1. Introduction
Among active galactic nuclei (AGN), blazars are objects that emit variable non-thermal radiation throughout the electromagnetic (EM) spectrum [
1] with their jets pointing at an angle of no more than ∼
from the observer’s line of sight [
2]. These extragalactic sources present total luminosities in the range of
–
(see, e.g., [
3]).
The light curve variabilities in blazars are commonly classified as follows: (a) intraday variability (IDV), corresponding to periods of over a day or less [
4]—they are also called intra-night variability or micro-variability [
5]; (b) short-term variability (STV), which corresponds to variability over days to several weeks; and (c) long-term variability (LTV) that takes place on timescales of months to years [
6].
These variabilities have many explanations that are related to whether the source is jet-dominated or not. In the first case, Marscher and Travis [
7] suggest that the variability in compact jets is justified because of the non-thermal emission in blazars. Furthermore, variability studies in the radio and optical wave bands by Camenzind and Krockenberger [
8] conclude that shock waves in relativistic collimated flows are responsible for the observations. This idea was further studied by Mohan and Mangalam [
9], proposing a general relativistic model of jet variability in AGN, incorporating orbiting blobs in a helical motion along a magnetic surface near the black hole. In this direction, de la Cruz-Hernández and Mendoza [
10] interpreted that shock wave emissions inside the jet are caused by a periodic injection velocity of the flow at the base of the jet.
For non-jet-dominated sources, McHardy et al. [
11] suggest that the differences in lag between different bands indicate that the variability is produced by the disk. Also, Edelson et al. [
12]’s account of rapid luminosity changes that indicates emission regions confined to the inner disk or corona. In this sense, Uttley et al. [
13] suggest that the variability is due to processes and accretion rate variations. Also, Stella and Vietri [
14] explain that the variability can be attributed to the relativistic dragging of inertial frames around a rapidly rotating disk.
A famous example of a non-jet-dominated source is represented by the quasar OJ 287, which was extensively studied by Lehto and Valtonen [
15], who found sharp flares within major outbursts of the optical light curve. The authors proposed a model in which a smaller black hole crosses the accretion disk of a larger black hole during its binary orbit. An extensive analysis of its optical light curve was used to infer this supermassive black hole binary system [
16]. The periodicity of this blazar was discovered by analyzing its historical optical light curve, which contains data from more than 100 years, showing repeated bursts at intervals of about
. The best-known model of this periodicity was constructed by Lehto and Valtonen [
15], and it consists of a primary black hole—the central engine, with a mass of ∼
, surrounded by an accretion disk and a secondary black hole with a mass of ∼
, orbiting the primary and intersecting the accretion disk on each orbit, causing tidally induced mass fluxes from the accretion disk to the primary black hole.
Of particular relevance to the studies carried out in the present article is the case for postulating the existence of a secondary supermassive black hole orbiting a primary one. The first proposal of this kind was made by Begelman et al. [
17] in order to account for periodic or quasi-periodic oscillations.
An extensive survey to find periodic light curves in optical light was carried out by Graham et al. [
18]. Of the
studied light curves, it was found that the one corresponding to PG 1302-102 shows a periodicity of
. The authors assumed that this was due to the existence of a secondary black hole “eclipsing” the primary one, and they concluded that the system is separated by less than a parsec. More recently, Tavani et al. [
19] found 2.2 yr QPOs in the
-rays band of the blazar PG 1553+113 and once again proposed it as a supermassive black hole binary system.
Relevant periodicities of other AGN appear in the literature. Quite important to mention is the work by Li et al. [
20], who report long-term variability of ∼14 yr in the optical continuum of the nucleus of NGC 5548. For this same object, Bon et al. [
21] found a periodicity ∼43 yr. Also, Li et al. [
22] studied a possible ∼20 yr periodicity in long-term optical photometric and spectral variations of the nearby radio-quiet Active Galactic Nucleus Ark 120, and Chen et al. [
23] published a sample of quasar candidates with periodic variations from the Zwicky Transient Facility.
A very interesting and quite well studied blazar is Markarian 501 (Mrk 501). It is a BL Lac object with several periodicities reported in the literature: (1) A periodicity of
was reported during a flare detected in X-rays and
-rays and modeled as a supermassive binary black hole (see, e.g., [
24] and references therein). (2) In the same frequencies, a periodicity of
was found by Rödig et al. [
25]. (3) A periodicity of
was discovered by Wang et al. [
26] in X-rays. (4) Finally, Bhatta [
27] found a
periodicity in the
Fermi-LAT
-rays light curve. All of these reported periodicities are not achromatic, and they represent different databases in time. Although interesting, they are not relevant to the study presented in this article, which used long-term databases in four different frequencies.
Mrk 501 has a redshift of
(∼
∼
) with R.A.
, Dec. =
. It was discovered by Quinn et al. [
28] using the Whipple Imaging Atmospheric Cherenkov Telescope (IACT). It has been monitored since 1996 in various frequencies: radio [
29], optical [
30], and
-rays [
31,
32].
In this article, we report a mean periodicity of
in multi-frequency observations of Mrk 501. The dataset in radio was obtained from the Owens Valley Radio Observatory (OVRO) (
https://sites.astro.caltech.edu/ovroblazars/, accessed on 27 June 2020), the optical dataset from the American Association of Variable Star Observers (AAVSO) (
https://www.aavso.org/, accessed on 12 September 2021), the X-rays dataset is from the Neil Gehrels Swift Observatory (Swift) (
https://swift.gsfc.nasa.gov/, accessed on 8 October 2021) and for
-rays, the data were taken from the
Fermi Gamma-Rays Space Telescope (FGST, also FGRST) (
https://fermi.gsfc.nasa.gov/, accessed on 13 March 2020). These datasets and their corresponding processing (reduction) is explained in
Section 2. In
Section 3, we describe different methods to find this multifrequency periodicity, and in
Section 3.4, we model this periodic behavior, assuming a supermassive binary black hole using Jacobi elliptical functions, which offer good representations of eclipses that produce occultations (as they occur for binary stars or exoplanets eclipsing their central star) and magnifications (such as the ones that are produced by massive relativistic objects that bend light and magnify its intensity). Finally, in
Section 4, we discuss our results.
2. Data and Light Curves
The radio dataset from 22 January 2009 to 27 June 2020 was obtained from the OVRO database. OVRO consists of a
telescope with a cryogenic receiver at a central
frequency, a
bandwidth and two symmetric off-axis beams. This observatory has been monitoring blazars since 2008 [
29], and one of its main targets is the search for QPOs and correlations between radio and
-rays in blazars [
33,
34]. The signal-to-noise level reported by the OVRO database is such that it produces a systematic flux uncertainty of about 5%.
The optical AAVSO database is public, and it offers long-term datasets. The institution is an international organization of variable star observers who participate in scientific discovery through variable-star astronomy. It was founded in 1910, and its observations of variable stars are collected and archived for worldwide access in collaboration with amateur and professional astronomers. Observations with errors of ≥1.0 magnitudes are rejected by the AAVSO community. An error of 1.0 magnitude represents a signal-to-noise ratio of 1, making it statistically insignificant. The light curve of Mrk 501 was built using the database from 24 June 1998 to 12 September 2021.
The optical and radio data reductions were processed via AAVSO and OVRO, respectively. The data were obtained by consulting the public databases of both observatories. For details of the reduction, calibration, correlation processes, etc., of the radio database, see Richards et al. [
29], and for AAVSO, see Kinne [
35].
For the X-ray light curve, we used the Swift database from 2 October 2008 to 8 October 2021. This dataset contains energies in the range of 0.3–10 keV. Swift has an X-ray telescope (XRT) with two important characteristics that makes it important for observations: a low background and a constant point spread function across the field of view [
36].
The bin size used for obtaining the data was the default time of 5 s, and the command line interface used was xselect, which works in conjunction with the Fermitools software version v11r5p3, specifically designed for astrophysical X-ray analysis. It offers convenient functions to organize input data using the observation catalog, applies various filters to the event data (such as intensity, phase, region, etc.) and creates good time intervals based on a chosen selection criteria. It serves as a valuable tool to work with X-ray data, providing an efficient and customizable way to manage and analyze astrophysical data.
In addition to its major functions, xselect also provides three commands that correspond to different time systems: universal time (UT), modified Julian days (MJD), and the SpaceCraft Clock. We constructed the X-ray light curve using MJD.
The
Fermi public database of
-rays has fluxes in the range of
–
. For Mrk 501, it covers a time interval from 4 August 2008 up to 13 March 2020. The analysis was performed using the public Fermi/LAT data corresponding to the P8R3 SOURCE
-ray event selection within a 15-degree range of search. To ensure data quality, only events corresponding to good time intervals with DATA_QUAL > 0 and LAT_CONFIG == 1 were retained, and a maximal telescope zenith angle of 90 degrees was applied. Data reduction was performed using the Fermitools package v2.0.8. Galactic and extragalactic diffuse emission was taken into account using the gll_iem_v07.fits and iso_P8R3_SOURCE_V2_v1.txt models, respectively. The spectral shape was assumed to follow a LogParabola model (
https://fermi.gsfc.nasa.gov/ssc/data/analysis/scitools/source_models.html, accessed on 13 March 2020).
The data from the Swift and
Fermi observatories were reduced by processing the files corresponding to both the photons and the position and the orientation of the satellite on Flexible Image Transport System-format files (FITS) with HEASoft version 6.26.1 (
https://heasarc.gsfc.nasa.gov/docs/software/lheasoft/, accessed on 13 March 2020) and Fermitools version v11r5p3 (
https://fermi.gsfc.nasa.gov/ssc/data/analysis/software/, accessed on 13 March 2020) software. In both cases, it is necessary to know the relevant parameters of the object in question, such as the right ascension, declination, energy interval, and starting and ending times of the data processing (see, e.g., [
37]). The satellite information was reviewed to make various time and position corrections, and with this, the analysis of the maximum likelihood estimation was carried out, in which the FITS files were used (see, e.g., [
38]). Finally, a table was built that contains the light curve information, i.e., a file that contains discrete data of time, luminosity and its uncertainty with a signal-to-noise ratio <2
.
The multifrequency light curves in
Figure 1 are the result of processing the obtained data described above at a
confidence level; thus, the accepted total fraction of bins is 99.7%.
Table 1 shows the processed data in a synthesized way for all the electromagnetic frequencies studied for Mrk 501.
4. Discussion
A different analysis of the long-term multiwavelength (from radio to -rays) light curves of Mrk 501 show a common achromatic periodicity of ≲. The Monte Carlo fitting technique, assuming a relativistic eclipse as the cause of this periodicity, shows a periodicity of ∼. The eclipse is produced by a secondary supermassive black hole orbiting the primary supermassive black hole, i.e., about the central engine of Mrk 501. The reasoning for this conclusion is summarized in the following paragraphs.
The periodicities found with the RobPer and L-S algorithms described in
Section 3.1 are all consistent in all frequencies of Mrk 501 with average values in radio, optical, X-rays and
-rays, respectively, given via
,
,
and
days, according to the results of
Table 2. The mean value for these periodicities is
.
With the use of the VARTOOLS software described in
Section 3.2, we found that, for the AoV, AoV-h, BLS and DFT routines, the periodicity value lies in the intervals 226.73–229.2, 219.7–228, 220.960–240.404 and 222.374–242.818 days, respectively, according to the results presented in
Figure 3 and
Figure 4.
The fact that the color of the signal noise in the light curves of Mrk 501 presented in
Section 3.3 is pink means that, for each particular waveband, there is a robust oscillation (Brownian noise color) with a periodicity accompanied by a random signal (white noise color).
Due to the achromatic nature of the found periodicity of ≲
, we modeled this periodicity as a relativistic eclipse caused by an orbiting supermassive black hole about the central engine of Mrk 501. The results of
Table 6 show that this model is quite coherent in the fitting of the long-term multi-frequency light curves. The only small inconsistency found is with the dimensionless brightness magnification
presented in X-rays and more prominent in
-rays. This is most probably due to the large errors reported in
-rays and the large gaps that appear in the X-ray light curve.
Figure 8 shows a time-folding of the multi-frequency light curves with the corresponding eclipse function using the results of
Table 6. The shaded region represents the time duration,
, of the eclipse. The dotted horizontal lines represent a
significance level. It is important to note that the radio, optical and X-ray panels in
Figure 8 do not contain all the used data points at a
confidence level. In other words, they represent zooming in to the light curves in order to emphasize the detected eclipse. The large error bars in the
-ray make the bump on the eclipse function cross the
confidence level.
As mentioned in the introduction, shockwave interactions and motions inside the jet could cause periodic oscillations on the light curves, and although the periodicity in this case may appear, in principle, as an achromatic effect, it is extremely unlikely that this is the case for Mrk 501 since the reported amplitudes,
A, in
Table 6 are quite similar to one another, a property unlikely to occur in periodic shock wave formation and interactions due to the complexity of the radiation produced at each particular frequency.
In this article, we found an achromatic periodicity of ∼224 d in the radio, optical, X- and -rays light curves of Mrk 501, which we modeled as an eclipsing event produced by a massive (secondary) supermassive black hole orbiting the (primary) supermassive black hole of Mrk 501. Since the eclipse is produced by a relativistic object (the secondary black hole), the radiation brightness is magnified (contrary to what occurs in, e.g., a solar eclipse on Earth).
Using the results of our eclipse model in
Table 6 and Kepler’s third law for a circular orbit given via
where v is the velocity of the orbiting test mass (secondary black hole),
and
p are the radius and period of the orbit,
is the mass of the central supermassive black hole and
G represents Newton’s constant of gravitation, the following conclusions can be drawn about the results obtained in this article:
The mass of the central (primary) black hole in Mrk 501 is
[
65], which means that its gravitational radius is
.
The radius of the orbit of the eclipsing (secondary) binary black hole is .
Using a full relativistic approach for Schwarzschild’s space–time, Equation (
8) can be written as follows [
66]:
The above equation is cubic for the radius
, for which the only real solution is given via the following:
where
By using the same input values as the ones we used for the non-relativistic Kepler formula, we obtain , which is of the same order of magnitude as the that we obtained with the simple Kepler relation.
The orbital period of the eclipsing binary black hole is ∼.
The orbital velocity of the eclipsing binary black hole is ∼ of the speed of light, i.e., ∼.
The brightness magnification of the radiation produced by an eclipse due to the secondary black hole is ≳.
The coalescence time of the binary system is approximately given via the following [
67]:
where
is the mass of the secondary or eclipsing black hole. Since the periodicity found corresponds to a few decades of observation—meaning that the orbit is sufficiently stable—it is reasonable to assume that
, which means that the mass of the secondary black hole
.
The results of (
9) imply that the orbit of the secondary black hole is half the value obtained from the Newtonian calculation, and so its orbital frequency is about
∼3 its Newtonian value. Since the frequency,
f, of the of the binary’s quadrupolar gravitational waves is about a factor of 2 larger than the orbital frequency (see, e.g., [
68]), then
where the total mass of the binary system is given via
. In other words,
which, for a value of
∼
, yields
f∼
. This suggests that the frequency of the emitted gravitational waves is just above
, which is within the range of the forthcoming LISA space-based interferomenter, and as such, the prediction of a binary black hole system in Mrk 501 should be considered a science target for future LISA observations.
The magnification
of the radiation is the ratio of the Einstein radius,
, to the distance,
, from the line of sight connecting the observer to the lens [
69]. Since the secondary eclipsing black hole mass ∼
, then
∼
, and so, in order to get a magnification of
, then
. In other words, it would seem that the primary and secondary black holes should be quite well aligned with our line of sight. However, since Mrk 501 is a blazar, we are observing the source within an angle of no more than
from the emitted jet (in fact, for Mrk 501, the observing angle is between
and
, as reported by Giroletti et al. [
70]), and so the radiation would be a combination of the radiation processes occurring close to the central engine plus the relativistic jet. This extended emission from the relativistic jet means that the chances of having an 10% amplification increase.