Main

In 2020, the Whole Earth Blazar Telescope (WEBT; https://www.oato.inaf.it/blazars/webt/) observed the highest-amplitude optical outburst of BL Lac (redshift z = 0.069 (ref. 13)) in 20 years (ref. 9), acquiring 16,497 flux-density measurements from 1 March 2020 to 31 December 2020 (modified Julian date (MJD) 58,909–59,215) in the optical Cousins R-band (λeff = 635 nm), using 37 telescopes (Extended Data Table 1). Five telescopes also conducted 1,285 R-band measurements of the degree, PR, and position angle, χR, of linear polarization. Figure 1a plots the R-band light curve (Methods). The dense sampling leads to the discovery of QPOs in flux, SR, with a duration <1 day. Figure 1b shows 14 pulses detected over a 2-week interval during the highest-flux state. The outburst began around 20 July (MJD 59,050) and then peaked on 21 August (MJD 59,082.98608) and again on 5 October (MJD 59,127.68480) at R-magnitude of about 11.9, with a high-flux plateau between the peaks (Fig. 1a); it ended by 17 October (MJD 59,140). During the plateau, PR (Extended Data Fig. 1b) underwent intraday, high-amplitude oscillations, from <1% to 20%, whereas χR rotated by approximately 200° over 0.3 days near the beginning of the outburst (when PR was low, implying turbulence) and then varied on day timescales. We find no statistically significant correlation between SR and PR, which suggests strong turbulence superposed on the kink instability (Methods). The variations of χR form arc-like structures in Fig. 2 on timescales ≤1 day. The high degree of polarization and similarity of the SR and PR QPO timescales imply that the QPOs occur in the jet rather than the accretion disc, as commonly inferred in X-ray binary systems14 and Seyfert galaxies15. BL Lac is a bright γ-ray source16,17 and Extended Data Fig. 1 presents the 0.1–300-GeV γ-ray light curve during the outburst constructed using the Fermi Large Area Telescope data (Methods). There is a strong correlation (coefficient ρ = 0.62 ± 0.04) between the optical and γ-ray light curves, with no statistically significant delay (\(-{0.02}_{-0.44}^{+0.05}\,{\rm{days}}\); Extended Data Fig. 2); this implies co-spatiality of the optical and γ-ray emission regions.

Fig. 1: Optical R-band light curves.
figure 1

a, For the whole period of monitoring from 1 March 2020 to 31 December 2020 (n = 16,497). The grey area denotes the time of ejection of superluminal knot K and its 1σ uncertainty. b, At the high state of the outburst from 15 to 28 August (n = 2,145). Different symbols and colours in both panels indicate observations conducted by different telescopes, designations of which are given in Extended Data Table 1. The error bars represent 1σ uncertainties, which are, in general, less than the symbol sizes.

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Fig. 2: Model-data comparison.
figure 2

a, Optical flux densities in R-band (after removal of long-term trend, n = 2,145). b, Degree of polarization (n = 309). c, position angle of polarization (n = 309). d, γ-Ray light curve during the high state of the outburst (n = 128). Black symbols correspond to the data (error bars represent 1σ uncertainties) and solid red curves show fitting of a kink instability model to the data using an MCMC code (see text). ph, photons.

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We have imaged BL Lac in total and polarized intensity monthly at 43 GHz with the VLBA at a resolution of about 0.1 milliarcseconds (mas)18 or 0.13 pc. The images (Fig. 3a, Extended Data Fig. 3 and Methods) feature a jet extending southward from a bright ‘core’ (A0, presumed stationary). Modelling of the images19 shows quasi-stationary components A1–A3 located 0.12 ± 0.03, 0.29 ± 0.05 and 0.38 ± 0.09 mas from A0, respectively. We interpret these features, which have persisted for >15 years (refs. 20,21), as a series of recollimation shocks resulting from pressure mismatches between the jet and its surroundings22,23. We also identify (Fig. 3b) a bright knot, K, moving at 3.32 ± 0.46 mas year−1 (apparent speed βapp = 15.2 ± 2.1c), which passed through A0 on 11 July 2020 (MJD 59,042 ± 13), when the outburst began (Fig. 1). The mean travel time of K from one stationary feature to another is about 14 days. According to previous observations since 1998 (refs. 19,21,24), the jet usually moves with a Lorentz factor Γ of about 6 and viewing angle Θo of about 5° (refs. 21,24); βapp of approximately 15c (the highest yet observed24) requires that the jet accelerated to at least Γ of about 15 in mid-2020 and changed its direction to Θo around 1/Γ corresponding to  3.8°. The Doppler factor δ = (Γ(1 − βcosΘo))−1 then increased from 9 to 15 after K crossed the core at the onset of the outburst. K passed through stationary feature A2 on MJD 59,075 ± 10 (Fig. 3b) during the first peak of the outburst and the start of QPOs (Fig. 2). If the distance of A0 from the vertex of the jet is around 0.5 pc (ref. 25), A2 is located about 5 pc (deprojected) from the black hole, where Cohen et al.26 determined that the plasma pressure is dominated by a helical magnetic field, a condition favourable for the development of current-driven kink instabilities in a jet6. Indeed, a kink is visible south-west of A2 during the outburst in Fig. 3a, which also indicates that the position angle of polarization at 43 GHz is along the jet, as expected for a tight helical field.

Fig. 3: Parsec-scale jet of BL Lac.
figure 3

a, VLBA total (contours) and polarized (colour scale) intensity image of BL Lac at 43 GHz on 7 September 2020, convolved with a circular beam of radius 0.1 mas at PA = 0° (bottom-left circle). The global intensity peak is 1.944 Jy beam−1 and contour levels start at 0.35% of the peak and increase by factors of √2. The red circles represent the FWHM of Gaussian components according to models of the flux intensity distribution. Black line segments in the image show the electric vector (polarization) direction. b, Distance of knots with respect to the core (dotted red line) versus time, with each vector indicating knot position angle at a given time (error bars correspond to 1σ uncertainties). The dashed lines show the average locations of stationary features A1, A2 and A3, and the solid green line approximates the motion of superluminal knot K.

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Visual inspection of Figs. 1b and 2 clearly identifies pulses that repeat on a timescale <1 day. We have used different methods of time series analysis—REDFIT27 periodogram, continuous wavelet transform (CWT)28 and weighted wavelet Z-transform (WWZ)29 (Methods)—to search for periodicity in the optical and γ-ray flux and PR data. The periodograms (Extended Data Fig. 4) indicate QPOs of around 0.5–0.55 days for all three datasets with ≥99% significance. Both wavelet methods show that the periodicity is transient (Extended Data Figs. 5 and 6). A QPO of 0.55 days is detected during MJD 59,076–59,112 at 99% significance in R-band (CWT, WWZ) and γ-ray (CWT) fluxes, and PR (WWZ). The CWT PR scalogram also possesses the most significant period, about 0.55 days, during the same time interval, but with 92% significance. There are other periods ≤1 day apparent at 99% significance, but they do not agree among the methods and different arrays. We have applied a trial-period method for the R-band measurements over MJD 59,076–59,112 using a sinusoidal function, finding that a period of 0.55 days yields the lowest χ2 value (Extended Data Fig. 7a and Methods). Therefore, the period of 0.55 days is the most consistent among the methods and datasets. Comparison of the average SR and PR pulses indicates that, although the average PR pulse has a wider profile than that of SR (Extended Data Fig. 7b,c and Methods), they are clearly similar, implying that the SR and PR oscillations are produced by means of the same process. In addition, the wavelet methods show QPOs of 4–5 days with 99% significance identified in the middle of the outburst (MJD 59,100–59,130) and a period of approximately 14–15 days, with similar significance during the first half of the outburst.

We attribute the 0.55-day QPOs to a current-driven kink instability, triggered when the jet is perturbed by a lateral displacement6. This random process occurred near stationary feature A2, associated with a recollimation shock where Alfvén waves are excited26. Growth of the kink creates two further effects: (1) development of contiguous regions in which magnetic field lines become oppositely directed, then reconnect to accelerate electrons, which produce electromagnetic radiation whose flux and polarization oscillate; and (2) driving of turbulence that disorders the magnetic field and accelerates electrons that generate radiation with very low polarization and random fluctuations. The multiwavelength outburst results from a combination of the increase in flow Lorentz factor, which sends a shock wave down the jet (knot K), a decrease in viewing angle, a kink in the jet and its magnetic field, and turbulence. The size of a kink grows with time30, which could explain the approximately 4-day QPOs in the second half of the outburst, whereas the QPO of about 14 days during the first half can arise from the passage of K through successive stationary features A0–A3 (see above).

We model the polarized optical flux as synchrotron radiation and the γ-ray flux as synchrotron self-Compton emission, using jet parameters derived from the VLBA data, Γ of about 15, Θo of about 3.8° and δ of about 15. The model assumes that the magnetic field is a superposition of constant toroidal, periodically fluctuating poloidal and turbulent components (Methods). We use Markov chain Monte Carlo (MCMC) fitting to constrain the toroidal and poloidal components, the fluctuation amplitudes of the optical and γ-ray flux, the quasi-period and the average turbulent contribution (Extended Data Fig. 8a). We find that the full period of the twist in the kink of about 1.1 days (involving only the poloidal component) corresponds to the period in χR, whereas the flux and PR have a period of half this value, 0.55 days (Extended Data Fig. 8a), which coincides with the result of the time series analyses. The QPO expected from a kink has an observed period of about \(\frac{\Delta r(1+z)}{2{v}_{\perp }\delta }\), in which Δr is the size of the kink and v⊥ is the transverse velocity, estimated at about 0.1–0.2c (ref. 6). This yields Δr of around 8 × 1015 cm (roughly 3 light-days), which is similar to the theoretical expectation, Δr approximately 1016 cm (ref. 6). Figure 2 compares SR, PR, χR and γ-ray flux versus time with the MCMC fit. Figure 4 sketches the development of the kink instability as a superluminal disturbance propagates through recollimation shocks, a scenario that results in short-timescale QPOs.

Fig. 4: Sketch of the development of the QPOs.
figure 4

Jet flow is from left to right (corresponding to north-to-south in BL Lac). Interaction of an off-axis moving shock with a standing recollimation shock triggers a kink instability, which grows, causes quasi-periodic structure to form, leading to QPO emission and disrupts the magnetic field, whose structure changes from helical to turbulent. The peak of the outburst occurs during times 5 and 6.

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Methods

Optical R-band photometric and polarimetric data reduction

Extended Data Table 1 presents the telescopes that participated in the campaign, along with the number of observations and designations used in Fig. 1 and Extended Data Fig. 1, and an average 1σ uncertainty of measurements provided by each telescope. Optical photometric data reduction and the R-band light curve compilation were performed following prescriptions described in detail in Villata et al.31 Out of 37 telescopes indicated in Extended Data Table 1, 23 have participated in many BL Lac WEBT campaigns, for example, refs. 31,32,33, and their photometric systems (BVRI) are very well aligned on the basis of comparison stars used for BL Lac (B, C and H)34. We have analysed whether measurements from each telescope show a constant shift (exceeding an average 1σ uncertainty) with respect to telescopes that are consistent with each other for observations simultaneous within 15–30 min. If so, we have corrected all measurements from this telescope according to the shift. Applied shifts range from 0.02 to 0.26 magnitude. For two telescopes that provided measurements with a 1σ uncertainty > 0.075 magnitude, we have performed weighted averaging of individual measurements over 10–15 min (observations were performed every 3–5 min) and used the corresponding standard deviation as the uncertainty. Of the 16,659 observations originally submitted, 0.2% were removed from the dataset based on a significant (>3σ) deviation from the general behaviour at a given time. We consider that our R-band light curve is very robust. The dataset was corrected for Galactic extinction and host-galaxy contamination, and transformed into flux density units following the technique described in Weaver et al.33. The polarization observations from different telescopes were aligned using standard polarized stars (VI Cyg #12, Hiltner 960, BD+64.106 and BD+59.389) from Schmidt et al.35 observed during the campaign. Data were corrected for instrumental and interstellar polarization using comparison stars B, C and H, and the contribution of unpolarized host-galaxy starlight was taken into account in the same manner as in Weaver et al.33. Extended Data Fig. 1 shows the R-band flux density and polarization parameters PR and χR versus time from 1 March 2020 to 31 December 2020, after applying all corrections.

γ-Ray data reduction

We have downloaded photon and spacecraft data of the Fermi Large Area Telescope provided by the Fermi Space Science Center. The γ-ray data were analysed using the FermiTools package version 2.0.8 installed with Conda (https://github.com/fermi-lat/Fermitools-conda/), with instrument response function P8R2_V6, Galactic diffuse emission model gll_iem_v06, and isotropic background model iso_P8R2_SOURCE_V6_v06. To build a γ-ray light curve, we applied an unbinned likelihood analysis in the 0.1–300-GeV energy range. The background model includes all sources from the 4FGL catalogue16 inside a 15° radius surrounding BL Lac. Fluxes of the sources within a 10° radius were set as free parameters of the model, whereas fluxes of more distant sources were fixed to their mean values according to the 4FGL catalogue. The flux of BL Lac itself was modelled using a log-parabolic spectral energy distribution with spectral parameters fixed to their catalogue values (αg = 2.14, βg = 0.06 and Eb = 796.15 MeV). To obtain the highest possible temporal resolution of the γ-ray light curve, we used an adaptive temporal binning strategy. We started the integration with a 1-h bin and increased it gradually by increments of 1 h until we reached a likelihood test statistic value TS ≥ 10 (which corresponds to an approximately 3σ detection level)36. This strategy allows one to attain the highest possible temporal resolution during active periods while still obtaining a robust signal level during quiescent states. Extended Data Fig. 1d shows the γ-ray light curve for the 2020 analysed time intervals, whereas Fig. 2d plots the γ-ray light curve during the peak of the outburst.

Correlation analysis

We used the z-transformed discrete correlation function37, with uncertainties derived by sampling errors based on the noise in the original data and calculated with 100 random Monte Carlo draws. To compute 1σ-like bounds on the derived time lag, we used the ‘peak likelihood’ algorithm (PLIKE)38, which estimates the probability of a correlation without any a priori knowledge about the shape of the correlation function or flux/polarization curves. To verify the significance of the correlations (or non-correlations), we used a bootstrap method of generating 3,000 pairs of artificial flux and polarization curves of the same duration and temporal cadence, using the algorithm suggested by Emmanoulopoulos et al.39, in which the power spectral density (PSD) and probability density function (PDF) of each curve correspond to those of the observed data. Specifically, the R-band flux and polarization PDFs were modelled by normal distributions, whereas the γ-ray PDF was modelled by a log-normal distribution (with means and standard deviations derived from the data in each case). The PSDs of the data were calculated using the power spectral response method40,41,42. We attempted to fit the PSD with a simple power-law model having a negative slope and obtained a satisfactory fit for each curve. Further attempts to fit with a bending power-law model did not notably improve the success fraction. During the fitting, we varied the slope from −0.5 to −2.5 in steps of 0.05 and determined the best-fit value of the slope and its uncertainty from the range of success fractions obtained in the above iterations. This resulted in slopes of −1.7 ± 0.3, −1.5 ± 0.3 and −0.9 ± 0.3 for optical and γ-ray light curves and fractional polarization, respectively. These artificial curves have been used throughout the paper to estimate significances of our time series analyses. Extended Data Fig. 2 presents results of the correlation analysis between the R-band and γ-ray flux, and between SR and PR for the entire period shown in Extended Data Fig. 1, between R-band residuals (after subtraction of a long-term trend; see Extended Data Fig. 1a) and γ-ray flux, between R-band residuals and PR, and between corresponding theoretical values of SR and Sγ, and SR and PR, shown in Fig. 2.

Imaging with the Very Long Baseline Array

BL Lac is one of 35 AGN observed in the BEAM-ME programme18 with the VLBA at 43 GHz, with roughly monthly cadence. The results presented here are based on 11 epochs of observations from June 2020 to May 2021. BL Lac was monitored with 12 scans of 4–6 min each, interspersed between observations of other sources from the sample, which provided the u-v spatial-frequency coverage needed to make images with a dynamic range of roughly 500:1. The data were calibrated and imaged following the same procedure as described in ref. 21, which uses the Astronomical Image Processing System (AIPS; provided by the National Radio Astronomy Observatory (NRAO)) and Difmap43 software packages. The images were modelled in Difmap by circular components with Gaussian brightness distributions. Moving knot K was detected at four epochs at a distance between 0.5 and 1.5 mas from the core (Extended Data Fig. 3). Usually, this region has very low intensity at 43 GHz (ref. 44), hence it is rare to detect such a feature there. The average flux density of the knot was about 100 mJy, with mean 1σ uncertainty of 13 mJy, which corresponds to detection of the feature at an approximately 8σ level.

Search for periodicity

We have performed a search for periodicity in the R-band and γ-ray flux and fractional polarization using the REDFIT method27, which is specifically designed for unevenly spaced time series affected by red noise. REDFIT fits the red-noise spectrum with a first-order autoregressive process and compares it with that calculated as the mean of a number of Monte Carlo simulations. The ratio between the simulated red-noise spectrum and the theoretical spectrum is then used to correct the Lomb–Scargle Fourier transform45 of the time series for the bias owing to the sampling. Extended Data Fig. 4a–c shows the results of the application of the REDFIT technique to the optical flux densities and γ-ray flux and optical degree of polarization of BL Lac, respectively.

We have used a magnitude scalogram, which is the CWT28 of a signal plotted in time–frequency space. A scalogram allows better time localization of high-frequency events, and better frequency localization of long-duration events, than traditional methods. We used the scalogram realization in MATLAB Wavelet Toolbox v.2021b using a Morlet wavelet with ωo = 6. Scalograms of the R-band and γ-ray flux and PR versus time are shown in Extended Data Fig. 5a–c, respectively. We used the wavelet-based software for MATLAB developed by Torrence and Compo46 (available at http://paos.colorado.edu/research/wavelets) along with artificial light curves (mentioned above) to analyse the significance of the periods seen in the scalograms, based on a standard χ2 test.

As well as the above methods to search for periodicity, we have used the WWZ29 to alleviate problems induced by large fluctuations of the local number density and edge effects47. We have used the publicly available Python script (https://github.com/skiehl/wwz) developed by O’Neill et al.48. We performed the WWZ using 150 frequency bins from 0.3 to 100 μHz, logarithmically spaced. We find that the recommended window decay rate of c = (8/π2)−1 is sufficient for our purposes29,46. Extended Data Fig. 6a–c shows the results of the WWZ for the R-band, γ-ray and fractional polarization data, respectively. To determine the significance of the peaks in the WWZ, we have used the 3,000 artificial curves generated for the correlation analysis and the estimate_significance subroutine of the WWZ script.

We have determined a long-term trend of the R-band light curve by constructing a spline drawn through the lowest points at intervals of around 0.5–1 days during the outburst. The spline is shown in Extended Data Fig. 1a. To analyse pulses, we subtracted the long-term trend from the R-band flux densities during the outburst; Extended Data Fig. 7a plots the residuals during MJD 59,076–59,112. We combined pulses 2, 5, 7, 8, 9 and 12, which have well-observed maxima (see Fig. 1b), and determined their average shape (Extended Data Fig. 7b), which is characterized by an asymmetric profile with amplitude 17 ± 4 mJy and full width at half maximum (FWHM) of about 7 h. We performed a similar analysis of pulses of the fractional polarization, combining them after normalizing each pulse relative to its maximum (Extended Data Fig. 7c), using all PR pulses plotted in Fig. 2. The average PR pulse has a similar shape as the average SR pulse with FWHM of about 9 h. We have approximated the R-band behaviour during MJD 59,076–59,112 by fitting the oscillations with a sinusoidal function with different trial periods running from 0.35 to 1.65 days, with a step of 0.05 days, and trial amplitudes from 14 to 21 mJy, with a step of 1 mJy. Extended Data Fig. 7a presents the best (lowest reduced χ2) sinusoidal approximation (solid red curve) of the observed oscillations, with a 0.55-day period and a 20-mJy amplitude. This supports the presence of a 0.55-day QPO during the outburst found by the time series analyses.

Kink instability model

We model the QPO patterns in the optical band with synchrotron emission from a kink in the jet and the γ-rays generated by Compton scattering of the synchrotron photons. Magnetohydrodynamic simulations30,49 have shown that kink instabilities can naturally develop in a jet with a strong toroidal magnetic field. The distorted field lines in the kinks give rise to magnetic reconnection, accelerating particles and further disordering the field. Both the poloidal field component and the number of accelerated particles increase in the same location50,51, which maintains the outburst of non-thermal radiation instigated by the moving shock crossing recollimation shock A2 (see Fig. 3a). The kink in the jet naturally becomes quasi-periodic, consisting of twisted magnetic field structures, referred to as kink nodes. The process results in a moving region (plasma ‘blob’) of enhanced emission, containing a few kink nodes6,30, which trails the moving shock. Owing to the quasi-periodic nature of the kink, this blob can exhibit QPO radiation patterns as long as there are no more than a few nodes52. If the kink is strong with a small number of nodes, its transverse displacement is roughly equal to the size of the emission blob, which is what we find in BL Lac.

We have carried out semianalytical numerical simulations of time-dependent emission from such a blob with a kink following the formalism of Dong et al.6. There are three contributions: a constant toroidal component, whose flux is normalized to unity in the code, a periodically fluctuating poloidal contribution characterized by a sinusoidal function with period T and amplitude Bp0, and turbulence, whose contribution averages to P0, with an amplitude of fluctuations of B0. The optical and γ-ray light curves follow the same temporal evolution. The average emission power in the code unit is normalized to Fo0 and Fγ0 for the optical and γ-ray bands, respectively. We use MCMC fitting to constrain the above six parameters based on all-optical and γ-ray data during the interval MJD 59,076–59,089, totalling more than 3,000 data points. All six parameters converge well into a small parameter range using 64 MCMC walkers with 60,000 iterations (Extended Data Fig. 8a), which implies that our model is robust.

Figure 2 presents the model, calculated with the 50% quantile (median) of the fitted values of each parameter (Fo0 = 21.05 mJy, Fγ0 = 9.48 × 10−7 photons cm−2 s−1 and T = 1.14 days, in the frame of the observer, and Bp0 = 1.93, B0 = 2.80 and P0 = 14.6 are in scalable code units, measured in the comoving plasma frame). Note that we fit the main characteristics of the observed variations rather than the actual details, which would take many more simulations to reproduce owing to the random nature of the turbulence. The turbulent contribution affects both SR (by addition) and PR (by dilution), but not χR. To constrain the time profile, we use all simultaneous optical and polarization measurements (about 600 data points) to constrain the strength of the turbulent contribution through the least-squares method. The result is consistent with a Gaussian distribution. Therefore, we characterize the turbulent contribution at different times by a random number drawn from a Gaussian distribution centred at P0, which is a new addition to the formalism of Dong et al.6. In the non-turbulent case considered in Dong et al.6, there is a strong anti-correlation between SR and PR (ρ ≈ −1). When turbulence is added to the code, the anti-correlation becomes statistically insignificant, in agreement with the observations (Extended Data Fig. 2). The model light curves show a positive correlation between optical flux densities and γ-ray fluxes without a delay, as the BL Lac data do. Magnetohydrodynamic simulations have shown that both the toroidal and the poloidal magnetic field components, as well as the non-thermal particle distributions, can exhibit considerable randomness48,49,50, which can contribute to the random behaviour in the evolution of χR. For the sake of simplicity, we do not include randomness in the toroidal and poloidal field components in our effort to capture the main physical characteristics of the variations. Extended Data Fig. 8b–e plots distributions of residuals between the data and the model for all four observables. The histograms have shapes close to a Gaussian distribution, with peaks near zero, as expected if the model represents the observed behaviour. The strength of our result comes from the extraordinary sampling, the number of QPO pulses and from the comprehensive interpretation of the multiwavelength behaviour, including polarization and VLBA images, which seem to be necessary components for understanding short-timescale QPOs in black-hole systems with relativistic jets.