Reverberation Mapping of the Broad Line Region: application to a hydrodynamical line-driven disk wind solution

2.3 MB·

by  Tim Waters

PreprinttypesetusingLATEXstyleemulateapjv.05/12/14
REVERBERATION MAPPING OF THE BROAD LINE REGION:
APPLICATION TO A HYDRODYNAMICAL LINE-DRIVEN DISK WIND SOLUTION
Tim Waters1,†, Amit Kashi2, Daniel Proga1, Michael Eracleous3,4, Aaron J. Barth5, and Jenny Greene6
1DepartmentofPhysics&Astronomy,UniversityofNevada,LasVegas,4505S.MarylandPkwy,LasVegas,NV,89154-4002,USA
2MinnesotaInstituteforAstrophysics,UniversityofMinnesota,116ChurchSt. SE.Minneapolis,MN55455,USA
3DepartmentofAstronomy&AstrophysicsandInstituteforGravitationandtheCosmos,
ThePennsylvaniaStateUniversity,525DaveyLab,UniversityPark,PA16802,USA
4DepartmentofAstronomy,UniversityofWashington,Box351580,Seattle,WA98195,USA
5DepartmentofPhysicsandAstronomy,UniversityofCalifornia,Irvine,Irvine,CA92697,USA
6DepartmentofAstrophysicalSciences,PrincetonUniversity,Princeton,NJ08544,USA
6
1
Abstract
0
2 The latest analysis efforts in reverberation mapping are beginning to allow reconstruction of echo
images (or velocity-delay maps) that encode information about the structure and kinematics of the
r
p broad line region (BLR) in active galactic nuclei (AGNs). Such maps can constrain sophisticated
A physical models for the BLR. The physical picture of the BLR is often theorized to be a photoionized
wind launched from the AGN accretion disk. Previously we showed that the line-driven disk wind

3 solution found in an earlier simulation by Proga and Kallman is virialized over a large distance
2 from the disk. This finding implies that, according to this model, black hole masses can be reliably
estimated through reverberation mapping techniques. However, predictions of echo images expected
]
from line-driven disk winds are not available. Here, after presenting the necessary radiative transfer
A
methodology, we carry out the first calculations of such predictions. We find that the echo images
G
are quite similar to other virialized BLR models such as randomly orbiting clouds and thin Keplerian
. disks. Weconductaparametersurveyexploringhowechoimages,lineprofiles,andtransferfunctions
h
depend on both the inclination angle and the line opacity. We find that the line profiles are almost
p
always single peaked, while transfer functions tend to have tails extending to large time delays. The
-
o outflow, despite being primarily equatorially directed, causes an appreciable blue-shifted excess on
r both the echo image and line profile when seen from lower inclinations (i(cid:46)45◦). This effect may be
t
s observable in low ionization lines such as Hβ.
a Keywords: accretion, accretion disks — hydrodynamics — (galaxies:) quasars: general
[

2
v 1. INTRODUCTION has a potentially major uncertainty associated with the
1 Broad emission lines have for decades been used as a valueoff,theso-calledvirialcoefficientthatdependson
8 basisforclassifyingactivegalacticnuclei(AGNs),yetthe the geometry and kinematics of the BLR. Furthermore,
1 structure and dynamics of the broad line region (BLR) therecanbesignificantuncertaintiesassociatedwiththe
5 around AGNs remains elusive. As it will be impossi- measurements of (cid:104)τ(cid:105) and ∆V (e.g., Krolik 2001), espe-
0 cially if (cid:104)τ(cid:105) is determined by first assuming a form for
ble for the foreseeable future to resolve a BLR via di-
. the transfer function (the approach taken in the code
1 rect imaging, we are left with only indirect methods to
Javelin, for example; Zu et al. 2011). Hence, even for
0 probe its spatial and kinematic properties. Temporal
this least demanding application of reverberation map-
6 monitoring observations can be used to obtain such in-
ping, it is necessary to look to physical models of the
1 formation using the technique of reverberation mapping
BLRthatobeyobservationalconstraintstobetterquan-
: (e.g., Blandford & McKee 1982; Peterson 1993; Ulrich
v tify the uncertainties associated with these quantities.
et al. 1997; Peterson & Wandel 1999, 2000; Kaspi et al.
i Severalmodelshavebeensuggested, includingrandomly
X 2000; Krolik 2001; Peterson 2001, 2006, 2013; Uttley et
orbiting clouds, inflowing and outflowing gas, rotating
al. 2014).
r
disks with thermal or line driven winds, and more (see,
a Assuming that the BLR is virialized, reverberation
for example, the review by Mathews & Capriotti 1985
mapping can be used to estimate the mass of the cen-
and a more recent summary in Section 5 of Sulentic et
tral supermassive black hole (SMBH), M . A measure
BH al. 2000).
of the time delay, (cid:104)τ(cid:105), for gas to respond to changes in
Although a great deal of work has been done to model
the continuum determines a characteristic BLR radius
thephotoionizationoftheBLRgas, relativelyfewcalcu-
R = c(cid:104)τ(cid:105) (where c is the speed of light), while the ve-
lations aimed at deriving line profiles and transfer func-
locity widths of broad emission line profiles are used to
tions have been performed, especially ones taking into
assign a characteristic velocity ∆V. The actual black
accountbothhydrodynamicsandradiativetransfer(e.g.,
hole mass measurement,
Chiang & Murray 1996). Indeed, the majority of these
R(∆V)2 modeling efforts employ stochastic methods (e.g., Pan-
M =f , (1)
BH G coast et al. 2011) that, while sophisticated,1 cannot eas-
1 We refer specifically to discrete particle, Monte-Carlo based
†email: [email protected] methods that model the BLR by prescribing probability distribu-
2 Waters, Kashi, Proga, Eracleous, Barth, & Greene
ily incorporate the extensive modeling capability offered ations in the ionizing continuum. We therefore calculate
byperformingcalculationsfromfirstprinciplesusingnu- the observables obtainable from reverberation mapping
mericalsimulations. Inthiswork,wethereforeadoptthe campaigns (namely echo images, emission line profiles,
complementaryapproachofcalculatingechoimages,line and transfer functions). Aside from qualitatively under-
profiles and transfer functions by post-processing grid- standing how echo images of line-driven disk wind so-
based hydrodynamical simulation data. lutions differ from the classic examples, it is important
Acceptable theoretical models for the BLR must be to quantify how the line profiles and transfer functions,
able to reproduce the profiles and relative strengths of as well as the echo images, depend on optical depth, in-
thebroademissionlines,aswellastheirvariabilityprop- clination angle, and kinematics. The main goal of this
erties in response to fluctuations of the ionizing contin- paper is to uncover this dependence after presenting the
uum on a variety of time scales. One of the suggested radiative transfer methodology necessary to perform re-
modelsisadiskwind(e.g.,Shields1977;Emmeringetal. verberation mapping calculations using hydrodynamical
1992; Chiang & Murray 1996; Bottorff et al. 1997), and disk wind solutions.
line driving (Castor et al. 1975) is one of the common Tothisend,weadoptverysimple,parametricprescrip-
mechanisms by which astrophysical objects can launch tions for the source function in order to compare our re-
winds. While line driving has been invoked to explain sultswithpastinvestigations. Inforthcomingpapers,we
AGN winds (e.g., Murray et al. 1995; Proga et al. 2000), will carry out detailed, self-consistent calculations of the
there is no consensus that it is the dominant mecha- photoionization structure of the wind in order to obtain
nism, as the wind may be over-ionized by X-ray radi- thesourcefunctionthroughoutitsvolumeandthedepen-
ation coming from the central engine, and in that case dence of the source function on the flux of the ionizing
theefficiencyoflinedrivingislow. However,Progaetal. continuum. Thus, we will be able to more realistically
(2000)showedthatclumpsforminginthevicinityofthe assess the short-term variability of the broad emission
SMBH can shield the other parts of the wind from the lines in response to a fluctuating ionizing flux from the
radiation, and enable line-driven winds (see also Proga central engine and produce suites of synthetic line pro-
& Kallman 2004, hereafter PK04). filesmeanttorepresentpopulationsofAGNs. Wedefera
Attributing the BLR to an accretion disk wind is ap- quantitative comparison of the model predictions to the
pealing because this type of model simultaneously pro- observations to these future papers.
vides a framework for understanding quasar broad ab- This paper is structured as follows. In §2, we present
sorption lines (BALs). Moreover, it does not require the our formalism to derive the impulse response function2,
existenceofdenseandhighlysupersoniccloudssurround- the fundamental quantity in reverberation mapping. In
ing the central engine. Such clouds were shown early on §3,wediscussthemethodsusedtoevaluateit. Weapply
to be prone to rapid destruction due to hydrodynamical our methods to the PK04 solution in §4. We summarize
instabilities (e.g., Mathews 1986; Krolik 1988), a find- and discuss our results in §5, and we conclude with a
ing supported by detailed numerical simulations (Proga mention of the limitations of this work and our opinion
et. al 2014; Proga & Waters 2015). Several previous on how to make further progress in §6.
investigations suggest that at least part of the observed
line emission originates in a virialized flow, such as a 2. FORMALISM
Keplerian disk or a rotating outflow (e.g., Kollatschny The classic work of Blandford & McKee (1982; here-
2003; Crenshaw & Kraemer 2007; Bentz et al. 2010b; afterBM82)waspublishedayearbeforetheappearance
Kollatschny&Zetzl2013; Pancoastetal.2014). Inview of a seminal paper by Rybicki & Hummer (1983; here-
of the promise that this family of models have shown so afterRH83),whopresentedthemethodologythatisnow
far, we have embarked on a more extensive investigation widely used to calculate line profiles in rapidly moving
of their observational consequences. media. Therefore, we first derive the impulse response
For any BLR model to permit the use of equation (1), function using the framework of RH83, showing how it
therespondinggasmustbevirialized. Hence,inthecase is consistent with the one first derived by BM82.
ofdiskwinds,theoutflowitselfmustbevirialized. Arig-
orous approach to testing this requirement was taken by 2.1. Derivation of the Impulse Response Function
Kashi et al. (2013), who analyzed various outflow solu- FromRH83,thespecificmonochromaticluminosityL
ν
tions and found that the line-driven wind solution pre-
due to line emission can be calculated by integrating the
sentedbyPK04isindeedvirializedouttolargedistances,
product of the monochromatic emission coefficient (or
owing to the dominance of the rotational component of
emissivity) j and the directional escape probability β
ν ν
the wind velocity. Formally, a system is virialized if the
over the volume V of the entire emitting region:
sum of the density-weighted, volume-integrated internal
(cid:90)
energyandkineticenergyisequalto-1/2thevalueofthe
L (t)= dV j (r,t)β (r,t). (2)
density-weighted,volume-integratedgravitationalpoten- ν ν ν
tial energy (see eqns. 2-3 in Kashi et al. 2013). Impor-
Here,bothj andβ dependonthedirectionofemission,
tantly, Kashi et al. (2013) found that the outflow in the ν ν
nˆ; only one direction, that pointing toward the distant
PK04solutionwillbeobservedasvirializedfromanyline
observer, contributes to L (t). The product j β can
of sight (LoS). ν ν ν
beconsideredaneffectiveemissivity,theroleofβ being
Inthispaper, weextendtheinvestigationofthePK04 ν
solution. It is not enough to show that the wind is
2 Whatwecalltheimpulseresponsefunctionisnormallytermed
virialized; we must quantify how gas responds to vari-
the 2-D transfer function, an echo image is its digital represen-
tation, and we reserve transfer function to explicitly denote the
tionsfortheparticles’emissionpropertiesandkinematics. frequency-integratedimpulseresponsefunction.
Reverberation mapping calculations using a hydrodynamical disk wind solution 3
to allow a unified treatment of optically thick and thin is computed as in equation (2), but in a time averaged
gas. In particular, as demonstrated by Chiang & Mur- sense, while the second equation reveals that ∂j /∂F ,
ν X
ray (1996), the escape probability formalism permits a termed the responsivity, is fundamental to reverberation
straight forward calculation of how optically thick re- mapping.
gions in rapidly moving media respond to variations in Since we are after the luminosity seen by a distant
the ionizing continuum (through the effects of velocity observer,weneedtoaccountfortheadditionaltimedelay
shear). In contrast, the response from optically thick foremittedphotonstotravelfromrtotheobserverplane
regions in static or slowly moving media is much more (i.e. animaginaryplaneorientedperpendiculartonˆ and
difficult to calculate on account of the extra time delays located beyond the outer edge of the emitting volume).
associated with multiple scatterings. We must further sum over all times t(cid:48) that contribute to
To proceed, a distinction must be drawn between observed emission at the distant observer’s time t:
steadyandvariablelineprofiles(e.g.,Kroliketal. 1991). (cid:90) (cid:20) (cid:18) r·nˆ(cid:19)(cid:21)
The variable line profile ∆L (t) can be defined as the ∆L (t)= dt(cid:48)∆L (t(cid:48))δ t− t(cid:48)− . (8)
component of the total obserνved line profile L (t) that ν ν c
ν
actually varies in response to continuum fluctuations,
Here, allofthebasicassumptionslistedabovewereonce
whilethesteadylineprofile(cid:104)L (cid:105)isatime-averagedback-
ν again invoked, and we additionally made the (standard)
ground component (that may or may not correspond to
assumption of negligible recombination times (because
the BLR gas); symbolically,
these times are typically very short). Replacing ∆F
X
Lν(t)=(cid:104)Lν(cid:105)+∆Lν(t). (3) with ∆LX/4πr2 in equation (7) and then substituting
equation (7) into equation (8) gives
The principle behind reverberation mapping is that the
variable line profile, as observed at time t, is caused by ∆L (t)=(cid:90) dt(cid:48)(cid:90) dV ∂jν ∆LX(t(cid:48)−r/c)β
ssommaelleflaurlciteurattiimonest−ofτth(teypcoicnatlinfruaucmtiolnigahltrmcusrvvaeriLabXiliatyt ν (cid:20) (cid:18) ∂FX (cid:19)(cid:21)4πr2 ν (9)
r·nˆ
amplitudes are (cid:46) 20%; e.g., De Rosa et al. 2015). Re- × δ t− t(cid:48)− .
worded from the standpoint of this paper, this principle c
implies that given the impulse response function Ψ(ν,τ)
Theimpulseresponsefunctionisbydefinitiontheratioof
(i.e. a model of the BLR) and the light curve of contin- ∆L to∆L foradelta-functioncontinuumfluctuation,
ν X
uum fluctuations, ∆L =L −L (with L a reference
X X 0 0
continuumlevel),wecanpredicttheshapeofthevariable Ψ≡ ∆Lν δ(t(cid:48)−r/c). (10)
line profile through the convolution ∆L
X
(cid:90) ∞ Making the substitution ∆L → ∆L δ(t(cid:48) − r/c) in
∆L (t)= Ψ(ν,τ)∆L (t−τ)dτ. (4) X X
ν X equation (9) collapses the dt(cid:48) integral, thereby defining
0
the total time delay
Returningtoequation(2),considertheresponseofthe
r
gastoachangeinionizingcontinuumflux∆FX asseenin τ(r)= (1−rˆ·nˆ), (11)
the rest frame of the source,i.e. accordingtoanobserver c
located at position r = 0 in a spherical coordinate sys- so that the impulse response function can be written as
temcenteredontheBLR.Thentheincreasedcontinuum
flux, ∆FX(t(cid:48)−r/c) = ∆LX(t(cid:48)−r/c)/4πr2, received by Ψ(ν,t)=(cid:90) dV ∂jν βν δ[t−τ]. (12)
a gas parcel at time t(cid:48) and position r is perceived by the ∂F 4πr2
X
observer to have been emitted by the continuum source
Equation(12)isseentobeconsistentwithBM82’sequa-
attheearliertimet(cid:48)−r/c. Hereweinvokedseveralofthe
tion (2.15). Specifically, the responsivity (which has
basic assumptions used in almost all reverberation map-
units cm−1 s) is analogous to their ‘reprocessing coef-
ping studies of the BLR: point source continuum emis-
ficient’ ε, while their factor g (the projected 1D veloc-
sion, straight line propagation from source to gas parcel,
ity distribution function) is unity in the hydrodynamic
andnoplasmaeffects(ensuringtheconstantpropagation
approximation. The only difference is our inclusion of
speed c). Provided ∆F is small relative to (cid:104)F (cid:105), the
X X the escape probability β to account for the effects of
emissivity can be expanded as ν
anisotropy using the formalism of RH83.
∂j
j ((cid:104)F (cid:105)+∆F (t(cid:48)−r/c))≈(cid:104)j (cid:105)+ ν ∆F (t(cid:48)−r/c).
ν X X ν ∂F X 2.2. Responsivity and opacity distributions
X
(5)
Thederivationleadinguptoequation(12)isquitegen-
Byinsertingthisrelationshipintoequation(2)andmak-
eralasfarastheradiativetransferisconcerned. Wenow
ing a comparison with equation (3), we identify
specializetotheSobolevapproximationbyfollowingRy-
(cid:90) bicki & Hummer (1978) and RH83, in which case
(cid:104)L (cid:105)= dV (cid:104)j (cid:105)β , (6)
ν ν ν (cid:104) ν (cid:105)
j (r)=kS δ ν−ν − 0v , (13)
and ν ν 0 c l
∆Lν(t(cid:48))=(cid:90) dV ∂∂Fjν ∆FX(t(cid:48)−r/c)βν. (7) wlinheeroepkac=ity(πofe2t/hme terca)fn1s2itnio1n[cwmit−h1oss−c1il]laistotrhsetrinenteggtrhatfe1d2
X andpopulationnumberdensityn ,S isthesourcefunc-
1 ν
The first equation just states that the steady line profile tion, ν is the line center frequency, and v ≡nˆ·v is the
0 l
4 Waters, Kashi, Proga, Eracleous, Barth, & Greene
line of sight velocity of the emitting gas which has bulk to the ansatz
velocity v. The delta-function here arises from the use S (r)=AFη(r), (14)
ν X
of the Sobolev approximation, for when it holds, locally
where A is a function of position, specified below, that
Gaussian line profiles will effectively behave as delta-
sets the overall response amplitude. Photoionization
functions (see, for example, §8.4 of Lamers & Cassinelli
modelingindicatesthatη typicallyrangesbetween0and
1999). Note that this statement is not equivalent to our
2 (see e.g., Krolik et al. 1991; Goad et al. 1993, 2012).
assumption that the intrinsic line profile is much nar-
For simplicity, we adopt η =1 in this work, which gives
rower than a Gaussian.
A units of seconds and defines our responsivity as
The argument of the delta-function accounts for a
non-relativistic Doppler shift only. There will also be ∂j (r) (cid:104) ν (cid:105)
a transverse redshift that can be of order 1.5(vt/c)2 × ∂Fν =kAδ ν−ν0− c0vl . (15)
105 km s−1, where v is the velocity component perpen- X
t
dicular to the LoS, as well as a gravitational redshift of Specifying the magnitude of A is only necessary when
order 1.5(r /r)×105 km s−1, where r =2GM /c2 is making quantitative comparisons with observed spectra.
s s BH
the Schwarzschild radius. Since the PK04 domain ex- We will use arbitrary flux units, allowing the constant
tends to a minimum radius rmin ≈30rs and the highest A0 in our fiducial relation,
velocities in the domain are ∼0.1c, either effect can po-
A(r)=A (r/r )2, (16)
tentially lead to shifts ∼1500 km s−1 at the base of the 0 1
profile. While acknowledging that these are important where r is one light day, to serve as a normalization
1
effects, we ignore both relativistic redshifts to first order factor. Our results are calculated using this heuristic
on the grounds that these estimates are still small com- prescriptionforA(r),whichwemotivatebelow,although
pared to the widths of our calculated line profiles and in§4.1wepresentanexamplecalculationwithA(r)=A
0
will apply mainly to the innermost gas, leading to a red instead.
wing. To obtain an expression for the responsivity that in-
The source function Sν in equation (13) describes all volves only hydrodynamical quantities, we estimate the
radiative processes responsible for the line emission and number density of the lower level of the transition in
ingeneralcanbedividedintotwocontributions: (i)local question in terms of the fluid density ρ through
intrinsic emission processes, and (ii) scattered emission.
ρ
We mention below how to realistically model (i), but in n (r)=A ξ , (17)
1 Z ionµm
thisworkweadoptsimplescalingrelationstoaccountfor p
(i) in a way that will enable us to compare our results
whereξ istheionfractionoftheemittingionwithele-
ion
with those from prior works. It is known that a proper
mentalabundanceA ,andµandm arethemeanmolec-
Z p
treatment of (ii) is important when calculating steady
ular weight and mass of a proton, respectively. These
line profiles, but it is beyond the scope of this work to
quantities are assumed to characterize the state of the
investigatetheimportanceofscatteringforshapingvari-
gas after the change in photoionizing flux. We can now
able line profiles.
define an effective opacity per unit mass as
Tocalculatethevariablelineprofile,weneedtospecify
the responsivity, ∂j /∂F . A self-consistent determina- (cid:18)πe2(cid:19)A ξ f
ν X κ= Z ion 12 [cm2g−1], (18)
tionoftheresponsivityrequiresdetailedphotoionization m c µm ν
e p 0
modeling coupled with radiation hydrodynamical simu-
lations. The former type of calculation has been fre- and in our calculations we take κ to be a spatially fixed
quently explored without regard to the latter (e.g., Du- quantity throughout the domain. Note that in writing
mont&Collin-Souffrin1990; Kroliketal. 1991; Goadet equation(15)wehaveassumedthatthefluxdependence
al. 1993; Korista & Goad 2000, 2004; Goad & Korista oftheemissivityisdominatedbythatofthesourcefunc-
2014). Here we take a first step in performing the latter tion, i.e. that k = κρν0 is insensitive to changes in the
type of calculation. In §4.5 we outline a basic modeling ionizing flux. This will not be true in general since κ
strategy that should be suitable for constraining BLR depends on the ion fraction, while hydrodynamic effects
models upon making a comparison with observations. can lead to changes in ρ. Ignoring the latter possibility
In essence, the velocity and density fields are found by (since it implies a nonlinear response; see §4.5) therefore
performing hydrodynamical simulations, and then sepa- implies that k is independent of FX when κ is treated as
rately the responsivity and opacity distributions are ob- a constant.
tainedbycarryingoutphotoionizationcalculationsusing As a very simple example of what the above scalings
the hydrodynamical simulation results as input. imply, consider a spherically symmetric, constant, high-
Forthisinitialinvestigation,weoptedforasimplerap- velocity outflow illuminated by an isotropic source at its
proachbyadoptingprescriptionsfortheresponsivityand center. By mass conservation, the density scales as r−2,
opacity distributions. To reach a common ground with andthereforesodoesk. ThenA∝r2 amountstoassum-
pastinvestigations,wenotethatitishasbeencommonto ingthattheemissivityofthegasisdirectlyproportional
adopt a power-law dependence for the responsivity (e.g., to the density, while the responsivity (∂jν/∂FX ∝κρr2)
Goad et al. 1993, 2012) similar to the one introduced by is constant with radius since the emissivity and flux
Kroliketal. (1991),whoassumedthepowercanberadi- both falloff as r−2. In contrast, taking A = A0 implies
allydependentandtakestheformη(r)≡∂lnSl/∂lnFX, jν ∝ r−4 and ∂jν/∂FX ∝ r−2; this scaling reproduces
where S is the local brightness of the line-emitting gas. the results of Chiang & Murray (1996), as shown in the
l
Phrasedintermsofthesourcefunction,thisisequivalent Appendix.
Reverberation mapping calculations using a hydrodynamical disk wind solution 5
2.3. The escape probability mentsasresonancepoints,andtotheequationgoverning
these locations as the resonance condition.
In equation (12), the escape probability, assuming a
For axisymmetric models, to which we confine our-
single resonant surface, is given by (RH83)
selves to in this work, the resonance condition is used
1−e−τν to solve for the resonant azimuthal angles φ˜correspond-
β (r)= . (19)
ν τ ing to each (r,θ) coordinate on the grid. It is clear
ν that dependence on φ enters through nˆ. Two angles are
Treating multiple resonant surfaces, which can arise for required to specify nˆ, namely the observer’s azimuthal
non-monotonic velocity fields, modifies equation (19) by and polar coordinates (φ ,θ ). Without loss of gener-
n n
an additional multiplicative factor of e−τν for each sur- ality we choose φ = 0, while θ is the same as the
n n
face, but we expect equation (19) to capture the domi- LoS inclination angle, hereafter denoted i. Then the
nantopticaldeptheffects. IntheSobolevapproximation, components of nˆ are n = sinθcosφsini + cosθcosi,
r
the optical depth is given by n = cosθcosφsini−sinθcosi, and n = −sinφsini,
θ φ
giving for the resonance condition the coupled algebraic
k c
τ (r)= , (20) equations
ν ν |dv /dl|
0 l
y =n v(cid:48)(r,θ)+n v(cid:48)(r,θ)+n v(cid:48)(r,θ);
where dv /dl≡nˆ·∇v is the line of sight velocity gradi- r r θ θ φ φ (24)
l l
ent, often denoted as Q: t=(r/c)(1−nr).
dvl ∼=Q(r)=(cid:88)1(cid:18)∂vi + ∂vj(cid:19). (21) Htheeryetahreepirnimuensitosnotfhecv(eclooncistiystceonmt pwointhenotsurindcoicnavteentthioant
dl 2 ∂r ∂r
i,j j i for v(cid:48) above). For analytic axisymmetric hydrodynamic
l
The components of Q in various coordinate systems can solutions, equations (24) can be easily solved for φ = φ˜,
be found in Batchelor (1967). Therefore, the product giveny, t, andi. However, thereisasubtletythatarises
kβ present in the integrand of equation (12) can be fordiscretizedsolutions,requiringfirstthesolutionofan
ν
written alternate form of the resonance condition, equation (25)
k(r)βν(r)= νc0 (cid:12)(cid:12)(cid:12)(cid:12)ddvll(cid:12)(cid:12)(cid:12)(cid:12)(1−e−τν). (22) bpreolocwed.urWeeinre§t3u.2rn. to this point and discuss our actual
2.5. Echo image sketches
Notice that this product is only dependent on the den-
sity and opacity through the optical depth. For τ (cid:29)1, Welsh & Horne (1991) derived simple equations relat-
ν
this dependence is very weak and the escape of photons ing the velocity field and the time delay for specific out-
is primarily governed by the local LoS velocity gradient. flow, inflow, and Keplerian velocity fields, which allowed
Onceτ (cid:46)0.1,ontheotherhand,β ≈1−τ /2,andthe them to sketch velocity vs. delay and thereby show the
ν ν ν
impulseresponsefunctionbecomesweaklydependenton possible outlines of echo images. A general equation for
|dv /dl|, instead depending primarily on the magnitude ‘echoimagesketches’ofaxisymmetricmodelsisfoundby
l
of k (i.e. the product of the density and opacity), which eliminating φ from equations (24); it is simplest to write
must be smaller than (ν /c)|dv /dl|. Thus, in general, downusingcylindricalvelocitycomponents,(v ,v ,v ):
0 l (cid:36) φ z
theresponsewillbeweakerforreprocessedphotonsemit-
(cid:34)
ted in an optically thin region compared to an optically r sinθ
t= 1−cosθcosi−
thick, rapidly moving region. c v(cid:48)2+v(cid:48)2
(cid:36) φ
(25)
(cid:32) (cid:33)(cid:35)
2.4. The resonance condition (cid:113)
× v(cid:48) y(cid:48)±v(cid:48) (v(cid:48)2+v(cid:48)2)sin2i−y(cid:48)2 ,
Having derived formulae for the quantities appearing (cid:36) φ (cid:36) φ
in the integrand of equation (12), we can express the
impulse response function in spherical coordinates as where
y(cid:48) ≡y−v(cid:48) cosi. (26)
1 (cid:90) rout (cid:90) π (cid:90) 2π z
Ψ(y,t)= dr sinθdθ dφA(r) Equation (25) reduces to the simpler ones presented in
4πc
rin 0 0 (23) Welsh & Horne (1991), i.e. the relationship for a spher-
(cid:12) (cid:12)
×(cid:12)(cid:12)(cid:12)ddvll(cid:12)(cid:12)(cid:12)(1−e−τν)δ[y−vl(cid:48)] δ[t−τ], ivcφal=invflzo=w/0o,ugtiflvoinwgis obtained by setting θ =−π/2 and
(cid:20) (cid:21)
wherer andr aretheinnerandouterradiiofthere- r y
in out t= 1+ , (27)
verberatingregionandwehavedefinedthedimensionless c v(cid:48)
(cid:36)
frequency shift y ≡ (ν −ν )/ν and denoted v /c = v(cid:48).
0 0 l l whereas a Keplerian disk satisfies,
The argument of the first delta function defines an iso-
fcroenqtureibnuctyestuorafagceivesnpefcriefqyuinegncayllshpihftysyi.caLlikloecwaitsieo,ntshethaart- (cid:20)t−r/c(cid:21)2+(cid:34) y (cid:35)2 =sin2i, (28)
gument of the second delta function defines an iso-delay r/c v(cid:48)
φ
surface, giving all points in the volume with nonzero re-
sponses at a given time t. Only the intersection of these obtained by setting θ =π/2 and v =v =0.
(cid:36) z
two surfaces contribute to the integral at a given (y,t). Figure1showsechoimagesketchesforthePK04solu-
We will refer to locations satisfying the combined argu- tion. From top to bottom, the first three rows show the
6 Waters, Kashi, Proga, Eracleous, Barth, & Greene
Figure 1. Echo image sketches of the PK04 solution for i=15◦ (1st column), i=45◦ (2nd column), and i=75◦ (3rd column). These
are plots of the two time delays, t+ and t− (green and black symbols, respectively, but note t+ = t− when vφ = 0), corresponding to
eachLoSvelocity,foundbysolvingequation(25)usingthevelocitycomponentsfromthePK04solution. Thelastcolumndisplaysmaps
ofthesevelocitycomponents. Thefirstthreerowsofechoimagesketchesshowstheeffectofzeroingthe(cylindrical)velocitycomponent
shown in the corresponding map. For example, the 1st row of sketches has nonzero vφ and vz. The sketch for i=75◦ in this row shows
a characteristic ‘virial envelope’, which is due to vφ alone; at lower inclinations contributions from vz become visible. In the 2nd row of
sketches there is no virial envelope, as only the poloidal velocity components are nonzero; comparison with the 1st row reveals that the
diagonal features are caused by v(cid:36). Vertical dashed lines are plotted at line center to highlight an overall blue-shift effect that is absent
in the 3rd row, which has vz =0 and hence lacks any shift caused by vzcosi in equation (26). This effect is best seen by comparing the
bottomrowofsketches,whichaccountsforthefullPK04velocityfield,withthe3rdrow. Weemphasizethatthesesketchescanbeusedto
assesswhereanechoimagecannot showaresponse,butelsewheretheyneednotresembletheactualimagesinceΨ(y,t)maybenegligible.
effectofzeroingeachvelocitycomponent,mapsofwhich velocity-delayspace,therebyshowingwhichregionsofan
are plotted in the right column. The top row lacks the echoimagecannot showaresponse. Mostofthefeatures
prominentdiagonalfeaturepresentintheotherrows,in- outside of the ‘virial envelope’ formed by the rotational
dicating that it is due to the v component. Note that velocitycomponentturnouttohavemuchsmallerfluxes
(cid:36)
diagonal features are expected for radial outflows (c.f. unlessthelinesoriginatinginthewindareveryoptically
Welsh & Horne 1991). thick.
The final row shows sketches with all velocity compo-
nents nonzero. A comparison with the third row high-
lights a tendency for echo images of outflows to exhibit
blue-shifted excesses. This effect is clearly revealed by
2.6. Transfer functions and line profiles
equation (26): the velocity shift y =(ν−ν )/ν is offset
0 0
by a factor of (v /c)cosi, so the vertical velocity com- Most reverberation mapping studies to date have pri-
z
ponent causes a blueshift for positive v and a redshift marily focused on two quantities derived from the im-
z
for negative v . This will only be significant at small pulse response function. The first is the transfer func-
z
inclinations (i (cid:46) 45◦) due to the factor of cosi. We will tion, which is the frequency-integrated impulse response
examine this result more closely in §4.3. function,
(cid:90) ∞
Thesignificantdifferencesbetweenthebottomandtop
Ψ(t)= Ψ(y,t)dy. (29)
rows of sketches hints that it may be possible to infer
−∞
the presence of a poloidal velocity field through observa-
In practice, the transfer function is the quantity used to
tionsofechoimages. However,thesesketchesaremainly
calculate mean time lags, and hence to measure a char-
usefulforvisualizingthemappingfromphysicalspaceto
acteristicradiusoftheBLR.Similarly,wecanalsodefine
Reverberation mapping calculations using a hydrodynamical disk wind solution 7
the line profile by Equation (31) becomes
(cid:90) ∞ (cid:90) 1 (cid:90) (cid:90) I
Φ(y)= Ψ(y,t)dt, (30) Ψ(y,t)= dµ dv(cid:48) dτ δ[y−v(cid:48)]δ[t−τ], (34)
0 −1 l |J| l
where the limits are (0,∞) since Ψ(y,t < 0) = 0. Note which evaluates to
that Φ(y) is not the same as the variable line profile de-
fined in equation (4). Rather, it is (to within a normal- (cid:90) 1 (cid:20) I (cid:21)
Ψ(y,t)= dµ . (35)
ization factor) the limiting case of a variable line profile |J|
−1 (r˜,µ,φ˜)
found by convolving Ψ(y,t) with a constant continuum
light curve. As such, the line profiles presented in this We use the subscript notation to indicate that for each
paper should be viewed as merely representative of the µ,theintegrandistobeevaluatedattheresonancepoint
line shapes expected for our disk wind models. Detailed (r˜,φ˜) corresponding to a given (y,t); geometrically this
predictions of variable line profiles are system specific, point will lie somewhere in a conical slice (r,φ) through
as they require carrying out the convolution with the thevolume. Itslocationisdeterminedbythesolutionto
observed continuum light curve ∆L (t). the resonance condition, equation (24). Assuming mo-
X
tion purely in the midplane (µ = 0), CM96’s result can
be obtained with the substitution I → Iδ[µ−0], as we
illustrate in the Appendix.
3. METHODS
Twoapproachesforcalculatingimpulseresponsefunc-
3.2. Numerical evaluation of the impulse
tions from models of the BLR were introduced early
response function
on. A stochastic approach was taken by Welsh & Horne
(1991) and P´erez et al. (1992), in which a domain was To numerically evaluate the remaining integral over µ,
populated with a large number (∼ 760,000 and 25,000, weemploythetrapezoidrule,leadingtothediscreteform
respectively) of points, satisfying some assigned veloc- N−1 (cid:34) (cid:12) (cid:12) (cid:35)
ity field, spatial distribution, and emissivity. These dis- 1 (cid:88) dΨ(cid:12) dΨ(cid:12)
crete particle models continue to provide intuition into Ψ(y,t)≈ 2 ∆µk dµ(cid:12)(cid:12) + dµ(cid:12)(cid:12) , (36)
the nature of the mapping between physical space and k=1 k+1 k
frequency-delay space. where we have used the simplifying notation
AnanalyticapproachwastakenbyBM82andlaterby
(cid:20) (cid:21)
dΨ I
Chiang & Murray (1996; hereafter CM96), whose BLR
= . (37)
model consisted of an axisymmetric Keplerian disk com- dµ |J|
(r˜,µ,φ˜)
bined with a simple radial wind prescription. Here we
adoptCM96’sapproach, extendingittoallowtheexplo- Note that for grid-based simulation data in spherical co-
ration of both 2-D analytic and numerical hydrodynam- ordinates, the native grid spacing can be used to arrive
ical models. directlyat∆µk =µk+1−µk. (Otherwise,thediscretized
solutionwouldneedtobeinterpolatedtoasphericalgrid
or a different Jacobian would need to be defined.)
Asmentionedin§2.4,whenappliedtosimulationdata,
3.1. Formal evaluation of the impulse
asubtletyarisesintheevaluationoftheintegrand,equa-
response function
tion (37). To clarify what is involved, it should first be
Simplifying equation (23) to its basic functional form emphasized that the goal is to arrive at a legitimate dig-
and changing integration variables to µ≡cosθ gives ital image to compare with echo images obtained from
observations. That is, we need to construct a 2-D array
(cid:90) rout (cid:90) 1 (cid:90) 2π
Ψ(y,t)= dr dµ dφIδ[y−v(cid:48)] δ[t−τ], of pixels with the center of each pixel at specified values
l of (y,t), and the magnitude of Ψ(y,t) determining the
rin −1 0
(31) value of the entire pixel. Ideally, we would like to di-
where rectly evaluate each of the N values of dΨ/dµ precisely
(cid:12) (cid:12)
I(r)= 1 A(r)(cid:12)(cid:12)dvl(cid:12)(cid:12)(1−e−τν). at (y,t). However, this cannot be done in practice. The
4πc (cid:12) dl (cid:12) reason is that with discretized data, it is impossible to
find resonance points exactly at the center locations of
To make use of the delta functions, any pair among pixels to an acceptable tolerance level. Indeed, as equa-
(dr,dµ), (dr,dφ), and (dµ,dφ) can be replaced by tion (25) reveals, there are only certain values of y that
(dvl(cid:48),dτ) using a Jacobian. For axisymmetric problems satisfy the resonance condition for a given t, and vice
in which the density and velocity fields are independent versa, when the grid coordinates (r,µ) and velocity fields
ofφ,itisnaturaltoreplaceeither(dr,dφ)or(dµ,dφ),so are given.
thatthetripleintegralcanbereducedtoasingleintegral Our procedure to generate an echo image therefore in-
over µ or r. To make a clear comparison with CM96, we volves interpolating from the resonant locations nearest
chose to use (dr,dφ), so the mapping reads the center of each pixel. For every value of y, i.e. for ev-
drdφ|J|=dv(cid:48)dτ, (32) erycolumnofpixelsinourimagearray, weloopthrough
l all grid points of our simulation and associate each one
where withaspecificvalue oftthatsatisfiesequation(25). We
|J(r)|=(cid:12)(cid:12)(cid:12)(cid:18)∂τ(cid:19)(cid:18)∂vl(cid:48)(cid:19)−(cid:18)∂τ(cid:19)(cid:18)∂vl(cid:48)(cid:19)(cid:12)(cid:12)(cid:12). (33) dthoatthceosrarmesepofonrdeatocharogwiveonfpti.xeFlso,rcoelalcehctipnigxeall,lwyevatlhueens
(cid:12) ∂r ∂φ ∂φ ∂r (cid:12)
8 Waters, Kashi, Proga, Eracleous, Barth, & Greene
evaluate dΨ(y ,t)/dµ, dΨ(y ,t)/dµ, dΨ(y,t )/dµ, and pendentandthereforeprovidedausefulmeanstobench-
L R A
dΨ(y,t )/dµ, where (y ,t), (y ,t), (y,t ), and (y,t ) mark the code used in this work (see the Appendix).
B L R A B
are the four locations nearest to (i.e. left of, right of,
above, and below, respectively) the center of the pixel. 4. RESULTS
Lastly,webilinearlyinterpolatethefourvaluesofdΨ/dµ
The above methods were implemented as a post-
to arrive at dΨ(y,t)/dµ. By adding up all such values of
processing routine and applied to the line-driven disk
dΨ(y,t)/dµ in accordance with equation (36), we finally
wind solution presented in PK04. The PK04 solution
arrive at Ψ(y,t), whose magnitude is assigned to that
is a hydrodynamic model of an outflow launched from
pixel.
a geometrically thin, optically thick disk accreting onto
108 M non-rotating SMBH at a rate of 1.8 M yr−1.
3.3. Direct vs. indirect calculation of the (cid:12) (cid:12)
For an accretion efficiency η = 0.06, this corresponds
transfer function and line profile
to a disk luminosity L = 0.5L , where L is the
D Edd Edd
If provided with an analytic hydrodynamical model Eddington luminosity. The numerical setup is similar
(e.g., that of CM96), there is no need to carry out the to that developed by Proga et al. (2000): for simplic-
interpolation procedure just described, since resonance ity and to reduce the computational time it was as-
points can be found for any (y,t). By summing over the sumed that X-rays and all ionizing photons are emitted
rowsandcolumnsofresultingechoimagewithasuitable by the central object, which in term was approximated
algorithm such as the trapezoid rule, excellent numeri- as a point source. Specifically, the central engine has
calapproximationstotheintegralsinequations(29)and L =0.05L anddoesnotcontributetotheradiation
X Edd
(30) can be obtained. We refer to this method of calcu- force acting on the wind.
latingthetransferfunctionandlineprofileasanindirect An important feature of the PK04 solution that indi-
one, since it first involves calculating Ψ(y,t). rectly contributes to the line-driving mechanism is self-
Thissummationcanalsobecarriedoutfordiscretized shieldingbythediskatmosphere: denseclumps(a“failed
solutions, using the non-interpolated values of dΨ/dµ. wind”) form at small radii as a result of over-ionization,
However,againasubtletyarises,whichisnoteasilydealt which shield the gas launched at large radii from ioniz-
with. The issue is the double-valued nature of the map- ingradiation. Theresultingline-drivendiskwindisvery
pingfrom(r,µ)to(y,t). Fromequation(25),weseethat fast(∼104 kms−1)atlowlatitudes, whereitisdirected
in general there can be two values of t for every y. Each primarily in the radial direction at small heights above
willhaveadifferentresonantφ˜coordinate,astheyphys- the disk (see the top panel in Figure 1). At somewhat
icallycorrespondtoemissionregionsonoppositesidesof greater heights, namely for 55◦ (cid:46) θ (cid:46) 70◦, the verti-
the BLR that have the same time delay. However, they cal component of the wind velocity also becomes large.
manifest as separate branches in a plot of Ψ(y,t) vs. t, It has been shown that this model can produce features
andwefindthatonebranch(correspondingtogasonthe observed in X-ray spectra of AGN (Schurch et al. 2009;
far side of the BLR) is sampled much less densely than Sim et al. 2010).
the other (due to the logarithmic grid spacing). Hence, The PK04 simulation was performed on a logarithmic
special integration routines are necessary to accurately grid with resolution [N ,N ] = [100,140]. Our results
r θ
carry out this indirect method, which will be needed to are calculated assuming emission from only the top half
calculate convolutions with observed light curves; they of the disk; the bottom half is assumed to be blocked
will be presented in a separate paper focused on making by the disk. Additionally, we exclude the polar region
a comparison with observations. from θ = 0◦ to 8◦, which is very hot and optically thin.
The direct method for calculating line profiles and Thus,itwilltypicallyshownegligibleresponseasithosts
transfer functions is to carry out the integrals over y few lines. Recall from §2.2 that the optical depth can
and t in equations (29) and (30) analytically. Using the be parametrized in terms of the opacity per unit mass
impulse response function in the form of equation (34), κ = k/(ρν ), giving τ = cρκ/|dv /dl|. We will explore
0 ν l
wefind,aftersomemanipulationoftheJacobiandefined the dependence on κ in §4.4, but elsewhere we adopt a
in equation (32), fiducial value of κ = 104κ with κ = 0.4[cm2g−1] the
es es
electron scattering opacity.
(cid:90) rout (cid:90) 1 (cid:88)2 (cid:20) I (cid:21)
Ψ(t)= dr dµ ;
|dτ/dφ| 4.1. Keplerian disk (no wind):
rin −1 i=1 (r,µ,φti) (38) Effects of varying the responsivity
(cid:90) rout (cid:90) 1 (cid:88)2 (cid:20) I (cid:21) We begin by analyzing a familiar case: an optically
Φ(y)= dr dµ .
|dv /dφ| thick, Keplerian disk, which is expected to show double-
rin −1 i=1 l (r,µ,φyi) peaked line profiles due to a lack of flux at line cen-
Thesubscriptnotationhereindicatesthattheintegrands ter. This calculation was done by only integrating from
aretobeevaluatedatthelocationwheret=τ(r,µ,φ)in θ = 89.75◦ to θ = 90◦, an interval that comprises about
thecaseofΨ(t)andy =v (r,µ,φ)inthecaseofΦ(y);in one third of the 140 grid indices due to the logarithmic
l
general there can be two such locations, φ and φ for PK04 grid, so there is ample resolution. While this re-
t1 t2
Ψ(t), and φ and φ for Φ(y), hence the summations. gionconstitutesthebaseofthewind, thepoloidalveloc-
y1 y2
We numerically evaluate these integrals (again using the ity components are very small relative to the azimuthal
trapezoid rule). For the technical reasons described in component; to strictly focus on the Keplerian velocity
the previous paragraph, our results only employ this di- field, we set v and v as well as their gradients to zero.
r θ
rectmethod. Nevertheless,wedrawattentiontothefact Ourobjectivehereistocomparethedifferencesbetween
that this and the indirect method are completely inde- a responsivity that is only proportional to the density,
Reverberation mapping calculations using a hydrodynamical disk wind solution 9
Figure 2. Two ‘disk only’ calculations for i = 75◦ performed Figure 3. Two disk wind calculations for i=45◦ performed by
by integrating over the PK04 solution only in the range θ = integrating over the PK04 solution in the range θ = 8.2◦−90◦.
89.75◦ −90◦. Top: A(r) = A0. Bottom: A(r) = A0(r/r1)2. Top: purely rotational case calculated by zeroing all quantities
ThenormalizationfactorA0 ischosentosatisfy(cid:82) Ψ(τ)dτ =1for involvingvr andvθ. Bottom: fullvelocityfi(cid:82)eldcase. Thenormal-
thetransferfunctioninthetoppanel. Bothcasesresultindouble ization factor A0 is again chosen to satisfy Ψ(τ)dτ =1 for the
peakedlineprofiles,consistentwithexpectations,withthelackof toptransferfunction. Thepurelyrotationalcaseresemblesthatof
line center flux clearly visible in the echo images. The prominent the disk-only calculation with A(r)=A0(r/r1)2: symmetric echo
spikesonthelineprofileat±23×103 kms−1 inthetopplotcoin- image,double-peakedlineprofiles,andatransferfunctiondisplay-
cidewithadarkringontheechoimage. Theyarealsopresentin ing an extended-response. Including the poloidal velocity field (i)
the bottom plot but are masked by the emission from outer radii changesthelineprofilefromdoubletosinglepeaked;(ii)broadens
causedbytheextrar2 dependenceintheresponsivity. thelineprofileoverall;and(iii)resultsinablue-shiftedexcess.
implying A(r)=A0, and one following our fiducial scal- dicatesthatonlytheinnerhighvelocity(andthushighly
ing with A(r) = A (r/r )2. The results are shown in broadened) portions of the flow give rise to the line pro-
0 1
Figure 2. These plots are akin to those presented by file. We call this deceptive since zooming in on the line
Welsh & Horne (1991): transfer functions are plotted to profileinthebottomplotwouldrevealspikesatthesame
the right of the echo image over the same range in time locations and equally broad emission, but these features
delay, while line profiles are plotted below. are dwarfed by the much higher flux contributed by the
These two cases are different in several respects. Most outer portions of the flow. This flux is in a relatively
strikingly, and rather deceptively, the line profile in the narrow velocity range around the core of the line profile
topplotisextremelybroad. Thisoccurrenceiseasilyex- since it originates from lower velocity gas.
plainedbylookingateithertheechoimageorthetransfer ItisusefultodrawacomparisonwithourCM96bench-
function. Bothshowasteepfalloffinresponse,whichin- mark solution (see the Appendix), which was generated
10 Waters, Kashi, Proga, Eracleous, Barth, & Greene
Figure 4. Mapsofthevelocityseenbytheobserver(i.e. negative v),averagedovereachquadrantofφ,inthePK04solutionfori=15◦
l
(left)andi=75◦ (right). Theobserverislocatedatφ=0,soineachcasetherighttwomapsrepresentthefrontsideoftheBLRandthe
left two the backside. Shades of blue denote regions with v >0, indicating that the gas is blue shifted and moving toward the observer,
l
while red shades denote receding gas that will contribute to the red side of line profiles. The portion of the domain with 55◦ (cid:46)θ (cid:46)70◦,
delineatedbythedashedlines,istheregionofthePK04solutionwithasubstantialpositivevz component. Itthusalwaysappearsblueto
thei=15◦ observer,butonthefarsideoftheBLRitappearsredtothei=75◦ observer. Thisimpliesthatechoimagesandlineprofiles
canacquirenoticeableblue-shiftedexcessesatsmallinclinations,asFigures5-7reveal. Referto§4.3fordetails.
using A(r) = A . Notice that its transfer function also depthsothatphotonscanmoreeasilyescape. Thereare
0
hasasecondsmallerpeaklikeinthetopplotofFigure2. two other noticeable effects: increased broadening in the
Such can also be seen in the Keplerian disk cases from lineprofileandanoverallexcessinblue-shiftedemission.
Welsh & Horne (1991) and P´erez et al. (1992) and cor- Thelatterisexpected,asweexplainin§4.3. Theformer
respond physically to the time at which the innermost is again attributable to enhanced velocity shear because
regions of the back side of the accretion flow ‘come into thepoloidalvelocityfieldofthewindaddsfluxtoawide
view’. The small dip right before this second peak fills range of LoS velocities.
in to become the only peak of the transfer function in While the transfer functions are rather similar in
the bottom plot of Figure 2. This can be understood by shape, there is overall more response (by roughly a fac-
picturing the time delay paraboloid as it sweeps toward tor of 3) for that of the bottom plot, which is again an
the back of the disk. Gas on the sides of disk residing indication of significantly enhanced emission due to the
at larger radii than the innermost far side gas dominate poloidal wind components. Additionally, the outlines of
over either peak due to the r2 weighting. theechoimagesarequitesimilar,themaindifferencebe-
ing a significant excess in blue-shifted emission for τ <3
days in the bottom plot. (This region is responsible for
4.2. Disk plus disk wind: the role of kinematics
the blue excess on the line profile; the diagonal feature
WenextpresentcalculationsforthefullPK04domain
on the blue edge of the echo image at 3 < τ < 12 days
(that is, from θ ≈8◦ to θ =90◦) in Figure 3. To explore
contributes negligibly.) Notice, however, the very differ-
thekinematiceffectsintroducedbythediskwind, inthe
ent shadings of the echo images. While the purely rota-
firstcase(topplot)wehaveagainzeroedoutthepoloidal
tional case is symmetric with distinct emission patterns,
velocity components and their gradients. The resulting
thebottomimageisblotchyandlacksanydistinguishing
echo image resembles the disk-only cases in Figure 2, al-
characteristics.
thoughitisobviouslynotaswidebecausetheinclination
anglehasbeenreducedtoi=45◦ (withtheconsequence
4.3. Dependence on inclination angle
that LoS velocities are smaller). While a purely rota-
tionalwindregionevidentlycontributessignificantemis- In §2.5, we analytically uncovered an effect of varying
sion to line center, the line profile is still prominently the inclination angle: an excess in blue-shifted emission
double-peaked. as i decreases, as would be expected for an outflow. To
The bottom plot in Figure 3 reveals the primary ef- betterillustratethispoint,inFigure4weshowmapsofv
l
fectofaddingapoloidalvelocityfield: thereisamarked averaged over each quadrant of φ for both a nearly face-
increase in flux at line center, so that the line profile on (i=15◦) and nearly edge-on (i=75◦) viewing angle.
is overall single-peaked. As pointed out by CM96, this The poloidal wind field of the PK04 solution is directed
effect is due to enhanced velocity shear. Specifically, re- nearlyradiallyoutwardforθ (cid:38)70◦, butthereisasignif-
gions contributing to line center for purely rotational icant positive v component in the region 55◦ (cid:46)θ (cid:46)70◦
z
flow (namely, gas residing φ = 0◦ and φ = 180◦) now (marked by dashed lines). This region will therefore ap-
have higher values of |dv /dl| due to the nonzero wind pear mostly blue-shifted at low inclinations (i (cid:46) 45◦),
l
components along the LoS, thereby reducing the optical even on the far side of the BLR, as shown in the left