February 2, 2008 Active Galactic Nuclei in the Sloan Digital Sky Survey: I. Sample Selection Lei Hao1,2, Michael A. Strauss1, Christy A. Tremonti3, David J. Schlegel1, Timothy M. Heckman4, Guinevere Kauffmann5, Michael R. Blanton6, Xiaohui Fan3, James E. Gunn1, Patrick B. Hall1, Zˇeljko Ivezi´c1, Gillian R. Knapp1, Julian H. Krolik4, Robert H. Lupton1, Gordon T. Richards1, Donald P. Schneider7, Iskra V. Strateva1, 5 0 Nadia L. Zakamska1, J. Brinkmann8, Robert J. Brunner9, Gyula P. Szokoly5 0 2 n ABSTRACT a J 4 We have compiled a large sample of low-redshift active galactic nuclei (AGN) iden- tified via their emission line characteristics from the spectroscopic data of the Sloan 1 v Digital Sky Survey. Since emission lines are often contaminated by stellar absorption 9 lines, we developed an objective and efficient method of subtracting the stellar con- 5 tinuum from every galaxy spectrum before making emission line measurements. The 0 1 distribution of the measured Hα Full Width at Half Maxima values of emission line 0 galaxies is strongly bimodal, with two populations separated at about 1,200km s−1. 5 0 This feature provides a natural separation between narrow-line and broad-line AGN. / h The narrow-line AGN are identified using standard emission line ratio diagnostic dia- p grams. 1,317 broad-line and 3,074 narrow-line AGN are identified from about 100,000 - o galaxy spectra selected over 1151 square degrees. This sample is used in a companion r t paper to determine the emission-line luminosity function of AGN. s a : v Subject headings: galaxies: active —galaxies: Seyfert—galaxies: starburst—galaxies: i X quasars: emission lines — surveys r a 1Princeton University Observatory,Princeton, NJ 08544 2Current address: Astronomy Department,Cornell University,Ithaca, NY14853; [email protected] 3Steward Observatory,University of Arizona, 933 North Cherry Avenue,Tucson, AZ 85721 4Department of Physics and Astronomy, Johns Hopkins University, 3400 North Charles Street, Baltimore, MD 21218 5Max-Planck Institut fu¨r Astrophysik,D-85748 Garching, Germany 6Center for Cosmology and Particle Physics, Department of Physics, New York University, 4 Washington Place, New York,NY 10003 7Department of Astronomy and Astrophysics, PennsylvaniaState University,UniversityPark, PA 16802 8ApachePoint Observatory,P.O. Box 59, Sunspot, NM88349-0059. 9DepartmentofAstronomyandNationalCenterforSupercomputerApplications,UniversityofIllinois,1002West Green Street, Urbana,IL 61801. – 2 – 1. Introduction Ever since the definition of Seyfert galaxies (Seyfert 1943) and the first recognition of quasars (Schmidt 1963), astronomers have putenormous effort into compiling large samples of active galac- tic nuclei (AGN, in this paper, “AGN” refers to active galactic nuclei at all luminosities, including quasars) and trying to understand the physics that powers them. This is not easy, since luminous AGN comprise only a few percent of normal galaxies. Based on the distinctive characteristics of AGN, different methods have been developed to search for AGN in various wavebands. In the optical, AGN show different colors from stars and normal galaxies, especially at high luminosity. Many surveys have used color selection (e.g. Schmidt & Green 1983; Boyle et al. 1990). In particular, the Sloan Digital Sky Survey (SDSS) (York et al. 2000) uses optical colors to identify quasar candidates, which are then observed spectroscopically (Richards et al. 2002). The color selection is very efficient but it requires that the optical luminosity of an AGN be at least comparable to the luminosity of its host galaxy for the color to be distinctive, and thus the color selection systematically misses less luminous AGN at low redshift. AGN also show strong optical and ultraviolet emission lines. Broadly speaking, AGN can be classified into two types: broad-line and narrow-line AGN. The former show broad permitted emission lines, with Full Width at Half Maxima (FWHM) of several thousand km s−1, while in narrow-lineAGN,boththepermittedandforbiddenemissionlines arenarrow, withFWHMs∼500 km s−1. This is comparable with emission lines in normal star-forming galaxies, but emission lines in narrow-line AGN have considerably greater ionization range. In particular, both high-ionization lines such as [NeIII]λ3869, [Ne V]λ3426 and [OIII]λ5007, andlow-ionization lines such as [OI]λ6300 and [NI]λ5200, are stronger in narrow-line AGN than in normal starforming galaxies. Based on this feature, narrow-line AGN can be identified by their distinctive emission line ratios. The first line ratio diagram was introduced by Baldwin, Phillips & Terlevich (1982), who suggested that AGN generically have greater [OIII]λ5007/Hβ (the flux ratio of [OIII]λ5007 to Hβ) than do galaxies whose emission lines are due to stellar processes. Veilleux & Osterbrock (1987) developed their idea and used diagnostic diagrams that consist of combinations of four line ratios: [OIII]λ5007/Hβ, [NII]λ6584/Hα, [OI]λ6300/Hα and [SII]λλ6716,31/Hα, and further developed a semi-empirical line on the diagrams to separate AGN and starburst galaxies. The emission-line pairs are chosen specifically so that the two lines in a ratio are at nearly identical wavelengths, therefore reddening and spectrophotometric uncertainties are not big effects. These diagnostic diagrams have been used ever since as a standard to identify narrow-line AGN. Kewley et al. (2001) developed a set of theoretical separation lines for AGN and starforming galaxies on the diagnostic diagrams. By constructing a detailed continuous starburst model with large realistic metallicity and ionization parameter ranges, they found that the model folds on the diagnostic diagrams and there exist upper limits for starforming galaxies. These upper limits can be used to separate AGN and starforming galaxies and they can be fitted to a simple rectangular – 3 – hyperbolic shape: [OIII]λ5007 0.61 log = +1.19 (cid:18) Hβ (cid:19) log([NII]/Hα)−0.47 [OIII]λ5007 0.72 log = +1.30 (1) (cid:18) Hβ (cid:19) log([SII]/Hα)−0.32 [OIII]λ5007 0.73 log = +1.33 (cid:18) Hβ (cid:19) log([OI]/Hα)+0.59 These theoretical separation lines between AGN and starforming galaxies are widely accepted to use to identify narrow-line AGN in the diagnostic diagrams. Kauffmann et al. (2003a), on the other hand, when studyinghost galaxy properties of narrow- line AGN in the SDSS, proposed an empirical and more lenient cut to identify AGN: [OIII]λ5007 0.61 log ≥ +1.3 (2) (cid:18) Hβ (cid:19) log([NII]/Hα)−0.05 This criterion will select many more galaxies as AGN than Kewley’s criteria. In this paper, we apply both criteria and discuss their differences in detail. There are several spectroscopic galaxy surveys from which people have tried to select AGN based on their emission lines. One is the CfA redshift survey (Davis, Huchra & Latham 1983; Huchra et al. 1983). Spectra of about 2,400 galaxies were taken to study their large scale distri- bution. Huchra, Wyatt & Davis (1982) used these spectra to select AGN by identifying emission lines indicative of nonstellar activity. As a result they found roughly 50 Seyfert galaxies, divided approximately equally between Seyfert 1 and Seyfert 2 galaxies, making the local Seyfert fraction ∼ 2% (Huchra & Burg 1992). AnotherexampleistheRevisedShapley-Amescatalogofbrightgalaxies (Sandage&Tammann 1981). Among its ∼1,300 galaxies, about 50 are Seyfert galaxies of luminosity comparable to those found in the CfASurvey. Ho et al. (1997a) took uniform high signal to noise ratio (S/N) spectra of 486 galaxies using very small apertures centered on the nuclei. 418 galaxies were found to contain emission-line nuclei, of which 206 are star-forming, and 211 show AGN activity. This demonstrates that low-luminosity AGN activity in the local universe is extremely common, as first discussed by Phillips, Charles, & Baldwin (1983). Hall et al. (2000) systematically selected high redshift AGN based either on their broad emis- sion lines or narrow [NeV] emission lines from the Canadian Network for Observational Cosmology field galaxy redshift survey (CNOC2; Yee et al. 2000). They found 47 confirmed and 14 candidate AGN in the redshift range 0.27 ≤ z ≤ 4.67. The SDSS has opened a new window to the study of AGN by providing a huge number of high quality spectraof galaxies. Therehave beenseveral studiesidentifying AGN spectroscopically from different subsamples of SDSS galaxies (Kauffmann et al. 2003a; Miller et al. 2003). Both studies – 4 – have focused on narrow-line AGN identified from the diagnostic diagrams. Miller et al. (2003) found AGN signatures in at least 20% of 4,921 galaxies in the redshift range 0.05 < z < 0.095, and studied the environment of these AGN. Kauffmann et al. (2003a) selected 22,623 narrow-line AGN from a parent sample of 122,808 galaxies and studied their host galaxy properties. The AGN detection rate, however, depends very much on the AGN selection criteria used and many other details in defining the sample. In this paper, we apply a systematic search for both broad and narrow-line AGN in a well-defined sky area in the redshift range 0 < z < 0.33. The framework of our narrow-line AGN selection is similar to Kauffmann et al. (2003a), but we differ in many details. In section 2, we give a brief overview of the SDSS, focusing on those aspects most relevant to our study. §3 introduces the parent sample from which we will select our AGN. In §4, we will discussthesubtractionofstellarabsorptionlinesfromthespectra. Theemissionlinemeasurements are discussed in §5. In §6, the selected AGN are presented and discussed. §7 gives a clean AGN sample and we summarize in §8. 2. The Sloan Digital Sky Survey The Sloan Digital Sky Survey (York et al. 2000) is an imaging and spectroscopic survey that will eventually cover approximately one-quarter of the Celestial Sphereand collect spectra of ∼ 106 galaxies and 105 quasars. It uses a dedicated 2.5m telescope at Apache Point, New Mexico, with a 3 degree field, and a mosaic CCD camera and two fiber-fed double spectrographs to carry out ′′ the imaging and spectroscopic surveys respectively. A separate 20 photometric telescope is used for photometric calibration (Smith et al. 2002; Hogg et al. 2001). The imaging camera (Gunn et ′′ al. 1998) consists of a mosaic of 30 imaging CCDs with 24µm pixels subtending 0.396 on the sky. The sky is observed through five broad-band filters (u,g,r,i,z) (Fukugita et al. 1996; Stoughton et al. 2002) covering the entire optical band from the atmospheric cutoff in the blue to the sensitivity limit of silicon CCDs in the red. The imaging is done in drift-scan mode and the total integration time per filter is 54.1s. The 50% completeness limits for point sources are 22.5, 23.2, 22.6, 21.9 and 20.8 magnitudes respectively and the photometric calibration is reproducible to 3%, 2%, 2%, 2% and 3% for the five bandpasses, respectively. The image data are processed by a series of automated pipelines (Lupton et al. 2001; Stoughton et al. 2002; Pier et al. 2003), which make various measurements of the flux of each detected object. The SDSS is obtaining spectra of complete samples of three categories of objects: Galaxies, Luminous Red Galaxies and Quasars. These spectroscopic targets are selected from the imaging dataviavarioustargetselectioncriteria. GalaxytargetselectionisdiscussedinStraussetal. (2002). Briefly speaking, galaxies are separated from stars by morphology. The magnitude limit cut for the galaxy sample was changed several times during commissioning, but is currently r = 17.77, where r represents the r band Petrosian magnitude. All magnitudes are corrected for extinction following Schlegel et al. (1998). These objects are the sample from which we will select AGN. – 5 – Quasar target selection (Richards et al. 2002) is based on the nonstellar colors of quasars and matching unresolved sources to the FIRST radio catalog. Luminous Red Galaxies are selected (Eisenstein et al. 2001) by a variant of the photometric redshift method, aiming to have a uniform, approximately volume-limited sample of highly luminous objects with the reddest colors in the rest frame to z = 0.5. In this paper, we will not discuss AGN selected from these two samples, unless they also satisfy the galaxy sample magnitude cut r = 17.77. The SDSS spectra are taken with two fiber-fed spectrographs, covering the wavelength range 3800-9200 ˚A over 4098 pixels. Each plate can hold 640 fibers, with a fixed aperture of 3′′. The plates are positioned by a tiling algorithm (Blanton et al. 2003) and fibers are assigned to targets. Galaxiesareamongthetiledtargetsthathavethehighestpriorityofhavingtheirspectrataken. The finitediameter ofthefibercladdingprevents fiberson anygiven platefrom beingplaced closer than ′′ 55 apart. The resolution λ/∆λ varies between 1850 and 2200. The relative spectrophotometry is accurate to about 20%. Each spectrum is accompanied by an estimated error per pixel, based on photon statistics and the amplitude of sky residuals. The typical S/N for galaxy spectra at the sample limit is 16/pixel. Thespectroscopicdataarereducedthroughthespectroscopicpipelines,spectro2dandspecBS. Spectro2d reduces the 2-dimensional spectrograms produced by the spectrographs to flux- and wavelength-calibrated spectra. SpecBS is different from the SDSS official pipeline, it determines classifications and redshifts via a χ2 fit to the spectrum in question with a series of rest-frame star, galaxy and quasar templates. The basic technique is described by Glazebrook et al. (1998) and Bromley et al. (1998). Even though specBS has made measurements on emission lines by fitting a single Gaussian at positions of each expected emission line, we carry out our own fits in our study. The main reason is that a single Gaussian fit is not adequate to model some AGN having both broad and narrow emission lines. The emission lines are measured directly from calibrated spectra, as we describe in detail in §5. 3. Parent Sample In order to do statistical studies of AGN, it makes sense to start with a complete sample of objects within a certain well defined area from the SDSS. We start from 129,625 target objects complete in 1151 square degrees. This is about 1/2 of the spectra available in the SDSS Second Data Release (Abazajian et al. 2004). Among these objects, there are 98,684 galaxies targeted by the main galaxy target algorithm (Strauss et al. 2002; §2) and 17,972 quasar target objects (Richards et al. 2002; §2). In this paper, we will focus on the galaxy target objects. However, there are 2057 extragalactic objects selected as quasar targets with r(Petrosian) ≤ 17.77, which were not targeted as galaxies because they are unresolved, and we include them in our galaxy sample. Among these 100,741 galaxies, most of them (99,990) have redshift z < 0.33, guaranteeing that the – 6 – Hα emission line lies in our spectral coverage. Hα is a very useful emission line in identifying AGN (Equation 2), and we limit our galaxy sample to these 99,990 objects. 4. Stellar Subtraction AGN are identified by their emission line characteristics. As described in §2, the SDSS galaxy ′′ spectraaretakenthroughafixed3 aperture,whichislargeenoughtoletthroughnotonlythelight from the nucleus but also substantial amounts of stellar light from the host galaxy. For example, at the median redshift of the sample (z = 0.1), a 3′′ aperture subtends about 4h−1kpc. Moreover, galaxies with higher redshift will have a larger host galaxy component in the observed spectra. Thus the nuclear emission lines are often contaminated by the stellar absorption lines of the host galaxy. For weak AGN, this contamination can be so severe that the interesting emission lines are completely submerged in the absorption lines. Thus before considering AGN selection, we have to develop a technique to properly remove the stellar absorption lines. The basic idea of stellar subtraction is to build a library of stellar absorption line spectra templates,andusethemasbuildingblockstosimulatethestellarspectrumoftheobjectinquestion. The library needs to be complete in the sense that it contains enough information on various absorption features to be able to simulate the stellar components of various galaxies with widely spreadmetallicities, ages and velocity dispersions. Thelibrary is typically composed of star spectra generatedfromapopulationsynthesiscode(Bica1988; Saraivaetal. 2001,Kauffmannetal. 2003b) or direct observational spectraof absorption linegalaxies (Ho et al. 1997a) or stars (Engelbracht et al. 1998). In this paper the library is constructed by applying the Principal Component Analysis (PCA) technique (e.g Connolly et al. 1995; Lahav et al. 1996; Bromley et al. 1998; Eisenstein et al. 2003; Yip et al. 2004) to a sample of pure absorption-line galaxies. The advantages of building stellar absorption templates via this method are two-fold: only the first few eigenspectra are significant, so we can limit the size of the library without losing much useful information. Moreover, the eigenspectra are orthogonal to each other, resulting in a unique solution to the stellar subtraction fit using these templates. 4.1. Preparing the Absorption Line Galaxy Sample We wish to identify a sample of pure absorption-line galaxies. In practice, we require that the Hα equivalent widthEW(Hα)<0 (positive EW correspondsto emission lines), and that [OII]λ3727 not be detected. [OII]λ3727 is used because it is always apparent even in very weak emission line galaxies. Due to the presence of complex absorption near [OII]λ3727, we measure this line after subtracting a preliminary PCA sample solely with the requirement that EW(Hα)<0. A Gaussian function is fit to the residuals between λ = 3700 and λ = 3754. If the χ2 of the fit is less than the χ2 of a linear fit minus 3, the line is considered significant, and the object is rejected as an – 7 – absorption-line galaxy. By limiting ourselves further to high S/N spectra, we defined three samples of several hundred pure absorption line galaxies grouped by their redshifts: 325 galaxies with 0.02 < z < 0.06; 338 with 0.06 < z < 0.12 and 372 with 0.12 < z < 0.22. PCA is done on each group separately, giving three sets of eigenspectra that each can be used to subtract stellar components for galaxies of similar redshifts. We divide the sample in three groups in order to obtain a larger wavelength coverage for each group as well as the resultant eigenspectra. Figure 1 shows the mean of the spectra (i.e., the firsteigenspectrum) in each galaxy group. They are essentially identical, although the lowest-redshift sample has slightly higher S/N. The galaxies within the three groups are shifted to fixed rest-frame wavelength bins using sinc interpolation. Afterwards, each spectrum is normalized to a constant flux value. Unlike some PCA analyses(e.g., Eisensteinetal. 2003), thecontinuaarenotsubtractedfromthespectrabeforePCA. Each sample includes several hundred normalized galaxy spectra, which we express as a matrix S of dimensions N ×M, where N is the total number of galaxies in each group and M is the total number of common wavelength bins. Singular Value Decomposition (SVD) is used to build the eigenspectra (Connolly et al. 1995; Bromley et al. 1998). 4.2. PCA Result and Stellar Subtraction The PCA analysis generates a set of eigenspectra, with main features of the absorption-line galaxies concentrated in the first few. In this study we will use the first eight eigenspectra as the stellar absorption-line templates. Empirical tests show that including more eigenspectra, which are basically just noise, does not improve the subtraction further. A χ2 minimizing algorithm was adopted to determine the synthetic stellar absorption spectrum for each galaxy. The minimizing is done over the entire observed wavelength range except regions around the strongest emission lines found in AGN and starforming galaxies: Hα, Hβ, [NII]λλ6548,84, [OIII]λ5007,λ4959 and [OII]λ3727. Since the stellar templates are eigenspectra of a sample of pure absorption line galaxies and galaxies having young stellar populations tend to have emission lines, the resultant eigenspectra mainly represent old stellar spectral features. Thus these eigenspectra are not representative of galaxies containing a young stellar component. One possible resolution would be to make sure that the PCA sample included enough E+A galaxies, which contain young stellar populations and do not have emission lines (Goto et al. 2003; Quintero et al. 2004). In this work, however, we simply add an A star spectrum selected from the SDSS spectroscopic data to the absorption-line template library to represent the young stellar population. Figure 2 shows an example of the stellar subtraction with and without this template, for a galaxy with a young star population. The improvement using the A star template is dramatic. The stellar subtraction is automatically done to all galaxies in our parent galaxy sample, – 8 – includingthosequasarsthatsatisfythegalaxy target selection magnitudecut(§2and3). However, doingstellarsubtractiontoabrightquasardominatedbyanon-thermalcontinuuminthespectrum will certainly be a disaster. We thus add a pure power-law spectrum, written as e ∝ λ−1.5, powerlaw to our template library. Figure 3 shows an example of stellar subtraction for a quasar with and without the power-law template. The power-law template significantly improves the subtraction. Since the spectrophotometry is not perfect, quasars have a range of intrinsic power-law slopes (Richards et al. 2002), and the power law fit can be systematically in error due to FeII emission (Figure 3 and Vanden Berk et al. 2001), the power-law template will certainly not be sufficient for clean continuum subtraction for all quasars. However we have found it to be adequate for our purposes. For galaxies thatdonothave anonthermalpower-law component, thistemplate canhelp compensate for continuum shape errors due to errors in spectrophotometry and internal reddening. Insummary,ourtemplatelibraryincludes8PCAeigenspectraofpureabsorptionlinegalaxies, a power-law continuum and an A star spectrum. As demonstrated in Figures 2, 3 and further in Figure 4, weak emission lines such as Hβ that were originally submerged in the stellar absorption line are successfully recovered. Moreover, emission lines such as [OI]λ6363 and [NI]λ5200 which are notapparentatallintheoriginalspectrumclearly standoutafter thesubtraction. Thesubtraction also helps correct the strength of Hα, which is very important in subsequent AGN identification using emission line ratios. The stellar subtraction is robust for galaxies of a range of velocity dispersions. Figure 4 shows two galaxies with very different velocity dispersions (Bernardi et al. 2003); note that in both cases, the subtracted spectrum shows no appreciable residuals of the strong absorption lines. The eigenspectra include terms that can give absorption lines of different width. This is another advantage of the PCA technique over the use of stellar libraries: we do not need to convolve each template with a Gaussian broadening function for each galaxy. The whole procedure of doing PCA and stellar subtraction using the resultant eigenspectra is quite straightforward. Once the stellar component is subtracted, the emission lines can be measured, as we now describe. 5. Emission Line Measurements Sincewearemainlyinterestedinemission-linegalaxies, wewillfirstsetupacriteriontoremove the pure absorption-line galaxies from the parent sample. We require that the EW of the Hα line (in the rest frame) be greater than 3˚A. In the rare cases in which the Hα line is saturated or affected by bad pixels, we examine the equivalent width of the [OIII]λ5007 and Hβ lines, requiring that one of them be greater than 3˚A. These equivalent widths are based on the line strength after stellar continuum subtraction, divided by the stellar continuum itself. This criterion also rejects weak emission line galaxies. This is acceptable since the S/N ratio for these lines will be low and thus our ability to distinguish AGN from starbursts will be poor. A – 9 – total of 42,435 galaxies, about half of the parent galaxy sample, pass the equivalent width cut. We refer to this sample as the emission line galaxy sample. All AGN are selected from this sample. Theintensity, fullwidthathalfmaximum(FWHM),centralwavelength andnearbycontinuum value of the main emission lines in these galaxies are measured via Gaussian fits weighted by the estimated errors per pixel. The following emission lines are needed for AGN selection: Hα, [NII]λ6584,48, Hβ, [OIII]λ5007, [SII]λ6716,31 and [OI]λ6300. The following lines are fit with a single Gaussian: Hβ, [OIII]λ5007 and [OI]λ6300. The [SII] doublet is fit with a two-Gaussian function model. The FWHMs and intensities of the two Gaussians are independent of each other, while the central wavelengths are correlated with a single variable z, and the continuum is shared by thetwo Gaussian functions. Sinceourfittingis donetoa stellar continuum subtracted spectrum that has been shifted to rest wavelength, the fitting results for z and the continuum are very close to zero. The Hα and [NII]λ6548,84 lines are fitted with three Gaussians. The FWHMs of the [NII] lines are kept the same and the intensity ratio of [NII]λ6584 to [NII]λ6548 is fixed to 3, as required by the energy level structure of the [NII] ion (Osterbrock 1989). Again, the central wavelengths of Hα and the [NII] doublet are correlated with a single redshift parameter and the three lines share the same continuum value. The pixels located within 100˚A of the central wavelength of the emission lines are used for Gaussian fits (adjacent emission lines in this range are shielded out). However, in order to be sensitive to a broad component of Hα, we fit the Hα, [NII] group to the range 6565˚A ± 300˚A (the adjacent [OI] and [SII] lines are masked in the fit). We will use the χ2 of the fit to test for the significance of the broad component, restricting ourselves to the range 6565˚A±80˚A to reduce the sensitivity to uncertainties in the continuum. Some AGN show both broad and narrow permitted emission lines. For these galaxies, a single Gaussian function for Hα is obviously not appropriate. To take this into account, we also fit the Hα and [NII] doublet with a four-Gaussian model: two for the [NII] lines and two for Hα. The two Hα Gaussian functions have the same central wavelength, but different intensities and FWHMs. The final decision of which model (the three-Gaussian model or the four-Gaussian model) to use for a given galaxy is done by comparing the χ2 of the two model fits, χ2 and χ2 respectively. In 3 4 particular, we choose the four-Gaussian model fit when: (χ2−χ2−2)/χ2 > 0.2 (3) 3 4 4 This criterion is empirical, but is inspired by a similar statistic for linear fitting models (Lupton 1993). If we would like to fit a data set (x ,y ) with a linear fitting model y = y(x), and the i i model errors σ are Gaussian, then i y −y 2 χ2 = i model (4) (cid:18) σ (cid:19) Xi i follows a χ2 distribution, where n and k are the numbers of the data points and the parameters n−k used in the model respectively. If there are two linear models M and N, with k and (k − r) parameters respectively (i.e. model M has r extra parameters), their χ2 functions χ2 , χ2 will N M then follow χ2 and χ2 distributions. It can be proved that for linear models, χ2 and n−k n−(k−r) M – 10 – χ2 −χ2 are independent, thus the quantity M N (χ2 −χ2 )/r f ≡ N M (5) χ2 /(n−k) M follows a Fr,n−k distribution. If f is large, then we can accept the hypothesis that the added parameters significantly improve the fit. In our case, our models are non-linear in the parameters, and χ2 and χ2 − χ2 will not M M N be independent. The four-Gaussian model has two more degrees of freedom than does the three- Gaussian model, so the numerator of the criterion is written as (χ2 − χ2 − 2). The limit “0.2” 3 4 in equation (3) is an empirical number and is demonstrated to be appropriate from our manual inspection. However it is not always unambiguous to identify the broad Hα component using the above criterion,sincenotallemissionlinesarewell-fitwithGaussians(Stratevaetal. 2003). Inparticular, narrow emission lines generally have extended wings at their bases (Ho et al. 1997b). If the narrow emission line is strong, this non-Gaussian feature will become prominent, and a 4-Gaussian model will be chosen by Equation 3. In this case, the height of the broader component h will be small 1 compared to the height of the narrow component, h and the Gaussian width of the broader 2 component σ will be relatively small. To not count such cases as broad line, we stick with the 3-Gaussian fit for those objects which satisfy, σ < 20˚A(∼ 2200km s−1) and h /h <0.1 (6) 1 2 no matter what the criterion of equation (3) might indicate. Wecompared ourempirical criterion (equation (6)) withthewell definedBayesian Information Criterion (BIC, Liddle 2004). We found that we are as efficient in choosing broad-line component as BIC, and at the same time, our criterion identifies far fewer fake broad-line components than BIC does. Afterfittingallrelevantemissionlines,thelinestrengthsarecalculated basedonthelinefitting parameters. Theobserved SDSSspectraareaccompanied by an estimated error perpixel, based on photon and read noise statistics and variation among sky spectra. These errors are approximately independent between pixels and are good to 8% (McDonald et al. 2004). In stellar continuum subtraction we assumed the stellar templates perfect since they are very high S/N. The errors in the line strengths are calculated using standard propagation of errors. Those emission lines that are less than 3σ detections are marked as weak. If they are needed in identifying an AGN, special care should be applied (§6.2).