The Fundamental Plane of QSOs and the Relationship Between Host and Nucleus
aa r X i v : . [ a s t r o - ph ] F e b The Fundamental Plane of QSOs and the Relationship BetweenHost and Nucleus
Timothy S. Hamilton, , , , , Stefano Casertano, , and David A. Turnshek , ABSTRACT
We present results from an archival study of 70 medium-redshift QSOs ob-served with the Wide Field Planetary Camera 2 on board the
Hubble Space Tele-scope . The QSOs have magnitudes M V ≤ −
23 (total nuclear plus host light) andredshifts 0 . ≤ z ≤ .
46. The aim of the present study is to investigate the con-nections between the nuclear and host properties of QSOs, using high-resolutionimages and removing the central point source to reveal the host structure. Weconfirm that more luminous QSO nuclei are found in more luminous host galax-ies. Using central black hole masses from the literature, we find that nuclearluminosity also generally increases with black hole mass, but it is not tightly cor-related. Nuclear luminosities range from 2.3% to 200% of the Eddington limit.Those in elliptical hosts cover the range fairly evenly, while those in spirals areclustered near the Eddington limit. Using a principal components analysis, wefind a kind of fundamental plane relating the nuclear luminosity to the size andeffective surface magnitude of the bulge. Using optical nuclear luminosity, thisrelationship explains 96.1% of the variance in the overall sample, while anotherversion of the relationship uses x-ray nuclear luminosity and explains 95.2% ofthe variance. The form of this QSO fundamental plane shows similarities to thewell-studied fundamental plane of elliptical galaxies, and we examine the possiblerelationship between them as well as the difficulties involved in establishing thisconnection.
Subject headings: quasars: general — galaxies: active — galaxies: fundamentalparameters Research performed while a National Research Council Associate at NASA/GSFC Dept. of Physics & Astronomy, University of Pittsburgh, Pittsburgh, PA 15260, USA Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA Present address: Shawnee State Univ., Dept. of Natural Sciences, 940 2nd St., Portsmouth, OH 45662 email: [email protected], [email protected], [email protected]
1. INTRODUCTION
The availability of high-resolution, space-based imagery has been of great help to thestudy of QSOs and their host galaxies. Many low-to-medium redshift QSO hosts have beenimaged with the Hubble Space Telescope in the past several years. At the same time,there has been a growing understanding of the workings of galactic nuclei and their centralblack holes. Magorrian et al. (1998) have found a proportionality between black hole andgalactic spheroid masses in normal galaxies, and this has been refined by Kormendy &Gebhardt (2001), H¨aring & Rix (2004), and McLure et al. (2006). McLure & Dunlop (2001,2002), Marconi & Hunt (2003), and Graham (2007) have studied this relation for activegalaxies. Black hole mass has also been found to be related to the central stellar velocitydispersion (Ferrarese & Merritt 2000; Gebhardt et al. 2000; Tremaine et al. 2002; Ferrarese &Ford 2005; Bernardi et al. 2007). The velocity dispersion is itself tied to larger-scale galaxyproperties through the fundamental plane of elliptical galaxies (Djorgovski & Davis 1987;Dressler et al. 1987). So there is now a wealth of evidence emerging on the individualrelationships between large-scale properties of galaxies and those of their central regions.In this paper we report on the connections between the nuclear and host properties ofa large sample of low-redshift QSOs observed with the Wide Field and Planetary Camera 2(WFPC2) on board the
Hubble Space Telescope ( HST ). This follows our study of the QSOhost luminosity function, described by Hamilton et al. (2002). We have collected and rean-alyzed wide-band archival images of 70 QSOs with M V ≤ −
23 mag (total nuclear plus hostlight) and redshifts 0 . ≤ z ≤ .
46. This includes images taken by several teams, especiallyBahcall et al. (1997), Hooper et al. (1997), Boyce et al. (1998), and McLure et al. (1999),among others. We have taken an inclusive approach in our sample selection, imposing noadditional selection criteria on the QSOs besides those of total absolute magnitude and red-shift. For each object we have subtracted the nuclear light component using two-dimensionalimage fits and have derived the luminosity and size of the underlying host galaxy by fittingan elliptical or disk-like light profile. A comparison of our results with other teams using thesame data is given in § A.We examine the host and nuclear luminosities in § §
4, we compare literature estimates of central black hole masses with our nuclear and bulgeluminosities. Nuclear luminosity generally increases with black hole mass, though there isnot a strong correlation. Bulge luminosity shows little trend at all. We find the nuclearbolometric luminosities span a wide range of Eddington fractions, except among QSOs inspiral hosts, which cluster near the Eddington limit. 3 –Using a principal components analysis, we demonstrate the existence of a fundamentalplane connecting the nuclear luminosity to the host bulge’s size and effective surface mag-nitude ( § § B.Throughout this paper, we adopt a Friedmann cosmology with H = 50 km s − Mpc − , q = 0 .
5, and Λ = 0, to be in keeping with Hamilton et al. (2002). The effects on the resultsof assuming a more recent cosmology, H = 65 km s − Mpc − , Ω Λ = 0 .
7, Ω matter = 0 .
3, arediscussed in §
2. DATA2.1. Sample
The sample is composed of 70 archival
Hubble Space Telescope ( HST ) images of low-redshift QSOs. The selection criteria are that they have redshifts between 0 . ≤ z ≤ . M V ≤ −
23. Furthermore,they must have been observed with the
HST ’s Wide-Field Planetary Camera 2 (WFPC2),using broad-band filters, and have images publicly available in the
HST archives as of 1999.This brings our sample to 70 QSOs.Rather than restrict our study to a specific class of QSOs, we impose no physical criteriaon the QSOs beyond those of magnitude and redshift. Thus we are able to study a broadrange of properties and draw general conclusions.
Even at
HST resolution, the light of the unresolved nuclear central source significantlyaffects the extended light distribution of the host galaxy. A careful subtraction of the centralpoint source is needed in order to measure the properties of the host accurately. The imagereduction is described in detail by Hamilton et al. (2002). The following is a brief summaryof the method, which is largely similar to that of Remy et al. (1997).Because of the complex structure of the
HST
WFPC2 Point Spread Function (PSF),our analysis procedure has three principal steps: (1) A two-dimensional model of the PSF isfitted to the central point source, in order to determine its subpixel position and the telescope 4 –focus, which affect the shape of the PSF. (2) The PSF and a galaxy model are simultaneouslyfitted to the entire image to distinguish the nuclear and host components. (3) The nuclearmagnitude is determined from the PSF model. Then the fitted PSF is subtracted, and themagnitude of the host is determined from the residual light.The model PSF is constructed from a set of artificial PSFs, created using the TinyTimsoftware (the latest version is described by Krist & Hook 2004). Pixel sampling is important(especially for the Wide Field images) because in the shape of the response function, thePSF structure changes significantly, depending on both the telescope focus and on exactlyhow the point source is centered with respect to the pixel grid. For instance, a point sourcecentered in the middle of a pixel will look different from one centered on a corner. If thiscentering is not accounted for, the model will not correctly apportion the light between thePSF and the extended source. The WFPC2 camera used for each observation is listed inHamilton et al. (2002), Table 1.The general PSF model is first fitted to the core of the QSO image, where the nuclearpoint source is the dominant feature, with the light of the extended galaxy treated as aconstant background term. Saturated pixels and those affected by CCD “blooming” aremasked from the fits. If the PSF is not saturated, we can achieve a precision of ≈ . ≈ µ m in the focus position, for QSOs dominated by their nuclei.Our technique works in the presence of saturation, although the focus is determined lessaccurately. Once the position and focus have been found, a PSF of angular size large enoughto cover the host image is created with these parameters, and it is used in the subsequentanalysis.A second two-dimensional fit distinguishes the light of the QSO from that of the resolvedhost galaxy, simultaneously fitting both parts. In this step, the model PSF’s brightness isscaled to match the QSO nuclear brightness, while a galaxy model is fitted to the host. Thehost model includes a simple morphological classification based on radial profile.For spiral hosts with a visible bulge, we use an elliptical mask to fit the bulge anddisk separately. The mask’s shape is determined by the QSO’s isophotal ellipticity andorientation. It exposes the region in which the bulge dominates the disk. The bulge+PSF isfitted first, using the mask. Next, the bulge model is subtracted from the entire image, andfinally the disk is fitted. Bars and any other irregular components are not modeled; theyhave masks drawn by eye. This includes the irregularly-shaped bulges in PG 0052+251,MRK 1014, PG 1001+291, LBQS 1222+1010, and PG 1402+261. Of the regular bulges, wenote that some are best modeled by a disk-like radial profile, while others are best fittedwith an elliptical galaxy type profile. 5 –Based on visual inspection, some hosts appear to have undergone recent, strong inter-actions that have severely distorted their appearance from those of a normal elliptical orspiral, and we have noted these in Table 1. The “interacting” category is rather subjective,and if seen at higher resolution, many others might have been classed this way.Using the fitted parameters, we then subtract the properly scaled PSF from the QSOimage, leaving the host galaxy. The magnitude of the nucleus is measured directly from thescaled PSF model. The host magnitude is measured from the PSF-subtracted image, withinan aperture large enough to encompass the visible extent of the host. Outside the aperture,we extrapolate the host model to a radius of infinity and add this contribution to the lightcontained within the aperture, yielding the apparent magnitude of the host galaxy. Thesize of the host is represented by the half-light radius, r / , in the case of ellipticals or theeffective radius, r eff , in the case of spirals. This parameter is taken directly from the fittedhost model.The absolute host and nuclear magnitudes are calculated as described in § V magnitude by applying a color correction. Colorsare taken from Cristiani & Vio (1990), for the nuclei, and from Fukugita et al. (1995), forthe hosts. Cristiani & Vio (1990) use the Johnson filter set, so we convert the nuclear colorsto the WFPC2 filters, using the IRAF synphot package and a power-law spectrum of theform f ν = constant. Once we have the apparent V magnitudes, the absolute V magnitudesare given by M V = m V − . − z − √ z ) − K ( V ) − A V , (1)where K ( V ) is the V -band K-correction, and A V is the Galactic extinction. Nuclear K-corrections are also taken from Cristiani & Vio (1990). For host K-corrections, we use thedata of Pence (1976), which assumes no intrinsic reddening in the host galaxies. Galacticextinction, A V , comes from Schlegel, Finkbeiner, & Davis (1998), by way of the Galacticextinction calculator on the NASA Extragalactic Database (NED).The error bars used in the plots and listed in Table 2 are the 1 σ statistical uncertainties.In looking at the systematic errors, tests on simulated QSOs (including elliptical, disk, anddisk+bulge hosts) show that we can recover their parameters rather well. In these tests,nuclear magnitudes are particularly accurate, with errors consistently less than 0.1 mag, andcomparing favorably with the statistical uncertainties of ≈ .
15 mag. Host magnitude errorsare . . ∼ . ∼ . − . Thedistribution of their statistical uncertainties peaks at ≈ .
12 mag arcsec − . And the errorsin log radius (log r / or log r eff ) are broadly distributed, almost all . .
02, ranging a bitlarger than the typical statistical error of ≈ . § A. The x-ray luminosities, listed in Table 2 for a subset of the objects, are obtained from
ROSAT and
Einstein satellite data. For the
ROSAT observations of Brinkmann et al. (1997)and Yuan et al. (1998), the data are provided as flux, f x [0 . , . − integrated over the energy range 0.1—2.4 keV in the observer’s frame. Also listed is theenergy spectral index, Γ, under the assumption of a power-law spectrum. With a spectrumof the form f ν = A x ν α x , (2)where A x is a constant and α x ≡ − (Γ − A x = f x [ E low , E high ] (1 + α x ) ν α x high − ν α x low , (3)where E low = 0 . E high = 2 . ν high = 1000 E high /h (similarly for ν low ), where h is Planck’s constant. The luminosities, νL ν | ν , are evaluated at a rest-frame energy of E = 0 . ν = 1000 E /h ). The x-ray luminosity for PG 1229+204 comes from the PSPC-pointed ROSAT observation reportedby Grupe et al. (2001). Reduction is performed as above, but over the 0.2—2.0 keV range.For x-ray data from the
Einstein satellite Imaging Proportional Counter (Wilkes etal. 1994; Margon et al. 1985), the energy range used is 0.16—3.5 keV. The luminosity of3C 93 is calculated from the flux listed by Wilkes et al. (1994) for a spectral index of α x ≈ − .
5, which is the closest to the average of our sample. Margon et al. (1985) actuallyprovide the Q 2344+184 x-ray luminosity itself, in the format desired here, but they do not 7 –quote a spectral index. Regardless of the source of data, we use the notation L X for thenuclear x-ray luminosities in ergs s − , evaluated at the rest frame energy of 0.5 keV. Radio-loudness classifications are collected primarily from Brinkmann et al. (1997) andYuan et al. (1998), both of whom use a loudness criterion that classifies an object as radio-loud if it has a radio-to-optical flux density ratio in excess of 10. Radio-loudness classificationsfor the remaining objects come from a variety of sources, with extensive use made of theNASA Extragalactic Database.In the analyses that follow, the QSO sample is subdivided into several categories, on thebasis of radio-loudness and host morphology. We refer to radio-loud (L) and radio-quiet (Q)QSOs, as well as those with elliptical (E) or spiral (S) hosts. These categories are combinedto describe radio-loud QSOs in elliptical hosts (LE), radio-louds in spiral hosts (LS), radio-quiets in elliptical hosts (QE) and radio-quiets in spiral hosts (QS). We note that the LSsubsample has just four members, so it is omitted from many of the statistical analyses thatfollow. The more detailed morphological descriptions applied to some hosts in Table 1 (suchas “interacting”) are not used in the analyses; only the E and S descriptions matter.
We find some redshift effects on the measured parameters. QSOs with nuclei muchfainter than the hosts (up to 3.5 mag fainter) are only seen at low redshifts ( z < . z ≥ . µ e , described in detail in § − as we go to z ≥ .
3, and we see somewhatfewer faint hosts at high redshift, though the trend is not strong. We do not, however, seea trend in log r / itself (when measured in kpc).The uncertainties in host and nuclear magnitudes show no change with redshift. Theuncertainties in log r / and µ e are larger, on average, for objects with redshift above 0.3.The median σ log r / rises from 0.007 (for z < .
3) to 0.023 (for z ≥ . σ µ e rises from 0.10 to 0.14. 8 –
3. NUCLEAR VS. HOST LUMINOSITIES3.1. Expectations
In addition to quantifying the behavior of nuclei and hosts separately, a major purposefor decomposing the QSO image into nuclear and host components is to look for relationshipsbetween the two. The correlations between nuclear and host parameters are importantbecause they shed light on the interplay between the central engine and the galaxy thatharbors it. One of the most straightforward questions is whether or not the nuclear and hostluminosities are correlated.In posing the question, we will assume the standard model of the central engine, asupermassive black hole surrounded by an accretion disk. Further assuming a fixed mass-to-light ratio for galaxies (following, for example, Jørgensen et al. 1996 and Magorrian etal. 1998), then the more massive a galaxy is, the more luminous it is. The mass of thegalaxy includes not only stars but also gas, in the form of nebulae and a diffuse interstellarmedium. Even in elliptical galaxies, which have a lower gas content than spirals, there canbe gas in the form of a hot interstellar medium given off by the winds of post-main-sequencestars. Some of this gas (of both types) eventually accretes onto the central engine as fuel(Di Matteo et al. 2000), so a more luminous galaxy has more fuel available for the nucleus.Under Bondi accretion (Bondi 1952), the gas accretion rate onto the disk is proportional tothe density of the gas far from the engine and to the square of the black hole mass, and theluminosity of the accretion disk is proportional to the accretion rate (Carter 1979). So giventhe model assumed here, a correlation between nuclear and host luminosities may naivelybe expected.Earlier investigations have shown evidence of such a relationship. Hutchings et al. (1984)studied 78 QSOs and lower-luminosity Active Galactic Nuclei (AGN) out to a redshift of z = 0 . H -band imaging ofQSOs, in which band the ratio of nuclear to host luminosity is diminished. They consideredtwo samples of low-redshift QSOs, one with low total (nucleus plus host) luminosities andone with high total luminosities, and they found that the hosts of the high-luminosity sampleare brighter than those of the low-luminosity sample. 9 – The extinction-corrected absolute nuclear and host magnitudes are plotted against eachother in Figure 1, with separate symbols for the LE, QE, LS, and QS categories. The objectMS 2159.5 − M V (total) = − .
0, the faintest allowed under theconventional definition of a QSO and therefore the faint limit for our sample. The sampleselection is made using published, ground-based data, so this limit is only approximate. Theregion in which the nuclear absolute magnitude is M V (nuc) > −
23 suffers from selectioneffects. We therefore perform the fits and correlation tests in this section only on thoseQSOs for which M V (nuc) ≤ − The Spearman correlation coefficients for M V (nuc) and M V (host) are listed in Table 3.The overall correlation is not strong ( ρ = 0 . ρ = 0 . ρ = 0 . M V (nuc) is used as the independent variable because it has a larger range than M V (host) does, even after restricting it to M V (nuc) ≤ − M V (host) = (0 . ± . M V (nuc) − (15 . ± .
2) . (4)That is, host luminosity varies with nuclear luminosity as a power law with an exponent of0.3. The slopes of most of the subsamples are similar to this, to within the errors. Thosesubsamples with especially low M V (host) to M V (nuc) correlations, QS and Q, are exceptions,as is the unreliable LS subsample. 10 – It appears from this analysis that QSOs with more luminous nuclei are slightly morelikely to have more luminous hosts. The correlation itself is rather weak ( ρ = 0 . ≈ . M V (nuc) ≤ −
23, is chosen to eliminate the stronger selection effects.
The extremely narrow range of M V (host) as compared to M V (nuc) is partly due to thesample selection, with objects being selected to have a combined M V (nuc + host) ≤ − M V (nuc) ≤ −
23 region is examined, the range of magnitudes spannedby the hosts (3.1 mag) is smaller than the range spanned by the nuclei (4.2 mag).If we instead used all of the objects lying outside the M V (nuc + host) ≤ −
23 line (ratherthan M V (nuc) ≤ − M V (nuc) vs. M V (host) fit would be slightly flatter (slope=0.24),but within 1- σ of our results here. The subsample slopes would show a similar change, exceptfor the radio-louds, which aren’t affected (they only exist at higher nuclear luminosities).The Spearman correlation would be stronger (from ρ = 0 .
350 to 0.451). The radio-loudsubsamples, again, would not be affected. But all of the others would be strengthened,except for the QE’s, which weaken (from ρ = 0 .
394 to 0.191). The correlations might beeven stronger if fainter classes of AGN (those with M V (nuc + host) > −
23) were included,but they are outside the scope of this study.There may be objects missing from the region of Figure 1 with high host luminosityand low nuclear luminosity (the upper left corner), due to sample selection effects. Opticalsearches for QSOs have typically stipulated that objects have a “stellar appearance” onphotographic plates. The dominance of the host here might keep this an empty region inoptically-selected QSO samples, but it is not a problem for x-ray and radio-selected samples.This study uses archival data from a variety of
HST proposals and, therefore, a mixof QSO selection methods. These are noted in Table 1. From the proposals and publishedarticles by the original proposers, we have determined that only one object, PG 1229+204,was chosen for its lack of an extended host in ground-based images. An additional 13QSOs were chosen from optically-selected catalogs, most commonly the Palomar-Green (PG)survey (Schmidt & Green 1983). Another 9 QSOs were chosen from the Large Bright QuasarSurvey (LBQS), which is optically selected but has safeguards to prevent the rejection ofobjects with detectable “fuzz” surrounding the nucleus (Foltz et al. 1987). The remaining47 QSOs in the sample were originally observed with
HST for reasons that do not have 11 –direct optical biases. This includes radio- and x-ray-selected QSOs. For those QSOs thatappear in multiple
HST proposals that were available for this study, we place them in themost “unbiased” category their proposals permit. The QSOs proposed for or selected inways most likely to be biased against the upper left corner of Figure 1 therefore number 14out of 70 (those marked with “B” or “C” in Table 1). Most of these 14 QSOs have nucleithat are more luminous than their host galaxies and tend to lie on the right-hand side ofFigure 1, as expected. (And so there is little overlap with those 14 QSOs that are fainterthan the M V (nuc) ≤ −
23 cut.) But two of them have approximately equal host and nuclearluminosities (PG 1229+204 and PHL 1093), and US 3498 has a nucleus 3.3 magnitudes lessluminous than its host.As for the effects of these potentially biasing QSOs on the M V (nuc)– M V (host) correla-tion, we note that the farthest outliers in the upper-right corner of Figure 1 are not amongthe 14 biased QSOs. Furthermore, the sample of radio-loud QSOs includes only 2 such bi-ased objects, and the correlation between M V (nuc) and M V (host) is actually stronger forthat sample. Other space-based studies have examined the relationship between host and nuclearluminosity, as well. We present the technical details of their observations and analysismethods here, but we will defer a comparison with their results until § HST to study a sample of 20 QSOs withredshifts z ≤ . − . ≤ M V ≤ − .
1. Allof these QSOs are included in our sample. Observations of four different stars are used forPSFs. The nucleus is subtracted from each image using both 1- and 2-dimensional methods,with the 2-D results being adopted in the end. In the 2-D method, the PSF is subtractedfrom the QSO first, with each of the four stars being tried in turn, and the one which leavesthe fewest residuals being adopted. A host model is then fitted to the residual image.Dunlop et al. (2003) also use the WFPC2 on
HST to study a sample of 10 radio-loudQSOs and 13 radio-quiet QSOs with redshifts 0 . ≤ z ≤ .
25. All of their QSOs are includedin our sample except for 5 objects (3 radio-louds and 2 radio-quiets) whose images were notpublicly available when we defined our sample and began analysis. Stellar observations areused to create a PSF that is subtracted from the QSO images using 2-dimensional modeling,with the host model being fitted simultaneously with the scaled PSF.Another study using
HST ’s WFPC2 is that of O’Dowd et al. (2002), who analyze the 12 –host and nuclear luminosities of 40 BL Lacs and 22 radio-loud QSOs with redshifts between0 . . z . .
5. They perform a simultaneous host plus PSF fit to the radial profiles ofthe BL Lacs. For the QSOs, they use the published results from other studies and thereforemostly overlap our sample.
4. NUCLEAR LUMINOSITY AND ITS RELATION TO BLACK HOLEMASS4.1. Black Hole Mass
Black hole masses have been variously estimated using H β line widths, galaxy bulgemasses, and stellar velocity dispersions, among other methods, since they cannot be measureddirectly. There are problems with applying some of these methods to QSOs, given their greatdistances and the glare from the nuclear emission. Not all are convinced by such methods,and it is with these caveats that we adopt the following black hole masses from the literature.But while the mass measurements themselves are subject to these systematic uncertainties,their use by other researchers does permit us to make self-consistent comparisons of theirQSO results with ours.Magorrian et al. (1998) study black holes in nearby, normal galaxies. Using HST pho-tometry and ground-based spectroscopy, they model the stellar orbital velocities for givenmass-to-light ratios and black hole masses. With the assumption that the black hole massis proportional to bulge mass, they find that M BH = 0 . M bulge , where M BH representsthe black hole mass and M bulge is the bulge mass. More recently, other groups (Kormendy& Gebhardt 2001; Marconi & Hunt 2003; McLure et al. 2006) have used improved measure-ments to revise the black hole to bulge mass ratio downward to M BH ∼ . M bulge . (5)H¨aring & Rix (2004) find evidence of a non-linear relationship, but their average ratio agreeswith this value.Without assuming that this sort of relation holds true for QSOs, we have taken the blackhole masses for 26 of our QSOs from McLure & Dunlop (2001) and list these in Table 4. Theycalculate the masses from the QSOs’ H β line width, a method that assumes the line widthcomes from the Doppler-broadened motion of clouds around the black hole. This methodrequires knowing the size of the Broad-Line Region (BLR), which they calculate from therelation of Kaspi et al. (2000).The results of McLure & Dunlop (2001) come from a combination of published data 13 –and new spectral observations. We do not undertake independent calculations of the blackhole masses from the variety of published line widths and BLR sizes because, as stressedby Laor (1998), published values of H β for the same QSO often vary greatly from onestudy to another. McLure & Dunlop (2001) have original H β observations for nearly half oftheir sample, make their calculations in a consistent manner, and demonstrate care wherethey add literature data to their own observations, making sure that these aren’t likely tointroduce large systematic errors. We use their black hole masses for those reasons andbecause of the large overlap (26 objects) between their sample and the QSOs in our study.As they list no uncertainties for the black hole masses, we adopt the high-end estimate ofVestergaard (2004) for this method, 0.4 dex. This 1 σ uncertainty is larger than what wecan obtain by propagating the known uncertainties. We would also like to know if these QSOs are radiating at close to their theoretical limits.The Eddington limit for the bolometric emission from the nucleus is given by (McCray 1979) L Edd ≃ . × M BH M ⊙ erg sec − . (6)We calculate the bolometric luminosities from M V (nuc) (listed in Table 1), using the datafrom Elvis et al. (1994), who provide the ratios between different monochromatic luminosities, νL ν , and the bolometric luminosity, L bol (their Table 17). For the V -band, (cid:28) νL ν L bol (cid:29) = 14 . ± . νL ν is evaluated at ν = 5 . × Hz, corresponding to a wavelength of λ = 5500 ˚A,the central wavelength of the V filter.We obtain L ν from our calculated M V (nuc), which are Vega-system magnitudes, usingthe zeropoints and photometric conversions of Voit (1997). In terms of M V (nuc),log f ν = − . M V (nuc) + 48 .
62] , (8)where f ν is the V -band flux density per unit frequency, in units of erg cm − s − Hz − .The V -band luminosity density, L ν , is then calculated from f ν and the QSO’s redshift, z . Since M V (nuc) is already K-corrected, then no further cosmological correction needs tobe applied, and the luminosity density is obtained from the simple geometric relation, L ν = 4 πd f ν , (9) 14 –where d is the distance. The bolometric luminosity can then be calculated from Table 17 ofElvis et al. (1994). The nuclear luminosity and the black hole mass can be used togetherto determine the Eddington fraction, L bol /L Edd , listed logarithmically in Table 4. Since theerrors in nuclear magnitude are much smaller than those assumed for the black hole masses,the propagated errors in log ( L bol /L Edd ) all come out to be 0.4. Beaming of the nuclearemission would create further uncertainties. We discuss this effect later.As our calculated Eddington fractions depend on the black hole mass estimates of § Out of the 26 QSOs that have black hole masses given by McLure & Dunlop (2001),we have modeled, spheroidal bulges for 20, whose absolute magnitudes are given in Table 4.We restrict consideration to the spheroidal (elliptical) bulges, as opposed to those that havedisk-like profiles, because they are more likely to have a dynamical relationship betweentheir bulge and black hole (Kormendy 2001). In Figure 2, the bulge magnitudes, M V (bulge),are plotted against the black hole masses. Figure 2 also shows the slope of any relationshiplike equation (5). If the QSO black holes have masses that are a constant fraction of thebulge mass, the distribution will lie parallel to this line. Assuming a constant mass–to–lightratio, then if black hole mass is a fixed fraction of bulge mass, an increase of one decadein black hole mass will correspond to an increase in one decade in bulge luminosity, or 2.5magnitudes.There is a great deal of scatter in the distribution of bulge magnitude with black holemass. The two parameters show only a weak anticorrelation overall (Table 3), just ρ = − . ρ = − . ρ = − . M V (bulge) = ( − . ± .
18) log ( M BH / M ⊙ ) − (10 . ± .
5) . (10)The slopes of most of the subsamples are similar, to within the errors, with the notableexceptions of the spiral and QS categories. This is not really parallel to equation (5), whichwould have a slope of 2.5 in this figure. The scatter is quite large, and they are about onestandard deviation different.
A plot of the nuclear luminosity vs. black hole mass is presented in Figure 3. Thedistribution of M V (nuc) vs. log M BH shows that there is a very general trend of luminosityincreasing with mass. The Spearman correlations, given in Table 3, include a fairly weakanticorrelation overall, with ρ = − . ρ = − . § M V (nuc) = − (1 . ± .
16) log ( M BH / M ⊙ ) − (6 . ± .
2) , (11)but it suffers from outlying points. The spiral and QS subsamples show positive slopes, withthe spirals having 2 . ± . − . ≤ (log L bol /L Edd ) ≤ .
3. Two objects (3C 273 and PG 1402+261)appear to have luminosities above the Eddington limit, but they exceed it by less than theuncertainties. PG 1302 − L bol /L Edd = 0 .
0, is also placed in the highest bin.We can see that the objects in spiral hosts tend to have higher Eddington fractions, -0.8to 0.3 (in the logarithm), while those in elliptical hosts are spread across -1.6 to 0.2 (againin the logarithm). No distinction is apparent between the Eddington fractions of radio-loudand radio-quiet objects. 16 –
We cannot say much about a fixed black hole to bulge mass ratio, such as equation (5).While it appears from Figure 2 that there might be some weak correlation between the bulgemagnitude and the black hole mass, the details of the relationship are hidden by the largeuncertainties in the masses. The range of masses is not much larger than their uncertainties,allowing for a wide range of possible slopes, so the results are simply inconclusive. That the7 QE objects form the tightest correlation is interesting. Since there are not many objectsin this category, not too much can be drawn from this result yet, but it presents an avenuefor follow-up studies.
We see that as black hole mass increases, the QSO’s nuclear luminosity also increases(Figure 3), although the trend is rather weak. A stronger trend might naively be expected, asthe more massive black holes have higher Eddington limits. Yet the QSOs do not all radiatenear their limits (Figure 4), although there is a clear division according to host morphology.Those in spiral hosts do tend to cluster at relatively high Eddington fractions and only goas low as 16% of Eddington. But those in elliptical hosts are more evenly distributed overa wider range, extending as low as 2.5% of Eddington. The difference may be caused byspirals typically having smaller black holes.
5. THE “FUNDAMENTAL PLANE” OF QSOS
Instead of only looking at correlations of two parameters at a time, we can analyzemultiple dimensions at once. Principal Components Analysis (PCA) provides a convenientmethod for identifying and examining multidimensional correlations. We use this method tosearch for further connections between the nuclear and host properties. 17 –
For our PCA, we use a restricted sample of those QSOs for which we have all of thefollowing parameters: M V (nuc), L X , r / , and µ e , where µ e is the effective surface magnitudeof the galactic bulge. We further require that each QSO have a modeled, spheroidal bulge(the entire galaxy, in the case of elliptical hosts). These qualifications restrict the sample tothe 42 QSOs listed in Table 2.In order to prevent the choice of units from artificially weighting some parameters morethan others, each parameter is first normalized by subtracting its mean and dividing byits variance. The PCA is then performed on these normalized variables. Note that thenormalization for each of the subsamples is carried out separately from the others. In theoutput, each of the eigenvectors is written as a linear combination of the original (butnormalized) parameters. The eigenvalues are scaled so that the sum of all eigenvalues equalsthe total number of eigenvectors (and therefore the total number of parameters as well). There is already a well-studied fundamental plane (FP) for normal, elliptical galaxies(Djorgovski & Davis 1987; Dressler et al. 1987) that incorporates galaxy size, r / , centralvelocity dispersion, σ c , and effective surface magnitude, µ e . Here we define µ e as the surfacemagnitude (expressed in magnitudes arcsec − ) of an elliptical galaxy or bulge at a distance r / from the center of the galaxy. Note that there is another form of effective surfacemagnitude, h µ i e , which is the average surface magnitude within the half-light radius. Thesetwo forms are related such that µ e − h µ i e = 1 .
39 (van Albada et al. 1993). We can derive anexpression for µ e by first noting that the total host luminosity in counts, L host (correspondingto the absolute magnitude, M V (host)), of a spheroidal galaxy following an r / profile andextended out to an infinite radius can be expressed as L host = κ L e . I ( r / ) r / (1 − ǫ ) , (12)where I ( r / ) is the galaxy’s surface brightness (in count-rate per unit area) at the half-lightradius, r / , and ǫ is the ellipticity of the isophotes. Then κ L is a constant, which we calculatenumerically as κ L = 0 . L host thatleads to κ L = 0 . − , we obtain µ e = M V (host) + 2 . (cid:2) θ / κ L e . (1 − ǫ ) (cid:3) , (13)where θ / is the half-light radius expressed in arc seconds. In the case of an elliptical bulge,we substitute M V (bulge) in place of M V (host). We can now look for a QSO fundamentalplane that involves the host parameters, log r / and µ e , as well as a measure of the nuclearluminosity, M V (nuc) or log L X . In this, we use the effective surface magnitude, rather thanthe bulge magnitude, so that we can make direct comparisons between our results and thewell-studied fundamental plane of elliptical galaxies. We can perform two PCAs, an optical one using M V (nuc), log r / , and µ e as theparameters, and an x-ray one that uses log L X for the nuclear luminosity. From the opticalPCA performed on this sample of 42 objects, we find that 96.1% of the variance can beexplained with just the first two eigenvectors ( e and e ), and the QSOs mostly lie in aplane within this parameter space. This we consider to be a fundamental plane for QSOs.For the corresponding x-ray results, the first two eigenvectors explain 95.2% of the variancein the sample, and we find here an x-ray QSO fundamental plane. See Table 5 for theindividual eigenvalues.We obtain the formula for the optical fundamental plane of the full sample, M V (nuc) = − . . µ e − . r / . (14)The x-ray fundamental plane for the full sample islog L X = 79 . − . µ e + 8 .
74 log r / . (15)Views of the optical and x-ray fundamental planes, with the QSO data points superimposed,are displayed in Figure 5. Note that the host properties describe the horizontal and thenuclear luminosity the vertical in this plot.The same analysis can also be made of the separate QSO subsamples. We save themathematics and analysis details for § B but discuss the results in more general terms in § § The precision of the fundamental plane can be judged from the percentage of the varianceexplained by the third eigenvector, as listed in Table 5. The third eigenvector describes thescatter perpendicular to the plane.We can obtain another measure of the fundamental plane’s errors by comparing themeasured nuclear luminosities with the values we calculate from the QSO FP. This can bedone when solving for any of the FP’s three variables in terms of the other two. The Pearsoncorrelation coefficients and the root-mean-squared differences between the actual values andtheir fundamental plane solutions are listed in Table 6. The Pearson correlation coefficientis useful for accounting for the size of the error, relative to the range of the variable.It is clear from these results that the QSO FP has both the smallest root-mean-square(RMS) errors and the highest correlation with the data when it is solved for the host galaxysize, log r / . The full-sample fundamental plane in this form islog r / = − . M V (nuc) + 0 . µ e − .
46 (16)for the optical, or log r / = 0 .
114 log L X + 0 . µ e − .
07 (17)for the x-ray. The measured host sizes are plotted against the fundamental plane in Figure 6.
The effect of the sample’s statistical errors on the measurement of the QSO fundamentalplane can be tested by creating an artificial FP, adding errors, and trying to recover the FPusing the PCA method described above. In this test, we first create a fake QSO FP equation.We then randomly place 42 objects within the limits of log r / and µ e covered by the 42 realQSOs of the PCA sample. Each object’s nuclear luminosity ( M V (nuc) or log L X ) is calculatedfrom the fake FP equation. We next add errors to each data point in the following way.The measurement uncertainties for each QSO are listed in Table 2, and we use them tothrow the errors. For each object in the fake FP, we use the uncertainties from a differentQSO of the actual sample. The error for a given parameter is thrown randomly, according toa two-sided Gaussian of width equal to the QSO’s uncertainty. The parameter value for thefake object is then shifted by this random error. In this way, the fake QSO FP is scrambledto simulate measurement errors. 20 –We then run a Principal Components Analysis on the fake, scrambled FP. After wederive the equation for it, we compare this with the equation we used to create the fakedataset in the first place. Using this technique, we not only test the actual optical and x-rayQSO FPs, but we also vary their coefficients. This lets us find out whether the statisticalerrors push the measured QSO FP into a particular direction. Starting from the QSO FPsin the form of equations (14) and (15), we individually vary the coefficients of log r / and µ e , replacing them with values 0.5, 1, and 2 times their actual amounts. It turns out thatthe statistical errors do not greatly affect the measured angle of the QSO FP, regardlessof how we vary the equation’s coefficients. The coefficients “recovered” from the scrambleddata are always within 2% of their assigned values and in most cases are within 1%. The existence of the QSO fundamental plane shows that there is a strong link between aQSO’s host and nuclear properties, although we cannot make a claim as to cause and effect.We find that the QSO subsamples have their own fundamental planes and that there is agreat difference in their slopes (we specifically refer to the gradient). Mathematically, the FPgradients calculated in § B describe the size of the dependence of the nuclear luminosity on thefeatures of the host. A large gradient, as in the x-ray form of the radio-quiet sample, meansthat a small change in the host size or effective surface magnitude is associated with a largechange in the nuclear luminosity. Because the FP is a function of two parameters, lookingonly at the gradient obscures the details of this relationship somewhat. But it is useful inthat it demonstrates the underlying similarity between the FPs of the different subsamples,a similarity that is not obvious from looking at the individual equations. Geometrically, thedifferent planes act almost as a single plane that pivots about its principal axis, changing itsangle relative to the host (log r / , µ e ) plane. That is, the gradients of most of the subsamplespoint along the same (or opposite) azimuth, and the major difference between their FPs isthe magnitude of their gradients. More interestingly, not only does the gradient changemagnitude, but it even reverses direction in some cases. In the optical FPs, only the QEsample slopes in the direction opposite the rest. In the x-ray FPs, all of the radio-quietsamples (Q, QE, and QS) exhibit this behavior. In fact, if we rank the planes by theirgradients, radio-loudness seems to have the biggest influence in how the subsamples aregrouped.The statistical errors in the final PCA dataset do not appear to affect the measuredorientation of the QSO FP significantly. While this does not itself discount the possibilitythat systematic errors could have an effect, we note the difference between the FPs of the 21 –radio-loud ellipticals (LE) and radio-quiet ellipticals (QE). These two FPs have slopes ofopposite sign, but they deal with hosts of the same morphology. Thus systematic errorsfrom the image fitting procedure are unlikely to create the QSO FPs artificially. Whilethey might still influence the measured slope, they are not enough to cover up these largedifferences between the subsample FPs. In addition to statistical and modeling errors, thenuclear luminosity is affected by QSO variability. This is another source of FP thickness.Finally, we have adopted a Friedmann cosmology with H = 50 km s − Mpc − , q =0 .
5, and Λ = 0 to be in keeping with Hamilton (2001) and Hamilton et al. (2002). Amore currently-accepted cosmology might have H = 65 km s − Mpc − , Ω matter = 0 .
3, andΩ Λ = 0 .
7. The fundamental plane variables can be converted into this cosmology for eachindividual QSO, but the inclusion of a non-zero cosmological constant makes the conversionsdepend on redshift.The differences are smallest for QSOs at the high-redshift end of the sample. Convertingto the updated cosmology lowers the values of log r / by 0.101, for QSOs at the low-redshiftlimit of z = 0 .
06, and it lowers them by 0.039, for QSOs at the high-redshift limit of z = 0 . M V (nuc) and µ e , fainter by 0.505 to 0.195 mag, andit lowers log L X by 0.202 to 0.007 (again, the low-redshift changes are listed first). Thesechanges are small compared to the ranges of the variables. More importantly, they do notproduce any significant effect in the QSO fundamental plane quality. In the optical case, thepercentage of the sample variance explained by the PCA would be lowered from 96.1% to95.6%, and in the x-ray case, from 95.2% to 94.7%.
6. DISCUSSION6.1. Host and Nuclear Luminosity
We have shown that host luminosity shows a shallow increase with nuclear luminosity.Some relationship of this type is expected because in more massive (and therefore moreluminous) hosts, there is more gas available to fuel the supermassive black hole thoughtnecessary to power a QSO.Bahcall et al. (1997) do not find convincing evidence in their results of a significantcorrelation between host and nuclear luminosities, and they do not provide a fitted relation,noting that much of the apparent correlation in their Figure 8 comes from 3C 273, thebrightest object in their sample.O’Dowd et al. (2002) find a shallow but statistically significant trend of host luminosity 22 –increasing with nuclear luminosity. For the combined sample, host luminosity increases by1 mag or less (depending on beaming corrections for the BL Lacs) for each 7 mag increasein nuclear luminosity. Their study uses a mixed sample of objects (BL Lacs and QSOs).For their radio-loud QSOs, Dunlop et al. (2003) find a shallow relationship, M R (host) =+0 . ± . M R (nuc) − . ± . M R (host) = − . ± . M R (nuc) − . ± . Without interpreting the weak bulge and black hole result too far, we note for compar-ison that our fitted relation, equation (10), is fairly consistent with that derived by McLure& Dunlop (2001). It corresponds to their equation (3), after transforming their M R C to M V with the elliptical galaxy colors from Fukugita et al. (1995). However, their newer resultswith an expanded sample (McLure & Dunlop 2002) find a steeper slope than our equa-tion (10). We should also note that a reanalysis of their data by Graham (2007), editingthe sample and using a different linear regression method, steepens it to almost twice ourslope. Graham finds close agreement between this and his similar reanalysis of Marconi &Hunt (2003).Neither McLure & Dunlop nor Marconi & Hunt restrict themselves only to QSOs, andso they have samples extending to much lower black hole masses than ours, which improvestheir correlations. The problem with looking at a QSO population alone, as we do, is thatthe short range of masses is not much greater than the measurement errors. More data areneeded for QSOs with less massive black holes, if true QSOs even exist with much smallermasses. And the real test will come when the uncertainties in black hole mass measurementshave been reduced. We see that nuclear luminosity generally increases with black hole mass, although thetrend is not strong. Some kind of relationship might be expected, since the gravity of theblack hole is the ultimate energy source for the QSO. For instance, as black hole massincreases, more material can be trapped by its gravity and be pulled into the accretion disk;the faster material accretes onto the disk, the more the disk heats up and light is produced. 23 –But any effect produced by this is rather weak.We can see some trends, though, when we look at the Eddington fractions of the QSOs.The QSOs in spirals tend to be radiating at higher Eddington fractions than those in ellipti-cals. While spirals generally have lower nuclear luminosities, they also tend to have smallerblack holes. This could be a result of the amount of gas available to fuel the QSO. If there isa fixed rate of gas flowing into the vicinity of the black hole, owing to the shape of orbits inthe galactic bulge, then it may be that a smaller black hole, with a lower Eddington limit,is able to capture enough material to keep it radiating near its limit. But as the black holegrows, although it is able to capture more of the material orbiting near it, there is not enoughavailable to keep it near the increased Eddington limit. It could also simply be that if, ratherthan limiting ourselves to traditionally-defined QSOs, we were to include lower-luminositySeyfert galaxies in our sample, we might find them populating the low Eddington fractionregion of the spiral host plot.A criticism of one of the trends described above comes from Woo & Urry (2002), whosuggest that what appears to be a correlation between black hole mass and nuclear luminositymight be explained by sample selection effects. They argue that there is not sufficientevidence for a correlation, because the lack of objects in the lower right side of Figure 3 is dueto the exclusion of lower-luminosity AGN from the sample. They state that radio galaxies,for instance, can have low nuclear luminosity but high black hole mass. The exclusion fromour sample of radio galaxies might not have exactly the same effect on our distributions asit does for Woo & Urry (2002), since they include radio galaxies but are not able to obtaingood bolometric measurements for the least luminous ones. The upper left side of the plotis real, however, because of the Eddington limit.Finally, we should also say that there is an added uncertainty from QSO variability andthe possible beaming of the nuclear emission. For instance, PG 1302-102 is variable (Eggers,Shaffer, & Weistrop 2000 show changes of about 0.2 mag), and its correlated optical andradio structure could indicate beaming (Hutchings et al. 1994b). There should be littlebeaming among the radio-quiets. But among the radio-louds, it could exaggerate somenuclear luminosities and Eddington fractions.
The fundamental plane for QSOs shows a relationship between the nuclear and hostfeatures that goes beyond the simple correlation of nuclear and host luminosities. Thisbehavior may be connected to other, known relations between the objects, such as the 24 –fundamental plane of normal, elliptical galaxies. The fundamental plane for elliptical galaxiesinvolves the galaxy’s size, effective surface magnitude, and stellar velocity dispersion. Itvaries quantitatively for galaxies in different environments and for observations in differentwavebands, but one measurement in the V -band (Scodeggio et al. 1998) islog r / = 1 .
35 log σ c + 0 . µ e + Constant , (18)where σ c is the galaxy’s central velocity dispersion.Although the ratio of the coefficients of the half-light radius to the effective surfacemagnitude is different from the QSO fundamental plane, there is a formal similarity betweenthe QSO and normal FPs, which might point to a link between the host galaxy’s centralvelocity dispersion and the nuclear luminosity of the QSO. It is therefore tempting to tryto derive the QSO fundamental plane directly from the elliptical galaxy fundamental plane,but we find two problems with this approach, both arising from the relation of black holemass to nuclear luminosity.By applying the correlation between the central velocity dispersion in elliptical galaxiesand the mass of their central black hole, we can put the elliptical galaxy fundamental plane,equation (18), in terms of black hole mass. We use the formula of Merritt & Ferrarese (2001), M BH = 1 . × (cid:18) σ c
200 km s − (cid:19) . M ⊙ . (19)Further substituting in equation (11) to put this in terms of nuclear luminosity, we obtainlog r / = − . M V (nuc) + 0 . µ e + Constant . (20)This is our attempt to derive the QSO optical fundamental plane from the normal galaxyFP, and we see that it differs from the actual QSO FP, equation (16).The coefficient we derive for M V (nuc) in the above equation depends upon the exponentof the velocity dispersion in equation (19) and on the slope of equation (11). Differentstudies have found exponents for the velocity dispersion (a hotly debated topic) that rangefrom about 3.75—5.1 (Ferrarese & Merritt 2000; Gebhardt et al. 2000; Tremaine et al. 2002;Ferrarese & Ford 2005; Bernardi et al. 2007), and the slope of equation (11) has a largeuncertainty. But we still cannot make the coefficient of M V (nuc) in equation (20) match thatof the QSO optical fundamental plane. Deriving the QSO fundamental plane directly fromthe normal galaxy FP would require a steep dependence of M V (nuc) upon log ( M BH / M ⊙ ),which we do not find.We can attempt to derive the x-ray QSO fundamental plane the same way, using thecorrelation between the x-ray luminosities and black hole masses in our sample. A linear 25 –fit using the BES method (in keeping with the derivation for the optical plane) gives uslog L X = 2 .
77 log ( M BH / M ⊙ ) + 19 .
8. This would lead tolog r / = 0 .
10 log L X + 0 . µ e + Constant (21)as our attempted derivation, which is close to the actual x-ray QSO FP. The use of x-rayluminosities normalized to 0.5 keV glosses over the complexities of QSO spectra, providingonly a first glimpse of the x-ray QSO fundamental plane. Follow-up analysis should accountfor the different spectral features of the individual QSOs and should use luminosities of ahigher energy band.There is both an interesting property and a problem in the QSO FP orientations, whichwe detail in § B. As we change from one QSO class to another, the fundamental plane essen-tially pivots about an axis, so the differences between the subsamples are mostly reduced toa single dimension, the slope (or gradient) relative to the µ e –log r / plane. Some subsampleFPs even slope in the opposite direction from the overall FP. This creates another problemfor deriving the QSO FPs (whether optical or x-ray) from the normal galaxy fundamentalplane. These opposite slopes are not reflected in our fits of nuclear luminosity against blackhole mass, and the poor correlation of these two quantities lies in contrast with the rela-tively thin QSO fundamental plane. This points to our need for a better understanding ofthe proper relationship between nuclear luminosity and black hole mass.Because the QSO FP mathematically describes a link between the host and the nucleus,its slope might depend on the physical nature of the link. It would be interesting to findif the different QSO FP orientations described in § B come about from different fuelingmechanisms that might be found in the various subsamples. We see, for instance, thatradio-quiet and radio-loud QSOs are characterized by very different slopes in their x-rayFPs, but the understanding of what makes these QSO types differ is still too limited tospeculate further here. In our ongoing research, we are expanding the fundamental planestudy to other types of AGN, such as Seyferts. We can then compare their FP orientationswith those of the different QSO subsamples, which may teach us more about the physicsunderlying the QSO fundamental plane.
7. CONCLUSIONS
1. QSO hosts cover a much smaller range of luminosity than do their nuclei. We find arelatively weak trend between host and nuclear luminosities.2. Host bulge luminosities do not show a significant correlation with H β -derived blackhole masses, but a Magorrian-type relationship cannot be ruled out. 26 –3. The correlations seen among nuclear luminosity, black hole mass, and Eddingtonfraction are subject to selection effects from the exclusion of lower-luminosity AGN.4. We show that there is a strong, 3-parameter relationship between host and nuclearproperties. The nuclear luminosity, size of the host, and effective surface magnitude of thehost together form a plane in which most of the QSOs lie. This fundamental plane for QSOsexplains 96.1% of the variance in the sample, in its optical form, and 95.2% in its x-ray form.While this plane shows some similarities to the fundamental plane for elliptical galaxies, wedo not find a clear, analytical derivation from one to the other.This research has been supported through a National Research Council Associateshipat NASA’s Goddard Space Flight Center, as well as through a Graduate Student Fellowshipand a grant from the Director’s Discretionary Research Fund, both from the Space Tele-scope Science Institute. We made use of the following databases: the HST data archive;NASA/IPAC Extragalactic Database (NED), operated by the Jet Propulsion Laboratory,California Institute of Technology, under contract with the National Aeronautics and SpaceAdministration; and NASA’s Astrophysics Data System Abstract Service. We would like tothank George Djorgovski for his comments on the interpretation of the fundamental plane.We are also grateful for the long support and encouragement of the late Elizabeth K. Holmes,a National Research Council Associate at JPL. A. COMPARISON WITH OTHER QSO IMAGING RESULTS
An idea of the systematic differences between our analysis method and others is gottenby comparing our QSO imaging results with those of other researchers using the very samedata sets (Table 7). The most useful comparisons are of the nuclear and host apparentmagnitudes, as well as the host effective radius. We include only those results that can bedirectly compared with ours–those reporting the apparent WFPC2 filter magnitudes in theVega magnitude system.As discussed in § ± .
3) times larger than those fitted by McLure et al. (1999), ignoringthose objects flagged in Table 7 where we have fitted models to different components of thehost. Note that although the apparent magnitudes are listed as Cousins R , they are, in fact,in the F W filter. McLure says that the R C − F W color conversions for these objectsturned out to be too small to make a difference.Larger differences, on the other hand, can be seen between our results and those ofBahcall et al. (1997) and Boyce et al. (1998). Their analysis methods are quite differentfrom ours. Bahcall et al. (1997) progressively subtract a stellar nucleus until the residualprofile turns over in the center, then fit a host model to the remaining light. We comparewith their 2-D model results. In principle, this method should somewhat overestimate thehost magnitude. Boyce et al. (1998) fit simultaneous nucleus and host models using thecross-correlation method described by Boyce, Phillipps, & Davies (1993). Host magnitudesare measured directly from the light remaining after PSF subtraction. All of their images aremarred by saturation in the innermost pixels, which they remove entirely from the analysisand set to null for the host magnitude calculation. Thus their host magnitudes are upperlimits, as they state.Lastly, Hooper et al. (1997) use the same cross-correlation method as Boyce et al. (1998),although they apparently take the host magnitudes from the models, rather than from thePSF-subtracted images. They report their results in the Johnson R filter, rather than theWFPC2 F675W in which they were taken (stated to be a < .
15 mag conversion), andthey apply other magnitude corrections ( ≤ . R . Sizeabledifferences with our results can be seen, but the magnitude differences should be treated asrough. 28 – B. SUBSAMPLE FUNDAMENTAL PLANESB.1. Subsample Forms
In addition to the full sample, we also look for fundamental planes in the individual QSOsubsamples; their PCA eigenvalues are presented in Table 8. As before, the subsamples arelabeled “L” for radio-loud QSOs, “Q” for radio-quiet, “E” for those in elliptical hosts, and“S” for those in spirals. Because the restricted sample includes only a single radio-loud QSOin a spiral host, there is no separate FP for the “LS” subsample.We find that, indeed, the subsamples are individually distributed into planes, and thethird eigenvalue (representing scatter perpendicular to the plane) in each case is about assmall as the full sample’s. Table 9 lists the coefficients for the subsamples’ fundamental planeequations. The equations are presented in the optical case as M V (nuc) = R coeff log r / + M coeff µ e + C and in the x-ray case as log L X = R coeff log r / + M coeff µ e + C , where R coeff and M coeff are coefficients, and C is a constant. The orientations of some of these planes areclearly different from the others. B.2. Gradients
The situation is more easily described geometrically, because it turns out that theirbiggest differences can be characterized by a single parameter, the slope (magnitude of thegradient). To visualize the relationship among the planes, let us imagine a 3-D plot similarto Figure 5, but rescaled, replacing M V (nuc) with M V (nuc) / . µ e with µ e / .
5, so thatall quantities are pure logarithms. This will allow a more straightforward comparison oflengths and angles. For the vertical axis ( k ), we will use the nuclear luminosity (log L X or M V (nuc) / . i axis will be log r / , and the j axis will be µ e / .
5. Again, the hostproperties describe the horizontal and the nuclear luminosity the vertical.In the following, let us represent the rescaled QSO fundamental plane by the function F = F (log r / , µ e / . ∇ F = (cid:0) ∂F/∂ log r / (cid:1) ˆ ı +[ ∂F/∂ ( µ e / . ˆ , where ˆ ı and ˆ are unit vectors. Note that since F is a plane, its gradient isthe same at any point we choose to evaluate it. We can look at the magnitude and directionof the gradient, where the magnitude is given by ||∇ F || = ((cid:18) ∂F∂ log r / (cid:19) + (cid:20) ∂F∂ ( µ e / . (cid:21) ) / , (B1) 29 –and the direction (azimuth, the angle counterclockwise from the + i axis) is given by α = arctan (cid:20) ∂F/∂ ( µ e / . ∂F/∂ log r / (cid:21) . (B2)These gradients are listed in Table 10. The full sample is listed first, while the sub-samples are grouped according to general azimuth and then ranked in order of gradientmagnitude. The subsample gradients tend to point either along the same azimuth as theoverall FP or in the opposite direction (180 ◦ away from it). This near-uniformity in azimuthlies in contrast with the wide range of magnitudes (from 6.45 to 65.9), and so the gradientmagnitudes seem to be the distinguising feature between the subsample planes.It should not be surprising that the azimuths are nearly uniform, as the first principalaxis of the PCA follows roughly the Kormendy relation (Kormendy 1977) between µ e andlog r / . What is interesting, then, is the large degree of freedom for the planes about thisaxis. B.3. Discussion
The x-ray FPs have an interesting symmetry in their gradients. We see the the Land Q classes are paired with each other in having the largest gradients (except for theS class, discussed below) but pointing in opposite directions. This opposition in gradientdirections is followed by the mixed classes, with the LE plane on one side and the QE andQS planes on the other. Note that the LS class is too small to derive a separate fundamentalplane for it. The S class is an exception to this symmetry. For the PCA sample, the onlydifference between the S and QS populations is a single object, 3C 351, which is radio-loudand apparently spiral. With the small populations of these two classes, this one QSO makesa real difference. In general, we find that the various planes for radio-loud objects point inone direction, while those for radio-quiets point in the opposite direction.The optical FPs all have approximately the same azimuth, with the exception of theQE class, which points in the opposite direction. But even among the optical planes, thereare distinct groupings (mostly by radio-loudness) that almost mirror those of the x-ray FPs.Aside from the QE class, the spirals and radio-quiet objects have the largest gradients, whilethe ellipticals and radio-loud objects have the smallest. It seems that radio loudness makesthe greatest difference in the gradient and azimuth, both in the x-ray and the optical FPs,while the host morphology appears to be less important. 30 –
REFERENCES
This preprint was prepared with the AAS L A TEX macros v5.2.
34 –Fig. 1.— Overall distribution of host and nuclear luminosities. The QSOs are marked asradio-loud (L) or quiet (Q) and with elliptical (E) or spiral (S) hosts. The dotted curve is theapproximate bound for objects with total (nucleus plus host) magnitude of M V (total) > − M V (nuc) ≤ −
23 to avoid anybias by the few objects with fainter nuclei. 35 –Fig. 2.— Bulge absolute magnitude vs. black hole mass. Objects with disk-like or unmodeledbulges are not included in the plot. The dashed line is not a fit but is drawn parallel to therelation of Magorrian et al. (1998). The correlation is weak and, given the large errors inthe masses, does not resolve the question of a black hole to bulge mass relation. 36 –Fig. 3.— Nuclear absolute magnitude vs. black hole mass. Nuclear luminosity generallyincreases with greater black hole mass, though the correlation is low. The lack of QSOsin the upper left corner is due to the Eddington limit, and there might be selection effectsresponsible for the lack of objects in the lower right corner. 37 –Fig. 4.— Distributions of Eddington fractions, with a bin width of 0.4 (equal to the un-certainties) and subdivided by morphology and radio loudness. The spirals tend to clusterat high Eddington fractions, while the ellipticals are spread over a wide range. Note thedifferent vertical scale used for the complete sample’s plot. 38 –
16 18 20 22 24 0 0.4 0.8 1.2 1.6-28-26-24-22-20-18 M V µ e log r M V
16 18 20 22 24 0 0.4 0.8 1.2 1.6 41 42 43 44 45 46 47 log L X µ e log r log L X Fig. 5.— Views of the optical (left) and x-ray (right) QSO fundamental planes, showing theindividual QSOs (filled and open circles) and the plane (grid) fitted to the overall sample.The host properties are the horizontal axes, while nuclear luminosity is vertical. In bothcases, nuclear luminosity increases upward. The plots are viewed from above the plane;filled circles lie above the plane, and open circles lie below. 39 –Fig. 6.— Overall QSO fundamental plane (vertical axis), plotted against the measured hostgalaxy size (log r / , horizontal axis). Points on the diagonal line show perfect correspon-dence. The left figure uses the QSO fundamental plane in its optical form, while the rightfigure uses the x-ray form. The QSO fundamental plane is most precise when solved for thehost size. 40 –Table 1. Observations and Data Name(1) z (2) m nuc (3) m host (4) M V (nuc)(5) M V (host)(6) Morphology(7) Radio(8) Selection(9)LBQS 0020+0018 0.423 19.30 19.34 -22.45 -22.77 E Q ALBQS 0021-0301 0.422 19.03 19.10 -22.74 -23.03 E Q APG 0043+039 0.385 16.04 19.03 -25.49 -22.67 E Q BPG 0052+251 0.155 16.04 16.83 -23.80 -23.08 S QPHL 909 0.171 15.97 16.89 -24.07 -23.29 E QUM 301 0.393 17.66 19.44 -23.92 -22.44 E Q A3C 47 0.425 17.82 18.75 -24.07 -23.37 E L3C 48 0.367 15.74 16.18 -25.72 -24.83 EI LPHL 1093 0.26 17.21 17.17 -23.47 -23.54 E L BMRK 1014 0.163 16.17 14.75 -23.30 -24.34 S a QPKS 0202-76 0.389 16.67 18.72 -24.98 -23.12 E LNAB 0205+02 0.155 15.40 18.08 -24.37 -21.77 S QQ 0244+194 0.176 16.80 17.54 -23.29 -22.47 E QUS 3498 0.115 19.30 15.87 -19.79 -23.10 S Q BPKS 0312-77 0.223 16.13 16.67 -24.40 -23.78 E LQ 0316-346 0.260 16.21 18.08 -24.68 -23.03 IS Q3C 93 0.357 18.58 18.49 -23.51 -23.73 E LIR 0450-2958 0.286 15.40 17.17 -25.41 -23.65 SI QPKS 0736+01 0.191 16.30 16.70 -24.03 -23.58 E LMS 07546+3928 0.096 14.26 14.37 -24.22 -23.47 E QIR 0759+6508 0.149 15.94 15.65 -23.57 -23.73 SI QMS 0801.9+2129 0.118 16.00 15.66 -22.91 -22.80 S Q3C 206 0.198 16.07 16.90 -24.03 -23.09 E L3C 215 0.412 17.71 18.23 -24.08 -23.08 E LPG 0923+201 0.19 15.53 17.46 -24.73 -23.00 E QMS 0944.1+1333 0.131 14.89 15.93 -24.16 -22.52 E QPG 0953+414 0.234 15.17 17.21 -25.23 -23.22 S Q BPG 1001+291 0.330 15.59 17.90 -25.58 -23.33 S Q BPKS 1004+13 0.24 15.15 17.00 -25.62 -24.06 E LPG 1012+008 0.185 16.22 16.76 -23.75 -23.19 SI Q BHE 1029-1401 0.086 13.84 15.86 -24.79 -22.74 E Q BMS 1059.0+7302 0.089 16.60 15.41 -21.65 -22.34 S QPG 1116+215 0.177 14.85 16.74 -25.19 -23.42 S Q BPG 1202+281 0.165 16.85 17.39 -23.03 -22.62 E Q BLBQS 1209+1259 0.418 19.35 19.38 -22.39 -22.71 E Q APG 1216+069 0.331 15.42 18.70 -25.76 -22.55 E QLBQS 1218+1734 0.444 18.33 19.01 -23.54 -23.26 E L AMS 1219.6+7535 0.071 15.06 14.56 -22.72 -22.52 ED QLBQS 1222+1010 0.398 18.38 18.62 -23.23 -23.24 S QLBQS 1222+1235 0.412 17.68 18.25 -24.04 -23.82 E L A3C 273 0.158 12.60 15.65 -27.19 -24.24 E LPG 1229+204 0.064 15.37 15.04 -22.27 -22.33 S Q CLBQS 1240+1754 0.458 17.98 19.31 -23.93 -23.02 E Q ALBQS 1243+1701 0.459 18.45 18.44 -23.49 -23.91 E Q A3C 277.1 0.321 17.97 18.35 -23.09 -22.76 S LPG 1302-102 0.278 15.19 17.35 -25.93 -24.14 E LPG 1307+085 0.155 15.46 17.47 -24.33 -22.42 E Q BPG 1309+355 0.184 15.56 16.61 -24.53 -23.62 S LPG 1358+04 0.427 15.96 18.02 -25.84 -24.02 E Q BQ 1402+436 0.323 15.15 17.42 -25.93 -23.73 EI QPG 1402+261 0.164 15.73 17.33 -24.12 -22.62 S QMS 1416.3-1257 0.129 15.83 16.90 -23.37 -21.70 E QB2 1425+267 0.366 15.88 17.47 -25.49 -23.45 E LMS 1426.5+0130 0.086 14.30 14.49 -23.87 -23.17 S QPG 1444+407 0.267 15.80 17.37 -25.14 -23.81 S Q BB2 1512+37 0.371 16.04 17.31 -25.38 -23.66 E LMS 1519.8-0633 0.083 16.01 15.07 -22.32 -22.75 S Q3C 323.1 0.264 16.07 18.01 -24.94 -23.33 E L BMC 1548+114A 0.436 18.27 19.92 -23.66 -22.20 SI LMC 1635+119 0.146 18.12 16.73 -21.38 -22.62 E Q3C 351 0.372 15.50 16.97 -25.96 -24.59 S LPKS 2135-147 0.200 16.21 16.91 -23.96 -23.23 E LOX 169 0.211 15.89 17.28 -24.59 -23.18 EI LMS 2159.5-5713 0.083 17.14 15.01 -20.91 -22.52 S ?Q 2201+315 0.295 15.46 16.75 -25.78 -24.50 E LLBQS 2214-1903 0.396 18.81 19.27 -22.81 -22.60 S Q AQ 2215-037 0.242 18.69 17.38 -22.06 -23.29 E Q
41 –Table 1—Continued
Name(1) z (2) m nuc (3) m host (4) M V (nuc)(5) M V (host)(6) Morphology(7) Radio(8) Selection(9)PKS 2247+14 0.237 16.65 17.22 -23.90 -23.32 E LQ 2344+184 0.138 20.22 16.68 -19.16 -22.60 S QPKS 2349-014 0.174 15.97 15.63 -23.82 -24.07 IE LNote. — Col. (3), apparent nuclear magnitude in filter. Col. (4), apparent host magnitude in filter. Col. (5), absolute V nuclear magnitude. Col. (6), absolute V host magnitude. Col. (7), host morphology: a) E=elliptical; b) S=spiral; c)EI=elliptical undergoing strong interaction; d) SI=spiral undergoing strong interaction; e) ED=elliptical with possibleinner disk; f) IE=irregular or interacting that is best fit with an elliptical model; g) IS=irregular or interacting that isbest fit with a spiral model. Col. (8), radio-loudness: Q = radio-quiet; L = radio-loud; ? = radio-loudness not available.Col. (9), original proposal selections: A = chosen from LBQS catalog; B = chosen from an optically-selected catalog; C= chosen for lack of extended host in ground-based images. a Classified here as spiral because it has a small central bulge on top of a larger component with apparently tidal arms.However, the tidal structure shows a de Vaucouleurs profile, and others (e. g., McLure et al. 1999) classify it as elliptical.
42 –Table 2. PCA Subsample
Name(1) log r / (2) σ log r / (3) µ e (4) σ µe (5) M V (bulge)(6) σ MV (bulge)(7) M V (nuc)(8) σ MV (nuc)(9) log L X (10) σ log LX (11) X-ray Ref.(12)LBQS 0020+0018 0.471 0.024 19.52 0.13 -22.77 0.06 -22.45 0.15 44.36 0.13 1PHL 909 0.919 0.005 20.66 0.07 -23.29 0.06 -24.07 0.14 44.60 0.05 1UM 301 0.643 0.045 20.69 0.24 -22.44 0.08 -23.92 0.15 44.66 0.08 13C 47 0.731 0.023 20.12 0.14 -23.37 0.08 -24.07 0.15 44.99 0.02 23C 48 1.290 0.004 21.06 0.10 -24.83 0.10 -25.72 0.14 45.34 0.01 2PHL 1093 1.164 0.006 22.01 0.07 -23.54 0.06 -23.47 0.15 44.45 0.01 2PKS 0202-76 0.572 0.013 19.53 0.12 -23.12 0.10 -24.98 0.14 44.93 0.09 2Q 0244+194 0.889 0.012 21.83 0.09 -22.47 0.06 -23.29 0.15 44.49 0.03 1PKS 0312-77 1.313 0.003 22.05 0.08 -23.78 0.08 -24.40 0.14 44.66 0.07 23C 93 0.441 0.025 18.28 0.15 -23.73 0.08 -23.51 0.17 44.73 0.04 3IR 0450-2958 0.896 0.010 21.83 0.15 -22.51 0.14 -25.41 0.14 44.70 0.02 2PKS 0736+01 1.074 0.005 21.70 0.06 -23.58 0.06 -24.03 0.14 44.22 0.03 2MS 07546+3928 0.568 0.007 19.25 0.10 -23.47 0.10 -24.22 0.15 43.58 0.15 2IR 0759+6508 0.961 0.004 21.29 0.14 -23.34 0.13 -23.57 0.14 42.28 0.15 2MS 0801.9+2129 0.568 0.012 20.49 0.17 -22.13 0.16 -22.91 0.14 43.40 0.22 13C 206 1.484 0.020 23.69 0.13 -23.09 0.08 -24.03 0.15 44.93 0.04 23C 215 0.891 0.047 20.94 0.25 -23.08 0.10 -24.08 0.14 44.57 0.03 2PG 0923+201 1.090 0.010 22.35 0.10 -23.00 0.08 -24.73 0.14 43.88 0.14 1MS 0944.1+1333 0.789 0.009 21.19 0.11 -22.52 0.10 -24.16 0.14 44.17 0.07 1PG 1012+008 0.939 0.040 21.61 0.31 -22.64 0.24 -23.75 0.15 44.04 0.13 1HE 1029-1401 0.915 0.004 21.59 0.08 -22.74 0.07 -24.79 0.14 44.83 0.01 1PG 1202+281 0.656 0.006 20.53 0.10 -22.62 0.09 -23.03 0.15 44.27 < < − ). Col. (6), absolute V magnitude of host bulge. Col. (10), 0.5 keV x-ray luminosity of nucleus (erg s − ).References. — X-ray literature; (1) Yuan et al. 1998; (2) Brinkmann et al. 1997; (3) Wilkes et al. 1994; (4) Grupe et al. 2001; (5)Margon et al. 1985.
43 –Table 3. Correlations and Linear Fits
Dependent Independent Subsample N obj ρ Probability Slope σ slope Intercept σ intercept M V (host) M V (nuc) All 56 0.350 0.01 0.30 0.09 -15.9 2.2LE 22 0.434 0.04 0.29 0.08 -16.6 1.9QE 16 0.394 0.13 0.26 0.17 -16.7 4.2LS 4 0.800 0.20 0.77 0.14 -4.5 3.5QS 14 0.002 0.99 -0.03 0.16 -23.9 4.0L 26 0.578 < M V (bulge) log ` M BH / M ⊙ ´ All 20 -0.263 0.26 -1.43 1.18 -10.5 10.5LE 10 0.103 0.78 -0.57 0.92 -18.6 8.2QE 7 -0.847 0.02 -1.86 1.41 -6.1 12.5LS 1 ... ... ... ... ... ...QS 2 ... ... 0.16 < M V (nuc) log ` M BH / M ⊙ ´ All 26 -0.378 0.06 -1.98 1.16 -6.9 10.2LE 10 -0.164 0.65 -2.75 4.09 0.1 37.0QE 7 -0.536 0.22 -5.16 4.98 22.4 44.8LS 1 ... ... ... ... ... ...QS 8 -0.643 0.09 3.05 4.03 -50.1 33.9L 11 -0.196 0.56 -2.27 2.23 -4.4 20.0Q 15 -0.318 0.25 -1.72 1.64 -9.2 14.3E 17 -0.472 0.06 -4.33 4.31 14.7 38.9S 9 -0.667 0.05 2.09 2.19 -42.0 18.4Note. — For elliptical hosts, we take the entire host as the “bulge.” For other host types, the bulge is the spheroidal component.These correlations include only those objects for which M V (nuc) ≤ − .
0. The number of objects in each sample varies from onetest to another, according to the available data. Col. (4), number of objects in subsample. Col. (5), Spearman rank correlationcoefficient. Col. (6), the significance, listed as the probability that | ρ | could be this large or larger by chance alone. Fits are ofthe form Dependent = slope × Independent + intercept.
44 –Table 4. Black Hole Masses, Eddington Fractions, and Bulge Magnitudes
Name log M BH / M ⊙ log L bol /L Edd M V (bulge) σ M V (bulge) PG 0052+251 8.8 -0.8 ... ...PHL 909 9.4 -1.4 -23.29 0.06PHL 1093 9.1 -1.3 -23.54 0.06MRK 1014 8.2 -0.5 ... ...NAB 0205+02 8.3 -0.2 ... ...Q 0244+194 8.5 -0.8 -22.47 0.06PKS 0736+01 8.5 -0.5 -23.58 0.06PG 0923+201 9.4 -1.1 -23.00 0.08PG 0953+414 8.9 -0.4 ... ...PKS 1004+13 9.6 -0.9 -24.06 0.12PG 1012+008 8.3 -0.4 -22.64 0.24HE 1029 − −
102 8.8 0.0 -24.14 0.09PG 1307+085 8.3 -0.2 -22.42 0.10PG 1309+355 8.5 -0.3 -23.62 0.17PG 1402+261 7.8 0.3 ... ...PG 1444+407 8.5 -0.1 -22.96 0.263C 323.1 9.4 -1.0 -23.33 0.08MC 1635+119 8.6 -1.6 -22.62 0.06PKS 2135 −
147 9.4 -1.4 -23.23 0.06OX 169 9.2 -1.0 -23.21 0.12PKS 2247+14 8.1 -0.1 -23.32 0.06PKS 2349 −
014 9.3 -1.3 -24.07 0.06Note. — For elliptical hosts, we describe the entire host as the “bulge.” Forother host types, the bulge is the spheroidal component. A standard devia-tion of 0.4 dex is used for all of the black hole masses, adopted from Vester-gaard (2004). Since the errors in M V (nuc) are much smaller, the propagatederrors in log ( L bol /L Edd ) all come out to be 0.4, as well.
45 –Table 5. PCA Results
Type Eigenvector λ (%) Cumulative %Optical e e e e e e Table 6. Fundamental Plane RMS Errors and CorrelationsVariable RMS Error CorrelationOpticallog r / µ e M V (nuc) 1.97 0.532X-raylog r / µ e L X Name ∆ m nuc ∆ m host ∆ log r e Morphology ReferenceLBQS 0020+0018 -0.22 0.01 ... E 4LBQS 0021 − a −
76 -1.23 -1.28 0.04 E 2NAB 0205+02 ... -0.92 0.30 S a
1Q 0244+194 0.00 0.04 -0.06 E 7US 3498 -0.20 -0.03 0.065 S a −
77 -1.77 -1.03 0.068 E 2PKS 0736+01 0.10 -0.20 -0.0496 E 7PG 0923+201 ... -0.04 0.00 E 1PKS 1004+13 ... 0.10 0.15 E 1HE 1029 − b −
102 ... -0.85 0.36 E 1PG 1307+085 ... -0.33 0.18 E 1PG 1309+355 ... -0.19 0.0 S c d E 2PG 1402+261 ... -0.97 0.04 S a e e − −
037 0.09 -0.82 0.28 E 3PKS 2247+14 -0.25 0.02 0.201 E 7Q 2344+184 1.02 -0.52 0.08 S a −
014 -0.03 -0.27 0.177 E 7Note. — In each of columns (2), (3), and (4), the difference is calculated as ourvalue - the literature value. Col. (2), difference in nuclear magnitude. Col. (3),difference in host magnitude. Col. (4), difference in log effective radius (log r / forellipticals or log r eff for spirals). Col. (5), host morphology, quoted in its simplestform: E (elliptical) or S (spiral). The number of significant figures in the table varieswith the precision of the literature values. Ellipses (...) are shown if no value is given.Values quoted from Bahcall et al. (1997) and Kirhakos et al. (1999) are their 2D modelresults.References. — iterature references; (1) Bahcall et al. (1997); (2) Boyce et al. (1998);(3) Disney et al. (1995); (4) Hooper et al. (1997); (5) Hutchings et al. (1994a); (6)Kirhakos et al. (1999); (7) McLure et al. (1999). a iterature size assumes a disk only, while we account for both a bulge (or bar) anda disk. For PG 0052+251, NAB 0205+02, and PG 1402+261, we mask the centralbulge or bar and fit only the disk, whereas Bahcall et al. (1997) fit one exponentialprofile to the entire host. For Q 2344+184 and US 3498, we fit both bulge and diskseparately and use our disk size here; McLure et al. (1999) fit one exponential profileto the entire host. b Hutchings et al. (1994a) do not list a morphology. c Size is r / . A de Vaucouleurs profile fits best, but we classify it as a spiral based
47 – on appearance. We compare with the de Vaucouleurs model of Bahcall et al. (1997). d We find the radial profile of PG 1358+04 to be a “broken” de Vaucouleurs profilewith two different effective radii, so we do not compare log r / here. e iterature size assumes a bulge model for the entire host. We use our bulge modelsize here but mask other features. For 3C 351, the bulge is surrounded by a ringfeature that we mask from the fit but Boyce et al. (1998) include. For OX 169, bulgeis crossed by a large, arm-like feature we mask from the fit; we do not know exactlywhich areas are fitted by McLure et al. (1999).
48 –Table 8. PCA Results by Subsample
Sample ( λ (%) Cumulative % λ (%) Cumulative %LE (19) e e e e e e e e e e e e e e e e e e e e e
49 –Table 9. Coefficients for FP FormsSample R coeff M coeff C OpticalAll -14.2 3.14 -77.5LE -14.7 3.14 -75.8QE 108 -22.4 361QS -29.5 5.55 -118L -15.3 3.29 -78.3Q -44.9 8.53 -168E -14.1 3.13 -77.1S -19.1 3.79 -88.1X-rayAll 8.74 -2.03 79.3LE 10.8 -2.39 84.1QE -11.8 2.55 0.0918QS -33.5 5.89 -54.9L 11.4 -2.51 85.9Q -59.3 11.5 -151E 8.14 -1.93 77.6S 42.1 -8.36 186Note. — Coefficients andconstants for the fundamentalplane equations: in the opticalcase, M V (nuc) = R coeff log r / + M coeff µ e + C , and in the x-ray case,log L X = R coeff log r / + M coeff µ e + C . 50 –Table 10. QSO FP GradientsSample Gradient magnitude ||∇ F || Azimuth α (deg)Optical All 6.49 151
Q 19.9 155QS 13.1 155S 8.52 154L 6.95 152LE 6.65 152E 6.45 151QE 48.7 -27X-ray
All 10.1 -30
S 47.0 -26L 13.0 -29LE 12.4 -29E 9.46 -31QE 13.4 152QS 36.6 156Q 65.9 154Note. — Full sample results are given in boldface. The subsampleQSO FPs are grouped by azimuth and ranked by magnitude. Thegradients are unitless, while the azimuthal directions are measuredin degrees counterclockwise from the + ii