history of these models can be found in various reviews such
as van der Klis (1989, 2000) and Ingram & Motta (2019).

$\it{RXTE}$ observations of microquasars found QPOs at nearly identical
frequencies: 67 Hz for GRS 1915+105 and 66 Hz for IGR J17091-3624.
Similarities in the complex light curves of these two sources prompted
investigation of whether the same physical mechanisms cause these behaviors
(Altamirano et al., 2011). A cardinal question in astrophysics is whether
galactic black holes in BHXRB and supermassive black holes (SMBHs) in AGN are
governed by shared physical processes. Merloni et al. (2003) investigated the
disk-jet connection, yielding tight correlations between BH mass, radio and
X-ray luminosity over many orders of magnitude, thus supporting the theory of
scale invariance. If there is a commonality in the underlying physics of disk-
accretion and jet-launching in BHXRB and AGN, observational evidence of timing
features would appear on much different time scales, often compared using the
mass ratio.

More recent studies into microquasars have given rise to new theories behind
QPOs and the newly named quasi-periodic eruptions (QPEs) (Giustini et al.,
2020; Miniutti et al., 2019), expanding the list of possible common
mechanisms. The same concept of shared processes applies to galactic BHs in
BHXRB and SMBHs in AGN. Similarly, Janiuk & Czerny (2011)’s expansion upon
Cannizzo (1992, 1993, 1998)’s work has led to the accretion disk limit-cycle
being considered in both AGN and BHXRB.

The widely known Schwarzschild radius is given by:

$R_{S}\equiv\frac{2GM}{c^{2}}$ (1)

For a non-rotating, or Schwarzschild, BH, the innermost stable circular orbit
(ISCO) for material in the accretion disk has a radius of $R_{ISCO}\equiv
3R_{S}$. For a rotating, or Kerr, BH the radius of the ISCO for a maximally-
rotating BH in both the co-rotating and counter-rotating cases is constrained
by $0.5R_{S}\leq R_{ISCO}\leq 4.5R_{S}$ (Jefremov et al., 2015).

Early searches for QPOs focused heavily on Seyfert galaxies, including studies
conducted by Gierliński et al. (2008); Middleton & Done (2010); Alston et al.
(2014, 2015); Smith et al. (2018); Edelson et al. (2014); Zhou et al. (2018);
Tarnopolski et al. (2020) and Smith et al. (2020). Successes in these
campaigns paved the way for similar searches in blazars. Much of the variable
emission in blazars, particularly BL Lacertae objects, comes from non-thermal
processes, particularly within the relativistic jet which is seen at very
small angles to the line of sight and therefore Doppler boosted (see, e.g.,
Giommi et al. (2021)). Nevertheless, recent work has begun to suggest that the
violent, variable emission in these objects can be traced to turbulence in the
jets launched by their accretion flows (O’Riordan et al., 2017), accentuated
by Doppler boosting. This has become the leading paradigm to explain some of
the long-term variability of blazars (Thiersen et al., 2019). Disk behavior
can produce some variability in the optical and X-ray. However, jet
variability must dominate in the radio and gamma-rays. The link between disk
behavior and jet variability is thus quite interesting.

In the radio, QPOs have been detected in the variability behavior of the FSRQ
J1359+4011 (King et al., 2013). In the gamma-rays, data taken with the Fermi
Gamma-ray Space Telescope have also proven fruitful in the search for QPOs.
These detections include a 34.5-day QPO detected in PKS 2247$-$131 and the
612-day QPO in PKS 2155$-$304 (Zhou et al., 2018; Tarnopolski et al., 2020).
Indeed, in a recent campaign on CTA 102, the gamma-ray and optical light
curves were both observed to have QPOs with a dominant period of 7.6 days that
lasted for about 8 cycles, a behavior attributed in that paper to enhanced
emission from a helical jet (Sarkar et al., 2020). In Mkn 421, two recent
works utilizing Fermi LAT data, (Bhatta et al., 2020; Ren et al., 2022),
claimed QPOs at $\sim$280 days and 308 $\pm$ 99 days respectively, although
the latter authors estimated a modest 2.9$\sigma$ significance. In contrast
Carnerero et al. (2017), Tarnopolski et al. (2020), and Sandrinelli et al.
(2017) did not find a QPO in Fermi LAT data of Mkn 421 despite their data
including the same time frame.

However, claims of quasi-periodic behavior in blazars in the X-ray and optical
have been less common. In Mkn 421, a recent power density spectrum (PDS)
analysis of its long-term X-ray variability using data from AstroSAT came to
the conclusion that the X-ray variability was due to changes in the accretion
disk translating into the jet (Chatterjee et al., 2018), due to a break in the
PDS. That work did not search for QPOs, but Li et al. (2016) did a combined
analysis of the Swift, Fermi LAT and Owen’s Valley Radio Observatory (OVRO)
data between 2008-2015. Those authors noted $\sim$300 day QPOs in all three
light curves, but offered little information on their overall significance or
persistence. In the optical, the Kepler observations of the blazar W2R1926+42
(Edelson et al., 2013) had significant PDS breaks, but no QPOs. On the other
hand, there is the notable case of the blazar OJ 287, where the optical light
curve shows evidence of a 12-year oscillation (Bhatta et al., 2016), believed
to be due to a binary SMBH (Valtonen et al., 2012).

In addition to QPOs, subsequent X-ray flares can appear in light curves. This
new phenomenon is known as quasi-periodic eruptions (QPEs). Giustini et al.
(2020) and Miniutti et al. (2019) define QPEs as rapid and recurrent X-ray
bursts above a stable level. Theorized causes of QPEs include radiation
pressure instability, the orbital motion of a second body, tidal disruption
events, and accretion disk emissions ”leaking” through windows. The first
confirmed QPEs are in the light curve of GSN 069, leading to a search for
similar AGN. The search led to RXJ1301.9+2747, chosen based on a shared set of
qualities with GSN 069. While Mkn 421 does not share these characteristics, it
is possible that some massive AGN may also exhibit partial QPEs (Miniutti et
al., 2019). Six similar flares are seen in Figure 4, though they are not
confirmed to be QPEs.

##  2\. Observations

To search for low-frequency (LF) QPOs, we investigated archival data from the
$\it{Rossi~{}X}$-$\it{ray~{}Timing~{}Explorer}$ ($\it{RXTE}$) satellite
collected between 1996 and 2011. AGN were targets for monitoring with RXTE
both with the Proportional Counter Array (PCA) and All-Sky Monitor (ASM). This
analysis uses data from the $\it{RXTE}$ PCA (Jahoda et al., 2006) and ASM
since Mkn 421 is nearly an order of magnitude brighter than any other AGN in
the 2$-$10 keV bandpass shared by the PCA and ASM.

###  2.1. ASM Observations

The All-Sky Monitor (ASM) contains three rotatable coded aperture cameras
called scanning shadow cameras (SSCs) (Levine et al., 1996). Each SSC has
position-sensitive proportional counters (PSPCs) that view the sky through
slits and measure the displacements and strength of shadow patterns cast by
X-ray sources in the field of view. The carbon-coated quartz anodes in the
PSPCs then create a pair of pulse heights used to determine event energy and
positions via the charge division technique. This technique can resolve the
position (the coordinates perpendicular to the slits) within 0.2 to 0.5 mm.
For each anode, the counts are binned as a function of position in three
energy bands and histograms are accumulated in ”dwells”. Dwells are the 90
seconds during which the spacecraft maintains a fixed attitude and the
orientation of each SSC is fixed with respect to the sky because the ASM’s
rotation drive is not active (Levine et al., 1996). These position histograms
are then used to calculate intensities using a least-squares fit. The fitted
intensities and their uncertainties are then used to calibrate the SSCs and
ensure that all three SSCs give the same results.

The ASM scans 80% of the sky every $\it{RXTE}$ orbit. The light curves
obtained by the ASM can be used in two formats: Dwell-by-Dwell or 1-Day
Average. The Dwell-by-Dwell format includes every ASM dwell for the object
while the 1-Day average format gives the average of all dwells on the given
date (determined by MJD).

For most AGN, the RXTE ASM data were of limited use since the fluxes were
below the sensitivity level of the ASM. However, Mkn 421 is by far the
brightest AGN in the RXTE band-pass, so it was easily detectable. Various
groups (Chitnis et al., 2009; Rani et al., 2009) have investigated other AGN
of lower flux by substantially binning the data.

###  2.2. PCA Observations

A total of 1182 pointed $\it{RXTE}$ observations were made of the BL Lac
object, Mkn 421. AGN monitoring was regularly performed for many sources, and
reduced 3-color light curves have been prepared and archived at the University
of California, San Diego (UCSD) (Rivers et al., 2011, 2013).

$\it{RXTE}$ conducted observations on a total of 153 AGN over its 16-year
lifespan. UCSD created the $\it{RXTE}$ AGN archive by collecting all the data
gathered by different groups of proposers for various scientific aims,
regardless of sampling strategy. Of the AGN in the UCSD database, we found 76
with enough separate observations and appropriate spacing to support the
search for low-frequency QPOs.

Details of the data reduction for the PCA data can be found in the description
of the UCSD archive. A standard screening procedure was to exclude data within
20 minutes of $\it{RXTE}$ passage through the South Atlantic Anomaly (SAA).
Observations made when RXTE was near the SAA could be expected to stand out
from other observations since, even if the background model is very good,
observations with higher background count rate will also have a larger
uncertainty in the background count rate.

The typical length of an observation is around 2000 seconds and is set by the
observable times for the object within an orbit. Each observation represents a
single data point in the light curve, regardless of its length. The mean flux
over the entire observation is the archived value, and the time tag is the
observation midpoint. For each AGN observation, the $\it{RXTE}$ AGN archive at
UCSD provides the mean count rate and its uncertainty in the 2$-$10 keV
spectral range, and for three sub-bands (2$-$4 keV, 4$-$7 keV, and 7$-$10
keV). Other preliminary results using the UCSD archive, both positive and
negative, have been reported (Smith et al., 2019, 2020; Hursh et al., 2020;
Robles & Perlman, 2020; Oramas et al., 2021).

##  3\. Data Analysis

###  3.1. Quality Assurance

A useful quality assurance technique is to compute the window transform, or
Lomb-Scargle periodogram (LSP) of the window function. This is done using the
time series from the actual observations, but with constant flux. This can
help rule out false peaks resulting from the data distribution (VanderPlas,
2018). We find it useful to plot the window LSP together with the data LSP,
allowing for easier visual comparison. Applying this technique to our analysis
of Mkn 421 helps establish that the 320 day period does not result from the
time spacing of the observations.

To formally establish a False Alarm Probability (FAP), we used the method of
Horne $\&$ Baliunas (Horne & Baliunas, 1986) to estimate the uncertainties in
the QPO frequency, as detailed in Smith et al. (2020).

###  3.2. RXTE Data Analysis

To analyze the power spectrum of Mkn 421, the $\it{RXTE}$ AGN light curve
files were run through a program to determine the LSP. The input data is
loaded into three arrays:

  1. 1.

x (the MJD midpoint of the observation),

  2. 2.

y (the flux in the selected band),

  3. 3.

dy (the uncertainty in the flux).

The program outputs the light curve and the LSP. In Figures 1-8, discussed
below, we show the light curves and LSP for the RXTE observations. We split
our analysis into the ASM data, discussed in Section 3.2.1, and the PCA data,
discussed in Section 3.2.2.

####  3.2.1 ASM Analysis

The ASM 1-Day Averages give a light curve spanning 5837 days from MJD
50091$-$55928 (1996 January 9 $-$ 2011 December 29). This is binning performed
by the ASM team.

We filtered flux values based on their signal-to-noise ratio to eliminate
outliers with higher errors and filter out negative flux values seen in the
raw data. A limiting signal-to-noise ratio of $S/N\geq 0.6$ accomplished both
goals. However, the same spectral features are seen with and without the
filtering process.

There were originally 5837 data points with a mean spacing of 1 day. After
filtering, we were left with a total of 3935 data points with a mean spacing
of 1.47 days. Our time frame of interest, MJD 50152$-$55924, contained 3928
data points (Figure 2). The LSP for these data (Figure 2), with window LSP in
red, shows a strong peak at 0.001708 $\mu$Hz, corresponding to a period of 650
days. This 37-$\sigma$ detection has a FAP $\sim 10^{-12}$. The peak to the
immediate left is almost as high and at the 3:1 resonance with a frequency of
0.00119 $\mu$Hz, corresponding to a period of 975 days. A third strong peak is
seen to right of the 0.001804 $\mu$Hz peak at 0.03552 $\mu$Hz, corresponding
to a period of 325 days. This 32-$\sigma$ detection has a FAP of $\sim
10^{-10}$.

Figure 1.— (left) The $\it{RXTE}$ ASM light curve for Mkn 421, 1-day averages,
where the flux is in $erg\cdot cm^{-2}s^{-1}$. The data have been filtered as
described in §3.2.1. The essentially continuous, 15-year-long duration of the
light curve is notable. Figure 2.— (right) Mkn 421 Lomb-Scargle power spectrum
(blue) for the ASM 1-day averages. The window power spectrum is shown in red.
Three strong peaks are seen: one at $0.036\mu$Hz, corresponding to a 325-day
period, another at 0.001781 $\mu$Hz (650 days), and a third at 0.00119$\mu$Hz
(975 days). These represent the primary, 2:1 and 3:1 resonances of a 325-day
QPO (§3.2.1). The false peak at 365 days appears only in the window power
spectrum.

Figure 3.— (left) Mkn 421 $\it{RXTE}$ ASM light curve, dwell by dwell, where
the flux is in $erg\cdot cm^{-2}s^{-1}$. While the sampling is much denser,
individual points have significantly lower signal-to-noise. No filtering has
been applied, as is evident due to the presence of negative flux values. Mkn
421’s ecliptic latitude of 30 degrees complicated the ASM dwell collection
when the source was closest to the Sun. See §3.2.1. Figure 4.— (right) Mkn 421
Lomb-Scargle power spectrum (blue) with an offset value of 10 has been added
to the LSP for ease of comparison and window power spectrum (red) for all ASM
dwells. All three peaks noted in Figure 2 are detected, and despite the lower
signal-to-noise on individual points, the significance of the features is even
higher.  Figure 5.— The Mkn 421 light curve of all $\it{RXTE}$ PCA
observations, where the flux is in $erg\cdot cm^{-2}s^{-1}$. The signal-to-
noise of individual points is much higher, but the data show fairly short
campaigns, separated by $\sim 1$ year. This is due to the target-of-
opportunity nature of the observations prior to 2009.  Figure 6.— The power
spectrum for Mkn 421 (blue) for the $\it{RXTE}$ ASM observations. The
regression line with a slope of -0.89 used for ANOVA is shown in orange.

Using the ASM Dwell-by-Dwell data, the obtained light curve spans the same
5837 days but contains 86935 data points with a mean spacing of 0.067 days or
about 15 points a day (Figure 4). The LSP for this data (Figure 4) shows the
same peak at the 2:1 resonance, corresponding to 0.001708 $\mu$Hz or a period
of 650 days. This 153-$\sigma$ detection with a FAP of $\sim 10^{-60}$ also
shows the same peak at the 3:1 resonance, corresponding to a period of 975
days. The 325-day QPO is also present, with a significance of 90-$\sigma$ and
a FAP of $\sim 10^{-33}$.

A previous study done by Chitnis et al. (2009) utilized the ASM dwell-by-dwell
data for Mkn 421 from MJD 50087 to MJD 54466 (1996 January 5 $-$ 2008 July
15), manually binning the data into 20-day averages. For our analysis, Mkn 421
was bright enough that we did not encounter problems with binning. Chitnis et
al. (2009) concentrated on the power spectrum of the X-ray variability and its
strength as a function of the break timescale. We found that with or without
the binning, our light curves and power spectrum show the same spectral
features. Additionally, cutting our data down so that it spanned the same
range as Chitnis et al. (2009)’s did not eliminate the spectral features we
see in our unbinned, original runs. Our analysis benefitted from the extra
data obtained after 2008 but ultimately shows that the QPO was present earlier
in the mission.

The overall power spectral density (PSD) between $2\times 10^{-9}$ and
$2\times 10^{-6}$ Hz is well-fit by a power law, linear on the log-log plot
with a slope of -0.89 (Figure 6). Analysis of variance (ANOVA) shows that
42.6% of the variance in the power values is explained by the regression line
(frequency dependent), while 57.3% of the variance is explained by intrinsic
random variation. When we repeat the false alarm probability (FAP)
calculations using only the residuals from the regression line, the QPOs are
still quite significant. The values for the FAPs calculated using the
residuals can be seen in Table 1. This table shows the values of the false
alarm probabilities (FAPs) calculated using the residuals from the regression
line for the 325-day, 650-day, and 975-day periods. While the values are
higher than the initially calculated FAPs, the new values show that the QPOs
are still quite significant.

We investigated regression fits for broken power laws. In particular, a fit to
the portion above 188 days, gives a slope of -0.43 on the log-log plot. Thus
the noise in the region of the reported QPO is closer to white than pink. See
also §4.2. Using the 20-day binned data for the regression line fit as in Rani
et al. (2009) gave a power law slope identical to -0.89. The LSP frequency
interval was shortened to above 40 days to avoid Nyquist-limited data.

Rani et al. (2009) found false periods of $\sim$365 days due to instrumental
effects. We see a similar phenomenon in Figure 2. The prominent peak in the
window LSP (in red) at 365 days does not seem to be influencing any of the
peaks in the data LSP (in blue). Recall that the window function assumes
constant flux and finds any Fourier signature due entirely to the uneven
spacing of the data points. Mkn 421 has an ecliptic latitude just under 30
degrees which complicated the ASM dwell collection when the source was closest
to the Sun. The false peak with a period of 1 year is commonly seen in RXTE
data.

Period (days) | Z | FAP (%)
---|---|---
325 | 16 | 0.376
650 | 16.8 | 0.157
975 | 13 | 6.66
Table 1Values of Z and FAP calculated using the values from the regression
line and ANOVA.

####  3.2.2 PCA Analysis

The PCA data cover MJD 50143-55926 (1996 March 1 - 2011 December 31), with an
average spacing of 4.9 days (Figure 5). There are substantial gaps. The
observing strategy for Mkn 421 also had systematic bias since prior to 2009
the observations were mostly triggered as targets of opportunity, such that
most observations occurred in periods when Mkn 421 was flaring. More intensive
observations were undertaken between MJD 55149-55926 (2009 November 14 - 2011
December 31) with an average spacing of 2.05 days. Figure 7 shows both the PCA
and ASM light curves for this interval. We note that the flares match up for
the two data sources, which strengthens the credibility of the ASM data where
there are many more points but lower signal-to-noise. However, the PCA data
have sparse coverage and a selection bias, while the ASM data do not. Despite
these contradictions, both sets of data give similar results. This data
segment is continuous, not biased to only flaring states. Analysis of the LSP
for the PCA data shows hints of peaks at the 325-day period and its harmonics
but at much lower significance. We believe that the bias toward flares, paired
with substantial gaps in the observations, is more likely to produce false
alarms. The LSP from the shorter light curve (2009-2011) shows a peak at
0.0429 $\mu$Hz, corresponding to a period of 270 days.This detection is in
good agreement with the 280-day period detected by Bhatta et al. (2016) and
Ren et al. (2022). The PCA data have a lower uncertainty (higher signal-to-
noise ratio) than the ASM data.

Figure 7.— This figure shows the light curves for both PCA (blue) and ASM
(orange) data from MJD 55200$-$55900\. Flux values for both light curves are
in $erg\cdot cm^{-2}s^{-1}$. The same peaks can be seen throughout the time
range shown.

##  4\. Quality Assurance

###  4.1. Censoring of Lower Limits

Figure 8 shows the QPOs detected in our current analysis, on an expanded scale
(including y-scale). We used a Gaussian fit to the conspicuous feature near
320 days to estimate its best fit value and error bar. The Gaussian fit gives
a best fit period of 325 days with an uncertainty of $-7$ and $+8$ days.
Subharmonics near 650 and 975 days also stand out. These are considerably
wider than the prime QPO, and the one near 650 days shows evidence of a side
lobe. A less significant peak appears at 270 days. The 325 and 270-day peaks
are in good agreement with previous findings discussed in the introduction.
However, the 270-day peak does not appear significant (9.8-$\sigma$) and is
not apparent in the light curve.

Outside of its original context, survival analysis can be thought of as a type
of cumulative probability (Feigelson, 1992, 2010). This allows survival
analysis to address censoring, or the bias introduced by excluding data. In
the ASM data, left-censoring is present since we detect negative flux for our
object, making it fainter than the expected limit of zero. One of the most
famous estimators in survival analysis is the Kaplan-Meier estimator (KME)
(Kaplan & Meier, 1958). In astronomy, it can be used to estimate the fraction
of objects above or below a certain flux level. In this case, we set the
limiting flux to zero in order to address the negative flux values seen in the
ASM data. This analysis showed that only two percent of all the flux values
obtained by the ASM were negative.

Traditional survival analysis, even when adapted to astronomical surveys,
typically only deals with various sources, not one source with varying flux
(Feigelson, private communication). We investigated other forms of survival
analysis, but concluded that they were not appropriate for this analysis. We
relied on our KME results and chose to leave our original FAP as the best
method of verification.

###  4.2. Uneven Sampling & Noise

Having uneven data spacing complicates some phases of the analysis, while
offering the potential for enhancement in other areas. Scargle (1998) points
out that binning data throws away a considerable amount of information and
introduces dependency of the results on the sizes and locations of the bins.
Binning means to divide the observation into equally spaced intervals, then
counting photons within each bin. On the other hand, it might be argued in
some cases that the portion of frequency space where information is lost may
not be one where detections are expected.

The RXTE PCA data from AGN monitoring campaigns tend to be unevenly sampled in
the extreme. There are gaps where no observations are proposed. For many
sources (ecliptic latitude less than 30 degrees), there are yearly gaps when
the source is too close to the Sun. And even where campaigns have a design
mean frequency, scheduling limitations lead to a standard deviation of the
spacing around the goal value. We found that the RXTE ASM data also had many
of these features. Mkn 421, with an ecliptic latitude of 29.5 degrees, just
missed the 30-degree PCA Sun exclusion constraint. Gaps of about 17 days are
evident in the PCA light curve. While the ASM could theoretically observe at
low sun angle, the Mkn 421 data also have yearly gaps. These were more
complicated, averaging about 40 days, surrounded by an additional period of
less than daily sampling. And the “daily averages” also use an average time
(so not truly binned), thus the spacing is distributed around a mean of 1 day
but with a substantial standard deviation.

Chitnis et al. (2009) did extensive research with the RXTE ASM database,
relying on binning at 20-day or 30-day intervals (depending on source flux)
the “dwell-by-dwell” data, but did not concentrate on searching for
periodicities in the data. Rani et al. (2009) used an approach more similar to
ours, relying on the “daily averages”, while noting that there were many
negative values, which were considered to be upper limits of 0 and omitted
from the analysis. Their structure function (SF) analysis rendered periods in
several AGN of close to 1 year (including a quoted period of 365 days for Mkn
421), which they concluded to be artifactual. See also §3.2.1.

Our use of the Lomb-Scargle periodogram (LSP) is guided by the suggestions in
VanderPlas (2018). One of our primary quality assurance tools is to compute
the window function and the window transform. This is to compute the LSP using
the data timestamps, but with a constant value of 1 in place of all data
values. (This is the red curve often displayed with our LSP results.) Then we
examine the window function for dominant features, such as daily or annual
aliases or Nyquist-like limits. This is where we find that a prominent peak in
the window LSP at 365 days is not seen in the data LSP. This peak is seen in a
constant flux (flat) light curve based only on the uneven spacing of the light
curve. So, this peak is definitely artifactual, but effectively screened out
by this technique.

The Lomb-Scargle approach has been criticized because it assumes Gaussian
noise, thus not appropriate for QPO/period searches in AGN light curves where
red noise type variability is known to be common. Intermittent time sampling
and low signal/noise of the data make it easier to mistake a phantom
periodicity for a real one. As discussed in Vaughan (2005), red noise in light
curves with a sinusoidal appearance are produced at a rate that is not
insignificant, and this rate depends strongly on the power spectrum. In this
context, we note that the slope of the power spectrum is not indicative of
strong red noise, but is pink, trending towards white, in the vicinity of our
QPO.

Another major complication from the unevenly spaced data is in the statistical
analysis of the significance of the detected peak. We implemented the False
Alarm Probability (FAP) using the method of Horne & Baliunas. The variability
properties of AGN have been characterized as red noise (Vaughan et al., 2003),
arising from a stochastic, rather than deterministic, process. Each individual
light curve is deemed as merely one realization out of an ensemble of random
light curves that might be generated by the underlying stochastic process.
Other realizations would look different due to the statistical fluctuations
inherent in any stochastic process. Two light curves will have different
characteristics (e.g., mean and variance) even when they are realizations of
the same process. However, data from a deterministic process (e.g., the light
curve of a strictly periodic source) should be repeatable within the limits
set by the measurement errors. That analysis found that the majority of the
variability in the ASM lightcurve was due to the 325-day QPO, while a smaller
but significant minority was due to the more quasi-random (white/pink noise
dominated) variability evidenced by the power-law slope of the PSD.

Vaughan et al. (2003) go on to describe simulations to study properties of the
variance of red noise data. Random artificial light curves were generated from
power-law (AGN-like) power spectral density (PSD), then using the Timmer-
Koenig algorithm (Timmer & Koenig, 1995), random time series are generated
with arbitrary broad-band PSD, to correctly account for the intrinsic scatter
in the powers. Large scatter in such periodograms is an intrinsic property of
stochastic processes and does not depend on the number of data points or the
Poisson noise in the data. Vaughan (2005) introduced a relatively simple
procedure to assess the significance of peaks in a periodogram when the
underlying continuum noise has a power law spectrum. However, it is only
applicable in the case of evenly spaced data.

We have found this to be a general problem and continue to look for other ways
to assess statistical significance, in particular implementing some of the
suggestions from VanderPlas (2018). In Figure 6, we note that the PSD does
have a linear trend on this log-log plot, matching the power-law dependence
often noted in these objects. The classical Horne-Baliunas approach to the
false alarm probability (FAP) assumes a flat PSD, or “white noise” spectrum.
It is fair to ask if some of the significance attributed to our QPO is due to
it sitting on the “up ramp”. The mean of the power values would be a straight
horizontal line that typically under-predicts the power at low frequency while
over-predicting it at high frequency. We computed the best fit line to the
power values, and repeated the FAP calculation using residuals from that line
rather than the raw values. We also did an analysis of variance (ANOVA) to
determine how much of the power variance is explained by the regression line,
and how much is explained by random variation. When repeating the regression
fit and ANOVA calculations for our data averaged over 20-day bins, mirroring
the analysis of Rani et al. (2009), we still get a slope indicative of pink or
white noise.

Figure 8.— This figure shows the LSP from the ASM 1-day averages. Peaks of
interest are labeled. See §3.2.1 for discussion.

##  5\. Modulation of Oscillation

###  5.1. Keplerian Orbital Period

The orbital timescale can be parameterized (Edelson & Nandra, 1999; Smith et
al., 2020) as:

$t_{orb}\approx 0.33M_{7}\left(\frac{r}{10\,R_{S}}\right)^{1.5}\text{days}.$
(2)

$M_{7}$ is the mass of the black hole in units of $10^{7}M_{\odot}$, $r$ is
the distance from the BH system barycenter, and Equation 1 is the
Schwarzschild radius.

This can be inverted to solve for $r$:

$r\approx 21\left[\frac{P(\text{days})}{M_{7}}\right]^{2/3}R_{S}.$ (3)

Using the previously scaled mass of 19.5 $M_{7}$ (Wang et al., 2004) as the
mass of the SMBH in Mkn 421 and a period of 325 days, interpreting the QPO as
a Keplerian orbital frequency gives a radial distance from the BH system
center of mass of 137 $R_{S}$.

###  5.2. Lense-Thirring Precession Period

The Lense-Thirring precession period is (Stella & Vietri, 1998):

$P_{LT}=\frac{8\pi GM}{c^{3}a}\left(\frac{r}{R_{S}}\right)^{3}.$ (4)

Parameterizing, as before, the radius of the hot spot becomes (Smith et al.,
2020):

$\frac{r}{R_{S}}=\sqrt[3]{\frac{P_{LT}c^{3}a}{8\pi
GM}}=4.117\;\sqrt[3]{\frac{P_{LT}(\text{days})\;a}{M_{7}}}.$ (5)

So, for this QPO at 325 days, and $M_{7}$ = 19.5 for Mkn 421, the hot spot
radius is 10.5 $R_{S}$ for a dimensionless spin parameter of $a\approx 1$. For
a=0.1, the radial distance is roughly half this value.

###  5.3. Other Possibilities

We have concentrated on Keplerian and Lense-Thirring frequencies as these are
common elements of many models. While we favor Lense-Thirring due to the
smaller radius, other interpretations are possible.

King et al. (2013) detected a QPO between 120 and 150 days over a 4-year
timescale in J1359+4011 using both radio and X-ray data. While Lense-Thirring
precession is considered a possibility, in this case, the timescale of the
observed oscillation points to Lightman-Eardley secular instability (King et
al., 2013). Pushkarev et al. (2017) suggest that jet geometry varies with AGN
type. Based on VLBI data used in MOJAVE studies, BL Lac objects are thought to
have conical jets while radio jets are closer to parabolic. Mkn 421 is one of
the few BL Lac objects where superluminal motion is not seen in VLBI
monitoring. Pushkarev et al. (2017) explain this phenomenon by stating that
superluminal features only take up a small piece of the entire jet cross-
section due to the precession of the jet.

##  6\. Mechanism of Oscillation

Simulations suggest that creating a relativistic jet requires a BH, hot inner
flow, large-scale poloidal magnetic field, and large BH spin (Blandford et
al., 2019). For these and other reasons, not all astronomical systems with
accretion disks develop jets. Based on Shakura and Sunyaev’s Keplerian disk
theory (Shakura & Sunyaev, 1973), both the disk and jet in such states are
periodic since the jet responds to the disk state.

Looking at disk-jet coupling in AGN has proven difficult because of the long
dynamic timescales associated with SMBH. Fender & Belloni (2004) shifted their
focus to binary systems since variability within these systems occurs on much
shorter timescales, and they still exhibit disk-jet coupling. Based on
observations of GRS 1915+105, the authors were able to create a
phenomenological model that works analogously for all systems with disk-jet
coupling. The model uses physical indicators to paint a quantitative picture
within BHXRBs that can be generalized to include AGN. These indicators focus
on the X-ray emission of the central object. They looked for prolonged hard
X-ray states with a steady production of self-absorbed jets. They additionally
searched for prolonged soft X-ray states with suppressed jet production and
optically thin ejection events occurring during the transition between hard
and soft states. The bulk velocities of transient jets associated with these
events are much larger than those of steady jets.

Fender & Belloni (2004) state that these physical aspects give an overall
picture, as follows. Jet production occurs above a certain spectral hardness,
with strength correlated to the X-ray strength nonlinearly. As the spectrum of
a source softens, the velocity of the jet increases in a way that is
significant for the highly relativistic outflow. The source then reaches a
peak in soft X-ray flux, followed by a rapid increase in jet velocity, and,
finally, the jet shuts off. The rapid increase in the jet’s velocity produces
a shock in the preexisting jet, creating an optically thin flare with large
bulk motions. A secondary flaring state in the same outburst is common. This
flare is associated with further optically thin radio flares of declining
strength.

A more recent review of QPO models by Ingram & Motta (2019) focuses on the
relationship between a moving, characteristic radius and changes in QPO
frequency. The authors favor precession as the cause of QPOs, with Lense-
Thirring precession as the leading theory based on observational evidence. For
this to be the case, a $5^{\circ}$ misalignment between the BH spin axis and
the binary rotation axis is required. The authors present evidence for
differential precession, meaning QPO frequency would increase closer to the
central accreting object, similar to the increase in jet velocity described by
Fender & Belloni (2004). While this adds to the case of Lense-Thirring
precession, Ingram & Motta (2019) call for further observations to confirm
their findings.

###  6.1. Disk-Jet Coupling

The main models considered for the X-ray emission in blazars (particularly BL
Lac objects) in the high state are jet based, while in the low state, blazar
X-ray emission can include a significant contribution from the accretion disk,
particularly for FSRQ (flat-spectrum radio quasars, Gupta (2018)). In high-
state blazars, Doppler beaming effects cause emission from relativistic jets
observed near our line of sight to dominate emissions over the whole spectrum.
VLBI studies have shown that the helical structure of jets can lead to a
misalignment between the core and jet of blazars. The misalignment would
result in rapid, large-amplitude variability as the jet moves toward or away
from the line of sight thus changing the Doppler beaming factor (Gupta, 2018;
Zhou et al., 2018).

While QPOs are rare in intra-day variability (IDV) blazars, Mkn 421 and PKS
2247$-$131 show IDV in X-ray observations (Gupta, 2018) as well as QPOs. The
Fermi Gamma-ray Space Telescope detected a 34.5-day QPO in PKS 2247$-$131,
thought to be caused by the helical structure of the jet. Based on Very Long
Baseline Array (VLBA) monitoring of Mkn 421, Lico et al. (2017) propose the
presence of two nested helical magnetic fields with opposite helicities in the
jet. The variability of the rotation measure (RM) over the year-long
observations implies that the jet precesses over a timescale of one year.
While multiple studies concur that helical jets and their motion are the most
likely cause of QPOs in BL Lacs, further observations over the whole spectrum
are needed in order to be certain (Carnerero et al., 2017; Lico et al., 2017;
King et al., 2013; Sandrinelli et al., 2017).

As noted in §1, three recent papers (Li et al., 2016; Bhatta et al., 2020; Ren
et al., 2022) have found QPOs with dominant periodicities at $\sim$300 days in
Mkn 421, using data from Fermi LAT, Swift and OVRO. In contrast Carnerero et
al. (2017), Tarnopolski et al. (2020), and Sandrinelli et al. (2017) did not
find QPOs in Fermi LAT data of Mkn 421 despite their data including the same
time frame. Nevertheless the claimed $\sim$300 day quasi-periodicities are
consistent with the 325-day QPO we report here. While the time windows of the
RXTE (1996-2011) and Fermi (2008-Present) observations have only modest
overlap, it seems likely that this QPO may have been present in both the X-ray
and gamma-ray light curves of Mkn 421 over an extended period of time, perhaps
dissipating only in 2014, as claimed by Ren et al. (2022). There are multiple
possible mechanisms for such behavior. One, cited by Li et al. (2016), is
nonballistic helical motion of the emitting material. Another possibility is
that low-frequency QPO behavior in jetted sources may be the disk response to
jet instabilities, a possibility raised by Ferreira et al. (2022). In this
mechanism, the assumption, a jet instability at some distance from the disk,
would lead to jet wobbling. This instability would transmit waves both
upstream and downstream. If the waves transmitted upstream, they would connect
to the disk at some point, acting like a hammer impacting the disk with a
characteristic frequency. Such an instability could transfer through the disk
and be modulated either at the Keplerian frequency or modulated by Lense-
Thirring precession, particularly in a spinning black hole system. While the
majority of the energy in the instability would advect downstream, the
instability would remain near the same locus, thus leading to a relatively
long-lasting and quasi-periodic disturbance. Such a wave could arise either
through pressure-driven, current-driven (kink) instabilities (e.g., McKinney &
Blandford (2009); Tchekhovskoy et al. (2011)) or Kelvin-Helmholtz
instabilities (McKinney & Blandford (2009); Porth & Komissarov (2015)). While
the Ferreira et al. (2022) model was developed for X-ray binary jets and lacks
detailed MHD modeling, there is clear applicability to AGN jets as well. At
least in AGN jets it would be much less likely that the instability would have
to do with jet recollimation, as in many jets that typically occurs at
distances of tens to hundreds of thousands of $R_{S}$ from the black hole
(Perlman et al., 2019). However, many other such loci for instabilities are
present and these modes are generally in jets.

It should be noted that helical jet models have been linked to blazar
variability in other objects, e.g., OJ 287 (Valtonen & Pihajoki, 2013), CTA
102 (Sarkar et al., 2020), AO0235+164 (Wang & Jiang, 2020), 0405-385 (Gong et
al., 2022), 0716+714 (Hong et al., 2018) and 3FGL J0449.4-4350 (Yang et al.,
2020). They have also been invoked to explain oscillations in the polarization
variability of the prototype radio galaxy M87 (Perlman et al., 2011). In
addition, helical jets have been seen on larger scales in VLBI images by
Larionov et al. (2013) for 0716+714, Gómez et al. (2016) in the case of BL
Lac, or Vega-García et al. (2020) in the case of 0836+710. Another
possibility, cited by Li et al. (2016), is nonballistic helical motion of the
emitting material. While such helical motion could be induced by a coupling
between the disk and jet, it could also be excited by the same current-driven
or pressure-driven kink modes described above (McKinney & Blandford, 2009;
Tchekhovskoy et al., 2011; Porth & Komissarov, 2015; Anantua et al., 2018).
The main difference here is that one does not necessarily need an oscillation
to transmit to the disk in order to obtain a quasi-periodic oscillation,
especially if the oscillation arises at some fairly large distance from the
central black hole. As above, while the wave would advect downstream, the
instability that generated it would be relatively long-lived and could
therefore lead to a quasi-periodic signal in the light curve.

###  6.2. Accretion Disk Limit Cycle

The accretion disk limit cycle was originally put forth to account for
outbursts in dwarf novae but can also be relevant to SMBH systems (Cannizzo,
1993). It is caused by an inherent, hysteretic relationship between vertically
integrated viscous stress and surface density at a fixed radius in the
accretion disk. Based on various modeling efforts (Cannizzo, 1992, 1993, 1998;
Janiuk & Czerny, 2011), it is believed that the limit cycle mechanism operates
in most AGN during the quasar era, solving the problem of feeding quasars at
the required rates.

In some light curves, outbursts aside from QPOs can be seen. Short-term UV
variations in AGN spectra have been attributed to Lightman-Eardley instability
- a phenomenon in which the accretion rate varies inversely with vertically
integrated disc surface density (Siemiginowska & Czerny, 1989; Lightman &
Eardley, 1974). Applying these models or assumptions to AGN can be quite
difficult since they don’t have such clearly defined boundaries as in binary
systems. Additionally, AGN which accrete at the Eddington rate during the high
state may utilize the disk instability only as a mechanism to store matter and
then trigger the instability when the surface density becomes greater than the
maximum surface density when the viscosity parameter is 0.01 (disk temperature
is lower) somewhere in the disk. Systems with lower supply rates may be in the
thin-disk regime during both quiescence and outburst (Cannizzo, 1992).

Janiuk & Czerny (2011) believe limit-cycle oscillations of the accretion disk
may be caused by ionization instability or radiation pressure instability,
with radiation pressure instability affecting galactic sources whose Eddington
ratios are above 0.15 and AGN whose Eddington ratios are above 0.025. Based on
their results, they tend to favor radiation and gas pressure in both galactic
sources and AGN. The authors argue that radiation pressure instability applies
to active galaxies since separations between outbursts are on the order of
$10^{6}$ years for a $10^{8}M_{\odot}$ BH, parametrization of viscosity is
consistent with lack of instability in FR II radio galaxies as their Eddington
ratio is less than 0.025 (lower limit for instability in active galaxies).
During the hot state, the luminous core produces a radio jet, while during the
cold state the radio activity ceases, and when scaled by BH mass
($10^{8}M_{\odot}$) the $10^{2}$-$10^{4}$ year long outbursts and amplitudes
are sensitive to the energy fraction deposited in the jet. This scaling gives
an additional, independent argument that the intermittency in quasars on the
timescales of hundreds/thousands of years is likely of a similar origin to
those in microquasars. Observationally, Janiuk and Czerny push for searches
for excess variability in galactic sources, outbursts that last tens to
hundreds of seconds. These would be caused by the radiation pressure
instability and may not be strictly periodic or clear in periodograms (Janiuk
& Czerny, 2011). We believe the additional fluctuations seen in our light
curves may be partially attributed to limit cycle oscillations in the disk,
perhaps propagating out into the jet or in response to jet instabilities. The
oscillations would not interact with Keplerian or Lense-Thirring produced
oscillations. Both would contribute to the light curve in different ways.

###  6.3. Future Work

Several ongoing missions ($\it{XMM}$-$\it{Newton}$, $\it{Chandra}$,
$\it{NuStar}$) perform long-stare observations that have facilitated the
search for HFQPOs in AGN. Conversely, the RXTE AGN archive offers long but
sparse monitoring campaigns providing the chance to find LFQPOs in AGN. We
suggest that more work on additional sources in the RXTE PCA archive is
warranted.

The RXTE ASM had a small collecting area compared to other ongoing missions
such as Swift BAT and MAXI. Additional work with these other all-sky monitors
is likely to also prove fruitful. Studying the QPO phenomenon in AGN and BHXRB
will further the understanding of both.

We thank the anonymous referee for many insightful comments that helped to
improve the quality of this paper. We thank Eric Feigelson, Jacob VanderPlas,
Richard Anantua, and Brandon Curd for valuable and informative discussions.
This work utilized light curves provided by the University of California, San
Diego, Center for Astrophysics and Space Sciences, X-ray Group (R.E.
Rothschild, A.G. Markowitz, E.S. Rivers, and B.A. McKim), obtained at
http://cass.ucsd.edu/~rxteagn/. We also thank the ASM team at MIT for the
light curves used in this analysis, obtained at
http://xte.mit.edu/ASM_lc.html.

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