The Quasar Mass-Luminosity Plane III: Smaller Errors on Virial Mass Estimates
aa r X i v : . [ a s t r o - ph . H E ] N ov Mon. Not. R. Astron. Soc. , 000–000 (0000) Printed 8 November 2018 (MN L A TEX style file v2.2)
The Quasar Mass-Luminosity Plane III: Smaller Errors onVirial Mass Estimates
Charles L. Steinhardt and Martin Elvis
Harvard-Smithsonian Center for Astrophysics, 60 Garden St, Cambridge, MA 02138
ABSTRACT
We use 62,185 quasars from the Sloan Digital Sky Survey DR5 sample to explore thequasar mass-luminosity plane view of virial mass estimation. Previous work shows de-viations of ∼ . .
13 and 0 .
29 dex for virial mass estimates. Across differentredshift bins, the maximum possible Mg II mass uncertainties average 0 .
15 dex, whileH β uncertainties average 0 .
21 dex and C IV uncertainties average 0 .
27 dex. Any phys-ical spread near the high-Eddington-ratio boundary will produce a more restrictivebound. A comparison of the sub-Eddington boundary slope using H β and Mg II massesfinds better agreement with uncorrected Mg II masses than with recently proposed cor-rections. The best agreement for these bright objects is produced by a multiplicativecorrection by a factor of 1.19, smaller than the factor of 1.8 previously reported asproducing the best agreement for the entire SDSS sample. Key words: black hole physics — galaxies: evolution — galaxies: nuclei — quasars:general — accretion, accretion discs
Strong correlations between the mass of supermassiveblack holes (SMBH) and the stellar velocity dispersion(Ferrarese & Merritt 2000; Gebhardt et al. 2000) and lumi-nosity (Ferrarese & Ford 2005) of their host galaxies ar-gue that our understanding of galactic formation is in-complete without an understanding of the SMBH foundat their centres. Our modern arsenal for learning aboutSMBH growth is predicated on four basic tools: (1) Largesamples containing ∼ quasars provided by mod-ern surveys (Schneider et al. 2007; Skrutskie et al. 2006;Croom et al. 2004); (2) Bolometric luminosity estimationcomparing a piece of the spectrum (Richards et al. 2006)to templates made from composite quasar spectra(Elvis et al. 1994); (3) The Soltan argument (Soltan 1982;Salucci et al. 1999; Yu & Tremaine 2002) that the inte-grated luminosity density of active galactic nuclei is consis-tent with the mass density in the local SMBH population;and (4) A series of SMBH mass estimation techniques com-prising the “BH Mass Ladder” (Peterson et al. 2004).Until recently, the first three of these tools mighthave been thought sufficient. The highest-redshift rung ofthe black hole “ladder”, virial masses, is the only op-tion at redshift z & .
2, and comparison with rever-beration mapping yields a ∼ . ∼ ∼ c (cid:13) Charles L. Steinhardt and Martin Elvis underlying physical causes. However, SMBH masses cannotsimply be well-estimated from the quasar luminosity, andso the investigation of SMBH evolution indeed requires allfour tools.Our initial work relied upon H β - and C IV -based virialmass estimates from Vestergaard & Peterson (2006) andMg II -based estimators from McLure & Dunlop (2004).There have been several attempts to improve thesevirial mass scaling relationships (Onken & Kollmeier 2008;Risaliti, Young, & Elvis 2009; Marconi et al. 2009). In thispaper, we reconsider virial mass scaling relations in light ofthese new boundaries in the quasar M − L plane.Every boundary in the SDSS quasar locus in the mass-luminosity plane is a combination of a real, underlying phys-ical boundary, SDSS selection, statistical uncertainty, andsystematic uncertainty. The Eddington luminosity and sub-Eddington boundary (SEB) characterise the quasars withthe highest Eddington ratios at each mass, or equivalently,quasars with the highest Eddington ratio at fixed luminos-ity. While we have termed this boundary “sub-Eddington”because of its behaviour at high mass, the SEB is flatterthan the Eddington limit and for the lowest-mass objectsat most redshift, the Eddington luminosity is the more re-strictive bound. As discussed in Paper I, the SDSS selectionfunction is not a factor in the location of the SEB. Therefore,the decline in number density is a combination of statisti-cal uncertainty, systematic uncertainty, and an underlyingphysical cutoff, and as such, its sharpness can be used toderive an upper bound on the maximum statistical uncer-tainty in virial mass estimation. The statistical uncertaintyhas been estimated as 0 . §
2, we use the sharpness of boundaries in mass toplace a tighter upper bound on the statistical uncertainty ofvirial mass estimates. In §
3, we show that proposed correc-tions to Mg II -based virial mass estimation are inconsistentfor the brightest quasars at 0 . < z < .
8. We discuss theimplications of these results in § We use black hole masses for 62,185 of the 77,429 SDSS DR5quasars (Schneider et al. 2007) as determined by Shen et al.(2008) using H β - and C IV -based virial mass estimators fromVestergaard & Peterson (2006) and Mg II -based estimatorsfrom McLure & Dunlop (2004). We divide the SDSS quasarpopulation into 14 redshift bins, of width 0.2 below z = 2and wider at higher redshift. In each bin, we consider themass distribution of quasars in a luminosity bin of width0.2 dex centred at the average bolometric luminosity for thecatalogue at that redshift. Choosing this fixed luminositygives us a large number of objects in our attempt to fit thedecay rate at low mass. The decay at low mass is more rapidthan at high mass. In each bin, we fit the decline in numberdensity as both a Gaussian and an exponential decay (Figure1), reporting the dispersion or e-folding in Table 1.What form should we expect from the decay in quasarnumber density? Let the true quasar mass distribution be ρ p ( M ) and let φ ( x ) be the probability distribution that avirial mass is incorrect by x . Then, the observed quasar mass Figure 1.
Top: The quasar M − L plane for MgII masses at0 . < z < .
6. We take the quasar number density as a functionof mass around the average luminosity (between the red, dashedlines). Quasar luminosity variability of 0 . .
15 dex (blue). Bottom: Thelow-mass decay in quasar number density as a function of mass at45 . < log L bol < .
66 is best-fitting by an exponential decaywith an e-folding of 0 .
15 dex (red). For comparison, the best-fitting decays using the average e-foldings for H β (cyan, 0 .
21 dex)and CIV (magenta, 0 .
27 dex) are also indicated.
Table 1.
Best-fitting forms for the decay in quasar number den-sity as a function of mass. These e-foldings imply maximum sta-tistical uncertainties of, on average, 0 .
21 dex for H β -based virialmasses, 0 .
15 dex for MgII, and 0 .
27 dex for CIV. z N h log L i σ (dex) χ ν e-folding (dex) χ ν H β distribution ρ o ( M ) is the convolution ρ o ( M ) = ( ρ p ∗ φ )( M ) = Z ∞−∞ ρ p ( µ ) φ ( M − µ ) dµ. (1)For example, if ρ p ( M ) were a step (Heaviside) function, i.e.,a constant number density at low mass and zero above theSEB, and the uncertainty φ ( x ) were Gaussian, we would see c (cid:13) , 000–000 he Quasar Mass-Luminosity Plane III: Smaller Errors on Virial Mass Estimates a decay taking the form of the error function, 1 + Erf ( x ) = R x −∞ e − t dt. In practice, the exact form of the tail is highlysensitive to ρ p , and the exponential decay described in Table1 is a better fit to the low-mass tail of ρ o than a Gaussian(Table 1), polynomial, or Erf ( x ). The exact form of ρ p ap-pears to be more complicated than a step function at massesabove the SEB (Figure 1). However, the convolution acts tospread out the signal, and therefore the dispersion of fea-tures in ρ o will be at least as large as those in φ ( x ). So, thee-foldings of best-fitting exponential decays in Table 1 areinconsistent with an 0.4 dex statistical uncertainty for virialmass estimation. Virial mass estimates take the formlog(
M/M ⊙ ) = A (2)+ log "(cid:18) FWHM(BLR line)1000 km/s (cid:19) (cid:18) λL λ ( B ˚A)10 erg/s (cid:19) C , for emission lines in the broad line region (BLR) and anearby continuum flux. For the virial mass estimates usedin the Shen et al. (2008) catalogue, C is between 0 . .
53 (McLure & Jarvis 2002; McLure & Dunlop 2004;Vestergaard & Peterson 2006). One possible explanation forthis discrepancy, then, is that the analysis above only con-siders quasars at fixed bolometric luminosity, while the adja-cent continuum is one component of virial mass estimation.Perhaps the statistical uncertainty in virial mass estimationis mostly caused by quasar variability, in which the bolo-metric luminosity (and adjacent continuum) change on atimescale of years while the black hole mass remains verynearly constant.Variability moves objects along a L = M line (Figure1), blurring each thin luminosity slice along the mass axisby 0.18 dex for typical long-term, 0.3 dex optical variations(de Vries et al. 2005). This blurring is larger than most ofthe Mg II decays, notably smaller than the C IV decays, butis consistent with the H β decays. The additional C IV scat-ter must have some other cause. Variability is wavelengthdependent in quasars, being stronger toward the UV, butthe difference between 2800 ˚A and 1500 ˚A is too small toexplain the different Mg II and C IV decaysSimilarly, the adjacent luminosity used to estimate theblack hole mass is based on the continuum local to each line,while the bolometric luminosity is based on the five SDSSphotometry points. Thus, systematic offsets between thesetwo luminosities could cause a scatter. The correction wouldhave to have a peculiar shape as the longest and shortestH β and C IV measurements both have larger scatter thanthe intermediate wavelength Mg II value. II VIEW OF THESUB-EDDINGTON BOUNDARY Mg II has the sharpest boundary in the M − L plane, andso appears to be the most precise mass indicator. Currently,though, H β -based masses are considered to be the most re-liable, because they have been calibrated directly against reverberation masses at low redshift. Certainly, several po-tential problems with C IV virial masses have been suggested(Shen et al. 2008; Marconi et al. 2009).Corrections have also been suggested to the Mg II massesderived by McLure & Dunlop (2004) (MD04). Onken &Kollmeier (2008) examine SMBH for which both H β andMg II masses are available and find that the Mg II -based M BH may be overestimated at high Eddington ratio and under-estimated at low Eddington ratio. Risaliti, Young, & Elvis(2009) (RYE09) quantify this correction empirically aslog[ M BH ( Hβ )] = 1 . × log[ M BH ( MgII )] − . . (3)In addition to correcting the central values of Mg II virialmasses, the RYE09 correction also increases the e-foldingdecay by the same factor of 1.8, to an average of 0 .
27 dex,which would make Mg II masses less precise than H β andcomparable to C IV .This multiplicative correction is surprising, because itrequires that the gas emitting either H β or Mg II lineshas a non-virial component. If H β is virial, as expectedfrom calibration between H β and reverberation masses(Vestergaard & Peterson 2006), then the mass-velocity re-lation for Mg II would need to be M ∝ v . . RYE09 proposethat this mismatch might instead be due to uncertainties inmeasurement of the Mg II line, mainly because of potentialFe II contamination.To test the multiplicative correction we examine theSEB more closely. The slope of the SEB is sensitive to themultiplicative correction to Mg II -based virial masses but notto any additive correction. As the SEB is composed of thebrightest quasars in each redshift bin they typically havethe best-measured spectra at each mass, minimising Mg II measurement errors. Both Mg II and H β masses are availableat the 0 . < z < .
8. Using the techniques detailed in PaperI we subdivided this bin into four redshift bins of width∆z= 0.1. We then calculated the slope of the best-fittingSEB in each bin using both the MD04 scaling relation andthe RYE09 correction. In Figure 2, we show the four H β SEB slopes (green) compared to slopes using MD04 (blue)and RYE09 (magenta).In each redshift bin, the deviations between MD04 andH β slopes are between 0.3 σ and 1.6 σ (MD04 averages 0 . σ higher). Deviations between RYE09 and H β are between0.9 and 2.7 σ (average 1.9 σ ), with the RYE09 slope alwayslower. If Mg II masses with the RYE08 correction producedidentical SEB slope estimates to H β , the probability that allfour measurements would be this far below the H β slope is0.2% (comparing to a Monte Carlo of normally-distributedmeasurements). A best-fitting correction between H β andMg II slopes (Figure 2) reduces the factor of 1.8 to one of1.19. The M − L plane view of Mg II virial masses for thebrightest quasars at each redshift is that corrections mightnot be necessary, and if necessary, are likely substantiallysmaller than previously proposed. Most of the RYE09 cor-rection may indeed be a result of measurement difficulties forMg II , which become unimportant for the brightest quasarswith the best-measured spectra. c (cid:13) , 000–000 Charles L. Steinhardt and Martin Elvis
Figure 2.
A comparison of best-fitting slopes to the sub-Eddington boundary in four redshift ranges using different virialmass scaling relations. The H β -based estimates (green) are a bet-ter fit for the original McLure & Dunlop (2004) MgII masses(blue) than the Risaliti, Young, & Elvis (2009) correction (ma-genta). The best match between slopes is produced by a smallerMgII correction (red). The black dashed line is drawn at the slopeof the Eddington luminosity The rapid falloff in quasar number density at fixed lumi-nosity near the sub-Eddington boundary (SEB), places astrong upper bound on the statistical uncertainty of virialmass estimation, of, on average, 0 .
21 dex for H β -based virialmasses, 0 .
15 dex for Mg II , and 0 .
27 dex for C IV . We consid-ered the possibility that the narrow spreads were induced byquasar variability. We find that variability can explain only ∼ II -based virial masses at theSEB implies, surprisingly, that the Mg II masses are morereliable than the H β -based masses. We also find that, at leastfor the most luminous objects in each redshift bin, any Mg II to H β mass corrections are likely smaller than previouslythought.We note that the ordering of the decay sizes is in-verse to the ordering of the emission lines by distance fromthe ionising continuum, as derived from reverberation map-ping (Peterson et al. 2004). This ordering suggests a phys-ical explanation in which the non-virial motions in thebroad-line region become relatively weaker with increas-ing distance from the continuum. An obvious candidateis a reduced role of radiation pressure at larger distances(Marconi et al. 2009), which would affect C IV most stronglyand Mg II the least.The measured decays provide a fixed error budget tobe distributed between non-virial motion and quasar lumi-nosity variability. If fluctuations in the adjacent continuumscale linearly with the quasar bolometric luminosity, the ∼ .
15 dex bound on Mg II masses limits quasar variabilityto ∼ .
33 dex. If the decay rate is due primarily to non-virial motion in the broad line region, then the contribution fromquasar luminosity variability is smaller.However, if the statistical uncertainty is truly smaller invirial masses than previously believed, then we must explainthe origin of the ∼ . .
09 dex for broad line region radii from reverber-ation mapping, with an 0 .
11 dex uncertainty for the H β line(Peterson, B. M., private communication). If the scatter inthe R − L relation is uncorrelated with other errors, thiswould restrict the remaining uncertainty to just ∼ . II . Further, these uncertainty measurements use animproved R − L relation (Bentz et al 2009), while the Shenet al. (2008) catalogue does not include these improvements.The two methods also differ in that virial estimates usethe entire emission line, while the best reverberation meth-ods use the rms line profile, i.e., the minority fraction of theemission line that responds to continuum changes. Korista etal. (2001) show that only the gas that is optimally emittinga given line will respond to continuum changes. The virialmethod averages over gas in the entire broad line region.In some manner, virial masses appear to be giving bettervalues than reverberation masses. We cannot satisfactorilyexplain this surprising result. We used the sub-Eddington boundary (SEB) in the quasarmass-luminosity plane to compare different virial mass es-timators for quasar black hole masses. In particular, theMg II estimator developed by McLure & Dunlop (2004)has been considered less reliable than the H β -based es-timator of Vestergaard & Peterson (2006), with sev-eral proposals for corrections (Onken & Kollmeier 2008;Risaliti, Young, & Elvis 2009; Marconi et al. 2009). Thequasar M − L plane indicates, surprisingly, that Mg II maybe the most reliable virial mass estimator. A decline in therelative importance of non-virial motions at large radii mayaccount for the differences in precision when using differentemission lines.Surprisingly, using the adjacent continuum as a proxyfor radius seems to be more precise than using time delays.If the statistical uncertainties of virial mass estimates arereally smaller than previously believed, this would be a sub-stantial improvement. It means that we produce better black c (cid:13) , 000–000 he Quasar Mass-Luminosity Plane III: Smaller Errors on Virial Mass Estimates hole mass determinations using one lower-resolution opticalspectrum than from a time-series of high-resolution spectra.Moreover, many of the conclusions drawn from quasarmass functions, in Papers I and II, and in other work relyon the ability to segregate quasars into mass bins. With anuncertainty of 0.4 dex, cross-contamination is a concern. Anuncertainty of 0.15 dex allows us to divide a quasar sampleinto 2-3 times as many independent bins. Given the largesystematic uncertainties in fitting bolometric luminositiesto templates, it is possible that quasar mass functions willbe more reliable than luminosity functions.The quasars that define the SEB are necessarily theclosest to the Eddington limit of their cohort. It could bethat this gives them a greater uniformity of properties, in-cluding more accurately virial motion in the broad-line re-gion, than quasars at lower Eddington rates. Lower Edding-ton rate quasars, such as those used in reverberation map-ping, might then have a wider dispersion in these same prop-erties, leading to the difference in observed spread. However,since the location SEB moves with redshift, something morecomplex than just the Eddington ratio would need to be re-sponsible for this possible greater uniformity in quasar prop-erties near the SEB.We also compared the SEB slopes using Mg II massesand H β . Proposed corrections to Mg II masses comparingthe two methods for the entire SDSS catalogue are quite se-vere, but these corrections might be due to uncertainties inthe measurement of Mg II lines parameters. For the brightestobjects at 0 . < z < .
8, the SEB produced by the Risal-iti, Young, & Elvis (2009) correction is a significantly worsematch for the SEB produced by H β than using uncorrectedMg II -based virial masses.The authors would like to thank Bradley Peterson, YueShen, Michael Strauss, and Jonathan Trump for valuablecomments. This work was supported in part by Chandragrant number G07-8136A. REFERENCES
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