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Two Transient Quasi-periodic Oscillations in γ-Ray Emission from the Blazar S4 0954+658

Two Transient Quasi-periodic Oscillations in γ-Ray Emission from the Blazar S4 0954+658 In this work, we report periodicity search analyses in the gamma-ray light curve of the blazar S4 0954+658 in monitoring undertaken by the Fermi Large Area Telescope. Four analytical methods and a tool are adopted to detect any periodic flux modulation and corresponding significance level, revealing: (i) a quasi-periodic oscillation (QPO) of 66 days with a significance level of >5σ spanning over 600 days from 2015 to 2016 (MJD 57,145–57,745), resulting in continuous observation of nine cycles, which is one of the longest cycles discerned in blazar gamma-ray light curves; (ii) a possible QPO of 210 days at a moderate significance of ∼3.5σ, which lasted for over 880 days from 2020 to 2022 (MJD 59,035–59,915) and for four cycles. In addition, we discuss several physical models to explain the origin of the two transient QPOs and conclude that a geometrical scenario involving a plasma blob moving helically inside the jet can explain the timescale of the QPO. Unified Astronomy Thesaurus concepts: Active galactic nuclei (16); Blazars (164); Jets (870); Period search (1955) 1. Introduction references therein). QPO phenomena are usually quite rarely detected and non-persistent for AGNs, but they seem to be It is generally believed that all active galaxies are powered relatively common in black hole X-ray binaries (Remillard & by the accretion of dense ionized gases onto a supermassive McClintock 2006; Gupta 2014). So far, more than 30 of 5064 6 10 black hole (SMBH) with a mass in the range 10 –10 M , and sources above 4σ significance are reported to have QPO ∼10% of them have jets of relativistic charged particles. Radio- phenomena based on time series data in the fourth Fermi loud active galactic nuclei (AGNs), with their jets pointing Gamma-ray Large Area Telescope (LAT) catalog of sources almost directly along the observer’s line of sight („10°), form a (4FGL; Abdollahi et al. 2020; Wang et al. 2022). special subclass called blazars (Antonucci 1993; Urry & Recently, Jorstad et al. (2022) claimed that the γ-ray flux, Padovani 1995). Moreover, blazars can be further divided into optical flux, and linear polarization of BL Lacertae all exhibited two subcategories based on the strength of emission lines ∼13 hr QPO variability during a dramatic outburst in 2020. emerging in optical–ultraviolet spectra: BL Lacertae (BL Lac) Such a short-term QPO is explained by the current-driven kink objects (very weak and narrow emission lines) and flat- instabilities near a recollimation shock ∼5 pc from the black spectrum radio quasars (FSRQs; broad and strong emission hole. In the same year, a quasi-periodic signal of approximately lines). Blazars usually manifest substantial variability over 420 days with >5σ significance was found in the measure- almost the whole electromagnetic spectrum; their emission is ments of the degree of optical linear polarization for the blazar dominated by nonthermal radiation and ranges from radio to γ- PKS 1222+216 and a helical jet model was employed to rays (Ulrich et al. 1997). explain the signal well (Zhang & Wang 2022). Furthermore, Observations from both ground-based and space telescope several models have been proposed by different authors to show that blazars have flux variability of the order of minutes explain periodic radiation from blazars at various frequencies to years in different electromagnetic wave bands, which may on diverse timescales, i.e., a hotspot orbiting near the innermost indicate that different physical mechanisms (intrinsic and stable circular orbit of the SMBH (Gupta et al. 2009, 2019; extrinsic) play a leading role. An interesting phenomenon Sarkar et al. 2021), the presence of the SMBH in a binary related to flux variability is quasi-periodic oscillation (QPO), system (Valtonen et al. 2008; Ackermann et al. 2015), although flux variability frequently exhibits nonlinear, stochas- precession of relativistic jets or a helical structure (Graham tic, and aperiodic characteristics (Kushwaha et al. 2017).So et al. 2015b; Sandrinelli et al. 2016), the existence of quasi- far, a large number of QPO behaviors with different timescales equidistant magnetic islands inside the jet (Huang et al. 2013; in multifrequency light curves have been reported by Shukla et al. 2018; Roy et al. 2022), and pulsational researchers using different detection techniques (e.g., Raiteri instabilities in accretion flow (Tavani et al. 2018). Hence, we et al. 2001; Liu et al. 2006; Gupta et al. 2009; Lachowicz et al. can analyze the quasi-periodic modulation in the blazar light 2009; King et al. 2013; Zhang et al. 2014; Graham et al. 2015a; curve to explore the accretion physics and the connection Ackermann et al. 2015; Bhatta 2017; Gupta et al. 2018; Zhou between accretion disk, jet, and central engine (Kushwaha et al. et al. 2018; Sarkar et al. 2020; Gong et al. 2022; Roy et al. 2020). 2022; Zhang et al. 2022; Otero-Santos et al. 2023, and S4 0954+658 (also referred to as QSO B0954+65) is one of the most well studied blazar sources with complex variability Original content from this work may be used under the terms and is situated at a redshift of z = 0.3694 ± 0.0011 (Becerra of the Creative Commons Attribution 4.0 licence. Any further González et al. 2021). Stickel et al. (1991) regard this source as distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. a BL Lac object in view of the small equivalent width of the 1 The Astrophysical Journal, 949:39 (8pp), 2023 June 1 Gong et al. emission lines of the spectrum. However, this target can also be region of interest having a radius of 12° centered at the source hm s classified as an FSRQ because the kinematic features of the (ad== 09 58 47.244, 6533¢54.8). At the same 2000.0 2000.0 radio jet belong to class II (Hervet et al. 2016). In 2021, time, we used a screening expression ‘(DATA_QUAL >0)&& Becerra González et al. (2021) detected an Mg II emission line, (LAT_CONFIG=1)” to select events with good time intervals whose equivalent width is close to 5 Å, which is commonly and set a zenith angle cut of 90° to suppress the γ-ray pollution taken as the limit to classify a blazar as an FSRQ. Therefore, it from the Earth’s limb. An XML file is generated through the seems more reasonable to consider this γ-ray emitter as a 4FGL catalog containing the γ-ray background emission transitional object. Wagner et al. (1993) investigated the optical templates ‘gll_iem_v07” and ‘iso_P8R3_SOUR- variability of this source for the first time and then Raiteri et al. CE_V2_v1.txt” for the Galactic and isotropic extragalactic (1999) detected fast large-amplitude variations using the 4 yr contributions respectively. We consider three commonly used light curve. Their results indicate that the long-term behavior of spectral models (power law, log-parabola, and power law with the source is not related to spectral variations. Continuous an exponential cutoff) for the whole time series. And we also observation of this blazar shows that the optical flux varies by test for the spectral curvature in the spectrum using more than 2.5 mag and the degree of polarization reached 40% ΔTS = TS − TS = 343 (Abdollahi et al. 2020). The LogPb PL (Papadakis et al. 2004; Hagen-Thorn et al. 2015). Additionally, results show that the log-parabola (LogPb) model is the most Gaur et al. (2019) found a positive correlation between color suitable for describing γ-ray emission from the target source. index and magnitude based on simultaneous data in the B and R The best-fit spectral parameters were α = 2.14 ± 0.01, bands. β = 0.53 ± 0.04, and E = 699.12 ± 29.46 MeV. In addition, MAGIC Collaboration et al. (2018) presented the first we selected low states (MJD 54,687–55,558 and detection of the blazar S4 0954+658 in very high-energy 55,778–56,702) and high states (MJD 56,918–57,160 and (…100 GeV) γ-rays, which was obtained through monitoring 59,619–59,894) to test the spectral shape of the time series. The with the Major Atmospheric Gamma Imaging Cherenkov results show that the fitting parameters (except β) and flux (MAGIC) Telescopes during an exceptional flare (2015 variability are close to the whole time series. February). In 2021, Raiteri et al. (2021) found a 31.2 day Based on the best fitting results mentioned above, we tested QPO behavior in the optical long-term variability through the the construction of a binned light curve for 1–30 days and observation of the Transiting Exoplanet Survey Satellite found that bins of 10 days are the most appropriate size because (TESS) and the Whole Earth Blazar Telescope (WEBT) they not only reveal the details of the flux variation, but also Collaboration, in which the rotation of an inhomogeneous ensure that the blazar S4 0954+658 can be detected in almost helical jet provides a reasonable explanation for this phenom- all bins (TS … 9). In addition, the 10 days binned light curve enon. It is worth mentioning that such a month-long transient also shows greater intensity in the calculation of the power QPO is also detected in the γ-ray band for PKS 2247–131 spectrum than other bins. In the 10 days binned light curve (see (Zhou et al. 2018). More recently, Kishore et al. (2023) report Figure 1), the average value and standard deviation are 1.11 −7 −2 −1 the discover of several QPOs around 0.6–2.5 days in the optical and 0.95 × 10 photons cm s , respectively. Detection of light curve of the blazar S4 0954+658 with data acquired in six periodicity in the light curve was based on the weighted sectors by the TESS. wavelet Z-transform (WWZ) method (see Figure 2).We Here, we are inspired by the QPO report on the optical selected panels B (segment 1) and C (segment 2) in Figure 1 as radiation, and try to analyze whether the ∼14.3 yr data the regions of interest for analysis of QPO variability. measured by Fermi-LAT also contain any QPO phenomena. The paper is structured as follows. In Section 2, we describe the 3. Periodicity Search for γ-ray Emission process of data analysis in the 0.1–300 GeV energy band. In Section 3, we present the QPO detection algorithm and our It is not rigorous enough to visually measure the QPO main results. In Section 4, we summarize our conclusions and variability in the unevenly sampled light curve, but many explore several models to explain the QPO results. methods have been proposed to detect periodic components and corresponding significance levels. Here, we applied four methods to analyze the light curves, i.e., epoch folding, 2. Fermi-LAT Data Analysis REDFIT, Lomb–Scargle periodogram (LSP), and WWZ, and a The LAT on board the Fermi observatory continually tool. i.e., light-curve simulations, was used to determine surveys the entire sky every 90 minutes in the energy range confidence levels. Although the γ-ray light curve obtained by from 20 MeV to >300 GeV (Atwood et al. 2009). Based on the us is evenly binned, we only consider data points with TS > 9, observation data from the first 12 years, the 4FGL incremental resulting in uneven sampling of data. version of γ-ray sources contains 6658 sources, including more Epoch folding is one of the most popular methods of light- than 100 newly classified blazars (Abdollahi et al. 2022). The curve analysis (Leahy et al. 1983; Davies 1991). This method is blazar S4 0954+658 (named as 4FGL J0958.7+6534) was insensitive to the modulating shape of periodic components and found in the first Fermi Gamma-ray LAT catalog, and has also the uneven sampling of time series data, which is different from been detected by various radio surveys and optical and the traditional discrete Fourier periodogram (Bhatta 2018).We millimeter surveys. In order to build the light curve of this computed χ values of the γ-ray light curve with a time step of source, we used the standard software package FERMITOOLS 6 days for trial periods ranging between 6 and 510 days using and the user-contributed tool make4FGLxml.py. Equation (1) of Bhatta (2018). The results show that maximum The data for the blazar S4 0954+658 were taken during the χ values of 225 and 172 correspond to trial periods of 66 days period 2008 August 4 (MET: 239557417) to 2022 December 5 and 210 days, respectively. In segment 1, we constructed a (MET: 691900553), covering ∼14.3 yr. We chose LAT folded light curve by binned likelihood analysis with a period 0.1–300 GeV Pass 8 (evclass = 128, evtype = 3) events of ∼66 days, where phase zero corresponds to MJD 57,145 and recommended by the Fermi-LAT collaboration from a circular 10 phase ranges are selected (upper left panel of Figure 3). 2 The Astrophysical Journal, 949:39 (8pp), 2023 June 1 Gong et al. Figure 1. The 10 days binned light curve of the blazar S4 0954+658 at γ-ray energies of 0.1–300 GeV obtained from Fermi-LAT. (A) The purple shaded region (marked as segment 1) covers MJD 57,145–57,745 and represents the first segment for QPO analysis. The gray shaded region is the epoch MJD 59,035–59,915 (marked as segment 2) where the QPO analyses were carried out. (B) Zoom-in of segment 1, where the red dashed–dotted line indicates the sine fitting result of the light curve. The orange histogram corresponds to the TS value of each data point. (C) Same as panel (B) but for segment 2. Similar to segment 1, the phase zero of segment 2 is set at MJD An additional method, REDFIT, is also used to calculate the 59,035 to complete the folding light curve with a period of bias-corrected power spectrum of the light curve and estimate ∼210 days (upper left panel of Figure 4). Both results show the significance level of the corresponding dominant period that the γ-ray flux obviously varies with phase. (Schulz & Mudelsee 2002). This method can calculate the 3 The Astrophysical Journal, 949:39 (8pp), 2023 June 1 Gong et al. Figure 2. Left panel: WWZ map of the S4 0954+658 light curve in MJD 57,085–57,795. The bright red patch represents a possible QPO in the interval MJD 57,145–57,745 (segment 1). Right panel: WWZ map of the γ-ray light curve of blazar S4 0954+658 in MJD 58,315–59,915. The bright red patch represents a possible QPO in the interval MJD 59,035–59,915 (segment 2). underlying red-noise spectrum by fitting the time series with a also shows the result of the analysis of segment 2, which first-order autoregressive process (AR1), which is caused by revealed a significant signal centered at 208 ± 43 days. some stochastic processes in the accretion disk or jet for blazars Further evidence for the two transient QPOs is provided by (Fan et al. 2014; Covino et al. 2019). In the AR models the the WWZ method. The WWZ method, first introduced by present emission is connected with the past emission. The Foster (1996), can identify the localized features in both time theoretical power spectrum of an AR1 model is given as and frequency domains, especially in unequally spaced data, based on three trial functions, i.e., f (t) = 1(t), 1 - q f ()tt=- cos[wt ( )], and f ()tt=- sin[wt ( )]. The calc- 2 3 Gf() =G ,1 () rr 0 ulation of WWZ power intensity can search for a periodic 12-+ qp cos() ff q j Nyq modulation signal with frequency ω and time shift τ in a where G is the average spectral amplitude, θ is the average statistical manner, which is described as autoregression coefficient, and f represents the discrete () NV - 3 eff y frequency up to the Nyquist frequency ( f ). We used the Nyq WWZ = ,2 () REDFIT3.8e program to estimates the power spectrum and the 2() VV - xy significance level of the corresponding peak based on the LSP where N denote the effective number density of data points in combination with Welch overlapped segment averaging eff contributing to the signal, and V and V are the weighted (Welch 1967). As can be seen from the upper right panel of x y variations of the nonuniform data x and the model function y, Figure 3, it is evident that there is a peak around the timescale respectively. For more details on the definition of these factors, of 65 ± 12 days with significance level of >99% in the power see Li et al. (2021) and references therein. For segment 1, we spectrum during MJD 57,145–57,745 (segment 1). The upper −1 set the frequency range from 0.005 to 0.08 day and the step right panel of Figure 4 shows that the periodic modulation in −1 size is 0.00005 day in WWZ analysis, which enables the MJD 59,035–59,915 (segment 2) is centered at 210 ± 55 days QPO timescale of the region of interest to be displayed as much with a significance level of ∼99%. We take the half-width at as possible. Furthermore, in order to balance the frequency and half-maximum (HWHM) of the power peak fitted by the time resolution, we set a decay constant of c = 0.001. The Gaussian function as the uncertainty of the periodic modulation color-scaled WWZ power of the 10 days binned light curve in signal. The LSP is one of the most common methods of finding the time–period plane is presented in the bottom panel of periodicities in time series with nonuniform sampling, and it Figure 3, which shows that the power for the characteristic can calculate the intensity of the power spectrum at different period centered around 66 days persists over the entire frequencies (Lomb 1976; Scargle 1982). This method is the observational period. The corresponding time-averaged WWZ projection of the light curve on sinusoidal functions and power is centered at the period of 66 ± 4.7 days, corroborating constructs a periodogram from the goodness of the weighted χ the LSP result. In segment 2, we adopted a limited frequency fit statistic (Ferraz-Mello 1981). Nevertheless, the aperiodic −1 range of 0.001–0.03 day in WWZ analysis, where the step part of time series data will reduce the goodness of the LSP size and decay constant are the same as for segment 1. As sinusoidal fit, which leads to a reduction in the transient shown in the bottom panel of Figure 4, the time-averaged periodic power. The bottom panel of Figure 3 shows the power WWZ power of the segment 2 light curve also shows a (black solid line) of the LSP for the extracted segment 1 data. significant peak lasting throughout the activity at One signal, at the period of 66 ± 4.8 days, reached that significance level. Meanwhile, the bottom panel of Figure 4 208 ± 40 days, which is similar to the feature in LSP analysis. 4 The Astrophysical Journal, 949:39 (8pp), 2023 June 1 Gong et al. Figure 3. The results of QPO analysis in segment 1 (MJD 57,145–57,745). Upper left: the folded γ-ray light curve, which is constructed from binned likelihood analysis of the nine cycles of segment 1. Phase zero corresponds to MJD 57,145 and 10 phase ranges are set. For clarity, we show two periods. The dashed blue horizontal line represents the mean flux. Upper right: the results of periodicity analysis from REDFIT. The solid black line indicates the bias-corrected power spectrum. The red dashed line represents the theoretical AR(1) spectrum. The blue, green, and purple dashed curves are 90%, 95%, and 99% confidence contours, respectively. Bottom: LSP and WWZ results of the γ-ray time series data. The left subpanel displays a two-dimensional contour map of the WWZ power spectrum and the horizontal red patch indicates a strong QPO signal of ∼66 days. The right subpanel shows the time-averaged WWZ (red solid line) as well as the LSP powers (black solid line). The blue, purple, and orange dashed curves are 3σ,4σ, and 5σ significance lines, respectively. The dominant period of ∼66 days can be clearly seen to cross the 5σ significance curve. The flux variability of blazars usually shows a frequency- a Monte Carlo method provided in Emmanoulopoulos et al. dependent colored-noise-like behavior, which is very likely to (2013). The underlying red-noise PSDs of blazar light curves lead to a pseudo-period in the identification of periodic are often reasonably approximated by a power-law form −α components of time series data, especially at lower temporal P( f ) ∝ f , where P( f ) is the power at temporal frequency f frequencies (Vaughan et al. 2003, 2016; Bhatta et al. 2016;Li and α is the spectral slope (Vaughan 2005). Then, we et al. 2017). The significance estimated by the REDFIT method generated 10 artificial light curves to estimate the significance is based on the χ distribution of periodogram points about the level of the LSP and WWZ periodic components. In segment 1, model, which can avoid underestimating the significance of the the significance level for the QPO signal was found to be >5σ peak in power spectral density (PSD). Here, the significance of (the bottom panel of Figure 3). In segment 2, the simulation of segments 1 and 2 obtained by using the REDFIT method the light curve shows that the periodic modulation of 210 days reveals a …99% level. Another way to estimate the significance seems to have a significance level close to 3.5σ (the bottom in the LSP and WWZ peaks is to simulate light curves with the panel of Figure 4). In the recent QPO search, a large number of same PSD and flux distribution as the original light curve using blazars are claimed to have periodic signals with a significance 5 The Astrophysical Journal, 949:39 (8pp), 2023 June 1 Gong et al. Figure 4. Same as Figure 3, but for the segment 2 light curve (MJD 59,035–59,915). that is usually greater than 3σ (Zhang et al. 2017a, 2017b; Peñil significance level during MJD 57,145–57,745, which lasted for et al. 2020; Zhang et al. 2020, 2021). Thus, the QPO signal nine cycles. Interestingly, the 66 days periodic modulation, similar with ∼3.5σ significance of segment 2 is sufficiently important to that of PKS 2247–131, also occurred after an outburst event to be reported. These two transient QPO signals may appear (2014 December) with multiwavelength observations (see again in the future, so it will be interesting to keep monitoring Figure 1;Zhouetal. 2018;Gauretal. 2019).For segment2, at the γ-ray frequency. we found a possible QPO of about 210 days with ∼3.5σ significance in the γ-ray light curve over 880 days. This signal is clearly visible for about four cycles and seems to continue to 4. Conclusions and Discussion appear after MJD 59,915 (2022 December). It is of interest to keep monitoring the source, checking whether or not the QPO We collected 0.1–300 GeV energy band data from the blazar signal of ∼210 days would appear again. Unfortunately, we S4 0954+658 from the Fermi-LAT archive and conducted a cannot verify the authenticity of the two transient QPOs in the temporal analysis in two interesting periods: segment 1 (MJD multiwavelength light curve due to a lack of good coverage of 57,145–57,745) and segment 2 (MJD 59,035–59,915).Four multiwavelength observations and data-point resolution during the analytical methods (i.e., the epoch folding, REDFIT, LSP, and WWZ) and a tool (light-curve simulations) areemployedtodetect period concerned. We expect that different telescopes will study the transient QPO in the 10 days binned light curve, revealing a the QPO signal of this source in the future. good consistency between the different methods. For segment 1, A variety of scenarios have been proposed to explain the QPO our results showed that there was a QPO of 66 days above 5σ phenomenon in blazar emission. One of the most interesting 6 The Astrophysical Journal, 949:39 (8pp), 2023 June 1 Gong et al. features of the accretion flow is that stable twin high-frequency the jet, which has recently been applied in many cases (Zhou QPOs often appear with frequency ratio 3:2 in the X-ray flux, e.g., et al. 2018; Li et al. 2021; Roy et al. 2022). In this model, as the plasma blob (containing higher particle and magnetic energy Sgr A and GRO J1655-40 (Abramowicz & Kluźniak 2001; densities) injected into jet enhances the emission, every plasma Török 2005). The scale of QPOs with two stable peaks indicates blob will change its orientation with respect to the line of sight that they can originate from some resonant process taking place in and this will produce a quasi-periodic flux modulation due to the accretion disk’s oscillations (Abramowicz et al. 2003;Horák the Doppler beaming effect. The helical motion of plasma et al. 2009). In the framework of the resonance model, the blobs within the jet may be a natural process in magnetically frequencies reflect epicyclic motion of perturbed flow lines in the dominated jets (Chen & Zhang 2021). In the helical motion of accretion disk, or combinations of these with a fixed perturbation the blob, θ (t) of a given emitting region depends on the pitch frequency (Rubio-Herrera & Lee 2005). The scaled similarity obs angle of the helix f and on the angle ψ of the axis of the jet between stellar-mass systems and AGNs indicates that resonances with respect to our line of sight according to are important for AGNs as well. No pairs of QPOs at that 3:2 ratio have been detected for S4 0954+658: 66 days and 210 days cosqf ()tt =+ cos cosy sinf siny cosw(). (3) obs −7 −7 correspond to frequencies of 1.75 × 10 Hz and 0.55 × 10 Hz, where ω(t) = 2πt/P is the variable azimuth and P is the obs obs respectively. Separate but related is the relativistic precession observed period. From θ (t), and adopting the bulk Lorentz obs model, which associates three different QPOs to a combination of factor Γ = 11.4 given by Jorstad et al. (2017), we calculate the the fundamental frequencies of particle motion (Motta et al. 2014). Doppler factor δ(t) from the equation d() t =G (11 / )( - While the higher-frequency QPOs correspond to the Keplerian frequency of the innermost disk regions, the lower-frequency bq cos () t ),where β = ν /c. Then, the periodicity in the rest obs jet QPOs correspond to the relativistic periastron precession of frame of the blob can be calculated (see Roy et al. 2022 for eccentric orbits and the Type-C QPOs in the nodal precession details).For thecaseofS40954+658, if we assume the para- (or Lense–Thirring precession) of tilted orbits in the same meters used in Jorstad et al. (2017) for the parsec-scale radio jet, regions (Stella & Vietri 1998; Stella et al. 1999).For theLense– i.e., the pitch angle f = 1°.75 (assumed to be half of the opening Thirring precession, the period can be expressed using t = LT angle), the viewing angle ψ = 1°.5, and P = 66 days, then the obs -1 0.18aM()// 10 M(r r) days, where a , M, r ,and r are the  g s g blob traverses about a distance D=» 9cPbf cos siny rest dimensionless spin parameter, the mass of the black hole (BH), 1.64 pc down the jet during nine periods (Zhou et al. 2018). Here, the gravitational radius, and the radial distance of the emission P is the physical period at the host galaxy: P = P / rest rest obs region from the BH, respectively. In such a scenario, taking the (1−βcosfcosψ). In addition, for P = 210 days, the blob obs spin parameter a = 0.9 and the BH mass M = 2.3 × 10 M s e travels ∼2.32 pc during four periods. As the blob is injected into (Becerra González et al. 2021),the timescaleofthe twoQPOs the jet (or dissipates), the periodic modulation tends to become places the emission region in the range from 10 to 15 r . Because more (or less) noticeable. This model has a defect in that it can the accretion disks are warped, the QPO phenomenon could be the result of jet precession, therefore resulting in a period only explain a QPO with almost constant amplitude. However, the of thousands of years (Bhatta 2018;Liska et al. 2018;Li etal. amplitude of the QPO is almost constant either in segment 1 or in 2023). Such a long timescale does not seem to apply to segment 2, but different in the two (see Figure 1). Hence, it is this case. reasonable that different plasma blobs of this model are used to A model of a binary SMBH system was proposed to explain explain the transient properties of QPOs with different timescales. the ∼2 yr periodic fluctuation in the multiwavelength light We expect the 210 day QPO behavior will continue to appear in curve of PG 1553+113 and later applied to interpret similar Fermi-LAT observations. Furthermore, we also hope that the fluctuation behavior of other blazers (Ackermann et al. 2015; multiwavelength campaign (i.e., TESS and WEBT) will pay Sandrinelli et al. 2018; Otero-Santos et al. 2020; Wang et al. attention to whether variability due to two transient QPOs also 2022). The orbital motion of this model may cause long-term appears and identify the underlying physical mechanism among periodic temporal signals, which is reflected in periodic different hypotheses. accretion perturbations, or jet-precessional and nutational motions (Liska et al. 2018). The observed period P is obs We thank anonymous referee for very helpful suggestions. corrected to the intrinsic orbital period in the local galaxy via This research or product makes use of public data provided the relation P = P /(1 + z), where z = 0.3694 is the int obs by Fermi-LAT. J.F. is partially supported by National cosmological redshift. Using the periods of 66 days and 210 Natural Science Foundation of China (NSFC) under grant days, we get the intrinsic orbital periods of 48 days and 153 U2031107, the Joint Foundation of Department of Science and days respectively. We assume that the mass ratio between the Technology of Yunnan Province and Yunnan University two SMBHs is 0.1 and take the central black hole to be the 8 (202201BF070001-020), the grant from Yunnan Province primary black hole with a mass of 2.3 × 10 M . We substitute (YNWR-QNBJ-2018-049) and the National Key R&D Pro- two transient QPO values into the formula given by Fan et al. gram of China under grant (No. 2018YFA0404204). Y.L.G. is (2010), and the results show a very tight orbit (0.001 pc and supported by Yunnan University Graduate Scientific Research 0.002 pc) and a short merging timescale (95 yr and 2048 yr) in Innovation Fund under grant KC-2222975. T.F.Y. is supported the gravitational wave-driven regime (Bhatta 2018). Never- by NSFC under grant 11863007. theless, the two transient QPOs we detected were too short compared to the period expected by this model. And a binary ORCID iDs SMBH system should produce a more stable/persistent periodic behavior, which is not observed. 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Two Transient Quasi-periodic Oscillations in γ-Ray Emission from the Blazar S4 0954+658

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© 2023. The Author(s). Published by the American Astronomical Society.
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10.3847/1538-4357/acca7b
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Abstract

In this work, we report periodicity search analyses in the gamma-ray light curve of the blazar S4 0954+658 in monitoring undertaken by the Fermi Large Area Telescope. Four analytical methods and a tool are adopted to detect any periodic flux modulation and corresponding significance level, revealing: (i) a quasi-periodic oscillation (QPO) of 66 days with a significance level of >5σ spanning over 600 days from 2015 to 2016 (MJD 57,145–57,745), resulting in continuous observation of nine cycles, which is one of the longest cycles discerned in blazar gamma-ray light curves; (ii) a possible QPO of 210 days at a moderate significance of ∼3.5σ, which lasted for over 880 days from 2020 to 2022 (MJD 59,035–59,915) and for four cycles. In addition, we discuss several physical models to explain the origin of the two transient QPOs and conclude that a geometrical scenario involving a plasma blob moving helically inside the jet can explain the timescale of the QPO. Unified Astronomy Thesaurus concepts: Active galactic nuclei (16); Blazars (164); Jets (870); Period search (1955) 1. Introduction references therein). QPO phenomena are usually quite rarely detected and non-persistent for AGNs, but they seem to be It is generally believed that all active galaxies are powered relatively common in black hole X-ray binaries (Remillard & by the accretion of dense ionized gases onto a supermassive McClintock 2006; Gupta 2014). So far, more than 30 of 5064 6 10 black hole (SMBH) with a mass in the range 10 –10 M , and sources above 4σ significance are reported to have QPO ∼10% of them have jets of relativistic charged particles. Radio- phenomena based on time series data in the fourth Fermi loud active galactic nuclei (AGNs), with their jets pointing Gamma-ray Large Area Telescope (LAT) catalog of sources almost directly along the observer’s line of sight („10°), form a (4FGL; Abdollahi et al. 2020; Wang et al. 2022). special subclass called blazars (Antonucci 1993; Urry & Recently, Jorstad et al. (2022) claimed that the γ-ray flux, Padovani 1995). Moreover, blazars can be further divided into optical flux, and linear polarization of BL Lacertae all exhibited two subcategories based on the strength of emission lines ∼13 hr QPO variability during a dramatic outburst in 2020. emerging in optical–ultraviolet spectra: BL Lacertae (BL Lac) Such a short-term QPO is explained by the current-driven kink objects (very weak and narrow emission lines) and flat- instabilities near a recollimation shock ∼5 pc from the black spectrum radio quasars (FSRQs; broad and strong emission hole. In the same year, a quasi-periodic signal of approximately lines). Blazars usually manifest substantial variability over 420 days with >5σ significance was found in the measure- almost the whole electromagnetic spectrum; their emission is ments of the degree of optical linear polarization for the blazar dominated by nonthermal radiation and ranges from radio to γ- PKS 1222+216 and a helical jet model was employed to rays (Ulrich et al. 1997). explain the signal well (Zhang & Wang 2022). Furthermore, Observations from both ground-based and space telescope several models have been proposed by different authors to show that blazars have flux variability of the order of minutes explain periodic radiation from blazars at various frequencies to years in different electromagnetic wave bands, which may on diverse timescales, i.e., a hotspot orbiting near the innermost indicate that different physical mechanisms (intrinsic and stable circular orbit of the SMBH (Gupta et al. 2009, 2019; extrinsic) play a leading role. An interesting phenomenon Sarkar et al. 2021), the presence of the SMBH in a binary related to flux variability is quasi-periodic oscillation (QPO), system (Valtonen et al. 2008; Ackermann et al. 2015), although flux variability frequently exhibits nonlinear, stochas- precession of relativistic jets or a helical structure (Graham tic, and aperiodic characteristics (Kushwaha et al. 2017).So et al. 2015b; Sandrinelli et al. 2016), the existence of quasi- far, a large number of QPO behaviors with different timescales equidistant magnetic islands inside the jet (Huang et al. 2013; in multifrequency light curves have been reported by Shukla et al. 2018; Roy et al. 2022), and pulsational researchers using different detection techniques (e.g., Raiteri instabilities in accretion flow (Tavani et al. 2018). Hence, we et al. 2001; Liu et al. 2006; Gupta et al. 2009; Lachowicz et al. can analyze the quasi-periodic modulation in the blazar light 2009; King et al. 2013; Zhang et al. 2014; Graham et al. 2015a; curve to explore the accretion physics and the connection Ackermann et al. 2015; Bhatta 2017; Gupta et al. 2018; Zhou between accretion disk, jet, and central engine (Kushwaha et al. et al. 2018; Sarkar et al. 2020; Gong et al. 2022; Roy et al. 2020). 2022; Zhang et al. 2022; Otero-Santos et al. 2023, and S4 0954+658 (also referred to as QSO B0954+65) is one of the most well studied blazar sources with complex variability Original content from this work may be used under the terms and is situated at a redshift of z = 0.3694 ± 0.0011 (Becerra of the Creative Commons Attribution 4.0 licence. Any further González et al. 2021). Stickel et al. (1991) regard this source as distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. a BL Lac object in view of the small equivalent width of the 1 The Astrophysical Journal, 949:39 (8pp), 2023 June 1 Gong et al. emission lines of the spectrum. However, this target can also be region of interest having a radius of 12° centered at the source hm s classified as an FSRQ because the kinematic features of the (ad== 09 58 47.244, 6533¢54.8). At the same 2000.0 2000.0 radio jet belong to class II (Hervet et al. 2016). In 2021, time, we used a screening expression ‘(DATA_QUAL >0)&& Becerra González et al. (2021) detected an Mg II emission line, (LAT_CONFIG=1)” to select events with good time intervals whose equivalent width is close to 5 Å, which is commonly and set a zenith angle cut of 90° to suppress the γ-ray pollution taken as the limit to classify a blazar as an FSRQ. Therefore, it from the Earth’s limb. An XML file is generated through the seems more reasonable to consider this γ-ray emitter as a 4FGL catalog containing the γ-ray background emission transitional object. Wagner et al. (1993) investigated the optical templates ‘gll_iem_v07” and ‘iso_P8R3_SOUR- variability of this source for the first time and then Raiteri et al. CE_V2_v1.txt” for the Galactic and isotropic extragalactic (1999) detected fast large-amplitude variations using the 4 yr contributions respectively. We consider three commonly used light curve. Their results indicate that the long-term behavior of spectral models (power law, log-parabola, and power law with the source is not related to spectral variations. Continuous an exponential cutoff) for the whole time series. And we also observation of this blazar shows that the optical flux varies by test for the spectral curvature in the spectrum using more than 2.5 mag and the degree of polarization reached 40% ΔTS = TS − TS = 343 (Abdollahi et al. 2020). The LogPb PL (Papadakis et al. 2004; Hagen-Thorn et al. 2015). Additionally, results show that the log-parabola (LogPb) model is the most Gaur et al. (2019) found a positive correlation between color suitable for describing γ-ray emission from the target source. index and magnitude based on simultaneous data in the B and R The best-fit spectral parameters were α = 2.14 ± 0.01, bands. β = 0.53 ± 0.04, and E = 699.12 ± 29.46 MeV. In addition, MAGIC Collaboration et al. (2018) presented the first we selected low states (MJD 54,687–55,558 and detection of the blazar S4 0954+658 in very high-energy 55,778–56,702) and high states (MJD 56,918–57,160 and (…100 GeV) γ-rays, which was obtained through monitoring 59,619–59,894) to test the spectral shape of the time series. The with the Major Atmospheric Gamma Imaging Cherenkov results show that the fitting parameters (except β) and flux (MAGIC) Telescopes during an exceptional flare (2015 variability are close to the whole time series. February). In 2021, Raiteri et al. (2021) found a 31.2 day Based on the best fitting results mentioned above, we tested QPO behavior in the optical long-term variability through the the construction of a binned light curve for 1–30 days and observation of the Transiting Exoplanet Survey Satellite found that bins of 10 days are the most appropriate size because (TESS) and the Whole Earth Blazar Telescope (WEBT) they not only reveal the details of the flux variation, but also Collaboration, in which the rotation of an inhomogeneous ensure that the blazar S4 0954+658 can be detected in almost helical jet provides a reasonable explanation for this phenom- all bins (TS … 9). In addition, the 10 days binned light curve enon. It is worth mentioning that such a month-long transient also shows greater intensity in the calculation of the power QPO is also detected in the γ-ray band for PKS 2247–131 spectrum than other bins. In the 10 days binned light curve (see (Zhou et al. 2018). More recently, Kishore et al. (2023) report Figure 1), the average value and standard deviation are 1.11 −7 −2 −1 the discover of several QPOs around 0.6–2.5 days in the optical and 0.95 × 10 photons cm s , respectively. Detection of light curve of the blazar S4 0954+658 with data acquired in six periodicity in the light curve was based on the weighted sectors by the TESS. wavelet Z-transform (WWZ) method (see Figure 2).We Here, we are inspired by the QPO report on the optical selected panels B (segment 1) and C (segment 2) in Figure 1 as radiation, and try to analyze whether the ∼14.3 yr data the regions of interest for analysis of QPO variability. measured by Fermi-LAT also contain any QPO phenomena. The paper is structured as follows. In Section 2, we describe the 3. Periodicity Search for γ-ray Emission process of data analysis in the 0.1–300 GeV energy band. In Section 3, we present the QPO detection algorithm and our It is not rigorous enough to visually measure the QPO main results. In Section 4, we summarize our conclusions and variability in the unevenly sampled light curve, but many explore several models to explain the QPO results. methods have been proposed to detect periodic components and corresponding significance levels. Here, we applied four methods to analyze the light curves, i.e., epoch folding, 2. Fermi-LAT Data Analysis REDFIT, Lomb–Scargle periodogram (LSP), and WWZ, and a The LAT on board the Fermi observatory continually tool. i.e., light-curve simulations, was used to determine surveys the entire sky every 90 minutes in the energy range confidence levels. Although the γ-ray light curve obtained by from 20 MeV to >300 GeV (Atwood et al. 2009). Based on the us is evenly binned, we only consider data points with TS > 9, observation data from the first 12 years, the 4FGL incremental resulting in uneven sampling of data. version of γ-ray sources contains 6658 sources, including more Epoch folding is one of the most popular methods of light- than 100 newly classified blazars (Abdollahi et al. 2022). The curve analysis (Leahy et al. 1983; Davies 1991). This method is blazar S4 0954+658 (named as 4FGL J0958.7+6534) was insensitive to the modulating shape of periodic components and found in the first Fermi Gamma-ray LAT catalog, and has also the uneven sampling of time series data, which is different from been detected by various radio surveys and optical and the traditional discrete Fourier periodogram (Bhatta 2018).We millimeter surveys. In order to build the light curve of this computed χ values of the γ-ray light curve with a time step of source, we used the standard software package FERMITOOLS 6 days for trial periods ranging between 6 and 510 days using and the user-contributed tool make4FGLxml.py. Equation (1) of Bhatta (2018). The results show that maximum The data for the blazar S4 0954+658 were taken during the χ values of 225 and 172 correspond to trial periods of 66 days period 2008 August 4 (MET: 239557417) to 2022 December 5 and 210 days, respectively. In segment 1, we constructed a (MET: 691900553), covering ∼14.3 yr. We chose LAT folded light curve by binned likelihood analysis with a period 0.1–300 GeV Pass 8 (evclass = 128, evtype = 3) events of ∼66 days, where phase zero corresponds to MJD 57,145 and recommended by the Fermi-LAT collaboration from a circular 10 phase ranges are selected (upper left panel of Figure 3). 2 The Astrophysical Journal, 949:39 (8pp), 2023 June 1 Gong et al. Figure 1. The 10 days binned light curve of the blazar S4 0954+658 at γ-ray energies of 0.1–300 GeV obtained from Fermi-LAT. (A) The purple shaded region (marked as segment 1) covers MJD 57,145–57,745 and represents the first segment for QPO analysis. The gray shaded region is the epoch MJD 59,035–59,915 (marked as segment 2) where the QPO analyses were carried out. (B) Zoom-in of segment 1, where the red dashed–dotted line indicates the sine fitting result of the light curve. The orange histogram corresponds to the TS value of each data point. (C) Same as panel (B) but for segment 2. Similar to segment 1, the phase zero of segment 2 is set at MJD An additional method, REDFIT, is also used to calculate the 59,035 to complete the folding light curve with a period of bias-corrected power spectrum of the light curve and estimate ∼210 days (upper left panel of Figure 4). Both results show the significance level of the corresponding dominant period that the γ-ray flux obviously varies with phase. (Schulz & Mudelsee 2002). This method can calculate the 3 The Astrophysical Journal, 949:39 (8pp), 2023 June 1 Gong et al. Figure 2. Left panel: WWZ map of the S4 0954+658 light curve in MJD 57,085–57,795. The bright red patch represents a possible QPO in the interval MJD 57,145–57,745 (segment 1). Right panel: WWZ map of the γ-ray light curve of blazar S4 0954+658 in MJD 58,315–59,915. The bright red patch represents a possible QPO in the interval MJD 59,035–59,915 (segment 2). underlying red-noise spectrum by fitting the time series with a also shows the result of the analysis of segment 2, which first-order autoregressive process (AR1), which is caused by revealed a significant signal centered at 208 ± 43 days. some stochastic processes in the accretion disk or jet for blazars Further evidence for the two transient QPOs is provided by (Fan et al. 2014; Covino et al. 2019). In the AR models the the WWZ method. The WWZ method, first introduced by present emission is connected with the past emission. The Foster (1996), can identify the localized features in both time theoretical power spectrum of an AR1 model is given as and frequency domains, especially in unequally spaced data, based on three trial functions, i.e., f (t) = 1(t), 1 - q f ()tt=- cos[wt ( )], and f ()tt=- sin[wt ( )]. The calc- 2 3 Gf() =G ,1 () rr 0 ulation of WWZ power intensity can search for a periodic 12-+ qp cos() ff q j Nyq modulation signal with frequency ω and time shift τ in a where G is the average spectral amplitude, θ is the average statistical manner, which is described as autoregression coefficient, and f represents the discrete () NV - 3 eff y frequency up to the Nyquist frequency ( f ). We used the Nyq WWZ = ,2 () REDFIT3.8e program to estimates the power spectrum and the 2() VV - xy significance level of the corresponding peak based on the LSP where N denote the effective number density of data points in combination with Welch overlapped segment averaging eff contributing to the signal, and V and V are the weighted (Welch 1967). As can be seen from the upper right panel of x y variations of the nonuniform data x and the model function y, Figure 3, it is evident that there is a peak around the timescale respectively. For more details on the definition of these factors, of 65 ± 12 days with significance level of >99% in the power see Li et al. (2021) and references therein. For segment 1, we spectrum during MJD 57,145–57,745 (segment 1). The upper −1 set the frequency range from 0.005 to 0.08 day and the step right panel of Figure 4 shows that the periodic modulation in −1 size is 0.00005 day in WWZ analysis, which enables the MJD 59,035–59,915 (segment 2) is centered at 210 ± 55 days QPO timescale of the region of interest to be displayed as much with a significance level of ∼99%. We take the half-width at as possible. Furthermore, in order to balance the frequency and half-maximum (HWHM) of the power peak fitted by the time resolution, we set a decay constant of c = 0.001. The Gaussian function as the uncertainty of the periodic modulation color-scaled WWZ power of the 10 days binned light curve in signal. The LSP is one of the most common methods of finding the time–period plane is presented in the bottom panel of periodicities in time series with nonuniform sampling, and it Figure 3, which shows that the power for the characteristic can calculate the intensity of the power spectrum at different period centered around 66 days persists over the entire frequencies (Lomb 1976; Scargle 1982). This method is the observational period. The corresponding time-averaged WWZ projection of the light curve on sinusoidal functions and power is centered at the period of 66 ± 4.7 days, corroborating constructs a periodogram from the goodness of the weighted χ the LSP result. In segment 2, we adopted a limited frequency fit statistic (Ferraz-Mello 1981). Nevertheless, the aperiodic −1 range of 0.001–0.03 day in WWZ analysis, where the step part of time series data will reduce the goodness of the LSP size and decay constant are the same as for segment 1. As sinusoidal fit, which leads to a reduction in the transient shown in the bottom panel of Figure 4, the time-averaged periodic power. The bottom panel of Figure 3 shows the power WWZ power of the segment 2 light curve also shows a (black solid line) of the LSP for the extracted segment 1 data. significant peak lasting throughout the activity at One signal, at the period of 66 ± 4.8 days, reached that significance level. Meanwhile, the bottom panel of Figure 4 208 ± 40 days, which is similar to the feature in LSP analysis. 4 The Astrophysical Journal, 949:39 (8pp), 2023 June 1 Gong et al. Figure 3. The results of QPO analysis in segment 1 (MJD 57,145–57,745). Upper left: the folded γ-ray light curve, which is constructed from binned likelihood analysis of the nine cycles of segment 1. Phase zero corresponds to MJD 57,145 and 10 phase ranges are set. For clarity, we show two periods. The dashed blue horizontal line represents the mean flux. Upper right: the results of periodicity analysis from REDFIT. The solid black line indicates the bias-corrected power spectrum. The red dashed line represents the theoretical AR(1) spectrum. The blue, green, and purple dashed curves are 90%, 95%, and 99% confidence contours, respectively. Bottom: LSP and WWZ results of the γ-ray time series data. The left subpanel displays a two-dimensional contour map of the WWZ power spectrum and the horizontal red patch indicates a strong QPO signal of ∼66 days. The right subpanel shows the time-averaged WWZ (red solid line) as well as the LSP powers (black solid line). The blue, purple, and orange dashed curves are 3σ,4σ, and 5σ significance lines, respectively. The dominant period of ∼66 days can be clearly seen to cross the 5σ significance curve. The flux variability of blazars usually shows a frequency- a Monte Carlo method provided in Emmanoulopoulos et al. dependent colored-noise-like behavior, which is very likely to (2013). The underlying red-noise PSDs of blazar light curves lead to a pseudo-period in the identification of periodic are often reasonably approximated by a power-law form −α components of time series data, especially at lower temporal P( f ) ∝ f , where P( f ) is the power at temporal frequency f frequencies (Vaughan et al. 2003, 2016; Bhatta et al. 2016;Li and α is the spectral slope (Vaughan 2005). Then, we et al. 2017). The significance estimated by the REDFIT method generated 10 artificial light curves to estimate the significance is based on the χ distribution of periodogram points about the level of the LSP and WWZ periodic components. In segment 1, model, which can avoid underestimating the significance of the the significance level for the QPO signal was found to be >5σ peak in power spectral density (PSD). Here, the significance of (the bottom panel of Figure 3). In segment 2, the simulation of segments 1 and 2 obtained by using the REDFIT method the light curve shows that the periodic modulation of 210 days reveals a …99% level. Another way to estimate the significance seems to have a significance level close to 3.5σ (the bottom in the LSP and WWZ peaks is to simulate light curves with the panel of Figure 4). In the recent QPO search, a large number of same PSD and flux distribution as the original light curve using blazars are claimed to have periodic signals with a significance 5 The Astrophysical Journal, 949:39 (8pp), 2023 June 1 Gong et al. Figure 4. Same as Figure 3, but for the segment 2 light curve (MJD 59,035–59,915). that is usually greater than 3σ (Zhang et al. 2017a, 2017b; Peñil significance level during MJD 57,145–57,745, which lasted for et al. 2020; Zhang et al. 2020, 2021). Thus, the QPO signal nine cycles. Interestingly, the 66 days periodic modulation, similar with ∼3.5σ significance of segment 2 is sufficiently important to that of PKS 2247–131, also occurred after an outburst event to be reported. These two transient QPO signals may appear (2014 December) with multiwavelength observations (see again in the future, so it will be interesting to keep monitoring Figure 1;Zhouetal. 2018;Gauretal. 2019).For segment2, at the γ-ray frequency. we found a possible QPO of about 210 days with ∼3.5σ significance in the γ-ray light curve over 880 days. This signal is clearly visible for about four cycles and seems to continue to 4. Conclusions and Discussion appear after MJD 59,915 (2022 December). It is of interest to keep monitoring the source, checking whether or not the QPO We collected 0.1–300 GeV energy band data from the blazar signal of ∼210 days would appear again. Unfortunately, we S4 0954+658 from the Fermi-LAT archive and conducted a cannot verify the authenticity of the two transient QPOs in the temporal analysis in two interesting periods: segment 1 (MJD multiwavelength light curve due to a lack of good coverage of 57,145–57,745) and segment 2 (MJD 59,035–59,915).Four multiwavelength observations and data-point resolution during the analytical methods (i.e., the epoch folding, REDFIT, LSP, and WWZ) and a tool (light-curve simulations) areemployedtodetect period concerned. We expect that different telescopes will study the transient QPO in the 10 days binned light curve, revealing a the QPO signal of this source in the future. good consistency between the different methods. For segment 1, A variety of scenarios have been proposed to explain the QPO our results showed that there was a QPO of 66 days above 5σ phenomenon in blazar emission. One of the most interesting 6 The Astrophysical Journal, 949:39 (8pp), 2023 June 1 Gong et al. features of the accretion flow is that stable twin high-frequency the jet, which has recently been applied in many cases (Zhou QPOs often appear with frequency ratio 3:2 in the X-ray flux, e.g., et al. 2018; Li et al. 2021; Roy et al. 2022). In this model, as the plasma blob (containing higher particle and magnetic energy Sgr A and GRO J1655-40 (Abramowicz & Kluźniak 2001; densities) injected into jet enhances the emission, every plasma Török 2005). The scale of QPOs with two stable peaks indicates blob will change its orientation with respect to the line of sight that they can originate from some resonant process taking place in and this will produce a quasi-periodic flux modulation due to the accretion disk’s oscillations (Abramowicz et al. 2003;Horák the Doppler beaming effect. The helical motion of plasma et al. 2009). In the framework of the resonance model, the blobs within the jet may be a natural process in magnetically frequencies reflect epicyclic motion of perturbed flow lines in the dominated jets (Chen & Zhang 2021). In the helical motion of accretion disk, or combinations of these with a fixed perturbation the blob, θ (t) of a given emitting region depends on the pitch frequency (Rubio-Herrera & Lee 2005). The scaled similarity obs angle of the helix f and on the angle ψ of the axis of the jet between stellar-mass systems and AGNs indicates that resonances with respect to our line of sight according to are important for AGNs as well. No pairs of QPOs at that 3:2 ratio have been detected for S4 0954+658: 66 days and 210 days cosqf ()tt =+ cos cosy sinf siny cosw(). (3) obs −7 −7 correspond to frequencies of 1.75 × 10 Hz and 0.55 × 10 Hz, where ω(t) = 2πt/P is the variable azimuth and P is the obs obs respectively. Separate but related is the relativistic precession observed period. From θ (t), and adopting the bulk Lorentz obs model, which associates three different QPOs to a combination of factor Γ = 11.4 given by Jorstad et al. (2017), we calculate the the fundamental frequencies of particle motion (Motta et al. 2014). Doppler factor δ(t) from the equation d() t =G (11 / )( - While the higher-frequency QPOs correspond to the Keplerian frequency of the innermost disk regions, the lower-frequency bq cos () t ),where β = ν /c. Then, the periodicity in the rest obs jet QPOs correspond to the relativistic periastron precession of frame of the blob can be calculated (see Roy et al. 2022 for eccentric orbits and the Type-C QPOs in the nodal precession details).For thecaseofS40954+658, if we assume the para- (or Lense–Thirring precession) of tilted orbits in the same meters used in Jorstad et al. (2017) for the parsec-scale radio jet, regions (Stella & Vietri 1998; Stella et al. 1999).For theLense– i.e., the pitch angle f = 1°.75 (assumed to be half of the opening Thirring precession, the period can be expressed using t = LT angle), the viewing angle ψ = 1°.5, and P = 66 days, then the obs -1 0.18aM()// 10 M(r r) days, where a , M, r ,and r are the  g s g blob traverses about a distance D=» 9cPbf cos siny rest dimensionless spin parameter, the mass of the black hole (BH), 1.64 pc down the jet during nine periods (Zhou et al. 2018). Here, the gravitational radius, and the radial distance of the emission P is the physical period at the host galaxy: P = P / rest rest obs region from the BH, respectively. In such a scenario, taking the (1−βcosfcosψ). In addition, for P = 210 days, the blob obs spin parameter a = 0.9 and the BH mass M = 2.3 × 10 M s e travels ∼2.32 pc during four periods. As the blob is injected into (Becerra González et al. 2021),the timescaleofthe twoQPOs the jet (or dissipates), the periodic modulation tends to become places the emission region in the range from 10 to 15 r . Because more (or less) noticeable. This model has a defect in that it can the accretion disks are warped, the QPO phenomenon could be the result of jet precession, therefore resulting in a period only explain a QPO with almost constant amplitude. However, the of thousands of years (Bhatta 2018;Liska et al. 2018;Li etal. amplitude of the QPO is almost constant either in segment 1 or in 2023). Such a long timescale does not seem to apply to segment 2, but different in the two (see Figure 1). Hence, it is this case. reasonable that different plasma blobs of this model are used to A model of a binary SMBH system was proposed to explain explain the transient properties of QPOs with different timescales. the ∼2 yr periodic fluctuation in the multiwavelength light We expect the 210 day QPO behavior will continue to appear in curve of PG 1553+113 and later applied to interpret similar Fermi-LAT observations. Furthermore, we also hope that the fluctuation behavior of other blazers (Ackermann et al. 2015; multiwavelength campaign (i.e., TESS and WEBT) will pay Sandrinelli et al. 2018; Otero-Santos et al. 2020; Wang et al. attention to whether variability due to two transient QPOs also 2022). The orbital motion of this model may cause long-term appears and identify the underlying physical mechanism among periodic temporal signals, which is reflected in periodic different hypotheses. accretion perturbations, or jet-precessional and nutational motions (Liska et al. 2018). The observed period P is obs We thank anonymous referee for very helpful suggestions. corrected to the intrinsic orbital period in the local galaxy via This research or product makes use of public data provided the relation P = P /(1 + z), where z = 0.3694 is the int obs by Fermi-LAT. J.F. is partially supported by National cosmological redshift. Using the periods of 66 days and 210 Natural Science Foundation of China (NSFC) under grant days, we get the intrinsic orbital periods of 48 days and 153 U2031107, the Joint Foundation of Department of Science and days respectively. We assume that the mass ratio between the Technology of Yunnan Province and Yunnan University two SMBHs is 0.1 and take the central black hole to be the 8 (202201BF070001-020), the grant from Yunnan Province primary black hole with a mass of 2.3 × 10 M . We substitute (YNWR-QNBJ-2018-049) and the National Key R&D Pro- two transient QPO values into the formula given by Fan et al. gram of China under grant (No. 2018YFA0404204). Y.L.G. is (2010), and the results show a very tight orbit (0.001 pc and supported by Yunnan University Graduate Scientific Research 0.002 pc) and a short merging timescale (95 yr and 2048 yr) in Innovation Fund under grant KC-2222975. T.F.Y. is supported the gravitational wave-driven regime (Bhatta 2018). Never- by NSFC under grant 11863007. theless, the two transient QPOs we detected were too short compared to the period expected by this model. And a binary ORCID iDs SMBH system should produce a more stable/persistent periodic behavior, which is not observed. 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