A geometrical interpretation for the properties of
multiband optical variability of the blazar S5 0716+714 ¹¹1©This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/

To detect IDV, we performed the analysis of variance (ANOVA) (de Diego et al., 1998; de Diego, 2010). To do this, we have divided the intra-day data into groups. One group included 5 consecutive in time measurements. If the last group had less than 5 measurements, these were added to the previous group. ANOVA was applied to both the object and the comparison star to test the brightness stability of the latter and the sky quality. Table 1 shows the results of ANOVA tests. The intra-night variability is present throughout 16 nights out of 23. Intra-day light curves for dates with the detected IDV are plotted in Fig. 1.

The variability amplitude was found by the formula (Heidt & Wagner, 1996)

where $m_{\text{max}}$ and $m_{\text{min}}$ are the maximum and minimum magnitudes of the object during the night, $\sigma$ is the mean error at this night. The obtained variability amplitudes in the optical bands B, V, R${}_{\text{c}}$, I${}_{\text{c}}$ are listed in Table 1. From the comparison of variability amplitudes in various bands for nights with the detected IDV (Fig. 2), it follows that amplitudes for different bands are almost equal for small brightness variations (up to $\approx 0.15$ of magnitude in the B band). For larger brightness variations, the variability amplitude is systematically greater in the filter having a higher effective frequency of the two compared. This can be explained by the presence of the bluer-when-brighter (BWB) chromatism in IDV.

The dependence of the color index on the magnitude for nights with the detected IDV is shown in Fig. 3. The index “int” indicates that the magnitude in the corresponding band was linearly interpolated at the moment of observation in the B band. For the interpolation, we used two adjacent data points in the given optical band. The corresponding Pearson correlation coefficient $r$ is shown in each plot. For most nights, there is a strong BWB trend. And the correlation between the color index and magnitude is slightly higher when comparing optical bands with a larger interval between their effective frequencies. A similar behavior was also detected by Dai et al. (2015).

To reveal conditions for the appearance of intra-day BWB chromatism, we plotted $r$ versus the overnight average magnitude in the B band, the color index B$-$I${}_{\text{c, int}}$, and the variability amplitude for the corresponding night (in Fig. 4). The presence of intra-night BWB chromatism is not clear dependent on the considered parameters. Poon et al. (2009) obtained a similar result. Based on data over 19 nights of observations on 01-02.2006, Wu et al. (2007) found that the greater variability amplitude, the bluer color (V$-$R) of the object is observed. In Sect. 3, we self-consistently explain the observed different, sometimes contrary, color index behavior during variability.

[h]

The long-term light curve of S5 0716+714 during 04.2014$-$04.2015 is shown in Fig. 5. To plot the light curve, we completed our data with observational data derived with the 70 cm telescope AZT-8 at the Crimean Astrophysical Observatory of RAS (Gorbachev et al. in prep.). We observed the faintest S5 0716+714 on JD 2456751 and JD 2456966, when its mean intra-night magnitude was B=15.31, V=14.78, R${}_{\text{c}}$=14.34, and I${}_{\text{c}}$=13.78 in the optical bands B, V, R${}_{\text{c}}$, and I${}_{\text{c}}$, respectively. The maximum brightness of the object $\text{B}=12.95$, $\text{V}=12.54$, $\text{R}_{\text{c}}=12.16$, and $\text{I}_{\text{c}}=11.74$ was registered on JD 2457129. These values almost correspond to the faintest and brightest states according to the AZT-8 data (Gorbachev et al. in prep.). Our IDV observations cover almost the entire range of changes in the object magnitude. But as seen from Fig. 5, the occurrence of IDV events does not obviously depend on the object brightness.

The long-term variability shows a strong BWB trend: the Pearson correlation coefficient is $r_{\text{lt}}=0.86$ for all the intra-night averaged data for (B$-$I${}_{\text{c}}$) vs B (Fig. 6). If we consider the nights with and without the detected IDV separately, the long-term BWB trend persists, but the correlation is slightly less ($r_{\text{lt}}=0.69$) for nights without IDV than for nights with IDV ($r_{\text{lt}}=0.92$).

In this subsection, we estimate whether the deviation of some part of the jet component from the general trajectory may lead to IDV events. For this aim, we adopt the helical jet model (Butuzova, 2018a, b), which succeeded in providing a mutually agreed description of several observed properties of S5 0716+714. We briefly recall the main points of this model. Emitting components of the jet are placed consequently, forming a helical line on the surface of the notional cone. We call these components main to emphasize that they are responsible for the long-term properties of the object. Each of the main components moves at the angle $p$ to the radial direction (Fig. 7). The component position on the helical line relative to the line of sight is described by the azimuth angle $\varphi$. When the main component moves outwards, its angle $\varphi$ changes, but for a few hours in the observer’s reference frame this change is negligible because the distance traveled by the component during this time is much less than that from the cone apex to the component. The angle between the line of sight and velocity vector, which characterizes the general motion direction of the component, is defined by Formulae (11)–(13) in (Butuzova, 2018a):

where $\xi$ is the half-opening angle of the cone, $\theta_{0}$ is the angle between the cone axis and the line of sight. Values of these parameters are obtained by Butuzova (2018a) and listed in Table 2. Note, the more arithmetically compact expression for this angle is obtained by Butuzova & Pushkarev (2020).

Let us find the angle between the line of sight and velocity vector of the sub-component. By a sub-component, we mean some part of the component moving with the speed $\beta_{1}$ (in units of the speed of light $c$) at the angle $\chi$ to the component’s trajectory (Fig. 7). Similarly to the angle $\varphi$, we introduce the azimuth angle $\varphi_{\chi}$ characterizing the location of $\beta_{1}$ relative to $\beta$ and the line of sight. For clarity, the line of sight in Fig. 7b passes through the jet component. Figure 7b is similar to the scheme in Figure 1a in (Butuzova, 2018a). Then Expression (5) in (Butuzova, 2018a) can be used to find the angle $\theta_{\chi}$ between $\beta_{1}$ and Oz:

where $\theta=\theta\left(\xi,\theta_{0},\varphi\right)$ is the specified by Formulae (2) angle between $\beta$ and Oz at the considered moment in time. Formulae (2) and (3) may be used regardless of the physical nature of sub-components. For example, if the jet main component is a region with a higher number density of emitting particles, then the sub-component is formed by those particles, whose motion direction approximately coincides between themselves and slightly deviates from the general trajectory of the main component. If the main component is a region containing ejected electrons after their re-acceleration on the shock front, then sub-component forms in that place, where the shock front deviates from its general motion direction. This deviation may arise due to a small difference in speed of some part of the shock front during shock propagation through the inhomogeneous plasma.

In the considered model of the helical jet rotating around its axis, the components consecutively pass the region in which the jet medium becomes transparent for the optical emission. And the angle $\varphi$ of each successive component differs from that of the previous one. This leads to a change of the Doppler factor

The Doppler factor of the main component, located at the distance $d$, versus $\varphi$ is plotted in Fig. 8a by the black line. Color lines indicate the Doppler factor $\delta_{\chi}$ of the sub-component moving at the angle $\chi$ to the trajectory of the component having the azimuth angle $\varphi$. Figure 8b shows the corresponding changes of $\theta$ and $\theta_{\chi}$ versus $\varphi$. Comparing them, one can see that, e.g., the value of $\theta_{\chi}$ for $\chi\approx 10^{\circ}$ at $\varphi=200^{\circ}-360^{\circ}$ is roughly 20 times smaller than $\theta$, but the corresponding Doppler factor is about 10 times higher.

As follows from Figure 8a, at a small difference of $\chi$ from $0^{\circ}$, the function of $\delta_{\chi}\left(\varphi\right)$ has two peaks located symmetrically relative to the peak value of the Doppler factor of the main component, $\delta_{\text{max}}$. With increasing $\chi$, the difference in the position of these peaks increases: $\varphi$ of the one peak decreases, and that of the other one increases. For $\chi\approx 9^{\circ}$, both peaks appear in the plot at $\varphi>180^{\circ}$. All the mentioned peaks in $\delta_{\chi}\left(\varphi\right)$ are almost equal to $\delta_{\text{max}}$. With further rise of $\chi$, the value of the single peak of function $\delta_{\chi}\left(\varphi\right)$ decreases rapidly. For $\chi\geqslant 15^{\circ}$, the Doppler factor of the sub-component is always $\delta_{\chi}<10$. Thus, for the continuous presence in the radiating region of sub-components with a high Doppler factor $\left(\approx 30-40\right)$, deviations in their trajectories from the general motion direction of the component up to $11^{\circ}$ with the constant velocity modulus $\beta=0.999c$ is sufficient. This conclusion is true for the adopted value of $\varphi_{\chi}=180^{\circ}$ under which, as follows from Formulae (3) and (4), the value of $\delta_{\chi}$ is maximum at the fixed other parameters. As $\varphi_{\chi}$ decreases, the value of peaks of function $\delta_{\chi}\left(\varphi\right)$ decreases for the fixed $\chi$ (see Fig. 9a). Also the value of peaks of the function $\delta_{\chi}\left(\varphi\right)$ at the fixed $\varphi_{\chi}$, for example, $120^{\circ}$, decreases with increasing $\chi$, and the peaks experience a smaller offset along $\varphi$ than at $\varphi_{\chi}=180^{\circ}$ (Fig. 9b). To found the range of values of $\varphi_{\chi}$, under which $\delta_{\chi}$ is at least more than 10, we did as follows. For each value of $\varphi_{\chi}$ varying from 0 to $359^{\circ}$ at $1^{\circ}$ intervals, we estimated the number $N$ of all the values of $\delta_{\chi}\left(\varphi\right)>10$ when changing $\varphi$ from 0 to $359^{\circ}$ at $1^{\circ}$ intervals. The results are shown in Figure 10a. The range of values of $\varphi_{\chi}$, under which the Doppler factor of sub-components reaches 10$-$20, seems to narrow with increasing $\chi$. Note that the frequent presence of sub-components with $\varphi_{\chi}=180^{\circ}$ may not be somewhat peculiar, but perhaps, for example, a consequence of the symmetric distribution of sub-components velocity vectors relative to the $\beta$. Only sub-components with $\varphi_{\chi}\approx 180^{\circ}$ may be registered as IDV events due to their high Doppler factor. Whereas sub-components with other values of $\varphi_{\chi}$ have a significantly lower Doppler factor and their radiation, although it makes some contribution to the observed radiation of the blazar, but does not lead to IDV events.

In the above discussion, we used the lower limit on the speed of main components obtained in (Butuzova, 2018a) as a speed of the sub-component $\beta_{1}$. Let us consider the variation of $\delta_{\chi}$ with changing $\beta_{1}$. For instance, we used speeds corresponding to the both $30\%$ increase and decrease in the value of the Lorentz factor for $\beta_{1}=0.999$. These values are equal to $\beta_{1\text{,s}}=0.998$ and $\beta_{1\text{,h}}=0.9994$, respectively. At $\varphi_{\chi}=180^{\circ}$ graphs of $\delta_{\chi}$ maintain their shape, only the value of peaks decreases to $\approx 30$ for $\beta_{1\text{,s}}$ and increases to $\approx 55$ for $\beta_{1\text{,h}}$ regardless of the value of $\chi$ (for $\chi<11^{\circ}$). For small values of $\chi\leq 1^{\circ}$ and, e.g., $\varphi_{\chi}=150^{\circ}$, the behavior of $\delta_{\chi}$ is to be similar to the described above. But as $\chi$ increases, the maximum value of $\delta_{\chi}$ decreases. This decrease for $\beta_{1\text{,h}}$ occurs more rapidly than for $\beta_{1\text{,s}}$ (see Fig. 11b). For $\chi\approx 5^{\circ}$, graphs of $\delta_{\chi}\left(\varphi\right)$ for various speeds are slightly different.

Thus, the observed optical radiation of the blazar S5 0716+714 is a superposition of radiation from sub-components and a main component, which have different Doppler factors and different geometrical sizes. It causes both the continuous strong variability of the object and the absence of prominent maxima on the light curve. Let us explain it in detail. Figures 8, 9, and 11 show that the main component has a maximum Doppler factor at $\varphi\approx 100^{\circ}$, while the Doppler factors of sub-components are noticeably smaller. If the case of the bulk motion of a main component without significant deviations of its parts was fulfilled, this would result in periodic high peaks on the long-term light curve with a low-brightness plateau between them, as observed, e.g., for CTA 102 (Raiteri et al., 2017; D’Ammando et al., 2019). For $\varphi\approx 0-50^{\circ}$ and $\varphi\approx 150-360^{\circ}$, the component has $\delta<10^{\circ}$. In the latter case, the high brightness of the object may be provided by the presence of sub-components with $\chi\approx 8-11^{\circ}$ and having $\delta_{\chi}>20$. Then on the long-term light curve, prominent peaks are absent, and variability is driven by the superposition of individual continuously arising and disappearing sub-components. It provides variability with a stochastic behavior noted in (Amirkhanyan, 2006; Bhatta et al., 2013). The continuous formation and variation/destruction of sub-components, which at some $\varphi$ are seen at the extremely small ($<1^{\circ}$) angle to the line of sight, leads to the facts that the object is always active and the possibility of detecting IDV event is high (e.g. Wagner et al., 1996; Hu et al., 2014; Bhatta et al., 2016; Agarwal et al., 2016; Hong et al., 2017). On the other hand, the presence of sub-components with a lower Doppler factor than that of the main component results in a decrease of the observed flux from the object. It may occur at $\varphi\approx 100^{\circ}$ when the Doppler factor of the main component is extremely high ($\delta\geq 30^{\circ}$). As a result, the observed flux from S5 0716+714 does not change as violently as could be expected only from the helical trajectory of components.

As seen from Figures 8, 9, 11, Doppler factors of both the main component and sub-components change rapidly in certain intervals of values of $\varphi$. Let us analyze whether a change of $\delta$ or $\delta_{\chi}$ caused by variation of $\varphi$ can lead to the geometrical origin of IDV. For this aim, we consider only changes in $\delta$ because, despite the fact that $\delta_{\chi}$ depends on several additional parameters, the fastest changes in $\delta$ and $\delta_{\chi}$ are comparable (Figures 8, 9).

Assuming the flux in the reference frame of emitting plasma is constant, the magnitude change is

where $\alpha$ is the spectral index of radiation characterized by the power-law spectrum $F_{\nu}=\delta^{3+\alpha}Q^{\prime}\nu^{-\alpha}$ ($Q^{\prime}$ is the proportionality coefficient of the spectral flux in the reference frame of the emitting plasma). Inserting Formula (2) into (4) and differentiating with respect to $\varphi$, we found that the fastest growth and decrease of $\delta$ occur at $\varphi\approx 84.7^{\circ}$ and $\varphi\approx 115.8^{\circ}$, respectively. From Equations (4) and (5) for $\alpha=1$ it follows that a change in $\delta$ caused by an increase in the mentioned values of $\varphi$ by $1-2^{\circ}$ leads to $|\Delta m|\approx 0.13-0.27$. These values are consistent with the observed amplitude of IDV. Then, if IDV is caused by geometrical effects, the corresponding changes in $\varphi$ should occur in a few hours. To calculate the time for which $\varphi$ changes by $1-2^{\circ}$, we use Equation (10) in (Butuzova, 2018a) assuming that the distance of the component from the active nucleus $d$ changes negligibly over $\Delta t$:

For $d=4$ pc and $\beta=0.999$ (Butuzova, 2018b) during $\Delta t=6$ hours $\Delta\varphi=0.017-0.033^{\circ}$. Therefore, the Doppler factor changes that lead to the observed variation of magnitude continue over tens of hours.

A change in the direction of $\beta_{1}$ may lead to a change in $\delta_{\chi}$ on the intra-day timescale. For example, consider $\delta_{\chi}$ for the sub-component that rotates around the local jet axis. This scenario can occur when $n>1$ modes of the Kelvin-Helmholtz instability develop. These modes produce “convexities” on the jet surface that rotate around the jet axis (see, for example, Hardee, 1982). By analogy with the non-radial motion described in (Butuzova, 2018a), a change of the angle $\theta_{\chi}$ between $\beta_{1}$ of the rotating sub-component and the line of sight is expressed by Equation 2 with the substitution of $\xi$ by $\chi$, $\varphi$ by $\varphi_{\chi}$, $p$ by $p_{\chi}$, $\theta_{0}$ by the angle between the line of sight and the component velocity vector $\theta=\theta\left(\xi,\theta_{0},\varphi\right)$ defined by equation 2. $p_{\chi}$ is the angle between motion directions of sub-components with constant and various $\varphi_{\chi}$ under the fixed $\chi$. The Doppler factors of sub-components rotating around the jet axis versus $\varphi$ are displayed in Fig. 12. Thus, we assumed that the change of $\varphi_{\chi}$ occurs, for example, 10 times faster than the change of $\varphi$ to change $\delta_{\chi}$ on the intra-day timescale. Figure 12 shows: 1) when $p_{\chi}$ increases, then peaks of $\delta_{\chi}$ shift to such values of $\varphi$ at which $\delta$ is small. 2) When $p_{\chi}$ increases from 5 to $10^{\circ}$, the values of $\delta_{\chi}$ peaks decrease for a high $\chi$ and increase for a small $\chi$. 3) When $p_{\chi}=15^{\circ}$ for all the considered $\chi$ the maximum $\delta_{\chi}\leq 10$ and $\delta_{\chi}$ decrease with a further $p_{\chi}$ growth. The described above behavior of $\delta_{\chi}\left(\varphi\right)$ leads to a follow results. The brightness of the main component having the maximum Doppler factor decreases because there are sub-components with a small $\delta_{\chi}$. While sub-components with a large $\delta_{\chi}$ increase the brightness of the object at a small $\delta$. Thus, the object brightness changes within a smaller range than that in the case of bulk motion of the component along the helical trajectory. Furthermore, the continuous presence, physical and geometrical evolution of sub-components lead to the permanent and strong variability of the blazar S5 0716+714.

As shown in Section 3.2, the sub-component, having the steady flux in the reference frame of the emitting plasma and the constant motion direction relatively to the main component trajectory, produces variability on the timescale of a few days due to the azimuth angle variation of the main component. Then the following processes can lead to IDV: (i) a physical change in the radiation flux of the sub-component; (ii) a change in the velocity of the sub-component and/or its motion direction relative to the trajectory of the main component. These factors can operate simultaneously. If the variability is formed only by the second factor, i.e., the variability is formed by geometrical effects, then there is no dependence of the color index on the object magnitude under the power-law emission spectrum. Figure 3 shows the observed dependencies of color indices B$-$V${}_{\text{int}}$, B$-$R${}_{\text{int}}$, B$-$I${}_{\text{int}}$ on the magnitude in the B band and their linear approximations. In most cases, there is chromatism. The BWB chromatism on the long timescale is interpreted in the framework of the shock-in-jet model (e.g. Marscher & Gear, 1985). Then the observed difference between the color indices of the two adjacent IDV events can be explained by the fact that these IDV events are caused by the passage of the shock front through different turbulent cells (see, e.g. Bhatta et al., 2013, 2016; Xu et al., 2019), which, as Marscher (2014) believes, can be present in the jet. We propose an alternative interpretation of the BWB chromatism in IDV.

In the radio range, most of the radiation detected by a single antenna is originated in the VLBI core (Kovalev et al., 2005). The VLBI core position shifts closer to the true jet base as the observed frequency increases (Pushkarev et al., 2012). Hence, we can expect a region upstream of the jet where the medium becomes transparent to optical radiation. By analogy with the radio range, we believe the bulk of the observed optical radiation comes from this region, which we called the optical core.

The spectrum of the optical core has the form schematically represented in Fig. 13a by the green line. The maximum flux is at the frequency $\nu^{\prime}_{\text{m}}=\nu^{\prime}_{1}\tau_{\text{m}}^{-2/(2\alpha+5)}$ (Pacholczyk, 1970), where the prime denotes values in the reference frame of the source, $\nu^{\prime}_{1}$ is the frequency for which the considered part of the jet has the optical depth $\tau=1$, $\tau_{\text{m}}$ is the optical depth at the frequency $\nu_{\text{m}}$, $\alpha$ is the spectral index of the optically thin spectral part. The overall spectrum of the blazar is formed by the total radiation of regions, in which the jet medium becomes transparent for radiation of different frequencies (Fig. 13b).

The marked in Figure 13a frequencies $\nu^{\prime}_{\text{I, 1}}$ and $\nu^{\prime}_{\text{B, 1}}$ correspond to the observed effective frequencies of I${}_{\text{c}}$ and B bands before a beginning of IDV flare caused by a sudden change in the Doppler factor of the sub-component $\delta_{\chi}$. At the maximum of the flare, $\delta_{\chi}$ reaches the highest value that is $\delta_{\text{SC}}$. As $\nu=\nu^{\prime}\delta_{\text{SC}}$, then the radiation from the sub-component observed in the maximum has the lowest frequencies in the source reference frame. Frequencies $\nu^{\prime}_{\text{I, 2}}$ and $\nu^{\prime}_{\text{B, 2}}$ schematically represented in Figure 13a correspond to the observed effective frequencies of the I${}_{\text{c}}$ and B bands for the highest $\delta_{\text{SC}}$. In other words, the frequencies $\nu^{\prime}$, between which the spectral index is determined when the radiation is approximated by the power-law, decrease with an increase of $\delta_{\chi}$. This fact and the concave spectral shape will lead to the spectral index decrease with the brightening, i.e., to the BWB chromatism.

Let us examine our assumption about the origin of the BWB chromatism in IDV. To this aim, consider the IDV flare that occurred on JD 2457130 (Fig. 1). For this flare, a strong BWB trend was detected, and there are quite a large number of data points (Fig. 3). Using the calibration of (Mead et al., 1990), which associates the magnitude with the flux, the difference in the color index in the bright and low states, for example, between bands B and I${}_{\text{c}}$, is expressed as:

where $m$ is the magnitude in the optical band specified in the index, $F$ is the basic spectral flux (index “1”) and the flux from the sub-component (index “SC”) in the corresponding optical bands. We took the flux corresponding to the minimum brightness of the object for this date (see Table 3) as an underlying basic flux. When getting expression 7 we assumed the peak flux to consist of the sum of $F_{1}$ and $F_{\text{SC}}$ fluxes. The total emission spectrum, which is the sum of radiation from different regions of the inhomogeneous jet, is well described by the power-law $F_{1}=Q\nu^{-\alpha}$. In the observer’s reference frame, the spectrum of the sub-component, localized in the region of the optical core, has the form (Pacholczyk, 1970):

where $Q_{\text{SC}}$ is the proportionality coefficient, $\nu^{\prime}_{1}$ is the frequency in the comoving to the optical core reference frame, $\alpha$ is the spectral index of the optically thin part of the optical core spectrum. The last parameter cannot be determined from observations, so we assume it equal to the spectral index of the total observed radiation. Inserting $F_{1}$ and $F_{\text{SC}}$ into Formula (7) we obtained:

where $C=10^{0.4\Delta\left(m_{\text{B}}-m_{\text{I}}\right)}$,

$b\left(\nu^{\prime}_{1}\right)$ is expressed similarly to $a\left(\nu^{\prime}_{1}\right)$ with the substitution of the effective frequency of the second optical band from the considered ones, in this case it is I${}_{\text{c}}$. The value of the color index change during variability is determined from the linear approximation of the observed points with the substitution of magnitudes in the lowest and brightest states (see Table 3). The dependence of $Q_{\text{SC}}/Q$ on $\nu^{\prime}_{1}$ given by expression (9) with different values of $\delta_{\text{SC}}$ for different pairs of optical bands is shown in Fig. 14. The area of intersection of curves plotted for different pairs of bands for each $\delta_{\text{SC}}$ allows us to estimate $\nu^{\prime}_{1}$. The present dispersion of curve intersection points is caused by both the errors in magnitude measurements and the fact that, in neighboring optical bands, the correlation between the color index and the magnitude is weaker than comparing optical bands with a large frequency interval between them (see Table 3). From Figure 14 it can be seen that for $\delta_{\text{SC}}=15$ and for Doppler factor of component $\delta=5$, $\nu^{\prime}_{1}$ corresponds to effective frequency of the filter I${}_{\text{c}}$. The higher sub-component Doppler factor, the lower $\nu^{\prime}_{1}$. And $\nu^{\prime}_{1}$ for $\delta_{\text{SC}}=35$ in the observer’s reference frame is smaller than $\nu_{\text{I}}$. Taking into account the fact that this IDV flare occurs during high state of the blazar (see Fig. 5), the high Doppler factor of sub-component, producing the observed flare, is expected. Thus, there is an agreement between our assumption of the BWB-chromatism and the obtained $\nu^{\prime}_{1}=\left(2-4\right)\cdot 10^{13}$ Hz. For $\delta=5$ and $\alpha=1.12$ it follows (see, e.g., Pacholczyk, 1970) that the frequency of the spectrum maximum in the observer’s reference frame is $\nu_{\text{m}}=\left(0.9-1.7\right)\cdot 10^{14}$ Hz. The frequency $\nu_{\text{m}}$ estimated based on the observed data is less than the $\nu_{\text{I}}=3.7\cdot 10^{14}$ Hz. It corresponds to our assumption that the observed optical radiation comes from the region where the jet medium becomes optically transparent for the considered frequencies.

Thus, with an increase of $\delta_{\text{SC}}$, frequencies in the source reference frame, which correspond to the observed frequencies, decrease. Different Doppler factors of sub-components cause the frequency shift in the source reference frame to occur by a different value. The sub-component with a relatively small Doppler factor and a larger occupied volume will result in an IDV flare with achromatic behavior. Meanwhile, a high $\delta_{\text{SC}}$ and a smaller volume of the sub-component cause a flare with a similar amplitude with the BWB behavior. The registered by Wang et al. (2019) redder-when-brighter trend can also be explained by the fact that all data for the particular date are used in (Wang et al., 2019) to plot the dependence of the color index on the magnitude. But these points can correspond to various features (a monotonous growth or decline of brightness, flares) such as on JD 2458139. In Section 3.4, we have shown by the example of our data that adjacent IDV events can be characterized by different color behavior in variability.

Similarly to the IDV flare, we consider a long-term BWB trend. In this case, the variability is formed by not one but many sub-components having different Doppler factors. We assume that the observed lowest state of the object corresponds to the level of the underlying flux. An increase in the object’s brightness is due to the formation of sub-components with higher Doppler factors. When the object has the highest brightness, its emission is the sum of radiation of the underlying flux and the flux of the brightest sub-component having the highest brightness due to the highest value of the Doppler factor $\delta_{\text{SC}}$. In the same way, as in the previous case, we use linear approximations of the dependencies of the color indices on the object’s magnitude to find a change of the color index in variability (Table 3). The $Q/Q_{\text{SC}}$ dependencies on $\nu^{\prime}_{1}$ obtained for different $\delta_{\text{SC}}$ are shown in Figure 15.

The same as in previous case, the large $\delta_{\text{SC}}$, the lower $\nu^{\prime}_{1}$. It seems unlikely that the sub-component with $\delta_{\text{SC}}=15$ is responsible for the observed variability with an amplitude of about 2.5 magnitudes. In the cases of $\delta_{\text{SC}}=25$ and 35, $\nu^{\prime}_{1}=\left(1.5-4\right)\cdot 10^{13}$ Hz, which at $\delta=5$ and $\alpha=1.46$ gives the frequency of the spectrum maximum in the observer’s reference frame $\nu_{\text{m}}=\left(0.6-1.6\right)\cdot 10^{14}$ Hz. The values of $\nu^{\prime}_{1}$ and $\nu_{\text{m}}$ obtained from the long-term variability are in good agreement with the same values obtained from the particular IDV flare. This can only be explained by the action of the common variability mechanism on both the long and short timescales, namely the scenario under consideration.

Our assumption about the origin of IDV can be confirmed by the fact that different features on the intra-day light curve have different dependencies of the color index on the magnitude. This possibility was mentioned by Amirkhanyan (2006); Wang et al. (2019), but was not investigated in detail. For example, two adjacent flares of approximately the same amplitude and duration were observed on JD 2456785 (Fig. 1). The light curve and dependence of the color index on the magnitude are shown in Figure 16. The color index obtained from all data for the night shows a weak correlation with the object brightness, the Pearson correlation coefficient $r\approx 0.5$. Meanwhile, there is a stronger correlation between the color index and the magnitude under separate consideration of these two flares. This observational result can be interpreted by the fact that if the flares are formed by a changing Doppler factor of the corresponding sub-components, the values of the maximum Doppler factors of sub-components differ. The difference in the value of $\delta_{\text{SC}}$ with the concave optical spectrum of the region from which the observed radiation comes may lead to various changes in the color index of the object in variability.

For flares that occurred on JD 2457074, the correlation coefficient between the color index and the magnitude is high Fig. 16. But separately for the first flare $r\approx 0.1$ for B$-$V${}_{\text{c}}$ and B$-$R${}_{\text{c}}$, and slightly higher ($r\approx 0.4$) for B$-$I${}_{\text{c}}$. Whereas for the second flare $r$ reaches 0.87 for B$-$I${}_{\text{c}}$. The absence of BWB for the first flare can be explained by the fact that either the flare is a superposition of several flares with different color index, or the flare is formed by a relatively small increase in the Doppler factor of the sub-component, which does not change the color index under the concave optical spectrum.

In JD 2457129, a decrease in brightness at a rate of 0.096 mag h^-1 replaced an increase in brightness at about the same rate. For all the data, the correlation between the color index and brightness is strong ($r\approx 0.8$). Under separate consideration, a strong correlation is maintained for the brightness decline and becomes less ($r\approx 0.5$) during an increase in brightness.

If the turn-over frequency of the synchrotron spectrum and the source size are known, it is possible to estimate the transverse to the line of sight component of a magnetic field $B_{\perp}$ (Slish, 1963). The well-known formula for $B_{\perp}$ (see, e.g., (Marscher, 1983) and detailed derivation in (Pushkarev et al., 2019)) uses the observed angular size of the source, which is unknown in our case. We assume that the size of the optical radiating region in the reference frame, comoving with the relativistically moving plasma, is comparable to the gravitational radius of a central black hole $R^{\prime}=R_{g}$. Using relativistic transformations (see Appendix C in (Begelman et al., 1984)) and cosmological reduction of the observed spectral flux, we obtained

where $d\Omega^{\prime}=\pi R^{\prime\,2}/(4D^{2}_{L})$ is the solid angle of the source in the comoving with relativistically moving plasma reference frame, $D_{L}$ is the luminosity distance of the object, the special intensity is (Pacholczyk, 1970)

where $c_{1}$, $c_{5}$, $c_{6}$ are the constant and functions tabulated in (Pacholczyk, 1970), $\tau_{\nu}=\left(\nu^{\prime}/\nu^{\prime}_{1}\right)^{-\alpha-5/2}$ is the Lorentz invariant optical depth at the frequency $\nu$, $\alpha$ is a spectral index of optically thin part of the spectrum. Combining the expressions (10), (11) and taking into account $\nu^{\prime}=\nu\left(1+z\right)/\delta$, we obtained the perpendicular to the line of sight component of the magnetic field in the source reference frame

Using in formula (12) the power-law approximation of the observed fluxes in different filters for the effective frequency, e.g., of filter I${}_{\text{c}}$, for IDV flare (JD 2457130) and long-term variability, we plotted the magnetic field versus $\nu^{\prime}_{1}$ (Fig. 17). It can be seen that for the size of the radiating region, which is equal to the gravitational radius of a black hole with a mass of $5\cdot 10^{8}$ solar masses (as expected for blazars in (Sbarrato et al., 2012; Titarchuk & Seifina, 2017) and references therein), the magnetic field strength is within the values estimated by other authors from independent models (e.g., Field & Rogers, 1993; Mondal & Mukhopadhyay, 2019, and references therein).