SIGRAV Graduate School in Contemporary

1
Laura Ferrarese Rutgers University Lecture 4
Beyond the Resolution Limit

• SIGRAV Graduate School in Contemporary
• Relativity and Gravitational Physics

2
Lecture Outline

• The Innermost Regions of Active Galactic Nuclei
• Variability in Active Galactic Nuclei
• Continuum variability
• Emission line variability
• Reverberation mapping
• making sense of the continuum-emission line
  connection
• Potential problems.
• and how to solve them the Transfer Function
• Where the Observations Stand

3
Schematic View of an AGN
4
Reverberation Mapping in a Nutshell

• Measuring masses of SBHs using Reverberation
  Mapping is based on the assumption that the size
  r and velocity ? of the Broad Line Region (BLR)
  clouds are connected by a simple virial
  relationship
• According to the standard model, Broad
• Line Region (BLR) clouds are
• many (107-8, Arav et al. 1997, 1998,
• Dietrich et al. 1999)
• small, dense (Ne 109-11 cm-3)
• cold (Te 2?104 K)
• photoionized (Ferland et al. 1992)
• localized within a volume of a few to
• several tens of light weeks in diameter
• around the central ionization source.
• As such, the BLR is, and will remain, spatially
  unresolved at optical wavelengths even using
  space based instrumentation, and its size cannot
  be determined using conventional images.

5
Reverberation Mapping in a Nutshell

• If the BLR is photoionized, the broad lines
  should respond to continuum variations. The line
  response contains a wealth of information
  regarding the spatial and kinematic structure of
  the BLR therefore monitoring programs for AGNs
  started in the early 80s in the attempt to
  quantify the nature of the continuum and emission
  line variations (if any!)
• If everything works as planned, the time delay
  between continuum and line variation is simply
  (?) related to the size of the BLR.

6
Reverberation Mapping in a Nutshell

• Advantages
• reverberation mapping probes regions very close
  to the central engine (103 RSch), a factor at
  least 1000 smaller than allowed by traditional
  methods which relay on resolved kinematics.
• This leaves little doubt that the measured mass
  (if correct!) is in the form of a supermassive
  black hole
• Disadvantages
• The observations are difficult. Close time
  monitoring at very closely spaced intervals and
  multiple frequencies is necessary.
• For the virial approximation to be applicable,
  the kinematics must be dominated by gravity. The
  presence of outflows or not gravitational motions
  (to which gas might be prone) would undermine the
  method entirely.
• The geometry of the BLR are not known.

7
Observational Requirements for Monitoring Programs

• Temporal Sampling
• observations must be closely spaced in time
  relative to the physical timescale of interest
  (generally a problem for the early monitoring
  programs)
• difficulties in scheduling the observations
• S/N of the data
• S/N must be high relative to the magnitude of the
  flux variations (i.e. Fvar must be gtgt 0)
• e.g. S/N30 is necessary to detect 10 variations
  in continuum flux at the 3? confidence level.
• Aperture effects
• variations in seeing at the time of the
  observations, as well as pointing and guiding
  errors can cause variations in the amount of
  light entering the spectrograph, since both the
  NLR and the host galaxy are extended. This can
  cause spurious spectral variations.
• Datasets must be as homogenous as possible,
  ideally using a single instrument in a stable
  configuration

8
Observational Requirements for Monitoring Programs

• Flux calibration since it is impossible to rely
  exclusively on photometric conditions, relative
  spectrophotometry can be achieved by
• simultaneous observation of the AGN and a nearby
  (non-variable) field star.
• the slit response function (sensitivity as a
  function of position in the slit) must be known
  very accurately
• 1-2 spectrophotometric accuracy
• using non-variable components of the AGN spectra
  (e.g. OIII??4959,5007 narrow line emission)
• aperture effects can be difficult to control
  since the NLR is extended - must know the surface
  brightness distribution of the extended component
  and the PSF (as a function of time)
• 2 spectrophotometric accuracy

9
Observational Requirements
N9000
N64
N25
Systematic errors as a function of width of the
radial emissivity distribution for several length
experiments. Solid line 900 data points, dashed
line 64 data points. Dotted line 25 data points
(Krolik 2001).
10
Observational Requirements
Anisotropic model, ?t (1/24)(r/c)
Anisotropic model, ?t (3/8)(r/c)
Isotropic model, ?t (3/8)(r/c)
Isotropic model, ?t (1/24)(r/c)
Systematic errors as a function of width of the
radial emissivity distribution for four
experiments, each with 64 data points, but
different time sampling (Krolik 2001).
11
Monitoring Programs

• UV/optical monitoring programs started in the
  early 80s
• International AGN Watch (Peterson 1993, PASP,
  105, 247)
• IUE (UV) and ground-based optical monitoring
  program of NGC 5548.
• Mostly ground based optical monitoring of 8
  nearby Seyfert 1 galaxies.
• (although only 5 have been monitored
  extensively see Wandel, Peterson Malkan 1999,
  ApJ, 526, 579 for a compilation)
• Ohio State University
• optical monitoring of 9 Seyfert
• 1s, manly observed in H?
• Wise/Steward (Kaspi et al. 2000,
• ApJ, 533, 631)
• optical monitoring of 17
• quasars (H? and/or H?).
• LAG (Lovers of Active Galaxies),
• led by M.V. Penston
• optical monitoring of 8 AGNs
• and quasars, 3 not previously
• observed (H? and/or H?).

Sampling patterns of previous and planned AGN
multiwavelength programs
12
Monitoring Programs

• The main results of these programs are
• AGN continua vary on timescales which can be as
  short as days.
• The UV and optical continuum vary with no
  apparent time lag between them.
• AGN continuum variability must not be caused by
  either mechanical disk instabilities or
  variations in the accretion rates, since these
  would propagate through the disk slowly enough
  that the inner, hotter (UV) part of the disk
  should be seen to vary before the outer, cooler
  (optical) part.
• UV/continuum variations might be caused by
  reprocessing by the disk of the hard X-ray
  produced along the disk axis.
• Variations in the emission line flux correlate
  with continuum variations.
• The highest ionization lines respond most rapidly
  to continuum changes, implying that there is
  ionization stratification in the BLR.
• The BLR is not in pure radial motion, since there
  is no obvious difference in the timescale of the
  response of the red and blue wings of the
  emission lines.

13
Variability in AGNs

• AGNs are variable at all wavelengths at which
  they have been studied, not only in the
  continuum, but also in the emission lines.
• Typical quasars vary at the 0.3 - 0.5 mag level
  over timescales of a few months, with extreme
  cases varying on timescales as short as a few
  days.
• The variability in Seyfert galaxies is less
  dramatic and was not discovered until the late
  60s.
• Causality arguments imply that the emitting
  region is less than a few light days across
• Periodicity in the light curve have been searched
  for but never found variations are aperiodic and
  have variable amplitudes

14
Variability in AGNs

• The continuum variability can be characterized by
  the mean fractional variation

Difference in optical flux (upper panel) and
variability parameter (lower panel) for the
Seyfert 1 galaxy NGC 5548, as a function of the
time interval between observations.
15
X-Ray Variability in AGNs

• Rapid X-ray variability is a staple in all AGNs
  (Mushotzky et al. 1993, ARAA)
• Since X-rays arise near the SBH event horizon,
  X-ray variability sets the tightest constraints
  on the AGN size
• the fastest possible variability timescale for a
  coherent, isotropically emitting region is the
  crossing time
• The variability timescale is characterized by the
  fluctuation power density spectrum (PDS), which
  is the product of the Fourier Transform of the
  light curve with its complex conjugate.
• Observationally, P(?) ? ??? , with index ? 1-2
  over timescales of hours to months.
• Since the total power (integral of the PDS over
  all frequencies) must be finite, the PDS must
  turn over (? lt 1) at low frequencies.

16
X-Ray Variability in AGNs
RXTE light curve (left) and PDS (up) for NGC
3516. The cutoff frequency n corresponds to a
timescale of 27 days, much longer than the light
crossing time for a SBH of plausible mass
(Edelson Nandra 1999, ApJ, 514, 682)
17
X-Ray Variability in AGNs

• The observed cutoff must be related to the
  fundamental physics that generate the variability
  and to the processes by which the X-rays are
  produced, e.g.
• Compton upscattering of ultraviolet "seed"
  photons that probably arise in an accretion disk
  (e.g., Haardt Maraschi 1991 ,1993 Zdziarski et
  al. 1994 Stern et al. 1995 ).
• optical depth effects
• size of the scattering region
• Bright spot model (e.g. Bao Abramowicz 1996)
  . In this model, active regions on the surface of
  a rotating accretion disk produce the observed
  variability. The relevant turnover timescale
  could perhaps be identified with the orbital
  timescale, marginally consistent with the
  observed cutoff for the extremes of parameter
  space mentioned above if the emission is produced
  very far out in the disk.
• acceleration mechanism
• nature of the instabilities that cause the bright
  spots to form
• To first order, the cutoff frequency scale
  depends on the relevant timescale which controls
  the variations (light crossing time, orbital
  timescale, thermal timescale, sound timescale,
  drift timescale), all of which depend on the
  physical size of the incriminated region and
  therefore on the black hole mass. For instance,
  the cutoff frequency in NGC 3516 is a factor
  105-106 shorter than in Cyg-X1 (MBH 10 M?),
  implying a central SBH of the order 106-107 M?

18
Emission Line Variability

• Broad Emission lines in AGN spectra can vary in
  both flux and profile.
• Narrow lines fluxes do not vary! This is due to
  the fact that in the NLR both the light crossing
  time and the recombination time are large (gt100
  years), therefore short-term variability is
  smeared out.

19
Emission Line Variability
Mean spectrum formed from 34 individual spectra
of NGC 5548 (upper panel). RMS spectrum formed
from the same data by computing the rms flux at
each wavelength. Constant features, such as
narrow emission line and the host galaxy
continuum, do not appear in the rms spectrum
(Korista et al. 1995, ApJS 97, 285)
20
Line-Continuum Variations
H? line flux against the continuum flux measured
at the same time (left) and 15 days earlier
(right), for the Seyfert 1 galaxy Mrk 335. The
emission line fluxes are better correlated with
the earlier rather than current continuum fluxes
(Peterson et al. 1998, ApJ 501, 82)
21
The Time Lag

• The time lag is commonly measured by cross
  correlating the line and continuum light curves

Emission line light curve
Continuum light curve
22
The Time Lag

• Calculating the cross-correlation function can be
  tricky, in particular
• for larger and larger lags, fewer points
  contribute to the CCF, since the points at the
  end of the series drop out. This implies that for
  the cross-correlation to yield statistically
  significant results, it is necessary to have a
  large number of datapoints.
• sparse datasets need to be interpolated, with all
  the uncertainties that follow.

23
Observational Results

• AGNs with lags for multiple lines show that
  highest ionization emission lines respond most
  rapidly ? ionization stratification

24
The Basis of Reverberation Mapping

• The fact that emission lines vary in response to
  the optical/UV continuum variation immediately
  implies that
• The line emitting clouds are close to the
  continuum source
• the line emitting clouds are optically thick
• The ionizing continuum (? lt 912Å) is closely
  related to the observable optical/UV continuum.
• Therefore, our hopes are realized by
  characterizing the emission line response to
  continuum variations, the kinematics and geometry
  of the BLR can be constrained
• the time delay between continuum and emission
  line variations are ascribed to light travel time
  effects within the BLR the emission lines echo
  or reverberate to the continuum changes
  (Blandford McKee 1982, ApJ, 255, 419).

25
Reverberation Mapping Assumptions

• The continuum originates in a single central
  source. Typical scalelengths are
• Accretion disk (for 107 108 M? SBH) 101314
  cm
• Broad Line Region 1016 cm
• To all effects, as seen from the BLR, the
  continuum source can be treated as point-like
• The continuum is not required to be emitted
  isotropically (although isotropy is usually
  assumed)
• The most important timescale is the light-travel
  time.
• the cloud response to a change in the continuum
  flux is instantaneous.
• Light travel time
• Timescale to re-establish photoionization
  equilibrium
• Timescale it takes a Lyman ? photon to diffuse
  outward through the BLR

26
Reverberation Mapping Assumptions

• The structure of the BLR does not change on the
  variability time scale (or the timescale over
  which the experiment is conducted).
• Dynamical (cloud-crossing) time
• There is a simple, though not necessarily linear,
  relationship between the observed continuum and
  the ionizing continuum.

27
Reverberation Mapping Assumptions

• Once the (responsivity weighted) size r of the
  BLR is known, the AGN central mass can be
  obtained through the virial relationship

• where f is a dimensionless factor of order unity
  that depends on the geometry and kinematics of
  the BLR, and ? is the emission line velocity
  dispersion.
• The velocity width ? of the lines is measured in
  the rms spectrum
• the rms spectrum only contains information on the
  variable part of the lines constant components
  do not contribute.

28
Potential Problems

• What does reverberation mapping measure?
• The flux variations in each line are
  responsivity-weighted
• Determined by where the physical conditions
  (mainly flux and particle density) give the
  largest response for a given continuum increase.
• This can vary with time.
• Also, this conditions might be verified at
  different depths for r and ? (Krolik 2001, ApJ,
  551, 72), potentially leading to systematic
  errors on the masses.

Systematic errors due to the differing moments
that define r and ?, as a function of the width
of the radial emissivity distribution. Solid
curve isotropic radiation. Dotted curve
anisotropic radiation (directed towards the
center). Masses are systematically overestimated
in the anisotropic case, since we see
preferentially the outer regions of the cloud,
the measured lag is greater than it would be if
the entire emitting region could be observed
(Krolik 2001).
29
Potential Problems

• What is f?
• circular, coplanar orbits mean-square
  line-of-sight velocity is GMsin2i/(2r), therefore
  f2/sin2i. f could therefore take any value
  between 2 and ?.
• random, isotropic circular orbits mean-square
  line-of-sight velocity is GM/(3r), therefore f3
• random, isotropic parabolic orbits mean-square
  line-of-sight velocity is 2GM/(3r), therefore
  f3/2
• These potential problems add to the systematics
  arising from the (generally) inadequate temporal
  sampling of the observations, and the (generally)
  short duration of the experiments.

30
Potential Problems the Virial Hypothesis

• How can we test the virial hypothesis?
• If the motion of the gas is gravitational, using
  BLR sizes and velocity derived from different
  emission lines in the same AGN must produce the
  same estimate of the central mass.
• NGC 5548 highest ionization lines have smallest
  lags and largest Doppler widths, such that
  virial product r V 2 is constant.

? 1989 data from IUE and ground-based telescopes.
? 1993 data from HST and IUE.
31
Potential Problems the Virial Hypothesis

• There are a total four AGNs for which lag
  measurements for multiple emission lines exist,
  all supporting the virial approximation (Onken
  Peterson 2002)
• NGC 7469 8.4 ?106 M?
• NGC 3783 8.7 ?106 M?
• NGC 5548 5.9 ?107 M?
• 3C 390.3 3.2 ?108 M?

32
Potential Problems the Virial Hypothesis

• In the case of NGC 5548 only, there is sufficient
  information on the long term behavior of a single
  line, H?, to monitor the variation of the line
  width and lag as a function of time.
• As the continuum brightens, the lag is expected
  to become longer (since the BLR can be ionized to
  a greater depth) and the line width is expected
  to become narrower. This agrees with the
  observations.

33
Potential Problems

• While the time lag measured for different lines
  agrees with the virial relationship, it does not
  necessarily exclude other models. For instance,
  special cases of cloud outflows or disk winds
  could also explain the observations (Blumenthal
  Mathews 1975, ApJ, 198, 517 Murray et al. 1995,
  ApJ, 451, 498 Emmering, Blandford Shlosman
  1992, ApJ, 385, 460 Chiang Murray 1996, ApJ,
  466, 704 Bottorff et al. 1997, ApJ, 479, 200)

34
Potential Problems

• To summarize, all of our problems would be solved
  if the geometry and kinematics of the broad line
  region were completely determined
• One of the major remaining mysteries of AGN
  astrophysics
• We need to know this to understand systematic
  uncertainties in AGN masses.
• Can we determine the BLR geometry and kinematics
  from the observations? YES!
• BUT this will require a leap in data quality.
• Accurate mapping requires a number of
  characteristics (nominal values follow)
• High time resolution (? 0.2 day)
• Long duration (several months)
• Moderate spectral resolution (? 600 km s-1)
• High homogeneity and signal-to-noise (100)
• Given these data, we could not just restrict
  ourselves to measuring time lags, but we could
  measure the complete transfer function.

35
The Transfer Function

• The transfer function determines the relation
  between continuum and emission lines variations

• The transfer function is simply the time-smeared
  emission-line response to a ? function outburst
  in the continuum. In other words, the transfer
  function can be interpreted as a velocity delay
  map.
• Solving for the transfer function is a classical
  inversion problem. In practice, it requires
  extremely well sampled, high quality data.

36
The Transfer Function
Integrate over time delay to get the line
profile
Integrate over velocity to get the delay map

• In the best case, the data so far only allows us
  to solve for the velocity independent (or 1-d)
  transfer equation where both ?(?) and L(t)
  represent integrals over the emission line width

37
The Isodelay Surface

• Suppose that
• the BLR consists of clouds in a thin spherical
  shell in orbit around the central source.
• the continuum light curve is a ? function
  outburst.
• Then
• the signal anywhere in the shell is delayed with
  respect to the continuum outburst by the light
  travel time r/c.
• An observer at the central source will record a
  variation in the emission lines with a delay 2r/c
  following the continuum outburst.
• An observer at any other location will record
  time delays in the emission lines which are
  different for different parts of the cloud.

Observer
38
The Isodelay Surface

• the isodelay surface is defined as the locus of
  points for which the pathlength to the observer
  is constant.
• Emission lines variations arising from the
  intersection of the isodelay surface to the BLR
  are recorded simultaneously by the observer.
• Isodelay surfaces are conic sections with the
  observer at one focus and the continuum source at
  the other the time delay at position (r,?) is ?
  (1 cos ?)r/c

Observer
Continuum
39
The Isodelay Surface

• If the observer is at infinity, the isodelay
  surfaces are parabolas

40
Examples of Transfer Functions

• Lets take a look at some specific transfer
  functions. Geometries of particular relevance
  are
• Clouds distributed in a shell and in random
  Keplerian orbits, illuminated by either an
  isotropic or bipolar continuum.
• BLR in biconical outflows these might be
  relevant, since they are seen in the NLR and
  might apply to at least a component of the BLR.
• Disks of random orientation in Keplerian motion.

41
Isodelay Surfaces and Transfer Function

• The transfer function measures the amount of line
  emission emitted at a given Doppler shift in the
  direction of the observer as a function of time
  delay ?.
• Consider a thin spherical shell. The time delay
  for a particular isodelay surface is simply the
  equation of a conic section is polar coordinates
• The intersection of the isodelay surface with the
  BLR is a ring of radius rsin?. The surface area
  element of the ring is 2?(rsin?)rd?. If the
  response of each area element has the constant
  value ?, then the transfer function of the ring
  can be written as
• ?(?) 2?? r2 sin? d?
• or, in terms of the time delay
• ? rather than ?
• ?(?)d? ?(?) (d? /d?) d?
• 2?? rc d?
• Thus the 1-D transfer function for a thin
  spherical shell is constant over the range 0lt ? lt
  2r/c.

42
Transfer Function of a Thin Spherical Shell
43
Examples of Transfer Functions

• A complication is introduced if the line emission
  is anisotropic.
• Physically this can happen if the BLR is
  optically thick to both continuum and line
  radiation. In this case, most of the line
  emission will be from the side of the cloud
  facing the central continuum source.
• The BLR optical depth for different emission
  lines is different.
• The degree of anisotropy in the line emission can
  be parameterized as
• ?(?) ?0(1Acos?)
• with A0 for isotropic emission, A1 for
  completely anisotropic emission.
• A1 for Ly?, A0.7 for CIV, A0.7 for Ly?.
• The main effect of anisotropic line emission is
  to increase the measured lag for a given geometry
  because the apparent response of the near side of
  the BLR is suppressed.

44
Examples of Transfer Function

• Transfer function for a thin, completely
  anisotropic (A1) shell

45
Examples of Transfer Functions

• Transfer function in the case in which the BLR
  (again assumed to be a spherical shell) is
  illuminated by a biconical beam with given
  opening angle.
• Only part of each orbit is illuminated by the
  continuum source.

46
Examples of Transfer Functions

• Consider the simple case of clouds in a circular
  orbit of radius r and inclination i 90 (edge
  on), and orbital speed Vorb.
• Clouds at the intersection of the isodelay
  surface and the orbital plane have line-of-sight
  velocities Vz Vorb sin?.
• Therefore, the circular orbit projects to an
  ellipse in the (Vz, ?) plane with semiaxes Vorb
  and r/c.

47
Examples of Transfer Functions

• When the orbit is inclined at i lt 90, the range
  in both time delay and line-of-sight velocity
  contract
• The range in time delays decreases from 0,2r/c
  to
• (1-sini)r/c,(1sini)r/c
• The line of sight velocity decreases from -Vorb,
  Vorb to
• -Vorb sini, Vorb sini
• In the limit of i 0, the velocity-delay map
  contracts to a single point at
• ? r/c, Vz 0.
• A complete thin shell can be constructed by
  integrating over all inclinations.

48
Transfer Functions Thick Geometries

• The generalization to a disk or thick shell is
  trivial - simply a matter of integrating over a
  series of circular orbits (disk) or spherical
  shells (thick shell).
• In thick geometries, the responsivity per unit
  volume is generally a function of distance,
  because of geometrical dilution of the continuum
  and the fact that the BLR covering factor will
  vary.
• ?(r) ?0 r?

49
Transfer Function Keplerian Disk

• Transfer function for a thin keplerian disk at a
  45 degree inclination.

50
Transfer Functions Thick Shells

• In thick geometries, the responsivity per unit
  volume is generally a function of distance,
  because of geometrical dilution of the continuum
  and the fact that the BLR covering factor will
  vary.
• ?(r) ?0 r?
• Transfer functions for two shells with the same
  geometry, but different responsivity function

rin2 ld rout10 ld ?-4
rin2 ld rout10 ld ?0
51
Transfer Functions Thick Shells

• Transfer functions for two thick shells with the
  same geometry and responsivity functions, but
  illuminated by a biconical outflow which does not
  (left) and does (right) point towards the
  observer.

to the observer
to the observer
rin2 ld rout10 ld ?-2 A0 w75º i15º
rin2 ld rout10 ld ?-2 A0 w30º i45º
52
Transfer Function BLR Outflows

• Transfer functions in the case of a BLR in
  spherical (left) and biconical (right) outflow.

53
Complex Transfer Functions

• Recovering complex transfer functions requires
  mapping at multiple emission lines.

54
Transfer Functions

• Very different scenarios can correspond to very
  similar 1-D transfer functions, but can be very
  easily distinguished using the 2D transfer
  function (or, equivalently, the time delay map
  and the line profiles).

55
Two Simple Velocity-Delay Maps
Randomly inclined circular Keplerian orbits
Inclined Keplerian disk
56
Recovering Velocity-Delay Maps from Real Data

• Transfer functions can be recovered from real
  velocity-delay maps by Fourier inversion.
• This requires
• High S/N in each spectrum
• High S/N in the light curve
• Moderately high spectral resolution
• Long monitoring duration
• Dense temporal sampling
• In no case to date has this been achieved, though
  in no case has it been a design specification!

C IV and He II in NGC 4151. The double peaked
appearance in the line is due to a strong
absorption feature (Ulrich Horne 1996, MNRAS,
283, 748)
57
Recovering Velocity-Delay Maps from Real Data
Transfer function recovered from the CIV emission
in NGC 5548. The data has been interpreted as 1)
evidence of no outflows 2) evidence of radial
outflows 3) evidence of radial inflow (!).
Transfer function recovered from the H? emission
in NGC 5548. Caution should be exercised since
the data spans a period longer than the BLR
dynamical timescale.
58
Observational Results

• Although no experiment yet has recovered a
  reliable velocity-delay map, emission-line lags
  have been measured in 37 AGNs, in some cases for
  multiple emission lines.

The H? response in NGC 5548 has been measured for
14 individual observing seasons. Measured lags
range from 6 to 26 days.
59
Reverberation Mapped AGNs
From Kaspi et al. 2000, ApJ, 533, 631
60
Mass-Luminosity Relationship

• The measured masses correlate, although with very
  large scatter, with the continuum luminosity, in
  the sense that brighter AGNs have larger SBHs.

M ? L0.30.1
? QSOs (Kaspi et al. 2000) ? Seyfert 1s
(Wandel, Peterson, Malkan 1999) ? Narrow-line
AGNs ? NGC 4051 (NLS1)
61
Secondary Mass Estimators

• Reverberation mapping opens the way to calibrate
  a secondary mass estimators since, to first
  order, we expect the broad line region size to
  correlate with the ionizing continuum luminosity
• Photoionization equilibrium models are
  parameterized by the shape of the ionizing
  continuum, the elemental abundances, and the
  ionization parameter U
• where Q(H) is the number of hydrogen ionizing
  photons (?13.6 eV) emitted per
• second by the central source
• U characterizes the ionization balance within the
  cloud, since Q(H)/r2 is proportional to the
  number of ionizations occurring per unit area,
  while ne is proportional to the recombination
  rate.
• To first order, AGN spectra all look alike, i.e.
  they have the same ionization parameter and
  electron density (typical values are Q(H) 1054
  h0-2 photons s-1 ne 1011 cm-3 U 0.1).
  Therefore, we expect

62
BLR Scaling with Luminosity

• This is close to what we observe! For the 37 AGNs
  which have been reverberation mapped, the BLR
  radius, measured from the H? time lag, correlates
  (although with large scatter) with the continuum
  luminosity.

r(H?) ? L0.60.1
? QSOs (Kaspi et al. 2000) ? Seyfert 1s
(Wandel, Peterson, Malkan 1999) ? Narrow-line
AGNs ? NGC 4051 (NLS1)
63
Suggested Readings

• Review Peterson, B.M. 2001, Variability of
  Active Galactic Nuclei, in The Starburst- AGN
  Connection, World Scientific (astro-ph/0109495).
• Criticism Krolik 2001, ApJ, 551, 72