Testing relativistic boost as the cause of gamma-ray quasi-periodic oscillation in a blazar

The mechanism for producing gamma-ray quasi-periodic oscillation (QPO) in blazar is unknown. One possibility is the geometric model, in which without the need for intrinsic quasi-periodic variation, the relativistic Doppler factor changes periodically, resulting in observed gamma-ray QPO. We propose a method to test this geometric model. We analyze the {\it Fermi}-LAT data of PG 1553+113 spanning from 2008 August until 2018 February. According to 29 four-month average spectral energy distributions (SEDs) in the energy range of 0.1-300 GeV, we split the {\it Fermi}-LAT energy range into three bands: 0.1-1 GeV, 1-10 GeV, and 10-300 GeV. The spectrum in each energy range can be successfully fitted by a power-law. The light curves and photon indices in the three energy ranges are obtained. Then, light curves in three narrow energy ranges, i.e., 0.2-0.5 GeV, 2-5 GeV and 20- 40 GeV, are constructed, and the relative variability amplitudes in the three narrow energy ranges are calculated. A discrete-correlation analysis is performed for the light curves. Our results indicate that (i) the light curves in the different energy ranges follow the same pattern showed in the light curve above 0.1 GeV; (ii) the three groups of photon indices in the energy ranges of 0.1-1 GeV, 1-10 GeV, and 10-300 GeV keep nearly constant; (iii) the ratio between relative variability amplitudes in different narrow energy ranges are equal (within their errors) to the prediction by the Doppler effect. Our results support the scenario of the relativistic boost producing the gamma-ray QPO for PG 1553+113.


INTRODUCTION
Blazars are the subclass of radio-loud active galactic nuclei (AGNs) with their relativistic jet pointing toward us (Urry & Padovani 1995). Multi-wavelength radiations covering from MHz, optical to TeV gamma-ray energies have been observed from blazars. Blazar emission is dominated by non-thermal radiation from the relativistic jet, therefore undergoing Doppler boosting. The Doppler effect leads to flux enhancement and contraction of the variability timescales.
The Large Area Telescope (LAT) on the Fermi Gammaray Space Telescope is providing continuous monitoring of the gamma-ray sky. In the analysis of the LAT data of PKS 2155-304 spanning from 2008 August until 2014 June, ⋆ E-mail: yandahai@ynao.ac.cn † E-mail: zjn@shao.ac.cn ‡ E-mail: zhangpengfee@pmo.ac.cn Sandrinelli et al. (2014) found a possible QPO signal with the ∼ 1.7 year period cycle. Latter, Zhang et al. (2017a) analyzed the LAT data of PKS 2155-304 spanning from 2008 August until 2016 October, and found that this QPO signal is strengthened, with the significance of ∼ 4.9 σ. Taking advantage of LAT data, Ackermann et al. (2015) reported the gamma-ray QPO in PG 1553+113. This signal is found in the LAT data covering from 2008 August to 2015 July with a 2.18±0.08 year period cycle. Very recently Tavani et al. (2018) confirmed the QPO signal in PG 1553+113 by updating the LAT data to 2017 September.
Possible gamma-ray QPOs with year-like timescales have been claimed in other several blazars (e.g., Sandrinelli et al. 2016a,b;Prokhorov & Moraghan 2017;Zhang et al. 2017b,c) and in one narrow-line Seyfert 1 galaxy (Zhang et al. 2017). The mechanism causing gamma-ray QPO in blazar remains unknown. A few possibilities have been proposed (e.g., Ackermann et al. 2015;Sobacchi et al. 2017;Caproni et al. 2017), like pulsational accretion flow instabilities and jet precession. The possible models can be divided into two classes: intrinsic origin and apparent origin. The intrinsic origin refers to the case that QPO also exists in the comoving frame of the relativistic jet, while in the scenario of the apparent origin the intrinsic variation does not present QPO, and QPO is caused by a periodically changing Doppler factor. Here we propose a method to test whether the gamma-ray QPO in blazar is resulted from a periodically changing Doppler factor. In Section 2 we describe our method, and results are presented in Section 3; In Section 4 we will give a brief discussion.

METHOD
The Doppler factor is where Γ = (1 − v 2 /c 2 ) −1/2 is the bulk Lorentz factor, v is the jet velocity, and v · cosθ is its line-of-sight component. One can infer that the periodic change of the viewing angle θ results in a periodic variation of δ D . If the radiation in the comoving frame of the relativistic jet is isotropic, and follows a power-law distribution of the form F ′ ν ′ ∝ ν ′−α , the flux density in the observer frame, F ν , is written as (e.g., Urry & Padovani 1995) δ α D is the ratio of the intrinsic power-law fluxes at the observed and emitted frequencies (i.e., ν and ν ′ ), and ν = δ D ν ′ . Even if the intrinsic flux is constant, the periodic time modulation in θ will produce a periodic variation in the observed flux. The variability amplitude is derived as (e.g., D'Orazio et al. 2015;Charisi et al. 2018) One can see that besides δ D the variability amplitude is also controlled by the spectral index α. The ratio between the variability amplitudes in low and high gamma-ray energies is Considering the broad energy range of Fermi-LAT, we can use eq. 4 to test whether the gamma-ray QPO in blazar is caused by Doppler boost. For instance we first analyze low and high energies gamma-ray light curves to determine the ratio of the observed variability amplitudes (i.e. the left side of eq. 4). Next, we calculate the spectral indices in two energy bands, and compute the ratio between the two indices (i.e. the right side of eq. 4). Obviously the two sides of eq. 4 can be obtained independently. We then check whether the above two ratios are consistent within their errors.
In the following the variability amplitude is defined as (Abdo et al. 2010) where S 2 is the variance of the light curve (Vaughan et al. 2003) and σ 2 err = σ 2 i + σ 2 sys . We use σ sys = 0.03 F i . The error in σ 2 is evaluated by using the formula given in Vaughan et al. (2003).

RESULTS
Taking PG 1553+113 as an example, we analyze its LAT data covering from 2008 August to 2018 February. We first construct the four-month binned light curve in the energy range of 0.1-300 GeV (Fig. 1), and also construct the SED in each time bin (Fig. 2). In Fig. 1 one can see five peaks and 4.5 period cycles. The pattern of this light curve is very similar to that reported in Tavani et al. (2018), and one can find more detailed discussions there. In Fig. 2 it can be found that in some stages/bins the spectrum becomes harder at ∼ 1 GeV and then becomes softer at ∼ 10 GeV (for example the SED in stage/bin 12). Accordingly, we then build fourmonth binned light curves in three energy bands, namely 0.1-1 GeV, 1-10 GeV and 10-300 GeV (Fig. 3). For each energy band we obtain the photon index in each time bin. In the three light curves, we also can find five peaks and 4.5 period cycles (Fig. 3), which are almost the same as the patterns in the light curve in the energy band of 0.1-300 GeV (Fig. 1). The photon indices in each energy range almost keep constant (Fig. 3). The average photon index is respectively Γ 1 = 1.63 ± 0.27 in the range of 0.1-1 GeV, Γ 2 = 1.58±0.11 in the range of 1-10 GeV, and Γ 3 = 1.98±0.20 in the range of 10-300 GeV.
Next, we construct light curves in the three narrow bands of 0.2-0.5 GeV, 2-5 GeV and 20-40 GeV (Fig. 4). These light curves also follow the pattern in the light curve in Fig. 1. The flux in such a narrow band can be considered as the differential flux at the central energy of the band. Then we use eq. 5 to calculate the variability amplitudes in the three narrow bands. It is respectively A 1 = 0.046 ± 0.021 in the range of 0.2-0.5 GeV, A 2 = 0.041 ± 0.012 in the range of 2-5 GeV, and A 3 = 0.068 ± 0.038 in the range of 20-40 GeV. We then obtain A 1 A 2 = 1.12 ± 0.61, A 2 A 3 = 0.61 ± 0.47, and A 1 A 3 = 0.69 ± 0.5. With the relation of photon index and spectral index Γ = α + 1 and using the values of Γ 1 , Γ 2 as well as Γ 3 , we have 3+α 1 3+α 2 = 1.02 ± 0.17, 3+α 2 3+α 3 = 0.80 ± 0.22, and 3+α 1 3+α 3 = 0.91 ± 0.16. It is clear that these values follow eq. 4 within their errors.
Looking at the above ratios, although the errors of the ratios between variability amplitudes are large, the most valuable point is that the central value of A 1 /A 2 , 1.12, is very close to the central value of (3 + α 1 )/(3 + α 2 ), 1.02. Even if the error of A 1 /A 2 is controlled at ∼10% level, A 1 /A 2 is still consistent with (3 + α 1 )/(3 + α 2 ).

DISCUSSION AND CONCLUSIONS
We propose a model-independent approach to test whether the gamma-ray QPO in PG 1553+113 is only related to the relativistic jet itself. According to the SEDs in the energy range of 0.1-300 GeV during different stages (Fig. 2), we split the LAT energy range into three energy bands, i.e., 0.1-1 GeV, 1-10 GeV and 10-300 GeV. We construct light curves in the three energy bands, and obtain the photon indices in the three energy bands. The three groups of photon indices all keep nearly constant in 10 years. Three narrow energy bands are respectively selected in the above three energy ranges, and they are 0.2-0.5 GeV, 2-5 GeV and 20-40 GeV. Light curves in the three narrow energy bands are built.
With the above observations we separately calculate the left and right side of eq. 4, and find that the values of the two sides are consistent with each other within the errors. This is in agreement with the prediction from the Doppler effect causing the gamma-ray QPO. Namely the intrinsic emission is not necessary to be periodic, and a periodic modulation in Doppler factor causes the observed QPO. Jet precession (e.g., Romero et al. 2000;Caproni et al. 2013), helical structure (e.g., Villata & Raiteri 1999;Ostorero et al. 2004;Raiteri et al. 209, 2017) or rotation of a twisted jet (e.g., Villata & Raiteri 1999;Hardee & Rosen 1999) could cause emitting region to change its orientation and hence the Doppler factor. These model are called geometrical models (Rieger 2004). For PG 1553+113, Raiteri et al. (2015) proposed an inhomogeneous curved helical jet scenario to explain its complex UV to X-ray spectrum.
In previous studies the gamma-ray QPOs are found in the integrated flux above 0.1 GeV. Here we also find similar QPO pattern in different energy bands. This is consistent with the geometrical models.