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First Steps Toward Constraining Supermassive Black-Hole Growth: Mass Estimates of Black Holes in Distant Quasars Marianne Vestergaard University of Arizona Collaborators: Alex Beelen, Misty Bentz, Frank Bertoldi, Chris Carilli, Pierre Cox, Xiaohui Fan, Shai Kaspi, Dan Maoz, Hagai Netzer, Chris Onken, Pat Osmer, Chien Peng, Brad Peterson, Rick Pogge, Gordon Richards, Francesco Shankar, Adam Steed, Fabian Walter, David Weinberg Drexel University, February 10, 2006 Active Galactic Nuclei • Bright galaxies with a point-source of non-stellar activity in nuclei • They are rare – comprise only a few percent of bright galaxies • The most powerful are called quasars. • Quasar nuclei outshine their host galaxy light (Elvis et al. 1994) ~10 17 cm -- scale of our solar system (Francis et al. 1991) Supermassive Black Holes • How are their mass measured? • How do they grow? • How are black holes and galaxies connected? Black Holes and Galaxy Formation • Black holes are likely ubiquitous in galaxy centers • MBH – σ* relationship The M – σ Relationship M σ4 (Tremaine et al. 2002; See also Ferrarese & Merritt 2000; Gebhardt et al. 2000) Black Hole Mass Bulge Velocity Dispersion Black Holes and Galaxy Formation • Black holes are likely ubiquitous in galaxy centers • MBH – σ* relationship – Formation and evolution of bulges and black holes must be intimately connected – When was it established? And how? – What came first, black hole or bulge (galaxy)? • Black hole/star-formation feedback (theory) – Negative feedback kills star formation and black hole growth by expelling gas (e.g., Springel, Di Matteo, & Hernquist 2005) Star formation activity Black hole activity Time (Gyr) (Springel et al. 2005) Black Holes and Galaxy Formation • Black holes are likely ubiquitous in galaxy centers • MBH – σ* relationship – Formation and evolution of bulges and black holes must be intimately connected – When was it established? And how? – What came first, black hole or bulge (galaxy)? • Black hole/star-formation feedback (theory) – Negative feedback kills star formation and black hole growth by expelling gas (e.g., Springel, Di Matteo, & Hernquist 2005) – Positive feedback stimulate star formation (Silk 2005) • Consequence: Galaxy bulges form later than supermassive black holes Talk Outline I. Black Hole Mass a. Determinations b. Distributions II. Black Hole – Galaxy Connection III. Black Hole Evolution Talk Outline I. Black Hole Mass a. Determinations b. Distributions II. Black Hole – Galaxy Connection III. Black Hole Evolution Black Hole Mass 2 mv – m GmMBH /R = 0 M Black Hole Mass MBH = 2 v M R /G Black Hole Mass MBH = 2 v R /G Black Hole Mass 2 MBH = v R /G R V Insert figure from HST/ MW? Why Study Quasar Black-Holes? HST/STIS • Quiescent black holes (in normal galaxies) can 109 only be studied in the nearby Universe 8m telescope 108 30m telesc. • Quasars are luminous and therefore ideal tracers of black holes to the highest observable redshifts Black Hole Mass • Their host galaxies are prime targets for studying galaxy evolution in the early(Ferrarese Universe 2003) Distance (Mpc) 10 100 How Can MBH be Determined for Active Black Holes? Local Universe Higher-z • Stellar kinematics (√) • Gas kinematics (√) • Reverberation mapping √ √ Possible Virial Estimators Source Distance from central source 3-10 RS X-Ray Fe K Broad-Line Region 600 RS Megamasers 4 104 RS Gas Dynamics 8 105 RS Stellar Dynamics 106 RS In units of the Schwarzschild radius RS = 2GM/c2 = 3 × 1013 M8 cm . Mass estimates from the virial theorem: M = f (r V 2 /G) where r = scale length of region V = velocity dispersion f = a factor of order unity, depends on details of geometry and kinematics Note: the reverberation technique is independent of angular resolution Virial Mass Estimates MBH = f v2 RBLR/G Reverberation Mapping: RBLR= c τ t = t3 + t = t3 t = t2 t1 – t2 = t = t1 Reverberation Mapping Results Continuum Light Curves Emission line NGC 5548, the most closely monitored active galaxy (Peterson et al. 2002) Virial Mass Estimates MBH = f v2 RBLR/G Reverberation Mapping: –RBLR= c τ t = t3 + t = t3 • vBLR t = t2 t1– t2= t = t1 Line width in variable (rms) spectrum Reverberation Mapping NGC 5548, the most closely monitored active galaxy (Peterson et al. 1999) Velocity Dispersion of the Broad Line Region and the Virial Mass • Velocity dispersion is measured from the 2R in the rms Mline = f v BH BLR/G spectrum. – The rms spectrum isolates the variable part of the f depends on lines. structure – Constant components and geometry of broad (like narrow lines) line region vanish in rms spectrum (based on Korista et al. 1995) MBH-: Comparison of Active and Quiescent Galaxies Mass • Reverberation masses appear to fall along the MBH - relation for quiescent galaxies • The scatter is also similar: ≲ a factor of 3 Gals AGNs Bulge velocity dispersion (Courtesy C. Onken) How Can Quasar MBH be Determined? Local Universe Higher-z • Stellar kinematics (√) • Gas kinematics (√) • Reverberation mapping √ √ • Scaling relations √ √ Virial Mass Estimates MBH = f v2 RBLR/G • Reverberation Mapping: RBLR=cτ, vBLR Radius – Luminosity Relation: (Kaspi et al. (incl MV) RBLR Lλ(5100Å)0.50 2005; Bentz,Peterson, RBLR Lλ(1350Å)0.53 Pogge,MV,Onken 2006, ApJ, submitted) • Scaling Relationships: MBH FWHM2 L β (see e.g. Vestergaard 2002) Single-Epoch Mass Estimates - CIV FWHM(CIV) λL λ (1350 ) 4.5 106 3 44 10 km/s 10 ergs/s 2 M BH 0.53 M • 1 scatter = factor 2.3 Log [VP(CIV, single-epoch)/M] (Vestergaard & Peterson 2006) Virial Mass Estimates: MBH=f v2 RBLR/G Scaling Relationships: (calibrated to 2004 Reverberation MBH) • CIV: FWHM(CIV) λLλ (1350 ) M BH 4.5 10 3 44 10 km/s 10 ergs/s 2 0.53 6 M 1σ uncertainty: factor ~3.5 • Hβ: FWHM(H β) M BH 8.3 106 3 10 km/s 2 λL λ (5100 ) 44 10 ergs/s 0.50 M ( Vestergaard & Peterson 2006) (see also Vestergaard 2002, and McLure & Jarvis 2002 for MgII) NGC 5548 Highest ionization lines have smallest lags and largest Doppler widths. R (M/V) -1/2 Filled circles: 1989 data from IUE and ground-based telescopes. Open circles: 1993 data from HST and IUE. … Dotted line corresponds to virial relationship with M = 6 × 107 M. Peterson and Wandel 1999 Virial Relationships • All 4 testable AGNs comply: – – – – NGC 7469: 1.2 107 M NGC 3783: 3.0 107 M NGC 5548: 6.7 107 M 3C 390.3: 2.9 108 M • Scalings between lines: vFWHM2(H) lag (H) vFWHM2(CIV) lag (CIV) • R-L relation extends to high-z and high luminosity quasars: Emission Radiuslines: – Luminosity Relation SiIV1400, HeII1640, (Data fromCIV1549, Kaspi et al. 2005) CIII]1909, H4861, HeII4686 – spectra similar (e.g., Dietrich et al 2002) – luminosities are not extreme al 2002)2002) (Peterson & Wandel 1999, 2000;(Dietrich Onken &etPeterson • R-L defined for 1042 – 1046 erg/s (Vestergaard 2004) Improving the Scaling Relationships Main goal: improve scaling laws by reducing scatter R-L relation scatter dominates scatter in mass scaling law Issues: • Host galaxy contamination – HST imaging • Accuracy of Single-epoch MBH estimates – HST & ground-based study (HST archive project, PI: MV) • Improved Masses and RBLR – Improved monitoring of nearby sources (Bentz, Peterson, Pogge, MV, Onken 2006) Talk Outline I. Black Hole Mass a. Determinations b. Distributions II. Black Hole – Galaxy Connection III. Black Hole Evolution Masses of Distant Quasars • Ceilings at MBH ≈ 1010 M LBOL < 1048 ergs/s • MBH ≈ 109 M beyond space density drop at z≈3 (Vestergaard 2004) (H0=70 km/s/Mpc; ΩΛ = 0.7) Quasars • Dramatic space density drop at z ≳3 • Very luminous AGNs were much more common in the past. • The “quasar era” occurred when the Universe was 10-20% its current age. (Peterson 1997) Masses of Distant Quasars • Ceilings at MBH ≈ 1010 M LBOL < 1048 ergs/s • MBH ≈ 109 M beyond space density drop at z≈3 (Vestergaard et al. in prep) (H0=70 km/s/Mpc; ΩΛ = 0.7) Masses of Distant Quasars • Ceilings at MBH ≈ 1010 M LBOL < 1048 ergs/s • MBH ≈ 109 M beyond space density drop at z≈3 (DR3 Qcat: Schneider et al. 2005) (Vestergaard et al. in prep) Using MgII line to Estimate Black Hole Mass • Bridge 0.8 ≲ z ≲ 1.3 gap • Will use SDSS to calibrate MgII scaling law • Complications: – FeII contamination of line and continuum Requires template fitting (Vestergaard & Wilkes 2001) Talk Outline I. Black Hole Masses II. Black Hole – Galaxy Connection III. Black Hole Evolution High Redshift Quasars and their Galaxies • UV, radio, X-ray properties similar at z > 3 (e.g., Constantin et al. 2002; Dietrich et al. 2002; Stern et al. 2000; Mathur et al. 2002) • Black holes of distant quasars are very massive ~ (1-5)x 109 M – Are their host galaxies also massive and old? • Circumstantial evidence for intense star formation on galaxy scales associated with quasars at z ≳ 4: – strong sub-mm/far-IR emission: ~108 M warm dust – strong CO emission: ~1011 M of cold molecular gas (Ohta et al. 1996; Walter et al. 2003) Dust and CO emission: large scale star formation rates 500 – 2000 M/yr (e.g., Omont et al. 2001, Carilli et al. 2001) High Redshift Quasars and their Galaxies • Some evidence for massive, old galaxies: – z~2 quasar hosts have bulge luminosity consistent with old passively evolving stellar populations (Kukula et al. 2001) – Low-z host galaxies are dominated by old (8-14Gyr) stellar populations (Nolan et al. 2001) Quasar Host Galaxies at High Redshift • Conclusive test: mean age and mass of stellar bulge • Study of the most massive black holes at z ≳ 4 – HST UV imaging: young stars L(1500Å) → star formation rate – HST Cy15 IR imaging: older stars – Spitzer mid-IR: warm dust – Sub-mm data: cooler dust – CO imaging: cold molecular gas • Goals: – Characterize stellar bulge: mean age, mean mass, and star formation rate – Determine MBH /MBulge Redshift → (Vestergaard 2004) (Data from Bruzual & Charlot 2003) Black Hole to Bulge Mass Ratio at High Redshift (Peng et al. 2006, in prep) Lensed Quasar Host Galaxy at Redshift 4.7 Original data PSF+Galaxy Model Galaxy residual HST ACS UV image Strong sub-mm source VLA CO (2-1) emission image with Einstein Ring (Carilli et al. 2003) Talk Outline I. Black Hole Masses II. Black Hole – Galaxy Connection III. Black Hole Evolution Black Hole Growth in the Early Universe Theoretical model predictions: • Accretion only – Radiatively efficient – Radiatively inefficient • Merger activity • Obscured growth • A combination of the above? (Steed & Weinberg 2003) Predicted evolution of black hole mass functions for different growth scenarios Preliminary Mass Functions of Active Supermassive Black Holes • Different samples show relatively consistent mass functions (shape, slope, normalization) (Vestergaard & Osmer, in prep.; Vestergaard, Fan, Osmer et al., in prep.) • Goal: constrain BH growth (with Fan, Osmer, Steeds, Shankar, Weinberg) • BQS: 10 700 sq. deg; B16.16mag (H0=70 km/s/Mpc; ΩΛ = 0.7) • LBQS: 454 sq. deg; 16.0BJ18.85mag • SDSS: 182 sq. deg; i* 20mag • DR3: 5000 sq. deg.; i* >15, 19.1, 20.2 Preliminary Mass Functions of Active Supermassive Black Holes • Different samples show relatively consistent mass functions (shape, slope) (Vestergaard & Osmer, in prep.; Vestergaard, Fan, Osmer et al., in prep.) • Goal: constrain BH growth (with Fan, Osmer, Steeds, Shankar, Weinberg) • BQS: 10 700 sq. deg; B16.16mag (H0=70 km/s/Mpc; ΩΛ = 0.7) • LBQS: 454 sq. deg; 16.0BJ18.85mag • SDSS: 182 sq. deg; i* 20mag • DR3: 5000 sq. deg.; i* >15, 19.1, 20.2 Preliminary Mass Functions of Active Supermassive Black Holes • Different samples show relatively consistent mass functions (shape, slope) (Vestergaard & Osmer, in prep.; Vestergaard, Fan, Osmer et al., in prep.) • Goal: constrain BH growth (with Fan, Osmer, Steeds, Shankar, Weinberg) • BQS: 10 700 sq. deg; B16.16mag (H0=70 km/s/Mpc; ΩΛ = 0.7) • LBQS: 454 sq. deg; 16.0BJ18.85mag • SDSS: 182 sq. deg; i* 20mag • DR3: 5000 sq. deg.; i* >15, 19.1, 20.2 Preliminary Mass Functions of Active Supermassive Black Holes • Locally mapped volume (R ≤ 100 Mpc): MBH ≤ 3x109 M • SDSS color-selected sample and DR3: (Fan et al. 2001, Schneider et al. 2005) ~9.5 quasars per Gpc3 with MBH ≥ 5x109 M → need ~25 times larger volume locally (R ≤ 290 Mpc) (H0=70 km/s/Mpc; ΩΛ = 0.7) Summary • >>> We can do physics with active galaxies and quasars <<< • MBH in Active Nuclei can be determined to within an accuracy: – Low-z: ~factor of 3 (measured) – Higher z: ~factor of 4 (estimated!!) • Black hole mass distributions: – <MBH> ≈ 109 M, even at 4 ≲ z ≲ 6 – Maximum black hole mass at ~1010 M • Black Hole Evolution and Galaxy Formation in Early Universe: – Ongoing study of galaxies at high redshift with the most massive black holes (~1010 M) – MBH /MBulge ratio – Mass functions of active black holes – Constrain growth of black holes and their galaxy bulges by comparing these data with theoretical evolutionary models Black Holes and their Implications for Galaxy Formation and Evolution? The blue and red galaxy sequences SDSS DR1 (Baldry et al. 2004) Masses of Distant Quasars Mass LBOL LBOL/LEdd LBOL= BC1 L(1350Å) = BC2 L(4400 Å) • BQS: Boroson & Green (1992) • z ≈ 2: Barthel, Tytler, & Thomson (1990), Vestergaard (2000) • z ≈ 4: Fan et al. (1999, 2000, 2001), Anderson et al. (2001), Constantin et al. (2002) (Vestergaard 2004) (H0=70 km/s/Mpc; ΩΛ = 0.7) Mass Estimates of Distant Quasars LBOL Mass LBOL =BC1L(1350Å) =BC2L(4400Å) BC’s from updated Elvis et al. (1994) radioquiet SED. • z ≈ 0.3: BQS: Boroson & Green (1992); 87 quasars; MBH(reverberation) & MBH (H); LBQS: Hewett et al.(1995); Forster et al. (2001) 145 quasars MBH (H) • z≈2: LBQS: Hewett et al. (1995); Forster et al. (2001) 483 quasars; MBH (CIV) • z≈4: Fan et al. (1999, 2001), 39 quasars; MBH (CIV) (H0=70 km/s/Mpc; ΩΛ = 0.7) (Vestergaard & Osmer, in prep) The M – σ Relationship • Vittorini, Shankar, & Cavalier 2005, astro-ph/0508640 (BH growth history from merger/feedback events; simulation) • Robertson et al. 2005, astro-ph/0506038 (mergers simulation) • Di Matteo, Springel, & Hernquist 2005, Nature, 433, 604 (merger induced BH growth and starformation; simulation) • Springel, Di Matteo, & Hernquist 2005, MNRAS, 361, 776 (BH/star formation feedback; simulations) • Miralda-Escude & Kollmeier 2005, ApJ 619, 30 (stellar capture) • Sazonov et al. 2005, MNRAS 358, 168 (radiative BH feedback) • King 2003, ApJ 596, L27 (supercritical accretion, outflows) • Adams et al. 2003, ApJ 591, 125 (rotating BH collapse model) • ….and many more….. MBH- Relation for Active and Quiescent Galaxies Mass Gals AGNs (Courtesy C. Onken) Bulge velocity dispersion Secondary Mass Estimation Methods Via MBH - *bulge Relation Measured *bulge : CaII 8498, 8542, 8662Å; z < 0.06 1 scatter ≈ 0.3 dex M 4.0 AGNs (Ferrarese et al. 2001) (Tremaine et al. 2002) Secondary Mass Estimation Methods Via MBH - *bulge Relation [OIII]5007 FWHM *bulge 1 scatter ≈ 0.7 dex (Nelson & Whittle 1996; Nelson 2000) Radio-louds • Line asymmetries • Outflows • Radio sources • (Interacting systems) Tremaine slope (Boroson 2003) Secondary Mass Estimation Methods Via MBH - *bulge Relation Fundamental Plane: e, re *bulge MBH • Possibly significantly uncertain - nuclear glare - bulge/disk decomposition 1 scatter = ? ( 0.7dex) Fundamental Plane (e.g., McLeod & Rieke 1995; Barth et al 2003) -FP scatter (~0.6dex for RGs; e.g. Woo & Urry 2002) - MBH - *bulge scatter log re FP(,<e>) (Barth et al. 2003) Secondary Mass Estimation Methods Via MBH - Lbulge Relation MR Lbulge 1 scatter ≈ 0.45 - 0.6 dex • Nuclear glare • Bulge/disk decomposition (e.g., McLeod & Rieke 1995; Barth et al 2003) • Scaling relation scatter ? MBH(dynamical) MBH(scaling) (McLure & Dunlop 2001, 2002) Where Do We Go From Here? Best Accuracy (dex) Reverberation Mapping Scaling Relations Via MBH - *bulge: 0.3 Future Work Zero-point, understand BLR f, odd objects (3C390.3class?) 0.5-0.6 R-L relationships, understand outliers 0.3 0.7 Extend to luminous quasars Understand scatter & outliers -- Fundamental Plane: e, re ? Quantify & establish higher accuracy Via MBH – Lbulge & scaling rel.: 0.6-0.7 -- *bulge -- [OIII] FWHM – MR Calibrate to reverberation mapped sources To first order quasar spectra look similar at all redshifts (Dietrich et al 2002) Radius – Luminosity Relations To first order, AGN spectra look the same Q(H) L U 2 4 r nH c nH r 2 Same ionization parameter Same density [Kaspi et al (2000) data] r L1/2 Radius-UV Luminosity Relationship for High-z Quasars M = VFWHM2 RBLR/G ↑ ↑ ↓ 0.1109 M 4500km/s 33 lt-days Ф RBLR-2 L <L> ≈ 1047 ergs/s Log Ф Log n(H) (Korista et al. 1997) Radius-UV Luminosity Relationship for High-z Quasars M = VFWHM2 RBLR/G Ф RBLR2 (Dietrich et al. 2002) Reverberation Mapping • Kinematics and geometry of the broad-line region (BLR) can be tightly constrained by measuring the emission-line response to continuum variations. • Can be done with UV/optical lines. NGC 5548, the most closely monitored Seyfert 1 galaxy Reverberation Mapping Results • BLR sizes are measured from the crosscorrelation time lags between continuum and emission-line variations. Continuum Emission line • This gives the first moment of the transfer function. NGC 5548, the most closely monitored Seyfert 1 galaxy Reverberation Mapping Assumptions 1 Continuum originates in a single central source. – Continuum source (1013–14 cm) is much smaller than BLR (~1016 cm) – Continuum source not necessarily isotropic 2 Light-travel time is most important time scale. • Cloud response instantaneous • rec = ( ne B)1 0.1 n101 hr • BLR structure stable • dyn = (R/VFWHM) 3 – 5 yrs 3 There is a simple, though not necessarily linear, relationship between the observed continuum and the ionizing continuum. The Transfer Equation • Under these assumptions, the relationship between the continuum and emission lines is: L(V, t ) (V, )C ( t )d Emission-line light curve “Transfer Continuum Function” Light Curve – Transfer function is line response to a -function outburst. • In practice, programs have concentrated on solving the velocity-independent (or 1-d) transfer equation: L( t ) ( )C ( t )d • It is most common to determine the cross-correlation function and obtain the “lag” CCF( t ) ( ) ACF( t )d The Transfer Equation • The aim of reverberation mapping is to solve for the transfer function from the observables, the continuum light curve C(t) and the emission-line light curve L(V,t). L(V, t ) (V, )C ( t )d Emission-line light curve “Transfer Function” Continuum Light Curve • As noted earlier, currently we have been able to get only the cross-correlation lag with any certainty. CCF( t ) ( )ACF( t )d Reverberation Mapping Results • AGNs with lags for multiple lines show that highest ionization emission lines respond most rapidly ionization stratification • Combine lag with line width to get a “virial mass”. Lyman Break Galaxies • Discovered by color-selection (Ly-break) • High-z equivalent of local star-forming galaxies • Star-formation rates: ≈ 4-25 M /yr ; ≈9 M /yr typical (Steidel et al. 1996) AGNs in z ≈ 3 Lyman-break Galaxies Broad-lined AGNs: 1% of Lymanbreak galaxies (z ≈3) FWHM (CIV) ≈4700 km/s (Steidel et al. 2002) AGNs in z ≈ 3 Lyman-break Galaxies Broad-lined AGNs: ∼1% of Lymanbreak galaxies FWHM (CIV) ≈4700 km/s MBH ≈ 108 M LBOL ≈1045 ergs/s LBOL/LEdd ≈ 0.2 (Vestergaard 2002b) Luminosities of Distant Quasars (Vestergaard 2004) Masses of Distant Quasars • Ceiling at MBH ≦ 1010 M ; LBOL < 1048 ergs/s • MBH ≈ 109 M beyond space density drop at z≈3 (Vestergaard 2004) Masses of Distant Quasars II (Vestergaard 2004) Luminosities of Distant Quasars II (Vestergaard 2004) Limitations of UV Scaling Relations NLS1s: low MBH high LBOL/LEdd Possible outflow component to CIV (Leighly 2001) Are Quasar CIV Profiles Problematic? ~15% (EW) (FWHM) (Richards et al. 2002) Virial Mass Estimates: MBH=f v2 RBLR/G Scaling Relationships: (now calibrated to 2004 Reverberation MBH) • CIV: 6 FWHM(CIV) λL λ (1350 ) M BH 4.5 10 3 44 10 km/s 10 ergs/s 2 0.53 M 1σ uncertainty: factor ~3.5 • Hβ: 6 FWHM(H β) λL λ ( 5100 ) M BH 8.3 10 3 44 10 km/s 10 ergs/s 2 FWHM(H β) M BH 4.6 10 3 10 km/s 6 2 L(Hβ) 42 10 ergs/s 0.50 M 0.63 M (H0=70 km/s/Mpc; ΩΛ = 0.7; Vestergaard & Peterson 2006) (see also Vestergaard 2002, and McLure & Jarvis 2002 for MgII) How Can AGN MBH be Determined? Local Universe Higher-z • Stellar kinematics (√) • Gas kinematics (√) • Reverberation mapping √ √ • Scaling relations √ √ Black Holes and their Implications for Galaxy Formation and Evolution? • Black holes are likely ubiquitous in galaxy centers • MBH – σ* relationship – Formation and evolution of bulges and black holes must be intimately connected – When was it established? And how? – What came first, black hole or bulge (galaxy)? • Black hole/star-formation feedback (theory) – Negative feedback kills star formation and black hole growth by expelling gas (e.g., Springel, Di Matteo, & Hernquist 2005) – Positive feedback stimulate star formation (Silk 2005) • Consequence: Galaxy bulges form later than supermassive black holes High Redshift Quasars and their Galaxies • Black holes of distant quasars are very massive ~ (1-5)x 109 M – Are their host galaxies also massive and old? • Dust and molecular gas emission indicate large scale intense star formation (500 – 2000 M/yr) • Some evidence for massive, old galaxies: – z~2 quasar hosts have bulge luminosity consistent with old passively evolving stellar populations (Kukula et al. 2001) – Low-z host galaxies are dominated by old (814Gyr) stellar populations (Nolan et al. 2001) Black Hole Mass 2 ½mvM- GmM /R 2 BH = v R /G BH =0 1 GmM 2 mv 0 2 R