Relations between the Spectral Indices and Flux Densities of Eight Blazars

Relations between the flux densities (F) and spectral indices (α) can help us analyze the emission process. In this paper, we choose 8 blazars (0235+164, 0430+052, 1156+295, 3C345, 1308+326, 1413+135, 3C454.3, and 1749+096) from the University of Michigan Radio Observatory (UMRAO) database to study the relations between the spectral indices (α) and flux densities at 14.5GHz (F14.5). The main results are the following. (1) There are strong anticorrelations between α and F14.5, with the correlation coefficient r in the range from −0.33 to −0.87. (2)The α−F14.5 distributions show elliptic appearance, which have been fitted by elliptic curves. (3) For most of the sources, the time intervals of α−F14.5 elliptic circle (Tc) are consistent with the quasi-periodicities calculated by the averaged light curves and spectral variances (PαF).


Introduction
Blazars are a subclass of active galactic nuclei (AGNs).They show some extreme properties, such as violent optical variability, core dominance, superluminal motion, and so on [1,2].Blazars can be divided into two subclasses, BL Lacerate objects (BL Lacs), and flat-spectrum radio quasars (FSRQs).BL Lacs are characterized by featureless optical spectra or weak emission lines [3].Flat radio spectrum and broad optical emission lines are typical for FSRQs [1].Generally, the division between the two subclasses is based on the equivalent width (EW) of the optical broad emission lines; BL Lacs show  < 5 Å [1,[4][5][6].
The time scale of optical variability is an important quantity which is often used to probe the physical processes in blazars.The time scales are in the range from minutes to years and can be divided into three types: intraday variability (IDV), short-term variability, and long-term variability.
For blazars, there are many papers discussing relationship between the spectrum and flux density [16,[19][20][21][22][23][24][25][26].Dai et al. (2009) [27] presented long-term  observations and discussed the correlations between color index and brightness.Beaklini & Abraham (2014) [28] used variability at 7 mm to find evidence of shocks and precession in the jet.Fan et al. (2014) [15] analyzed the relation between V-band flux density (  ) and spectral index () for two nearby quasars and found two different relations below and above   = 28 mJy.When   < 28 mJy,   , and  showed anticorrelation, when   > 28 mJy,   , and  showed positive correlation.Yuan & Fan (2015) [29] found that, in 3C273 and 3C446, there was elliptic structure in the distributions between the flux densities and spectral indices.Wierzcholska et al. (2015) [22] analyzed the color-magnitude correlations of 30 blazars and observed the bluer-when-brighter behaviors.Carnerero et al. (2017) [23] pointed out that the flux densities and spectral variability can be compatible with jet models including at least two emitting regions that can change their orientation with respect to the line of sight.Isler et al. (2017) [24] provided an explanation of the long-term optical-infrared 2 color variabilities in some blazars and presented a general scheme which can apply to these variabilities.

Advances in Astronomy
The spectral indices are associated with the emission properties, and the flux densities demonstrate the optical variability properties.The study about the relations between the spectral indices and flux densities can combine the emission properties and optical variabilities.Generally, for BL Lacs, when the sources become brighter, the spectrum becomes flatter and when the sources become fainter, the spectrum becomes steeper.
This paper is arranged as follows.In Section 2, we calculate the spectral indices; in Section 3, we analyze the relations between the radio spectral index  and the 14.5 GHz radio flux density  14.5 ; in Sections 4 and 5, we give the discussion and conclusions, respectively.

The Spectral Indices
Based on the UMRAO (University of Michigan Radio Astronomy Observatory) database (https://dept.astro.lsa.umich.edu/datasets/umrao.php), we collected the radio flux densities at 4.8 GHz, 8 GHz, and 14.5 GHz and used the following method to calculate their averaged flux densities and spectral indices, similarly to Yuan & Fan (2011) [30] and Yuan et al. (2014) [31].Firstly, at each band (4.8 GHz, 8 GHz, and 14.5 GHz), we average the flux densities within the same bin and obtain N sets of data:   (  ,  4.8| ,  8| ,  14.5| ) ( ∈ ), which represent the flux density at each band.Secondly, based on the relation, (] = 4.8 GHz, 8 GHz and 14.5 GHz), we obtain and use linear fit to calculate the spectral indices ().
There are 8 blazars (0235+164, 0430+052, 1156+295, 3C345, 1308+326, 1413+135, 3C454.3, and 1749+096) with known quasi-periodicity and enough  values (N>50).We use them to build a sample, for which, the spectral indices () and the averaged flux densities within the respective bins are displayed in Figure 1.The upper panels show the averaged flux densities at 4.8 GHz, 8 GHz, and 14.5 GHz, and the lower panels show the spectral indices.The detailed descriptions about  have been listed in Table 1, where Col. 1: Name, Col.

The Relations between the Spectral Indices and Flux Densities
. .Methods.For two variable data sets,   ,   ( = 1, ), we use the linear fitting to analyze their correlations,  = ( ± Δ) + ( 0 ± Δ 0 ), where k is the slope,  0 is the intersection,  is the Student's t probability, and n is the number of points in the data set.The Pearson's correlation coefficient r is expressed as [32][33][34] where  is the averaged value of   and  is the averaged value of   .When x, y show elliptical appearance, we use the following elliptic curve to make the elliptic fit: with a, b, c, d, e, f being free parameters and || being normalized.Considering x as  14.5 and y as , we make the following calculations.
. .e Linear Correlations.We use the linear correlations to analyze the relations between  14.5 and .The results are shown in Figure 2, where the solid lines indicate the fit between  14.5 and .The slope k, the intersection  0 , the correlation coefficient r, and the chance probability  are listed in Table 2, where Col. 1: Name, Col. 2: the slope , Col. 3: the slope error Δ, Col. 4: the intersection  0 , Col. 5: the intersection error Δ 0 , Col. 6: the correlation coefficient , and Col. 7: the chance probability .Among the eight target sources,  values are in the range from −0.869 to −0.328, so  and  14.5 show strong anticorrelations.
. .e Elliptic Fitting.For every targeted source, based on time sequence, and the variable trend of spectral indices () dependent on flux densities, we can find the elliptic appearances in the  −  14.5 distributions, as seen in Figure 3, where   the filled circles, crosses, and open circles stand for the different elliptic appearances.For the whole elliptic appearances, the variable trends are anticlockwise.We use the elliptic curve to make fit, and the results are shown in Figure 3 and listed in Table 3.
In Table 3

Discussion
Phenomenon that there are cycle structures in blazar variabilities can help us study the theoretical models in both optical and radio light curves.This method has been used in analysis of the quasi-periodic behavior in some blazars So, studies about the activity cycles are very important for testing the theoretical models, the jet structure, the properties of central engine, and predicting the likely outbursts.
According to the time interval calculated by the elliptic fitting, we can divide the total light curves into different parts, as seen in Figure 4, and then compare the time intervals of elliptic circle with the quasi-periodicities calculated by the averaged light curves and spectral variances.
The most common tool for periodicity analysis of both evenly and unevenly sampled signals is the periodogram method, which is an estimator of the signal energy in the frequency domain [39].Lomb (1976) [40] introduced a modified form of this method.
We used the Lomb's periodogram method to calculate the periodicity and used the half width at half maximum (HWHM) to calculate the corresponding error.For every targeted sources, based on the averaged light curves and spectral indices, we calculate the two periodic signals, then place them together, and choose the common signal as the last periodicity,   .e period signals are displayed in Figure 5, where the upper panels show the averaged flux density, and the lower panels show the spectral variances.The red noise [41] with the noise levels 80%, 90%, 95%, and 99% is also shown in Figure 5.For every targeted source, the time interval of elliptic circle (  ), the periodicities calculated from the averaged light curves, and spectral variances (  ) are listed in Table 4.In order to make comparison, we quoted the quasi-periodicities (  ) calculated by Fan, Liu & Yuan (2007) [14], which were calculated from the whole light curves and listed in Table 4.
We use a linear fitting to compare   ,   , and   .The results show that   = (0.90 ± 0.52)  − (0.88 ± 3.63), with r=0.58, p=13.3%;  = (0.85 ± 0.27)  + (0.63± 1.92), with r = 0.78, p=2.1%, as seen in Figure 6, with the black line standing for   vs.   and the red line standing for   vs.   .The two fitting results   vs.   and   vs.   can be consistent with each other, which show strong correlations between the time internal of elliptic circle and quasi-periodicities not only calculated from the whole light curve   , but also calculated from the averaged light curves and spectral variances (  ).
Quasi-periodicity is a hot research topic in radio variability of blazars, but the reason for the periodicity is unclear.There are many models proposed to explain this phenomenon, for example, the binary black-hole model, the thermal instability model, and the perturbation model [14].Many authors [42][43][44][45] apply the model of a moving shock in a relativistic jet with a helical magnetic field to explain the flux     density variances.The  −  14.5 circles might come from the helical jet produced by the binary black holes.

Conclusion
In this paper, we choose eight blazars from the UMRAO data base to calculate the spectral indices.We average the light curves with the fixed interval, calculate the averaged flux densities () and spectral indices (), and then compare the relations between  and .
For each source, we obtain strong anticorrelations between  and  14.5 , which can combine the emission properties and radio variabilities.
Based on time sequence, the - 14.5 distributions showed elliptic appearances and the variable trends are anticlockwise.We use the elliptic curve to make fit, find that the variation directions of elliptic cycle are anticlockwise, and then obtain the cycle durations.Based on the averaged light curves and spectral variances, we calculate the quasi-periodicities and compare them with the cycle durations.e results show that they are consistent with each other, so the elliptic appearances should come from the helical structures of jets or the jet models containing at least two emitting regions.

４Figure 3 :Figure 4 :
Figure 3: The  −  14.5 circles of the target sources.The elliptic solid lines, dotted lines, and dashed lines indicate the elliptic fit results based on the time sequences.

Figure 5 :
Figure 5: The quasi-periodicities of 8 targeted sources.The upper panels are derived from the light curves and the lower panels are from spectral indices.The black lines indicate periodic signals.The red, green, blue, and cyan lines show 80%, 90%, 95%, and 99% red noise level, respectively.

Table 1 :
The spectral indices of 8 blazars.

Table 2 :
The linear correlations between the spectral indices and flux densities.

Table 3 :
The elliptic fitting of 8 blazars.  , and the time scale for the evolution of the jet,  V .They also proposed that   determines an 'activity cycle' for the source.

Table 4 :
The time spans and long-term quasi-periodicities of 8 blazars.Figure 6: Comparison between   and   ,   .The black line stands for   vs.   , and the red line stands for   vs.   .