Dependence of NLRs sizes on [OIII] luminosity in low redshift AGN with double-peaked broad Balmer emission lines

In the manuscript, the simple but interesting results are reported on the upper limits of NLRs sizes of a small sample of 38 low redshift ($z<0.1$) AGN with double-peaked broad emission lines (double-peaked BLAGN), in order to check whether the NLRs sizes in type-1 AGN and type-2 AGN obey the similar empirical dependence on \o3~ luminosity. In order to correct the inclination effects on projected NLRs sizes of type-1 AGN, the accretion disk origin is commonly applied to describe the double-peaked broad H$\alpha$ leading to the determined inclination angles of central disk-like BLRs of the 38 double-peaked BLAGN. Then, considering the fixed SDSS fiber radius, the upper limits of NLRs sizes of the 38 double-peaked BLAGN can be estimated. Meanwhile, the strong linear correlation between continuum luminosity and \o3~ luminosity is applied to confirm that the \o3~ emissions of the 38 double-peaked BLAGN are totally covered in the SDSS fibers. Considering the reddening corrected measured \o3~ luminosity, the upper limits of NLRs sizes of the 38 double-peaked BLAGN are within the 99.9999\% confidence interval of the expected results from the empirical relation between NLRs sizes and \o3~ luminosity in the type-2 AGN. In the current stage, there are no clues to challenge the Unified model of AGN, through the space properties of NLRs.


INTRODUCTION
Strong [O ] emission lines are fundamental characteristics of both broad line Active Galactic Nuclei (type-1 AGN) and narrow line AGN (type-2 AGN). Under the current framework of the preferred being-improved Unified Model (Antouncci 1993;Netzer 2015), type-1 AGN and type-2 AGN have totally the same central structures of narrow emission line regions (NLRs) and broad emission line regions (BLRs) (the two essential structures of AGN), but the BLRs are seriously obscured by central dust torus leading to no apparent broad optical emission lines detected in type-2 AGN. The well detected polarized broad emission lines in type-2 AGN, such as the results well discussed in Antonucci & Miller (1985); Tran (2001Tran ( , 2003; Nagao et al. (2004); Cardamone et al. (2007); Shi et al. (2010); Barth et al. (2014); Pons & Watson (2016); Savic et al. (2018), provide robust evidence to strongly support the hidden BLRs in type-2 AGN. Certainly, there is one precious subclass of true type-2 AGN: type-2 AGN without hidden BLRs. More recent arguments on the very existence of true type-2 AGN can be found in Bianchi et al. (2019); Zhang et al. (2021). We mainly focus on properties of NLRs sizes of AGN, therefore, we do not discuss true type-2 AGN in the manuscript.
Basic physical sizes of NLRs in type-2 AGN and of BLRs in type-1 AGN have been well known. Sizes of BLRs ( distances between BLRs to central black hole (BH)) can be well determined by the well-known reverberation mapping technique (Blandford & McKee 1982), through long-term variabilities of broad emission lines and continuum emissions. Peterson et al. (2004) have reported the measured sizes of BLRs in the sample of nearby broad line AGN (BLAGN) in the AGNWATCH project (see dozens of published papers listed in the project). Kaspi et al. (2000Kaspi et al. ( , 2005 have reported the measured sizes of BLRs of nearby 17 quasars . Bentz et al. (2010); Barth et al. (2015); Williams et al. (2018) have reported the measured sizes of BLRs in the sample of nearby Seyfert galaxies in the LAMP project . Wang et al. (2014); Du et al. (2016) have reported the measured sizes of BLRs in the sample of nearby Seyferts with high accretion rates. Shen et al. (2015); Grier et al. (2017) have reported the measured sizes of BLRs in the sample of SDSS QSOs . The reverberation mapping technique determined sizes of BLRs leads to the well accepted empirical relation between and continuum luminosity of broad line AGN discussed and shown in Kaspi et al. (2000Kaspi et al. ( , 2005; Bentz et al. (2013).
Unlike the measurements of BLRs sizes in type-1 AGN, longer extended structures of NLRs indicate reverberation mapping technique can not be applied to measure NLRs sizes ( ) due to no variabilities in narrow emission lines (expected variability timescale around hundreds of years). However, in some nearby type-2 AGN or nearby high luminosity type-2 QSOs, the NLRs sizes have been well determined by properties of spatial resolved [O ] emission images, such as the measured NLRs sizes in Seyfert 2 galaxy NGC1386 in Bennert et al. (2006a), in the small sample of in Seyfert-2 galaxies in Bennert et al. (2006b), in the sample of 15 obscured QSOs in Greene et al. (2011), in the radio quiet type-2 QSOs in Liu et al. (2013a,b), in the luminous obscured QSOs in Hainline et al. (2013Hainline et al. ( , 2014, in the obscured QSOs in Fischer et al. (2018), etc. Then, based on the measured sizes of NLRs, there is one empirical R-L relation between NLRs sizes and [O ] luminosity ( [ ] ) (in the manuscript, the R-L relation is for the relation on NLRs sizes, unless with specific statements) discussed and shown in Liu et al. (2013a); Hainline et al. (2013Hainline et al. ( , 2014 , especially for AGN with 8 m luminosity smaller than 10 45 erg/s as discussed in Hainline et al. (2013); Dempsey & Zakamska (2018). In order to explain the R-L relation of NLRs, Liu et al. (2013a) have proposed a model of [O ] emission clumpy nebulae in which clouds that produce line emission transition from being ionization-bounded at small distances from central power source to being matterbounded in the outer parts of the clumpy nebulae. Similar model was also discussed in Greene et al. (2011). More recently, Dempsey & Zakamska (2018) have re-constructed the R-L relation of NLRs by the proposed model that the NLRs as a collection of clouds in pressure equilibrium with the ionizing radiation.
https://www.physics.uci.edu/~barth/lamp.html https://www.sdss.org/dr15/algorithms/ancillary/reverberation-mapping/ Actually, NLRs sizes in type-1 AGN are hardly to be measured mainly due to inclination effects, although Bennert et al. (2002); Schmitt et al. (2003a,b) have reported NLRs sizes in several nearby broad line AGN. However, the measured NLRs sizes of the type-1 AGN in the literature, such as the results in Schmitt et al. (2003b), are about 1-2 magnitudes smaller than the expected results from the empirical relation between NLRs sizes and [O ] luminosity in the type-2 AGN in Liu et al. (2013a), which could be mainly due to inclination effects and/or quite shallow observations. Therefore, NLRs sizes of type-1 AGN in Schmitt et al. (2003b) are not considered in the manuscript. However, there is one subclass of type-1 AGN, the broad line AGN with double-peaked broad low-ionization emission lines (called as double-peaked BLAGN) of which inclination angles can be reasonably estimated, accepted accretion disk origin of the double-peaked broad emission lines as well discussed in Chen & Halpern (1989); Eracleous et al. (1995); Storchi-Bergmann et al. (2003); Strateva et al. (2003); Flohic & Eracleous (2008); Storchi-Bergmann et al. (2017), etc. In the manuscript, considering the fixed diameter of SDSS fibers, upper limits of NLRs sizes in low redshift double-peaked BLAGN in SDSS can be well estimated, which will be applied to test the R-L empirical relation of NLRs in type-1 AGN. The tests can provide further clues on whether there are difference of spatial distances of NLRs between type-1 AGN and type-2 AGN, to challenge or to support the Unified model of AGN, which is the main objective of the manuscript. Section 2 presents our hypotheses on estimating sizes of NLRs of double-peaked BLAGN. Section 3 shows our main sample of double-peaked BLAGN. Section 4 shows our main results and necessary discussions. Section 5 gives our final summaries and conclusions. We have adopted the cosmological parameters 0 = 70km · s −1 Mpc −1 , Ω Λ = 0.7 and Ω m = 0.3 in the manuscript.

MAIN HYPOTHESES ON ESTIMATING UPPER LIMITS OF SIZES OF NLRS OF DOUBLE-PEAKED BLAGN
In order to explain the double-peaked broad emission lines of AGN, besides the accretion disk origin (Chen & Halpern 1989;Eracleous et al. 1995;Storchi-Bergmann et al. 2017), the theoretical binary black hole model (BBH model) (Begelman et al. 1980;Di Matteo et al. 2005;Pfister et al. 2017;Sayeb et al. 2021) has been well proposed, such as the BBH model once applied to describe variability properties of the double-peaked broad Balmer lines of 3C390.3 in Gaskell (1996) and theoretical BBH model applied to generate double-peaked broad lines in Shen & Loeb (2010), etc. Once accepted the BBH model to explain the double-peaked broad emission lines, quasi-periodic oscillations (QPOs) could be expected, such as the well discussed in Graham et al.  Table 1) and the best descriptions to the stellar lights by the SSP method. In each panel (except the ones without apparent stellar lights), solid dark green line shows the SDSS spectrum, solid red line shows the best descriptions to the SDSS spectrum with the emission lines being masked out, solid purple line shows the SSP method determined stellar lights, and solid blue line shows the determined power law AGN continuum emissions. In each panel, from left to right, the vertical red lines mark the following emission features masked out, including [ Zhang (2022) with QPOs signs applied to detect candidates for BBH systems. However, through the study of long-term variabilities of double-peaked BLAGN in Eracleous et al. (1997); Shapovalova et al. (2001);Storchi-Bergmann et al. (2003); Lewis, Eracleous & Storchi-Bergmann (2010); Zhang (2013); Zhang & Feng (2017), quite few QPOs can be commonly detected in long term variabilities of doublepeaked BLAGN. Moreover, among the 38 collected lowredshift double-peaked BLAGN in our final sample in the following section, 37 double-peaked BLAGN have their longterm light curves collected from CSS (Catalina Sky Survey ) (Drake et al. 2009). Long-term variability analysis of our sources does not show any QPOs signs as shown in section 4. Therefore, in the manuscript, the accretion disk origin has been well accepted to explain the double-peaked broad emission lines, and no further discussions are shown on the BBH model.
There are different kinds of relativistic accretion disk models in the literature, such as the circular accretion disk model http://nesssi.cacr.caltech.edu/DataRelease/ in Chen & Halpern (1989), the improved elliptical accretion disk model in Eracleous et al. (1995), the model of circular disk with spiral arms in Storchi-Bergmann et al. (2003), a warped accretion disk model in Hartnoll & Blackman (2000), and a stochastically perturbed accretion disk model in Flohic & Eracleous (2008), etc. Here, the elliptical accretion disk model (without contributions of subtle structures) well discussed in Eracleous et al. (1995) is preferred, because the model can be applied to explain almost all observational double-peaked features of the double-peaked BLAGN in our sample. There are seven model parameters in the elliptical accretion disk model, inner boundary 0 and out boundary 1 in the units of (Schwarzschild radius), inclination angle of disk-like BLRs, eccentricity , orientation angle 0 of elliptical rings, local broadening velocity , line emissivity slope ( ∝ − ). Meanwhile, we have also applied the very familiar elliptical accretion disk model, see our studies on double-peaked lines in Zhang et al. (2005); Zhang (2013aZhang ( , 2015; Zhang et al. (2019). More detailed descriptions on the applied elliptical accretion disk model can be found in Eracleous et al. (1995); Storchi-Bergmann et al. (2003); Strateva et al. (2003), and there are no further discussions on the elliptical accretion disk model in the manuscript.

ZHANG
Then, the elliptical accretion disk model can be well applied to describe double-peaked broad emission lines, leading to the well determined inclination angle sin( ) of central accretion disk.
Considering the SDSS fiber diameter of 3 arcseconds (2 arcseconds for eBOSS, the Extended Baryon Oscillation Spectroscopic Survey (detailed descriptions can be found in the web ) as the projected space distance in units of , and considering the inclination angle sin( ) determined by the elliptical accretion disk model, the upper limits of sizes of the [O ] emission regions, as the upper limits of NLRs sizes , , should be estimated as .
Before the end of the section, it is interesting to check whether the equation above can be reasonably applied to estimate upper limits of NLRs sizes of AGN. For the 14 obscured quasars well discussed in Liu et al. (2013a), their upper limits of NLRs sizes can be estimated as SDSS fiber radius of 1.5 arcseconds. Here, for the obscured quasars (type-2 quasar), we simply accepted ( ) ∼ 1. Due to the high redshift around 0.5 of the 14 obscured quasars, their upper limits of NLRs sizes are about 2.06 × 10 4 , quite larger than their measured NLRs sizes in Liu et al. (2013a), indicating the method to estimate upper limits of NLRs sizes are reasonable to some extent.

SAMPLE OF THE DOUBLE-PEAKED BLAGN
In the manuscript, we mainly consider the low redshift double-peaked BLAGN with redshift smaller than 0.1 by the following main reason. For redshift at 0.1 (distance about 460.3Mpc), the corresponding space distance of the SDSS fiber radius (1.5 arcseconds) is about 3350pc (10 3.53 pc), similar as the mean value about 10 3.5 pc of the measured NLRs sizes of type-2 QSOs in Liu et al. (2013a); Hainline et al. (2013);Fischer et al. (2018). For objects with redshift quite larger than 0.1, the corresponding space distance by the SDSS fibers should be clearly larger than the expected sizes of NLRs, indicating that it is meaningless to check properties of upper limits of NLRs sizes of type-1 objects with redshift quite larger than 0.1.
We collect double-peaked BLAGN with redshift smaller than 0.1 as follows. On the one hand, there are 4 doublepeaked BLAGN with redshift smaller than 0.1 in the sample of double-peaked BLAGN listed in Strateva et al. (2003), SDSS 0721-52228-0600 (SDSS PLATE-MJD-FIBERID), SDSS 0725-52258-0510, SDSS 0746-52238-0409 and SDSS https://www.sdss.org/surveys/eboss https://www.sdss.org/dr16/spectro/spectro_basics/ 6139-56192-0440 (spectrum from eBOSS with fiber radius 1arcseconds). On the other hand, the other 11 appropriate double-peaked BLAGN are collected by the following two steps. First, all the 1752 SDSS QSOs with signal-to-noise larger than 10 and redshift smaller than 0.1 are firstly collected from SDSS DR16 (Data Release 16 (Ahumada et al. 2020)) by the following SQL query SELECT p l a t e , f i b e r i d , mjd FROM S p e c O b j a l l WHERE c l a s s = 'QSO' and z < 0 . 1 and z w a r n i n g = 0 and s n m e d i a n > 10 . Second, emission lines of all the 1752 SDSS QSOs are measured after host contributions are determined, as what are discussed in the following section. And then, we mainly check the QSOs with the broad H described by more than 2 broad Gaussian components one by one by eyes, and collect 11 double-peaked BLAGN based on the criterion that there is at least one apparent hump included in the red-side or blue-side of broad H . Unfortunately, it is hard to provide a standard criterion which can be well described by a formula or formulas to collect candidates of AGN with double-peaked broad emission lines, therefore, the collected double-peaked BLAGN are identified by eyes.
Besides the double-peaked BLAGN collected from SDSS quasars in DR16, there is a sample of candidates of doublepeaked BLAGN collected from SDSS main galaxies and reported in Liu et al. (2019). Then, based on the same criteria of < 0.1 and signal-to-noise of SDSS spectrum larger than 10, there are 106 candidates of double-peaked BLAGN collected from the sample of Liu et al. (2019) with the flag MULTI_PEAK=2. Furthermore, two additional criteria are accepted to collect double-peaked BLAGN from the sample of Liu et al. (2019). On the one hand, there should be apparent narrow Balmer emission lines detected in the SDSS spectra, in order to correct reddening effects on observed [O ] emission intensities. On the other hand, there should be apparent both broad H and broad H detected in the SDSS spectra, in order to ignore seriously obscurations on central BLRs, and in order to confirm the collected double-peaked BLAGN from SDSS main galaxies are also type-1 AGN (not type-1.5 nor type-1.9 AGN) totally similar as the double-peaked BLAGN (type-1 AGN) collected from SDSS quasars. Based on the two additional criteria, 23 double-peaked BLAGN are finally collected from the sample of Liu et al. (2019). In the following section, two examples are shown on an AGN with doublepeaked broad H but no apparent narrow Balmer emission lines, and on an AGN with only double-peaked broad H http://skyserver.sdss.org/dr16/en/tools/search/sql.aspx but no apparent double-peaked broad H , from the sample of Liu et al. (2019), after subtractions of host galaxy contributions from the SDSS spectra. Finally, there are 38 doublepeaked BLAGN with redshift smaller than 0.1 included in our sample, 15 double-peaked BLAGN collected from SDSS quasars and 23 double-peaked BLAGN collected from the sample of Liu et al. (2019). The basic information of the 38 double-peaked BLAGN are listed in Table 1 Zhang (2021a,b) to describe the stellar lights included in SDSS spectra of broad line AGN, we here exploit a power law component × applied to describe the AGN continuum emissions and the 39 simple stellar population templates from the Bruzual & Charlot (2003) which include the population age from 5Myr to 12Gyr, with three solar metallicities (Z = 0.008, 0.05, 0.02). Then through the Levenberg-Marquardtleast-squares minimization method applied to the SDSS spectra with both narrow and broad emission lines being masked out, host galaxy contributions and the intrinsic AGN power law continuum components in the SDSS spectra can be clearly determined. Here, when the SSP method is applied, the emission lines are being masked out by line width about 400km/s at zero intensity, and the broad H , H and H are being masked out by line width about 3000km/s at zero intensity, and the probable optical Fe emission lines (Kovacevic-Dojcinovic et al. 2010) are being masked out with rest wavelength range from 4400Å to 5600Å. Meanwhile, when the model functions are applied, there are 42 model parameters, 39 strengthen factors with zero as the starting values for the 39 SSPs, the broadening velocity with 100km s −1 as the starting value, and for the power law component with zero as the starting values. The SSP method determined stellar lights and corresponding residuals are clearly shown in Figure 1 for the 30 double-peaked BLAGN with apparent stellar lights. Here, the residuals shown in Figure 1 are calculated by SDSS spectrum http://classic.sdss.org/dr1/algorithms/reflines.dat minus sum of the stellar lights and the power law continuum emissions. The values of 2 (summed squared residuals divided by degree of freedom) are listed in Table 1 for the best fitting results to the SDSS spectra with emission lines being masked out.
Before subtracting the stellar lights from the SDSS spectra, one point is noted. Among the 30 double-peaked BLAGN of which SDSS spectra include apparent contributions of stellar lights, there are six objects, SDSS 1393-52824-0216, SDSS 1624-53386-0032, SDSS 0492-51955-0273, SDSS 1229-52723-0299, SDSS 2022-53827-0286 and SDSS 2419-54139-0083, of which intrinsic AGN continuum emissions are determined and described by red power law functions ( ≥ 0) not by commonly blue power law functions ( < 0). The red power law AGN continuum emissions are probably due to intrinsic reddening effects of host galaxies and/or central high density clouds. The intrinsic reddening could have effects on strengths of broad emission lines but have few effects on line profiles of broad emission lines. Whereas, the main consider parameter of inclination angle is determined through properties of double-peaked line profiles. Therefore, although in the procedure above to determine contributions of stellar lights without considerations of intrinsic reddening effects, there are few effects on our final results on the upper limits of NLRs sizes of the collected low redshift doublepeaked BLAGN.
After subtractions of the contributions of stellar lights, the emission lines are modeled by multiple Gaussian functions, in order to obtain more accurate line parameters (or line profiles) of both narrow and broad emission lines. Similar as what we have recently done in Zhang (2021a), the following model functions are applied to describe the emission lines around H within rest wavelength range from 6250Å to 6850Å, four broad Gaussian functions (second moment larger than 400km/s) applied to describe the broad H , one narrow Gaussian function (second moment smaller than 400km/s) plus one broad Gaussian function applied to describe the narrow and extended (if there was) components of narrow H , two narrow Gaussian functions applied to describe the [N ] doublet, two narrow Gaussian functions applied to describe the [S ] doublet, and two narrow plus two broad Gaussian Similar as the extended components in [O ] doublet, there are also extended components in the narrow Balmer lines, such as the shown results in SDSS 0721-52228-0600 (PLATE-MJD-FIBERID). Here, we do not discuss the physical properties and origin of the extended narrow Balmer components, but the applied extended components can lead to better descriptions to the emissions lines shown in Figure 2.   Then, through the Levenberg-Marquardt least-squares minimization method, the best-fitting results and corresponding residuals (line spectrum minus the best fitting results) to the emission lines can be well determined by the model functions above and shown in Figure 2. The values of 2 (summed squared residuals divided by degree of freedom) for the best fitting results to the emissions lines by the model functions above can be well determined and listed in Table 1 components in narrow Balmer lines in SDSS 0721-52228-0600, therefore, the determined extended components in narrow Balmer lines are accepted from narrow Balmer emission regions in SDSS 0721-52228-0600. Although the multiple Gaussian components can not provide physical properties of the expected disk-like BLRs, the results in Figure 2 can provide more accurate measurements of the narrow emission lines, especially the line luminosity of [O ] 5007Å. The determined narrow emission lines around H can be subtracted from the line spectrum to obtain the pure double-peaked broad H which will be described by the elliptical accretion disk model.
Finally, through the Levenberg-Marquardt least-squares minimization method, the pure double-peaked broad H are best fitted by the elliptical accretion disk model and shown in Figure 4. When the elliptical accretion disk model is applied, the seven model parameters have restrictions as follows. The inner inner boundary 0 is larger than 15R G and smaller than 1000R G . The out boundary 1 is larger than 0 and smaller than 10 6 R G . The inclination angle of disklike BLRs has sin( ) larger than 0.05 and smaller than 0.95. The eccentricity is larger than 0 and smaller than 1. The orientation angle 0 of elliptical rings is larger than 0 and smaller than 2 . The local broadening velocity is larger than 10km/s and smaller than 10 4 km/s. Here, two points are noted on properties of the local broadening velocity. On the one hand, it is not totally clear on the origin of the local broadening velocity. As discussed in Chen & Halpern (1989), the most likely interpretation to the local broadening is due to the effects of electron scattering in a photoionized atmosphere of the disk. On the other hand, as the reported model fitting results to the double-peaked broad emission lines, such as in Chen & Halpern (1989); Eracleous et al. (1995); Strateva et al. (2003) etc., the local broadening velocity could be round a few hundreds to a few thousands of kilometers per second. Therefore, in the manuscript, the upper limit 10 4 km/s is accepted to the local broadening velocity in the model. The line emissivity slope ( ∝ − ) is larger than -7 and smaller than 7. Fortunately, the doublepeaked broad H of the 38 double-peaked BLAGN can be well described by the elliptical accretion disk model, therefore, the other accretion disk models with improved subtle structures are not discussed any more in the manuscript. The determined seven model parameters and corresponding uncertainties are listed in Table 2 for the double-peaked broad H in each double-peaked BLAGN. Here, the uncertainty of each model parameter is the formal 1sigma error computed from the covariance matrix, through the Levenberg-Marquardt least-squares minimization technique (the MPFIT package ) (Markwardt 2009). The determined 2 values for the best descriptions to the double-peaked broad H by the elliptical accretion disk model are listed in Table 1.
Based on the procedures above, the measured continuum intensity at 5100Å, the measured [O ] 5007Å line flux, the measured fluxes of narrow Balmer emission lines and the determined inclination angle of the disk-like BLRs are listed in Table 1. Certainly, based on each measured inclination angle of the disk-like BLRs in central accretion disk and the redshift, the expected upper limit , of NLRs sizes can be calculated and also listed in the final column of Table 1 In one word, NGC5548 is not included in our final sample, however, the estimated , and [ ] (corresponding [O ] line luminosity 2.67 × 10 41 erg/s) will be applied in the following results. On the other hand, when the elliptical accretion disk model is applied, the model parameter 1 and sin( ) have probably similar effects on features around the peaks of double-peaked line profiles, therefore it is necessary to check whether the parameter 1 and the parameter sin( ) can be clearly and solely determined through the model applied to describe the double-peaked broad lines. In other words, it is necessary to determine that the parameters sin( ) and the other model parameters are not completely degenerate together. Here, a simple procedure is applied as follows. Based on randomly collected values of the model parameters within their limits as described above (input values of the model parameters) in the elliptical accretion disk model, 400 simulating double-peaked broad H are created with considerations of signal-to-noise about 15. Left panel of Figure 6 shows an example on the simulating double-peaked broad H . Then, the same procedure considering the elliptical accretion disk model is applied to fit the simulating double-peaked broad H through the Levenberg-Marquardtleast-squares minimization https://pages.physics.wisc.edu/~craigm/idl/cmpfit.html method with [200 , 1000 , 0.5, 2, 1000km/s, 0.2, 0.] as the starting values of the model parameters, leading to the measured values of the model parameters. An example on the best-fitting results to the simulating double-peaked broad H is shown in the left panel of Figure 6. If the model parameter of sin( ) and the other model parameters are degenerate together, the input values of sin( ) should be quite different from the measured values of sin( ). However, as shown in the top second panel of Figure 6, there is a strong linear correlation with Spearman Rank correlation coefficient 0.928 ( < 10 −15 ) between the input sin( ) and the measured sin( ). Moreover, besides the model parameter of sin( ), there are apparent linear correlations between the input model parameters and the re-measured model parameters determined by best fitting results to the simulating double-peaked broad H , with Spearman Rank correlation coefficients about 0.66 with < 10 −15 , about 0.46 with < 10 −15 , and 0.83 with < 10 −15 , about 0.88 with < 10 −15 , and 0.71 with with < 10 −15 for the correlations between between input 0 and re-measured 0 , between input 1 and re-measured 1 , between input and re-measured , between input and re-measured , between input and re-measured , respectively. Therefore, not only the model parameter sin( ) but also the other model parameters can be solely determined in the elliptical accretion disk model, as long as the double-peaked line profiles are clean enough.
Then, Figure 7 shows the dependence of , on the [O ] line luminosity of the 38 low redshift double-peaked BLAGN. Here, based on the intrinsic flux ratio of narrow H to narrow H as 3.1, reddening effects have been corrected to calculate the intrinsic [O ] line luminosity for the objects with flux ratios of narrow H to narrow H larger than 3.1, based on the well applied Galactic extinction curve in Fitzpatrick (1999). The applied values of ( − ) are also listed in Table 1 Figure 7 as solid purple circles.
However, considering the quite large uncertainties of NLRs sizes and the NLRs sizes as upper limits, it is hard to conclude that SDSS 0581-52356-0463, SDSS 1196-52733-0639 SDSS 6139-56192-0440 has its SDSS spectrum from the eBOSS with fiber diameter of 2 arcseconds, and the value , of SDSS 6139-56192-0440 is estimated by the fiber radius of 1 arcseconds. * * means that the value is the one determined by the Levenberg-Marquardt least-squares minimization method, although the uncertainty is larger than the value.      Strateva et al. (2003). The fifth row to the fifteenth row are for the double-peaked BLAGN collected from SDSS quasars. The other rows are for the double-peaked BLAGN collected from the sample of Liu et al. (2019) The listed sin( ) in the fourth column are the same as the ones listed in the seventh column of Table 1. Here, based on the intrinsic flux ratio of broad H to broad H as 3.1, intrinsic reddening effects have been corrected to calculate the intrinsic continuum luminosity for the objects with flux ratios of broad H to broad H larger than 3.1, based on the well applied Galactic extinction curve in Fitzpatrick (1999). The line fluxes of the broad Balmer lines are calculated by sum of the fluxes of the broad Gaussian components shown as solid lines in Figure 2 and listed in Table 3. The applied ( − ) to correct continuum emissions are also listed in Table 3. The correlations are shown in left panel of Figure 8 between the reddening corrected continuum luminosity and [O ] line luminosity for the type-1 AGN discussed in Zhang et al. (2017b) and for the quasars in Shen et al. (2011) and for the quasars in rakshit et al. (2020). Based on the shown results in left panel of Figure 8, two points can be found. On the one hand, the results in Zhang et al. (2017b) are well consistent with those in Shen et al. (2011)  Before given the final conclusions, it is necessary to check whether randomly collected values of sin( ) can lead to similar results as those in Figure 7  It is clear that there are more than one third of the 500 simulating data points lying out of the 99.9999% confidence interval of the expected results from the empirical relation between NLRs sizes and [O ] luminosity in the type-2 AGN. However, among the 38 low-redshift double-peaked BLAGN in our final sample, only three objects lying out of the 99.9999% confidence interval. Therefore, the results of the 38 double-peaked BLAGN in Figure 7 have real physical properties.
Moreover, as we have discussed in Section 2, the longterm variabilities of the 38 double-peaked BLAGN should be checked, in order to ignore the probable BBHs model applied to describe the double-peaked broad H . The long-term light curves have been collected from the CSS and shown in Figure 9 for the 37 double-peaked BLAGN, besides the SDSS 2123-53793-0443 without light curve provided by the CSS. The commonly accepted generalized Lomb-Scargle technique (GLS) (Lomb 1976;Scargle 1982;Zechmeister & Kurster 2009;Zheng et al. 2016) is applied to check the probable QPOs signs. Here, as shown in Figure 9, there are no signs for apparent QPOs in the 37 double-peaked BLAGN. So that, there are no further discussions on the results in Figure 9, and the BBH model is not preferred in the collected doublepeaked BLAGN. The results shown in Figure 7 are the basic and natural results on the upper limits of NLRs sizes after considerations of inclinations.

SUMMARIES AND CONCLUSIONS
Finally, we give our main summaries and conclusions as follows.
• Double-peaked broad H of 38 low redshift ( < 0.1) double-peaked BLAGN can be well described by the elliptical accretion disk model, accepted the accretion disk origin of double-peaked broad emission lines leading to the well determined inclinations of central line emission regions.
• Considering the fixed SDSS fibers and the determined inclinations, upper limits of NLRs sizes in the 38 double-peaked BLAGN can be well estimated.
• • There are no QPOs signs in the long-term light curves of the collected low redshift double-peaked BLAGN, indicating the proposed BBH model is not preferred to explain the double-peaked broad H of the low redshift double-peaked BLAGN.
• The 38 double-peaked BLAGN have their upper limits of NLRs sizes not statistically against the expected results through the R-L relation for NLRs in type-2 AGN, indicating that the current results can not provide clues to challenge the Unified model through the space properties of NLRs. the National Science Foundation and the U.S. Department of Energy Office of Science. The manuscript has made use of the long-term variability data from the CSS (http://nesssi.cacr.caltech.edu/DataRelease/).

DATA AVAILABILITY
The data and program underlying this article will be shared on reasonable request to the corresponding author (xgzhang@njnu.edu.cn). Figure 9. Light curves of the 37 double-peaked BLAGN (panels in the first and the third column) and the corresponding GLS properties applied to detect QPOs signs ( panels in the second and the fourth column). In each panel for the GLS power properties, horizontal dashed red line shows the 99.99% confidence level (0.0001 as the false-alarm probability) for the probable periodicity.