1. Introduction
Blazar OJ 287 is the first active galactic nucleus in the center of which it was assumed a system of two supermassive black holes. This assumption was made by [
1] to interpret the 100-year optical light curve of OJ 287, which shows powerful flares that repeat approximately every 12 years. These flares were explained by tidal interaction of the secondary black hole with an accretion disc of the primary one at their maximum approach. The idea of the binary black hole in the center of OJ 287 was further developed in [
2,
3,
4,
5,
6], which suggested that the accretion disc of the primary is not in the orbital plane of the secondary component. In these models, the orbit of the secondary black hole is precessing, thus allowing to explain the observed difference in time intervals between the 12-year flares, which is about a year.
The assumption about the binary black hole system in the center of OJ 287 was made before the unified scheme of the active galactic nuclei (AGN) was proposed [
7], which later became widely used in the interpretation of the properties of active nuclei. According to this scheme, blazars are a class of active galactic nuclei with relativistic jets directed close to the line of sight. The blazar OJ 287 shows strong variability in the entire observed spectral range, a high polarization degree of optical (see, e.g., [
8] and references therein) and radio emission [
9]. Therefore, we suppose that the observed optical emission is synchrotron and comes from the parsec-scale jet. The optical emission of the accretion disc is low enough to be distinguished in the total spectrum of blazars, as it was recently proven [
10,
11]. Due to relativistic effects, the radiation formed in the approaching blazar jet increases in the observer’s reference frame and de-boosts the emission from the receding jet. The relativistic boosting depends on the speed of emitting plasma and the angle of the motion with respect to the line of sight. Thus, the periodicity in the light curve can be created by the motion of the radiating plasma along a helical trajectory, as discussed in [
12]. This scenario was applied to OJ 287 [
13]. Namely, authors assumed the presence of two approaching relativistic jets, twisted together. These jets are formed in a system of two black holes of the same mass. Flares with a two-peaked profile occur at times when the first jet and later the second jet are at a very small viewing angle [
13]. An alternative explanation is that a secondary black hole passes through the accretion disc of the primary component twice during the orbital period [
2]. Taking into account the relativistic amplification of the radiation generated in the jet, we think that models explaining the 12-year optical flares by a helical jet are more likely.
Recently, Villforth et al. (2010) [
14] argued the existence of a single supermassive black hole in the centre of OJ 287 based on the analysis of optical photo-polarimetric observed data for the 12-year flares. Under this assumption, the 12-year flares arose from avalanche-like accretion of the magnetic field. Britzen et al. (2018) [
15] had shown that the observed period of radio flux variability is about 25 years and that the changes of the inner jet position angle can be explained by the jet precession, which can be a result of either the orbital motion in the binary black hole system or the precession of the outer part of the accretion disc of the single supermassive black hole. However, there is no explanation for the difference of the variability periods observed in the radio and optical ranges.
Direct observations of parse-scale jets of active galactic nuclei can be made by very large baseline interferometry (VLBI). For most sources, including OJ 287, VLBI maps show a typical morphology manifesting one-sided jet, the apparent origin of which is the brightest compact feature, called the VLBI core. Due to energy losses on synchrotron radiation and adiabatic expansion, the jet quickly dims and becomes undetectable at a distance of few milliarcseconds from the core. As a result of the analysis of polarimetric observations of two hundred active galactic nuclei performed with the very long baseline array quasi-simultaneously at frequencies from 8.1 to 15.4 GHz, it was showed that the VLBI core observed at a certain frequency is a part of a jet in which the medium becomes optically thick for radiation at this frequency due to opacity caused by synchrotron self-absorption [
16,
17]. Due to the fact that the magnetic field strength and density of radiating particles decrease with the distance from the true jet base according to the power-law dependence [
18,
19], the lower the frequency of observations, the further from the true jet base the absolute position of the VLBI core. This implies that the observed optical radiation comes from a region close to the true jet base. Therefore, if we associate the 12-year optical flares with a cyclic change in the angle between the velocity vector of the radiating region and the line of sight, we can expect similar periodicity in the data of radio observations. A comparison of the flux density of AGN measured with VLBI and single-dish observations showed that VLBI core emission significantly contributes and often dominates in a total source radiation [
20]. Hence, in measurements of the total radio flux density from the entire source, the periodicity cannot be strongly obscured by radiation from the optically thin part of the jet. Statistical analysis performed in [
21,
22] based on data from the Owens Valley Radio Observatory (OVRO, 2008–2018) and Radio Astronomy Observatory of Michigan University (UMRAO, 1974–2011) shows that the period of long-term radio variability is about 25 years. This period washed out the period of about 6 years, obtained previously using shorter data set [
23]. We also note that the periods obtained from the total radio flux density (≈25 years [
21,
22]) and from the motion of the jet ridgeline (≈22 years [
15]) are consistent. The periodicity of ≈1 year is found in both optical and radio domains [
15,
22]. We do not consider this short periodicity here, though it might reflect the rotational motion of the plasma in the jet flow [
24] or could be caused by nutation-like motion in the precessing jet [
3,
15]. We believe that the issue of the absence of a prominent variability in the radio range with a period clearly seen in the optical domain is fundamental for understanding both the processes operating in OJ 287 and for determining the properties of the central object.
The layout of this paper is the following. In
Section 2, we derive geometrical and kinematic parameters of the OJ 287 jet assuming that it has a helical shape and using 15 GHz VLBA observations since 1994 through 2019, performed in the framework of the Monitoring Of Jets in Active galactic nuclei with VLBA Experiments (MOJAVE
1) project; we also provide interpretation of the difference between the variability periods in the radio and optical ranges. In
Section 3, we assume that different values of the maximum flux density of the 12-year flares can be caused by precession of the helical jet. We showed that the period of this precession is consistent with the Lense–Thirring precession of a single supermassive black hole or the inner part of its accretion disc. Discussion of the obtained results and conclusions are presented in
Section 4.
3. Helical Jet Precession
If the 12-year flares are formed only due to the increase in
, then, according to Formula (
2), the maximum value of
determines the flux peak value. Inserting Formula (
5) into (
3), we found that the function
has extremum at
For and , the Doppler factor has minimum and maximum , respectively. Hence, for constants p and , the values of and are invariable and occur at a fixed and , correspondingly. If the flux density in the reference frame of the jet is constant, the flux peaks in the observer’s reference frame must be equal as well.
Since
at the maximum of the flux density reaches a fixed value, the difference in peak fluxes of 12-year flares can be caused either by an internal change in the jet or by a change in the angle of the cone axis with the line of sight. In the first case, the ratio of peak fluxes at the maximum Doppler factor (e.g.,
for the flares of 1983 and 1995), would have to be maintained at other values of
, e.g., at
. The observed minimum flux density
between the flares differs only slightly [
14]. Therefore, in order to agree with the observed long-term light curve, the increase in the flux density in the reference frame of the jet would have to occur at a time when the Doppler factor for an earth’s observer would be close to the maximum value. There is no physical ground for this condition.
In our view, the most possible explanation for the observed difference in the peak fluxes of the 12-year flares is a change in the angle , which, for example, may be associated with precession.
Let us check this assumption. The scheme of the helical jet precession is presented in
Figure 8. The dependence of
on the azimuth angle of precession
is expressed as
where
is the half-opening angle of the precession cone and
is the angle of the precession axis with the line of sight.
As it follows from Equation (
11), the azimuth angle values, for which the Doppler factor achieves the maximum and minimum values which do not depend on
. Assuming that the flux density in the reference frame of the emitting plasma is constant, we obtained from Equation (
2)
Inserting into Equation (
3) the expression for
(
5), in which
and
are given by Formulae (
12) and (
11), correspondingly, and solving Equation (
13) with respect to
, substituting the observed flux densities of the 12-year peaks and minima between them, we obtained
where
We have taken into account that
for the range of azimuth angle from 0 to
. The parameters used in Formula (
14) and derived viewing angle
values for each peak are listed in
Table 3. Fitting the
values by the expression (
12), we obtained
,
and the precession period in the observer’s reference frame
years. The dependence of
with time is shown in
Figure 9.
In order to find the precession period
in the reference frame of the source, it is necessary to take into account not only the cosmological time change but also that the time interval in the source reference frame is
. Since
changes over a wide range (see
Figure 5), we have divided the whole precession period in the source reference frame into equal time intervals. Assuming that the changes of
in the jet reference frame occur uniformly and
changes are 7 times faster than
, we took the time for which
changes by
as a time interval unit. Then,
where
is the Doppler factor at the end of the time interval. From (
15), we obtained
years.
Thus, by substituting
from (
12) into Formula (
5), we can obtain an analytical expression to describe the change in the Doppler factor of the radiating region in the jet. Assuming that the 12-year flares are caused by a change in
, we modeled the light curve. For this aim, we took the 1983 flare as a reference point. From Formula (
11), we found the azimuth angle of the optical radiating region
. Using it together the data from
Table 3, we found the azimuth angle for the precession motion
by Formula (
12). The Doppler factor at a given time moment depends on
and
. The rate of
change was determined from
, and it is approximately
for 5 years in the source reference frame.
Figure 9 shows that
changes about 7 times faster than
. The flux density was found by Formula (
2) and normalized to the flux density of the 1983 flare.
Figure 10 shows the simulated light curve and observational data from the literature and the archive of the Crimean Astrophysical Observatory. As seen, since 1940, the observational data are in a good agreement with the simulated flux density. Some difference between the observed and modeled flux densities at the maxima may be caused by a small physical change in the flux density in the source reference frame compared to that of the 1983 flare maximum. Here, we do not model the radio light curve because it has no prominent maxima as in the optical range. There is no reliable reference point for which the values
and
can be confidently found. Since the optical and radio radiating regions are spatially separated, the azimuth angles of the radiating regions are different within the framework of the considered model.
Let us check whether the Lense–Thirring precession with such a period can occur in the accretion system of the single supermassive black hole. In this system, the precession of the black hole together with the inner part of the accretion disc can be originated in result of misalignment of the black hole and accretion disc outer part rotation axes, as described in [
43]. This precession period can be expressed as
where
is the dimensionless parameter expressing a black hole’s specific angular momentum,
is the viscosity parameter, and
and
are the mass and mass accretion rate of the black hole, respectively. We expressed the Formula (
16) as
where
is the mass accretion rate in Eddington units. Using the obtained period
in Formula (
17), we plotted a dependence of
for different masses and mass accretion rates of the black hole (
Figure 11). The range of possible values for the black hole mass is large enough. According to the broad-band observations, black hole masses for BL Lacertae objects were estimated as ∼(10
8–10
9) M
⊙ ([
44,
45], and references therein). Valtonen et al. [
42] derived the primary black hole mass of
under the assumption of the binary black hole system in the OJ 287 centre. This mass estimation is model-dependent, but we took it as an upper limit. The mass accretion rate we accepted is in the range
= 0.001–0.1, according to the values reported in [
46,
47]. As seen, the dependence on the black hole mass is weak. The precession of a slowly rotating black hole with a high mass accretion rate and a low
(as confirmed by observations [
48]) can occur with the found period.
On the other hand, it is possible that misalignment of the black hole and accretion disc rotation axes can cause precession of the outer part of the accretion disc with a period [
49]:
where
r is a radial distance of the accretion disc precessing part from the black hole and
is the gravitational radius of the black hole. From Formula (
18), it follows that, for
0.01–1, the regions of the accretion disc precessing with the found period are located at distances of (30–300)
from the black hole with
(10
8–10
10) M
⊙. These distances seem to be reasonable to form the jet by the Blandford–Znajek mechanism [
50].
Note that, here, we investigate the possibility of the formation of the 1200 year period in the accreting system of a single black hole. More precise determination of the parameters of this system, such as the mass, mass accretion rate, and viscosity, should be based on the broad-band photometric and VLBI data.
4. Discussion and Conclusions
OJ 287 is the first blazar for which periodicity in the optical light curve was detected, revealing prominent flares that occur approximately every 12 years [
1]. At that time, the unified scheme of active galactic nuclei [
7] had not yet been proposed. Therefore, early investigations of OJ 287 were predominantly focused on the interpretation of optical periodicity. It was supposed that such a short variability period can be formed in a binary black hole system [
1]. The secondary component moves along its elliptical orbit and passes the pericenter every 9 years (in the rest frame), invoking a tidal effect on the accretion disc of the primary black hole. As a result, the matter of the accretion disc flows into the black holes, leading to the observed flare on the light curve [
1]. This model does not take into account the structure of the 12-year flares, namely, that there are two peaks in each flare. The interpretation of this fact given in [
2] is that the secondary black hole passes through the accretion disc of the primary one twice during the orbital period. This model also provided the explanation for some of the observed properties of optical polarization, but it was focused on orbital motion. After the next 12-year flare occurred during 1995–1996, this model has been refined [
4,
5,
6]. The authors of [
6] predicted future evolution of the VLBI jet position angle observed at 15 and 43 GHz, but it has not been confirmed by later observations (here and [
28,
36]).
An alternative interpretation of the presence of two peaks in a 12-year flare was proposed in [
13] considering two supermassive black holes of the same mass, with accretion discs and jets. These jets have a helical shape and are twisted together. Doppler beaming of a part of each of two jets, responsible for optical emission, can lead to the 2-peaked 12-year flares.
As it is noted in [
1], due to the radiation of gravitational waves, a separation between the pericenters of the primary and secondary black holes is expected to decrease, leading to an increase in the intensity of next 12-year flares. The flares observed in 1995–1997 and 2005–2007, on the contrary, had significantly less maximum flux density (≈10–15 mJy in V band) than the previous flares in 1973 and 1984 (55 and 30 mJy in V band, respectively) [
14]. Interpretation of the decrease in the maximum flux density of the 12-year flares as well as prediction of the absence of the flares in the future is proposed in [
14]. These are explained by something, that occurred before the ≈1970 event, causing an avalanche-like accretion of the disc magnetic field, which repeats every ≈12 years [
14]. After each act of this accretion, the poloidal magnetic field near the black hole becomes stronger and more strongly hinders accretion of the poloidal magnetic field next to an act of accretion. This model requires further verification by magnetohydrodynamic simulations. The maximum flux density of the last 12-year flare occurred in 2016 was 17 mJy in B band [
42]. It is slightly higher than the ones for two previous flares (
Table 3).
None of the above models can consistently explain the main observable facts: (i) the sudden change in PA
in which was first detected at 43 GHz in 2004 [
28,
36] and about six years later at 15 GHz; (ii) the difference of the long-term variability period in optical (≈12 yr [
1]) and radio (≈25 yr [
21,
22,
23]) flux densities.
In this paper, we explain most of the observed long-term properties of the blazar OJ 287, while other properties can also be interpreted within the framework of the proposed model. First of all, we assume that the OJ 287 jet has a helical shape because this shape can be naturally originated by the jet nozzle precession or development of hydrodynamic instabilities. For the certain relation between the angle of the helix axis to the line of sight and the half-opening angle of the cone, the sudden change in the inner jet position angle occurs under the outward motion of the helical jet
. As it was shown in [
26], for the permanent geometrical parameters of the jet helix, the sudden change in PA
in occurs at a certain fixed azimuth angle. For the case of OJ 287, PA
in has a minimum and maximum value at
of about
and
, correspondingly. Therefore, the fact that the sudden change in PA
in occurred at a higher frequency first and then at a lower one indicates that the 43 GHz emitting region reached this value of
earlier than the 15 GHz emitting region. This means that the regions responsible for dominant radiation at different frequencies are spatially separated. This fact follows from the jet model [
18] and has been confirmed by VLBI observations for a large number of AGN jets [
17]. The higher the observed frequency, the closer to the true jet base is the location of the region, in which the observed radiation is generated. Hence, the region from which the observed optical radiation comes is located much closer to the true jet base. According to this, the changes in the flux density in the optical and radio ranges, caused by the geometric effects, do not coincide in time, since the Doppler factor of the emitting regions differs. The absence of sharp and prominent flares in radio similar to those present in optical range can be explained by a significantly larger size of the radio-emitting region compared to that in optics. This difference in the size of the regions can naturally be caused by different cooling time scales of ultrarelativistic electrons producing synchrotron radiation in the radio and optical ranges.
Within the framework of the described jet model, the difference between the periods of long-term optical and radio variability occurs when the jet components move along non-radial trajectories. Such motion is observed [
28,
36] and most simply explained by the development of hydrodynamic instabilities in the jet, for example, Kelvin–Helmholtz instability [
37]. Moreover, we have shown that the derived parameters of the helical jet can also explain the observed variability periods. These parameters are attributed to the jet, along which the helical mode of the Kelvin–Helmholtz instability is developed [
37].
In the framework of our model, the Doppler factor changes with distance from the true jet base, and for each jet component, it varies with time. Thus, for each jet component, the equal time intervals in the observer’s reference frame are different. Also equal time intervals in the observer’s reference frame for simultaneously observed jet features are different. These facts lead to different apparent speeds for different jet features, as it is observed for many AGN jets including OJ 287 [
33].
With the geometrical and kinematic parameters of the OJ 287 jet unchanged for decades and under the interpretation of the 12-year optical flares by geometry effects, the difference in peak fluxes in these flares may be due to the fact that the helical jet axis precesses. This precession in the our geometrical model of the observed optical flux density variations can additionally explain different flux densities between the different 12-year flares and different time interval in the observer’s reference frame between the 12-year flares. We found that the jet precession period in the source reference frame is ∼1200 years and showed that jet precession can be formed by the Lense–Thirring precession of a single supermassive slowly rotating black hole with a high accretion rate and a small viscosity parameter or by precession of the accretion disc at distances of 60–300 gravitational radii from a single supermassive black hole. The observed properties of OJ 287 are explained in the assumption of a single supermassive black hole in the center of the active nucleus, an important result for interpretaion of the blazar long-term flux periodicity. The latter one is often explained by the binary black hole system in AGN, by analogy with OJ 287.
The interpretation of the double-peaked structure of the 12-year optical flares is beyond the scope of this article. However, the discussed above possible explanations for this observed fact do not contradict the proposed model. Namely, the double-peaked flares can occur if some parts of the jet move at a small angle to the general helical trajectory [
24], or there are two jet regions radiating in the optical range, located at slightly different distances from the true jet base. The Doppler factor of the more distant region changes as well as the Doppler factor of the region closest to the true jet base, only with a certain time delay. Indications for the existence of extended optical jets have been recently found [
10,
11,
51]. With our model of the helical jet, we can make the following predictions for the blazar OJ 287: (i) an increase in the maximum flux density in the next three-four 12-year optical flares; (ii) a gradual decrease in the position angle of the observed at 15 GHz inner jet until
, when again there will be a sudden increase in the position angle (it will happen a few years earlier at 43 GHz).