MHD Disc Winds and Linewidth Distributions
aa r X i v : . [ a s t r o - ph . C O ] M a y Mon. Not. R. Astron. Soc. , 1–18 (2012) Printed 12 March 2018 (MN L A TEX style file v2.2)
MHD Disc Winds and Line Width Distributions
L. S. Chajet ⋆ and P. B. Hall Department of Physics and Astronomy, York University, Toronto, Ontario M3J 1P3, Canada
12 March 2018
ABSTRACT
We study AGN emission line profiles combining an improved version of the accretion disc-wind model of Murray & Chiang with the magneto-hydrodynamic model of Emmering et al.We show how the shape, broadening and shift of the C IV line depend not only on the viewingangle to the object but also on the wind launching angle, especially for small launching angles.We have compared the dispersions in our model C IV linewidth distributions to observationalupper limit on that dispersion, considering both smooth and clumpy torus models. As the torushalf-opening angle (measured from the polar axis) increases above about 18 ◦ , increasinglylarger wind launching angles are required to match the observational constraints. Above ahalf-opening angle of about 47 ◦ , no wind launch angle (within the maximum allowed by theMHD solutions) can match the observations. Considering a model that replaces the torus by awarped disc yields the same constraints obtained with the two other models. Key words: galaxies: active – galaxies: nuclei – (galaxies:) quasars: emission lines
Broad Emission Lines (BELs) are one characteristic feature of thespectra of Type 1 Active Galactic Nuclei (AGNs). Lines arisingfrom high-ionization species are generally blueshifted and single-peaked (e.g. Sulentic et al. 1995, 2000; Vanden Berk et al. 2001).Quasars (and AGN in general) are powered by accreting mass intoa central super-massive black hole (SMBH). The accreting mass isassumed to form a disc-like structure, responsible for most of theultraviolet (UV) and optical continuum emission. This continuumemission illuminates and ionizes dense gas surrounding the centralengine and forms the Broad Line Region (BLR), where the BELsoriginate. The black hole, disc and BLR are embedded in a dustytoroidal structure that obscures some lines of sight to the nucleus(e.g., Elitzur 2008). In the model of Lawrence & Elvis (2010), thetorus is replaced by a warped disc.Currently there is no consensus on the nature of the BLR andnumerous models have been developed to explain it. The spectro-scopic characteristics of the BEL can be explained by lines aris-ing from either an approximately spherical distribution of discreteclouds, with no preferred velocity direction (e.g., Kaspi & Netzer1999) or at the base of a wind from an accretion disc (e.g.,Murray et al. 1995; Murray & Chiang 1997; Bottorff et al. 1997).In the cloud scenario, the BLR is described as composed of nu-merous optically-thick clouds that, photoionized by the continuum-source emission, are the emitting entities responsible for the ob-served lines. Although this model can explain many observed spec-tral features, it also leaves several unsolved issues, such as the for-mation and confinement of the clouds (e.g., Netzer 1990). The tworelevant time-scales for these clouds are the sound crossing time t sc ⋆ E-mail: [email protected] and the dynamical time t dyn . According to the models, the massesof individual BLR clouds are below their Jeans mass, thereforewithout a confinement mechanism such clouds will disintegrate ona time-scale t sc ≪ t dyn , in which case they would need to becontinuously produced. In addition, the number of clouds neededto reproduce the observed smoothness of BEL profiles (Arav et al.1998; Dietrich et al. 1999) is implausibly large. Furthermore, evenif the clouds are confined, cloud-cloud collisions would destroy theclouds on a dynamical timescale (e.g. Mathews & Capriotti 1985),again requiring a high rate of cloud formation or injection.One approach aimed at solving the discrete-cloud modeldifficulties was proposed by Emmering, Blandford & Shlosman(1992). In their model, the BLR is associated with disc-driven, hy-dromagnetic winds and the lines are formed by clouds which areconfined by the magnetic pressure. Low-ionization line profiles,(e.g., Mg II ), and high-ionization line profiles (e.g., C IV ) are pro-duced in the wind at different latitudes and radii. Within that frame-work, the estimated values of parameters such as ionizing flux,electron density, cloud filling factor, column density, and velocityare in agreement with values for these quantities inferred from ob-servations. Emmering et al. (1992) consider emission models withand without electron scattering and attempt to construct a typicalC IV emission profile. Different blueshifts and line asymmetriesare obtained by varying model parameters.Murray et al. (1995) and later Murray & Chiang (1997, 1998)proposed a wind model motivated by the similarities between broademission lines in AGNs and other astrophysical objects, such as cat-aclysmic variables, protostars and X-ray binaries. They made theassumption that the outflow is continuous instead of being com-posed of discrete clouds and showed that such a continuous, opti-cally thick, radiatively driven wind launched from just above the ac-cretion disc can account for both the single-peaked nature of AGN c (cid:13) L. S. Chajet and P. B. Hall emission lines and their blueshifts with respect to the AGN sys-temic redshift, although not for the magnitude of these shifts. In aaccelerating wind, the wind opacity in a given direction depends onthe velocity gradient in that direction. The larger radial gradient ofvelocity in a radially accelerating wind means that the opacity seenby radially-emitted photons will be lower than the opacity seen byphotons emitted in other directions. Thus, photons will tend to es-cape radially and the resultant emission lines are single-peaked. Inaddition, the model high-velocity component of the wind naturallyexplains the existence of blueshifted broad absorption lines seenin an optically selected subset (15-20 % ) of the quasar population,known as broad absorption line quasars.Here we combine an improved version of theMurray & Chiang (1997) model with the Emmering et al.(1992) model and analyse the dependence of the resulting emissionline profiles on several parameters. The plan of the paper is asfollows. In Section 2 we review the Murray & Chiang (1997)disc-wind model and the modifications that we have introducedin a companion paper (Hall et al., in preparation). In Section 3we outline the basics of MHD winds and show how we combinedthe two models. In section 4 we calculate a key quantity of themodel, Q : the line-of-sight gradient of the line-of-sight velocity.The line profile results are presented in Section 5. In Section 6we follow Fine et al. (2008) and study the predicted BEL widthdistribution incorporating two models for the escape probabilityfrom the BLR, including the clumpy torus model of Nenkovaet al. (2008a, 2008b), and use the results to obtain constraintson the torus parameters. We analyse the warped disc models ofLawrence & Elvis (2010) in a similar fashion in Section 7. Wepresent our conclusions in Section 8. In a companion paper (Hall et al. 2012, in preparation) we extendthe disc-wind model of Murray & Chiang (1997, MC97 hereafter)to the case of non-negligible radial and vertical velocities. The newtreatment retains a number of factors neglected in MC97 and in-troduces the ‘local inclination angle’ to account for the differenteffective inclinations to the line of sight of different portions of theemitting region. Below we summarize these modifications.As shown in Figure 1, we assume the SMBH is at the origin ofa cylindrical coordinate system ( r, φ, z ) with the z axis normal tothe accretion disc and the observer, in the xz plane, making an an-gle i with the disc axis. At any r , the azimuthally symmetric emit-ting region has its base at z em = r tan β ( r ) above the disc planeand has a density which drops off above the base as a Gaussian withcharacteristic thickness that satisfies l em ( r ) ≪ r . Because the pho-tons originate in a narrow layer above the disk, the emission regioncan be approximated as an emitting surface with a source function S ν that is a function of radius only. The wind streamlines make anangle of ϑ ( r, z ) relative to the disc plane.Under these assumptions, the specific luminosity for a givenline in the direction of the observer, L ν ( ˆn ) , is given by L ν ( ˆn ) = Z r max r min S ν ( r ) a ( r ) r dr Z π [1 − e − τ ν ( r,φ, ˆn ) ] cos ι ( r, φ, ˆn ) dφ (1)where I ν = S ν ( r )[1 − e − τ ν ] is the specific intensity, a ( r ) r dr dφ is the area of the emitting surface between cylindrical radii r and r + dr , τ ν is the optical depth from z em ( r ) = r tan β ( r ) to infin-ity along the direction ˆn from the location ( r, φ ) , and ι ( r, φ, ˆn ) isthe local inclination angle between ˆn and the local normal to the rz i Β z em l em J = J = Figure 1.
Streamlines for two different launching angles: ϑ = 20 ◦ , ◦ .The blue line represents the base of the emitting region, tilted by an angle β with respect to the disc plane. surface at radius r and azimuthal angle φ . For i < ◦ − β ( r ) , thearea factor and the local inclination angle are given by a ( r ) = 1cos β ( r ) (2)and cos ι = cos i cos β ( r ) − cos φ sin i sin β ( r ) , (3)respectively.To evaluate the optical depth τ ν , MC97 expanded the projec-tion of the wind velocity along the line of sight, ˜v · ˆn , in terms of z − z em to first order to obtain (their equation 12) ˆn · ˜v ( r, φ, z ) ∼ = ˆn · ˜v ( r, φ, z em ) + ˆn · Λ · ˆn ( z − z em )cos ι/ cos β ≡ v D ( r, φ, z em ) + ( z − z em ) l em v sh ( r, φ, z em ) . (4)The zeroth order term in Equation 4 is the Doppler velocity v D : v D = − v φ sin φ sin i + v p (cos φ cos ϑ sin i + sin ϑ cos i ) (5)where v φ and v p are, respectively, the azimuthal and poloidal ve-locities of the wind at z = z em .The first order term in Equation 4 involves the shearvelocity v sh , defined as v sh = l em Q cos β/ cos ι where Q (Rybicki & Hummer 1978, 1983) is the line-of-sight gradient of theline-of-sight wind velocity: Q ≡ ˆn · Λ · ˆn (6)where ˆn is the unit vector in the line of sight direction and Λ isthe strain tensor. The entries of the strain tensor Λ consist of spatialderivatives of velocity components. It is symmetric ( Λ ij = Λ ji )and its elements are given in cylindrical coordinates by (see e.g.Batchelor 1967): Λ rφ = 12 (cid:18) r ∂v r ∂φ − v φ r + ∂v φ ∂r (cid:19) , Λ rz = 12 (cid:18) ∂v r ∂z + ∂v z ∂r (cid:19) , Λ φz = 12 (cid:18) ∂v φ ∂z + 1 r ∂v z ∂φ (cid:19) , Λ rr = ∂v r ∂r , Λ φφ = 1 r ∂v φ ∂φ + v r r , Λ zz = ∂v z ∂z . (7) c (cid:13)000
Streamlines for two different launching angles: ϑ = 20 ◦ , ◦ .The blue line represents the base of the emitting region, tilted by an angle β with respect to the disc plane. surface at radius r and azimuthal angle φ . For i < ◦ − β ( r ) , thearea factor and the local inclination angle are given by a ( r ) = 1cos β ( r ) (2)and cos ι = cos i cos β ( r ) − cos φ sin i sin β ( r ) , (3)respectively.To evaluate the optical depth τ ν , MC97 expanded the projec-tion of the wind velocity along the line of sight, ˜v · ˆn , in terms of z − z em to first order to obtain (their equation 12) ˆn · ˜v ( r, φ, z ) ∼ = ˆn · ˜v ( r, φ, z em ) + ˆn · Λ · ˆn ( z − z em )cos ι/ cos β ≡ v D ( r, φ, z em ) + ( z − z em ) l em v sh ( r, φ, z em ) . (4)The zeroth order term in Equation 4 is the Doppler velocity v D : v D = − v φ sin φ sin i + v p (cos φ cos ϑ sin i + sin ϑ cos i ) (5)where v φ and v p are, respectively, the azimuthal and poloidal ve-locities of the wind at z = z em .The first order term in Equation 4 involves the shearvelocity v sh , defined as v sh = l em Q cos β/ cos ι where Q (Rybicki & Hummer 1978, 1983) is the line-of-sight gradient of theline-of-sight wind velocity: Q ≡ ˆn · Λ · ˆn (6)where ˆn is the unit vector in the line of sight direction and Λ isthe strain tensor. The entries of the strain tensor Λ consist of spatialderivatives of velocity components. It is symmetric ( Λ ij = Λ ji )and its elements are given in cylindrical coordinates by (see e.g.Batchelor 1967): Λ rφ = 12 (cid:18) r ∂v r ∂φ − v φ r + ∂v φ ∂r (cid:19) , Λ rz = 12 (cid:18) ∂v r ∂z + ∂v z ∂r (cid:19) , Λ φz = 12 (cid:18) ∂v φ ∂z + 1 r ∂v z ∂φ (cid:19) , Λ rr = ∂v r ∂r , Λ φφ = 1 r ∂v φ ∂φ + v r r , Λ zz = ∂v z ∂z . (7) c (cid:13)000 , 1–18 HD Disc Winds and Line Width Distributions In terms of these Λ ji , the quantity Q is: Q = sin i (cid:2) Λ rr cos φ + Λ φφ sin φ − rφ sin φ cos φ (cid:3) +cos i [2Λ rz sin i cos φ + Λ zz cos i − φz sin i sin φ ] (8)Assuming azimuthal symmetry, all the ∂/∂φ = 0 and the simpli-fied expressions for the different Λ ij are: Λ rφ = 12 (cid:18) ∂v φ ∂r − v φ r (cid:19) , Λ rz = 12 (cid:18) ∂v r ∂z + ∂v z ∂r (cid:19) , Λ φz = 12 ∂v φ ∂z , Λ rr = ∂v r ∂r , Λ φφ = v r r , Λ zz = ∂v z ∂z (9)In the above, the novel element introduced in Hall et al. (2012, inpreparation) is the dropping of the assumption of v r ≪ v φ , thusallowing for non-negligible radial and vertical velocities.Including that and several factors that have been omitted orconsidered negligible in the original MC97 work, the final expres-sion for the specific luminosity in a line of central frequency ν emitted from a disc with a disc wind is given by L ν ( i ) = Z r max r min S ν ( r ) a ( r ) r dr Z π cos ι ( r, φ, i ) × (1 − exp[ − τ ( r, φ, i ) × e ν ( r, φ, i ) × e − x ν ( r,φ,i ) ]) dφ (10)where τ ( r, φ, i ) ≡ ck ( r ) / ν p Q ( r, φ, i ) + q ( r, φ, i ) (11) e ν ( r, φ, i ) ≡ erfc − ν − ν D ( r, φ, i ) √ ν tt p q ( r, φ, i ) /Q ( r, φ, i ) ! (12) x ν ( r, φ, i ) ≡ ν − ν D ( r, φ, i )∆ ν tt p Q ( r, φ, i ) /q ( r, φ, i ) ! (13)and erfc is the complementary error function. In the above expres-sion, the k ( r ) is the integrated line opacity (units of Hz/cm) at z em , and we have defined the Doppler-shifted central frequencyof the line emitted towards the observer from location ( r, φ ) onthe emitting surface, ν D = ν (1 + v D /c ) , and the ‘thermal Q ’,the ratio of the characteristic thermal plus turbulent velocity ofthe ion to the thickness of the emitting layer along the line ofsight, q tt ( r, φ, i ) = v tt cos ι ( r, φ, i ) /l em ( r ) cos β ( r ) , with v ≡ v + v . The effective frequency dispersion of the line is givenby ∆ ν tt = ν v tt /c . The z -dependent quantities are evaluated at z = z em where applicable. The emission region thickness is givenby l em ( r ) = 0 . z em (cid:20) v tt + v p ( r, z em ) v tt + v ∞ ( r, z em ) (cid:21) . (14) The presence of an ordered magnetic field threading anAGN accretion disc has been suggested by several authors(e.g., Blandford & Payne 1982; Contopoulos & Lovelace 1994;K¨onigl & Kartje 1994) as a mechanism able to either confine theclouds (as in Emmering et al. 1992) or direct the outflow velocityfield (e.g., Everett 2005). Along with most works in the field, we do not discuss here the origin of the magnetic field, but assume it ispresent and study its effects within the postulated framework.The standard magneto-hydrodynamic (MHD) wind equationsare: ∂ρ∂t + ∇ · ( ρ v ) = 0 (15a) ρ ∂ v ∂t + ρ ( v · ∇ ) v = −∇ p − ρ ∇ Φ g + 14 π ( ∇ × B ) × B (15b) ∂ B ∂t = ∇ × ( v × B ) (15c) ∇ · B = 0 , (15d)where B and v are respectively the magnetic and velocity fields, ρ is the mass density, p is the thermal pressure and Φ g is the gravita-tional potential.We look for steady state wind solutions for this model. In thatcase (i.e. when ∂/∂t = 0 ), there are conserved quantities alongeach magnetic field line (e.g., Mestel 1968), such asmass to magnetic flux ratio, k π = ρv p B p (16)specific angular momentum, l = r (cid:18) v φ − B φ k (cid:19) (17)specific energy e = v h + Φ g − r Ω B φ k (18)where h is the specific enthalpy, and, owing to the axisymmetryof the problem, we have separated the velocity and magnetic fieldsinto their poloidal and azimuthal components: v = v p + v φ B = B p + B φ (19) Solutions of the steady, axisymmetric, non-relativistic ideal MHDequations assuming a spherically self-similar scaling were obtainedby e.g. Blandford & Payne (1982, BP82 hereafter) for the coldplasma outflow from the surface of a Keplerian disc. This solutionfor the field can be written in terms of variables χ , ξ ( χ ) , φ , and r ,which are related to the cylindrical coordinates via r ≡ [ r, φ, z ] = [ r ξ ( χ ) , φ, r χ ] , (20)where the adopted independent variables ( r , χ ) are a pair of spa-tial coordinates analogous to ( r, z ) . The function ξ ( χ ) describesthe shape of the field lines and, in the general case, is not a priori known, but found as part of a self-consistent solution to the MHDequations. The flow velocity components are given by v = (cid:2) ξ ′ ( χ ) f ( χ ) , g ( χ ) , f ( χ ) (cid:3) r GMr , (21)where a prime denotes differentiation with respect to χ , and G and M are respectively the gravitational constant and the mass of thecentral black hole.In this self-similar model, the scaling of the speed v , mag-netic field amplitude B , and gas density ρ with the spherical ra-dial coordinate r is determined from the relation B/ √ ρ ∝ r − / and from the assumption that r ρv is independent of r , fromwhere ρ ∝ r − / and B ∝ r − / . Other authors (e.g., c (cid:13) , 1–18 L. S. Chajet and P. B. Hall
Contopoulos & Lovelace 1994; Emmering et al. 1992) generalizedthis class of self-similar solutions by considering winds with a den-sity scaling ρ ∝ r − b , for which B ∝ r − ( b +1) / . Note that, in thiscontext, the BP82 solution corresponds to b = 3 / . The magneticfield and density at arbitrary positions can be then written, in accor-dance with the self-similarity Ansatz (20), as B = B ( r ) b ( χ ) and ρ = ρ ( r ) ̺ ( χ ) . On the disc plane the rotational velocity, v φ , isKeplerian and scales as v φ ∝ r − / . The functions ξ ( χ ) , f ( χ ) and g ( χ ) have to satisfy the flow MHD equations subject to the abovescalings of ρ , B , and v φ and boundary conditions. In particular, atthe disc surface ξ (0) = 1 , f (0) = 0 and g (0) = 1 .Following BP82, we introduce the dimensionless expressionsof the integrals of motion defined in equations 16-18, in terms ofwhich the solutions are defined: κ = k (1 + ξ ′ ) / ( GM/r ) / B (22) λ = l ( GMr ) / (23) ǫ = e ( GM/r ) (24)The parameters of the model are ǫ , λ and κ and ξ ′ . However, due tothe regularity conditions that must be satisfied, these parameters arenot independent. Combining equations 23 and 24 gives ǫ = λ − . The value of ξ ′ ≡ ξ ′ ( χ = 0) must be chosen to ensure theregularity of the solution at the Alfv´en point. The solutions aretherefore parametrized only by two numbers, which can be chosento be κ and λ (e.g. BP82).The Alfv´en speed, v A , is the characteristic velocity of thepropagation of magnetic signals in an MHD fluid and is definedby: v A = B µ ρ , (25)where µ is the vacuum permeability. Another important charac-teristic quantity in magnetized fluids is the Alfv´en Mach number ateach position. The square of this quantity is expressed in the presentmodel as m ( χ ) = v v = 4 πρ v B = κ f ( χ ) ξ ( χ ) J ( χ ) , (26)where J ( χ ) = ξ ( χ ) − χ ξ ′ ( χ ) (27)is the determinant of the Jacobian matrix of the transformation ( r, z ) → ( r , χ ) and v pA is the poloidal component of the Alfv´envelocity.The function g ( χ ) can be expressed in terms of the function m and the specific angular momentum, λ : g ( χ ) = ξ ( χ ) − λ m ( χ ) ξ ( χ ) [1 − m ( χ )] . (28)From this expression we can see that the point corresponding to m = 1 is a singular point of the problem. In particular, to avoidunphysical solutions there, we must have ξ A = ξ ( χ A ) = λ / ,where the subscript A refers to the Alfv´en point.Expressing f and g by Eqs (26) and (28) in terms of the The Alfv´en point is where the poloidal velocity of the fluid is equal tothe poloidal component v pA of the Alfv´en velocity v A defined in Eq. 25 Alfv´enic Mach number and of the function ξ ( χ ) and its deriva-tives, Eq. (18) is transformed into a fourth degree equation for thefunction f ( χ ) : T − f (1 + ξ ′ ) = (cid:20) ( λ − ξ ) mξ (1 − m ) (cid:21) , (29)where T = ξ + 2 p ξ + χ − . (30)Using the differential form of equation 29 combined with the z -component of the momentum equation (Eq. 15b), BP82 obtain asecond-order differential equation for ξ ( χ ) . The flow is then fullyspecified by that equation and equation 29, plus the boundary con-ditions, ξ (0) = 1 and ξ ′ (0) = ξ ′ .The model of Emmering et al. (1992, hereafter EBS92) rep-resents a simplified version of BP82 solution. The EBS92 solutioncorresponds to the case in which the solution asymptotically ap-proaches n = 1 as χ → ∞ , where n is the square of the Machnumber for the fast magnetosonic mode for an arbitrary scaling ofdensity ρ ∝ r − b and magnetic field B ∝ r − ( b +1) / . The quan-tity n is given by (e.g. BP82, BPS92): n = 4 πρ v B = κ ξ f J (1 + ξ ′ ) T . (31)While BP82 found their solutions by integrating a second-order differential equation, EBS92 impose a priori the functionalform of the solution so that it will asymptotically tend to the BP82solution. In their equation (3.19), EBS92 give an explicit form forthe function ξ ( χ ) : ξ = (cid:18) χc + 1 (cid:19) / , (32)where c = tan ϑ was chosen to ensure that the field lines makean initial angle ϑ with the disc plane, so that cot ϑ = ξ ′ and thesubscript means that the quantities are evaluated at the disc plane.It can be demonstrated (e.g. BP82, Heyvaerts 1996) that there is anupper limit for this angle: ϑ < ◦ .EBS92 found that to satisfy the condition that n → when χ → ∞ , the parameters κ and λ must be related by: κ = 2 (cid:18) λ − (cid:19) / (33)In addition, the asymptotic value of the function f is given by f ∞ = (cid:18) λ − (cid:19) / . (34)Thus, in this model the solutions depend on λ and ϑ . Figure 1depicts the streamlines for two different launching angles.Note that in the general case, in order to find the flow vari-ables, we should have solved a second-order differential equation ξ ′′ = ξ ′′ ( χ, ξ, ξ ′ , f ( χ )) , with f ( χ ) given implicitly by equation29. However, by using the EBS92 model we could evaluate f ′ , m and g ′ from an analytic estimate for f . The point of having usedan analytical functional form for f ( χ ) has to do with the inclusionof the velocity field in the MC97 model. The opacity depends onthe projection of the line of sight (LOS) component of the gradientof the LOS velocity through the quantity Q (equation 8), which in-volves the spatial derivatives of velocity components. Combiningthe EBS92 functional form for ξ ( χ ) and ξ A = λ . it is straight-forward to obtain χ A = ( λ − c . In the general case this quantitymust be found numerically as part of the solution. However, in the c (cid:13)000
Contopoulos & Lovelace 1994; Emmering et al. 1992) generalizedthis class of self-similar solutions by considering winds with a den-sity scaling ρ ∝ r − b , for which B ∝ r − ( b +1) / . Note that, in thiscontext, the BP82 solution corresponds to b = 3 / . The magneticfield and density at arbitrary positions can be then written, in accor-dance with the self-similarity Ansatz (20), as B = B ( r ) b ( χ ) and ρ = ρ ( r ) ̺ ( χ ) . On the disc plane the rotational velocity, v φ , isKeplerian and scales as v φ ∝ r − / . The functions ξ ( χ ) , f ( χ ) and g ( χ ) have to satisfy the flow MHD equations subject to the abovescalings of ρ , B , and v φ and boundary conditions. In particular, atthe disc surface ξ (0) = 1 , f (0) = 0 and g (0) = 1 .Following BP82, we introduce the dimensionless expressionsof the integrals of motion defined in equations 16-18, in terms ofwhich the solutions are defined: κ = k (1 + ξ ′ ) / ( GM/r ) / B (22) λ = l ( GMr ) / (23) ǫ = e ( GM/r ) (24)The parameters of the model are ǫ , λ and κ and ξ ′ . However, due tothe regularity conditions that must be satisfied, these parameters arenot independent. Combining equations 23 and 24 gives ǫ = λ − . The value of ξ ′ ≡ ξ ′ ( χ = 0) must be chosen to ensure theregularity of the solution at the Alfv´en point. The solutions aretherefore parametrized only by two numbers, which can be chosento be κ and λ (e.g. BP82).The Alfv´en speed, v A , is the characteristic velocity of thepropagation of magnetic signals in an MHD fluid and is definedby: v A = B µ ρ , (25)where µ is the vacuum permeability. Another important charac-teristic quantity in magnetized fluids is the Alfv´en Mach number ateach position. The square of this quantity is expressed in the presentmodel as m ( χ ) = v v = 4 πρ v B = κ f ( χ ) ξ ( χ ) J ( χ ) , (26)where J ( χ ) = ξ ( χ ) − χ ξ ′ ( χ ) (27)is the determinant of the Jacobian matrix of the transformation ( r, z ) → ( r , χ ) and v pA is the poloidal component of the Alfv´envelocity.The function g ( χ ) can be expressed in terms of the function m and the specific angular momentum, λ : g ( χ ) = ξ ( χ ) − λ m ( χ ) ξ ( χ ) [1 − m ( χ )] . (28)From this expression we can see that the point corresponding to m = 1 is a singular point of the problem. In particular, to avoidunphysical solutions there, we must have ξ A = ξ ( χ A ) = λ / ,where the subscript A refers to the Alfv´en point.Expressing f and g by Eqs (26) and (28) in terms of the The Alfv´en point is where the poloidal velocity of the fluid is equal tothe poloidal component v pA of the Alfv´en velocity v A defined in Eq. 25 Alfv´enic Mach number and of the function ξ ( χ ) and its deriva-tives, Eq. (18) is transformed into a fourth degree equation for thefunction f ( χ ) : T − f (1 + ξ ′ ) = (cid:20) ( λ − ξ ) mξ (1 − m ) (cid:21) , (29)where T = ξ + 2 p ξ + χ − . (30)Using the differential form of equation 29 combined with the z -component of the momentum equation (Eq. 15b), BP82 obtain asecond-order differential equation for ξ ( χ ) . The flow is then fullyspecified by that equation and equation 29, plus the boundary con-ditions, ξ (0) = 1 and ξ ′ (0) = ξ ′ .The model of Emmering et al. (1992, hereafter EBS92) rep-resents a simplified version of BP82 solution. The EBS92 solutioncorresponds to the case in which the solution asymptotically ap-proaches n = 1 as χ → ∞ , where n is the square of the Machnumber for the fast magnetosonic mode for an arbitrary scaling ofdensity ρ ∝ r − b and magnetic field B ∝ r − ( b +1) / . The quan-tity n is given by (e.g. BP82, BPS92): n = 4 πρ v B = κ ξ f J (1 + ξ ′ ) T . (31)While BP82 found their solutions by integrating a second-order differential equation, EBS92 impose a priori the functionalform of the solution so that it will asymptotically tend to the BP82solution. In their equation (3.19), EBS92 give an explicit form forthe function ξ ( χ ) : ξ = (cid:18) χc + 1 (cid:19) / , (32)where c = tan ϑ was chosen to ensure that the field lines makean initial angle ϑ with the disc plane, so that cot ϑ = ξ ′ and thesubscript means that the quantities are evaluated at the disc plane.It can be demonstrated (e.g. BP82, Heyvaerts 1996) that there is anupper limit for this angle: ϑ < ◦ .EBS92 found that to satisfy the condition that n → when χ → ∞ , the parameters κ and λ must be related by: κ = 2 (cid:18) λ − (cid:19) / (33)In addition, the asymptotic value of the function f is given by f ∞ = (cid:18) λ − (cid:19) / . (34)Thus, in this model the solutions depend on λ and ϑ . Figure 1depicts the streamlines for two different launching angles.Note that in the general case, in order to find the flow vari-ables, we should have solved a second-order differential equation ξ ′′ = ξ ′′ ( χ, ξ, ξ ′ , f ( χ )) , with f ( χ ) given implicitly by equation29. However, by using the EBS92 model we could evaluate f ′ , m and g ′ from an analytic estimate for f . The point of having usedan analytical functional form for f ( χ ) has to do with the inclusionof the velocity field in the MC97 model. The opacity depends onthe projection of the line of sight (LOS) component of the gradientof the LOS velocity through the quantity Q (equation 8), which in-volves the spatial derivatives of velocity components. Combiningthe EBS92 functional form for ξ ( χ ) and ξ A = λ . it is straight-forward to obtain χ A = ( λ − c . In the general case this quantitymust be found numerically as part of the solution. However, in the c (cid:13)000 , 1–18 HD Disc Winds and Line Width Distributions adopted framework, all related quantities at the Alfv´en point areeasily found (because m A ≡ m ( χ A ) = 1 ). In particular, f A = 1 κ √ λJ A = 1 κ √ λ [ ξ ( χ A ) − χ A ξ ′ ( χ A )] . (35)The derivative of f ( χ ) at χ = 0 , f ′ ( χ = 0) = f ′ , is given byBP82 (their equation 2.23c), reproduced here: f ′ = (3 ξ ′ − / κ [( λ − + (1 + ξ ′ )] / . (36)We thus adopt f ( χ ) = f ∞ e k χ − e k χ + k (37)and look for k and k such that the conditions for f A and f ′ aresatisfied. Once f ( χ ) is found, m ( χ ) and thus g ( χ ) are obtained.We then have the three wind velocity components expressed in an-alytical form.Note that, as already mentioned, EBS92 model postulates thatthe emission lines arise in clouds confined by an MHD flow. How-ever, we follow MC97 and Murray & Chiang (1998, MC98 here-after) in assuming that the lines form in a continuous medium. Aswill be discussed in section 5, we consider line emissivity obtainedby CLOUDY photoionization model, different from either of theemissivity laws adopted by EBS92. Two of those emissivity modelsinclude electron scattering, which is not considered in our model.In EBS92 the dimensionless angular momentum λ and the launchangle are fixed, while in our work the former is still fixed but thelatter is varied to study its effect on the profiles. It is important tonote that, while EBS92 obtains the line luminosity integrating inthe two poloidal variables, we include the z -integral in the opticaldepth expression. Q FOR SELF-SIMILAR MHDWINDS
For self-similar solutions of MHD winds, the derivatives needed toobtain the different Λ ij that appear in the quantity Q have to beevaluated using the rules for changing variables (e.g., K¨onigl 1989) ∂∂r = 1 J ∂∂r − χr J ∂∂χ , (38) ∂∂z = − ξ ′ J ∂∂r + ξr J ∂∂χ , (39)where J ( χ ) has been defined in Eq. (27). Thus, we have the fol-lowing expressions, where for clarity we omit the functional de-pendence of the dependent variables: Λ rr = − J s GMr (cid:18) ξ ′ f χ (cid:0) ξ ′′ f + ξ ′ f ′ (cid:1)(cid:19) (40) Λ φφ = s GMr ξ ′ fξ (41) Λ zz = 1 J s GMr (cid:18) ξ ′ f ξf ′ (cid:19) (42) Λ rz = 12 J s GMr (cid:20)(cid:18) ξ ′ ξξ ′′ − (cid:19) f + (cid:0) ξξ ′ − χ (cid:1) f ′ (cid:21) (43) Λ rφ = 12 s GMr (cid:20) − J (cid:16) g χg ′ (cid:17) + gξ (cid:21) (44) Λ φz = 12 J s GMr (cid:18) ξ ′ g ξg ′ (cid:19) (45)For the particular form of ξ ( χ ) given by EBS92 the corre-sponding expressions for the strain tensor entries are: Λ rr = − s GMr (cid:20) fc + 2 χ f ′ + 2 χf ′ c χ + c ) ( χ + 2 c ) (cid:21) (46) Λ φφ = s GMr (cid:20) f χ + c ) (cid:21) (47) Λ zz = s GMr (cid:20) f + 4 f ′ ( χ + c )2( χ + 2 c ) (cid:21) (48) Λ rz = s GMr " (cid:0) χ + c − c χ − c χ (cid:1) f ′ − (cid:0) c χ + 4 c (cid:1) f √ c ( χ + 2 c ) √ χ + c (49) Λ rφ = − s GMr √ c (cid:20) (2 χ + 3 c ) g + 2 χ ( χ + c ) g ′ χ + 2 c ) √ χ + c (cid:21) (50) Λ φz = − s GMr (cid:20) g + 4 g ′ χ + 4 g ′ c χ + 2 c ) (cid:21) (51) We evaluated the line luminosity (Eq. 10) using the EBS92 solu-tion to estimate the quantities included there and in the associatedequations 11 and 13. In summary, v D and Q are computed as func-tions of position ( r, φ, z ) from the velocity field given by the EBS92model. Then, these two quantities and the ‘thermal Q ’, q tt , are usedto evaluate the optical depth τ ( r, φ, i ) (Eq. 11) and the quantities e ν ( r, φ, i ) (Eq. 12) x ν ( r, φ, i ) (Eq. 13). We emphasize again thatthe integral in the z direction is included in the optical depth ex-pression. We then calculate L ν ( ˆn ) by integrating over all ( r, φ )(Eq. 10). The process is repeated for different ν values to build upthe profile of the given emission line for the given input parameters.We have computed the C IV line profile for different combi-nations of inclination angle, i and initial angle, ϑ . We also studiedthe results of changing the initial density and the exponent of thepower law that governs the radial behaviour of the density. The spe-cific luminosity from each component of the C IV doublet is com-puted separately, and then the results added together. In Table 1 welist the meaning and adopted values of the main parameters in themodel. The fiducial values adopted for the density, density power-law exponent and thermal plus turbulent velocity are n = 10 cm − , b = 2 and v tt = 10 cm s − , respectively.We determine the source function for our simulations by ap-plying the reverberation mapping results of Kaspi et al. (2007) tothe radial line luminosity function L ( r ) calculated by MC98 for a c (cid:13) , 1–18 L. S. Chajet and P. B. Hall
Table 1.
Set of parameters used in the simulation.Variable Value Explanation M BH M ⊙ Black hole mass L UV erg s − Quasar ionizing luminosity S ν ( r ) CLOUDY results Source function r min q L UV erg s − × cm Inner BELR radius. r max q L UV erg s − × cm Outer BELR radius. n − cm − Hydrogen number density at r min , declining as r − b thereafter b n ( r ) ∝ r − b i ◦ − ◦ Observer inclination angle ϑ ◦ , ◦ , ◦ , ◦ , ◦ , ◦ Streamline launch angle tan β ( r ) z em = r tan β ( r ) = r tan(6 ◦ ) v tt − cm s − Thermal+turbulent speed of ion χ i solar Abundance of element η i λ
10 Specific angular momentum quasar with L ≡ νL ν (1350 ˚ A) = 10 erg s − and shown intheir Figure 5b. According to that figure, the peak C IV emissionis reached at R CIV = 10 cm, but the Kaspi et al. (2007) resultsshow that R CIV is smaller for a quasar of that luminosity. TheirEquation 3 gives R CIV = 6 . × cm (cid:18) L erg s − (cid:19) α , (52)where α = 0 . ± . in the original formulation and for simplic-ity we have adopted α = 0 . . Eq. 52 gives R CIV = 2 × cmfor a quasar with L = 10 erg s − . We therefore empiricallyadjust all the radii in the MC98 Figure 5b line luminosity functiondown by a factor of five. Each point in the line luminosity functionnow gives the line luminosity L ( r i ) in a logarithmic bin spanninga factor of √ in radius centred on adjusted radius r i for a quasarwith L = 10 erg s − . v th For a thermal velocity of v th = 10 cm s − and no turbulence, theprofiles of the individual components of the doublet are very nar-row (FWHM < inter-component separation) for small inclinationangles. As a result, the combined profiles are double- (or multiple-)peaked. The effect is less pronounced for higher ( & ◦ ) i values.These results suggested considering different velocities byincorporating the effect of turbulence. Bottorff & Ferland (2000)studied how microturbulence can affect the lines and showed that itaffects more the far UV lines. Figure 2 shows the lines correspond-ing to four different values of v turb , for two different inclinationangles, i = 15 ◦ (left panels) and i = 75 ◦ (right panels). The pro-files in the upper row correspond to ϑ = 10 ◦ and in the lowerrow, to ϑ = 57 ◦ . In general, the lines become smoother and moresymmetric with increasing v turb . A much higher v turb does makea noticeable difference, as expected. However, note that the upperright panel seems to represent an anomalous situation, as the pro-files become narrower as the turbulent velocity increases. The reader should be cautioned that, strictly speaking, this translationof the line-continuum lag measured in reverberation mapping experimentsand the peak of the radial emissivity distribution of the line is not straight-forward in the general case.
To investigate the apparently anomalous situation in the up-per right corner of Fig. 3 we plotted the same profiles as above, butnormalized to the values corresponding to our fiducial turbulent ve-locity ( cm s − ). Note that in all cases, even in the apparentlydeviant case, the flux increases when the turbulent velocity does. For the fiducial values of density ( n = 10 cm − ) and turbulentvelocity ( v turb = 10 cm s − ), we studied the effect(s) of changingthe launching and viewing angles, in the ranges ϑ = 5 ◦ − ◦ and i = 10 ◦ − ◦ , respectively. The results are shown in Figures 4 and5. In each panel of Figure 4 the profiles are plotted versus veloc-ity for a given launching angle, to enhance the effect of changingthe viewing angle. Zero velocity is the average of the two doubletwavelengths. The velocities plotted represent velocities from theobserver’s point of view, therefore negative velocities correspondto blueshifts. In all cases the profiles are slightly asymmetric, withincreasing degree of asymmetry with decreasing inclination. Theblue wings change less than the red wings, so that as the inclinationangle approaches to smaller values, the red wings are increasinglyweaker. In Figure 4, the effect is hard to notice for the lowest ϑ (the two upper panels), due to the shift to the red in the peak of theprofiles when the inclination increases. We will discuss the effectof launching angle dependency below.A way to see this is by noting that, for a given launching angle,when the object is seen face-on, the projection of the velocity intothe line of sight is towards the observer for any azimuthal angle,while for objects seen edge-on, that projection is towards the ob-server for part of the emitting region, and receding from them forthe rest. For intermediate cases, the closer the object’s LOS is tothe face-on case, the more the red wing of its profile is weakened,explaining why the lines are less symmetric for smaller inclinationangles. The effects of changing the launching angle ϑ are shown in Fig-ure 5, where in each panel we have plotted the profiles for a giveninclination angle and different launching angles. The actual angleto be considered is the angle ϑ at which a line launched with some ϑ crosses the base of the emitting region (when ϑ increases, sodoes ϑ ). For instance, for our chosen value of tan β , for ϑ = 20 ◦ , ϑ ∼ . ◦ and for ϑ = 45 ◦ , ϑ = 48 ◦ When ϑ increases, the projection of the wind velocity ontothe LOS is towards the observer in a portion of the emission region(i.e., for some azimuths) and is also towards the observer in the restof the region as long as ϑ > i . In the cases ϑ < i , that projectionis receding from the observer. As the wind velocity decreases withincreasing ϑ , so does the magnitude of its projection for given i ,and thus the blueshift decreases for increasing ϑ . However, it isthe Doppler velocity, including a contribution from the rotationalvelocity, which is the velocity relevant for producing the observedline profiles. Thus, as the wind velocity decreases with ϑ , then notonly is the blueshift reduced, but the rotational velocity is increas-ingly dominant and the profiles become more symmetric. For anylaunch angle, the relative importance of the receding term with re-spect to the approaching term increases with increasing viewingangle, but the effect is larger for smaller ϑ .We also analysed how strongly this broadening of line pro-file with decreasing ϑ depends on the density profile. Note that, c (cid:13) , 1–18 HD Disc Winds and Line Width Distributions - - - - @ km s - D L H Ν L (cid:144) L M a x J = ë i = ë - - - - @ km s - D L H Ν L (cid:144) L M a x J = ë i = ë - - - - @ km s - D L H Ν L (cid:144) L M a x J = ë i = ë - - - - @ km s - D L H Ν L (cid:144) L M a x J = ë i = ë v turb = cm s - v turb = cm s - v turb = cm s - v turb = cm s - Figure 2. L ν /L max versus velocity for 4 different values of the thermal plus turbulent velocity: v turb = 10 , . , , . cm s − and two differentinclination angles: i = 15 ◦ (left panels) and i = 75 ◦ (right panels). Upper panels correspond to ϑ = 10 ◦ and lower panels, to ϑ = 57 ◦ . The initial densityis n = 10 cm − in all cases. The two extra curves shown in the upper right panel correspond to v turb = 10 . cm s − (dashed blue) and . cm s − (dashed red). - - - - @ km s - D L H Ν L (cid:144) L M a x H v t u r b = c m s - L J = ë i = ë - - - - @ km s - D L H Ν L (cid:144) L M a x H v t u r b = c m s - L J = ë i = ë - - - - @ km s - D L H Ν L (cid:144) L M a x H v t u r b = c m s - L J = ë i = ë - - - - @ km s - D L H Ν L (cid:144) L M a x H v t u r b = c m s - L J = ë i = ë v turb = cm s - v turb = cm s - v turb = cm s - v turb = cm s - Figure 3. L ν /L max ( v turb = 10 cm s − ) versus velocity for four different values of the thermal plus turbulent velocity: v turb = 10 , . , , . cm s − and two different inclination angles: i = 15 ◦ (left panels) and i = 75 ◦ (right panels). Upper panels correspond to ϑ = 10 ◦ and lower panels, to ϑ = 57 ◦ .c (cid:13) , 1–18 L. S. Chajet and P. B. Hall - - - - @ km s - D L H Ν L (cid:144) L M a x J = - - - - @ km s - D L H Ν L (cid:144) L M a x J = - - - - @ km s - D L H Ν L (cid:144) L M a x J = - - - - @ km s - D L H Ν L (cid:144) L M a x J = - - - - @ km s - D L H Ν L (cid:144) L M a x J = i = = = = = = = Figure 4. L ν /L max versus velocity for i = 10 ◦ − ◦ . Looking clockwise from the upper left: ϑ = 10 ◦ , ◦ , ◦ , ◦ , ◦ . The latter two cases do notdiffer significantly. in principle, the smaller ϑ is, the larger the radii at which thestreamlines intersect the base of the emission region, that is, theselines will form at radii where the density (that goes as ∼ r − b ) issmaller, affecting the optical depth. To that end, we compared theinverse square power-law to other (less steep) density power-laws( b = 0 . , ) for different launching angles, and found that the de-pendence is negligible. That is, the broadening found in the small ϑ cases depends mainly on the velocity projection.In summary, the relevant quantity is neither of the angles, buta combination of them. This can also be seen by considering theexpression of the optical depth (Eq. 11) where the frequency de-pendance is encompassed in the exponent x ν , dependent on theDoppler velocity v D . The latter includes the wind contribution,which ranges from v p sin( ϑ + i ) when φ = 0 , to v p sin( ϑ − i ) when φ = π . Note also, that the FWHM increases with increas-ing inclination angle (see Fig. 4), but for small launching angles itreaches its maximum at i < ◦ , while the maximum is reached at i = 84 ◦ for larger ϑ . In the smaller ϑ cases, the broadening andsubsequent decrement is accompanied by a shift in the peak, frombluer (at smaller inclinations) to redder velocities (at larger incli-nations). This is due to the fact that the observer sees the base ofthe conical emission region from an almost edge-on perspective, sothat the part of the cone with φ ≃ ◦ (which produces blueshiftedemission) has very small projected surface area. c (cid:13) , 1–18 HD Disc Winds and Line Width Distributions - - - - @ km s - D L H Ν L (cid:144) L M a x i = - - - - @ km s - D L H Ν L (cid:144) L M a x i = - - - - @ km s - D L H Ν L (cid:144) L M a x i = - - - - @ km s - D L H Ν L (cid:144) L M a x i = - - - - @ km s - D L H Ν L (cid:144) L M a x i = - - - - @ km s - D L H Ν L (cid:144) L M a x i = - - - - @ km s - D L H Ν L (cid:144) L M a x i = J = J = J = J = J = Figure 5. L ν /L max versus velocity. In each panel, we have plotted the profiles corresponding to i from ◦ to ◦ for ϑ = 10 ◦ − ◦ . For given i , thoseprofiles for which ϑ > i are bluer, while those corresponding to ϑ < i are redder. Here, ϑ > ϑ is the angle between a line launched with some ϑ and thebase of the emitting region.c (cid:13) , 1–18 L. S. Chajet and P. B. Hall
We also looked at the effects of varying the initial density. Althoughwe adopted n ∼ cm − as the “standard density”, we alsochose to check the effect of even lower and higher densities. Inprinciple, one would expect broader profiles for smaller initial den-sity. In fact, that is what is found when running simulations that donot include the terms and factors introduced in Hall et al. (2012). Inthat case, the results showed that the profiles become broader as theinitial density decreases. In effect, as the density decreases, so doesthe opacity and, in that case there will be less photons absorbed inthe line wings and this translates into broader lines. However, theinclusion of these previously neglected terms and factors modifiesthe behaviour of the profiles, in such a way that the effect of chang-ing the initial density is much less important (negligible, in somecases). In the current model, the velocity field (which depends onboth inclination and launching angles) dictates the optical depth be-haviour. This is somewhat similar to the broadening of the low- ϑ case that we discussed above.Figure 6 shows the profiles obtained for a fixed inclination i = 15 ◦ (left panels) and i = 75 ◦ (right panels) and launchingangles ϑ = 15 ◦ (upper panels) and ϑ = 57 ◦ (lower panels) whenthe initial density, which declines radially according to n ∼ r − is varied. The results of the density analysis also show that forgiven ϑ , the smaller the inclination angle, the bluer the maximum.Note also that we have included two extra profiles in the upper rightpanel, to illustrate that in this particular case ( ϑ = 10 ◦ , i = 75 ◦ )the profiles do indeed converge at lower densities. The effect of even smaller launching angles is shown in Figure 7for the cases i = 5 ◦ , ◦ , ◦ . Included, for comparison, are theprofiles for the same inclination angles, but with ϑ = 10 ◦ . In theleft panel, each profile is normalized with respect to the maximumof the i = 5 ◦ profile for each launching angle, whereas in the rightpanel the normalization is with respect to its own maximum. Twoobservations can be made from the figure. First, the differences be-tween ϑ = 5 ◦ and ϑ = 10 ◦ profiles are much larger than thosebetween ϑ = 10 ◦ and ϑ = 15 ◦ profiles. Second, when the view-ing angle is large and ϑ = 5 ◦ , the profile is double-peaked, whichis not observed in C IV lines.To study these issues we analysed the evolution of the pro-files, for two different inclination angles ( i = 5 ◦ , ◦ ), when thelaunching angle changes between ϑ = 5 ◦ and ◦ . Figure 8 showsthat in both cases, as the launching angle increases the profiles be-come increasingly narrower. Here we see the same trend shown inFigure 5 and discussed in subsection 5.3: for smaller launch andviewing angles (left panel), most of the flux is due to motion to-wards the observer and the blueshift decreases with increasing ϑ .For larger i (right panel) the contribution of the receding term ofthe Doppler velocity dominates but also decreases with increasing ϑ , leading to an emission peak that approaches the systemic red-shift with increasing ϑ . Also noticeable is the fact that the double-peaked feature is only present when ϑ = 5 ◦ (and to lower extentwhen ϑ = 6 ◦ ). That suggests that we can impose an empiricalrestriction on ϑ and consider only those that satisfy the condition ϑ > ◦ . To determine whether this is a general constraint, valid Note that the velocity ranges are not the same in all cases: the left panelsshare the velocity range, but that differs from the range in either of the tworight panels. for any given β , would require simulations for different values ofthat parameter, which is beyond the scope of this work. From a set of profiles obtained for different inclination angles i and launching angles ϑ we study how the FWHMs are distributedas a function of the angles i at which quasars are visible. To doso, we use an approach similar to that of Fine et al. (2008), whoconstrained the range of possible AGN viewing angles by usinggeometrical models for the BLR and comparing the expected dis-persion in linewidths at each viewing angle to their observationaldata. We extend their analysis by also considering the clumpy torusmodel of Nenkova et al. (2008a, 2008b; hereafter N08), as con-strained by Mor et al. (2009) using infrared observations of lumi-nous AGN.Fine et al. (2008) measured the linewidth of the Mg II line in32214 quasar spectra from the Sloan Digital Sky Survey (SDSS)Data Release Five, 2dF QSO Redshift survey (2QZ) and 2dF SDSSLRG and QSO (2SLAQ) survey and found that the dispersion inlinewidths strongly correlates with the optical luminosity of QSOs.Fine et al. (2010) used 13776 quasars from the same surveys tostudy the dispersion in the distribution of C IV linewidths. In con-trast to their findings for the Mg II , they found that the dispersion inC IV linewidths is essentially independent of both redshift and lu-minosity. Fine et al. (2008, 2010) used the fact that if the linewidthmeasured from a spectrum depends on the viewing angle to theobject, the linewidth dispersion for a model for the BLR can bycalculated by ‘observing’ that model over different ranges of view-ing angles. Combinations of models and viewing angle ranges thatgive dispersions larger than the observed dispersion it can be re-jected. Fine et al. assumed a coplanar obscuring torus surroundingthe central SMBH and BLR with an opening angle i max (measuredfrom the vertical axis), so that the viewing angle i should satisfy i i max . If the FWHM of a BEL varies with i , the dispersion inthe FWHM distribution of that BEL should vary with i max . Following Fine et al. (2008), we compare the dispersion of ob-served log(
FWHM ) values with the dispersion of our simulated log( FWHM ) as a function of i max and launching angle ϑ to see ifwe can constrain i max or ϑ .Using the launching angle as a parameter, we evaluate the dis-persion of the function f ( i ) = log( FWHM ( i )) . As all the variablesare, in fact, continuous, we interpolated each set of FWHMs to ob-tain the corresponding continuous functions. For a given i min , themean and the variance of the FWHMs are functions of i max , ac-cording to ¯ f ( i max ) = R i max i min sin i P ( i ) f ( i ) di R i max i min sin i P ( i ) di , (53) σ f ( i max ) = R i max i min sin i P ( i ) (cid:2) f ( i ) − ¯ f ( i max ) (cid:3) di R i max i min sin i P ( i ) di , (54)where P ( i ) is a weighting factor, equal to 1 in the Fine et al. (2008)approach, that measures the probability of not having obscurationin the LOS direction. We first present results using P ( i ) = 1 and then turn to a more complex case. Both Fine et al. (2008) andMor et al. (2009) have i max = 90 ◦ as the upper limit for that angle. c (cid:13) , 1–18 HD Disc Winds and Line Width Distributions - - - -
500 0 500 1000 15000.00.20.40.60.81.0 v @ km s - D L H Ν L (cid:144) L M a x J = ë i = ë - - - - @ km s - D L H Ν L (cid:144) L M a x J = ë i = ë - - - -
500 0 500 1000 15000.00.20.40.60.81.0 v @ km s - D L H Ν L (cid:144) L M a x J = ë i = ë - - - - @ km s - D L H Ν L (cid:144) L M a x J = ë i = ë n = cm - n = cm - n = cm - n = cm - n = cm - n = cm - n = cm - n = cm - Figure 6. L ν /L max versus velocity. Here, the profiles correspond to fixed i ∼ ◦ (left panels) and i ∼ ◦ (right panels) and ϑ = 10 ◦ (upper panels) and ϑ = 57 ◦ (bottom panels) but different initial densities: n = 10 cm − (blue), n = 10 cm − (red), n = 10 cm − (green), n = 10 cm − (black), n = 10 cm − (cyan), n = 10 cm − (magenta), n = 10 cm − (dashed gray) and n = 10 cm − (orange). The extra lines in the upper rightpanel correspond to n = 10 cm − (dashed blue) and n = 10 . cm − (dashed red) and were included to show that, although slower than in other cases,the low-density profiles also converge. The effect of a lower density on the opacity, and thus on the line broadness, is surpassed by the effect of the velocityfield, leaving a weak dependence on density, especially at lower inclinations. For fixed ϑ (i.e., looking along rows) the spread is larger for higher inclinationangles, while for fixed i (i.e, looking along columns), it is larger for smaller launching angle, with the trend being more pronounced with decreasing n . - - @ km s - D L H Ν L (cid:144) L M a x H i = ° L - - @ km s - D L H Ν L (cid:144) L M a x i = J = = J = = J = = J = = J = = J = Figure 7.
Two different normalizations of the profiles for ϑ = 5 ◦ , ◦ and i = 5 ◦ , ◦ , ◦ . In the left panel, the profiles are normalized to the maximumof the i = 5 ◦ for the corresponding launching angle. In the right panel, the normalization is with respect to each profile’s own maximum. - - @ km s - D L H Ν L (cid:144) L M a x H J = ° L - - @ km s - D L H Ν L (cid:144) L M a x H J = ° L J = J = J = J = J = Figure 8.
Normalized profiles for ϑ = 5 ◦ - ◦ , ◦ , ◦ and i = 5 ◦ (left panel), ◦ (right panel).c (cid:13) , 1–18 L. S. Chajet and P. B. Hall
However, we have an extra limitation, set by the inclination of thebase of the emitting region, chosen to be β = 6 ◦ . Therefore, ourupper limit is i max = 84 ◦ .Noting that Fine et al. (2008, 2010) have employed inter-percentile values (IPVs) rather than FWHMs to characterize theline widths, we also investigated the behaviour of this line measurefrom our results. For a given percentage p , the definition of IPV p suggested by Whittle (1985) is the separation between the median(where the integrated profile reaches the of the total flux) andthe positions where p % and (100 − p )% of the total flux are reached.Thus, calling d and d the distances between the median and p and (100 − p ) respectively, IPV p = d + d . Note that EBS92 definethe analogous quantity W x (half-width at x ), where x is defined asa given fraction of the peak flux.Figure 9 shows the averages (top panel) and dispersions (bot-tom panel) obtained for ϑ = 5 ◦ , ◦ , ◦ , ◦ , ◦ , ◦ for dif-ferent i = i max ranging from ◦ to ◦ when i min = 2 . ◦ . Solidlines correspond to FWHM and dashed lines to IPV line width mea-surements. For most maximum inclination angles, for fixed i max ,the dispersion of the FWHMs decreases with increasing ϑ . In par-ticular, the dispersion of FWHM for the smallest launching angle issystematically larger than for all others. We can see that the generaltrend is lower dispersion for higher ϑ , except for the smaller i max ,where σ ϑ =45 ◦ departs from it. We also compared these results to a simple model which as-sumes g ( i ) = FWHM ( i ) = i ( 1000 km s − ), merely to see howthe dispersion in log(FWHM) behaves for a model with knownvariation of FWHM with i . The dotted line in Figure 9 correspondsto this simple model. For the i min considered in the figure, thismodel departs from the observational results at any ϑ . We foundthat when i min = 7 . ◦ , the model approximately matches the resultfor ϑ = 5 ◦ in the range ◦ . i max . ◦ . In the general case,it can be inferred that a more sophisticated model is needed. Sucha model probably has to include information about the launchingangle.The dispersions obtained for the FWHMs are increasing func-tions of the parameter i max , although they show a mild decreasingtrend at i max & ◦ . Similarly, the dispersions of the IPVs are in-creasing functions of i max . Also included in Figure 9 is the 0.08dex dispersion line reported by Fine et al. (2010) as the observa-tional upper limit on the dispersion measured from their sample.We can see that as the torus half-opening angle (measured from thepolar axis, and represented by i max ) increases above about ◦ , thewind launch angles required to match the Fine et al. (2010) con-straints are increasingly larger.Figure 10 shows the allowed region in the i - ϑ plane whenanalysing the dispersion of FWHMs (left panel) and of IPV25s(right panel). In both cases, i min = 2 . ◦ . Our results give, withinthe ϑ < ◦ range allowed by the MHD solutions, a maximumhalf-opening angle of about ◦ , above which no wind launch an-gle matches the observations. This maximum torus half-openingangle has a somewhat different behaviour if the IPVs are consid-ered, reaching a maximum at ϑ ∼ ◦ and declining for larger ϑ . However, note that the “absolute” maximum is similar in bothcases.In section 5 we showed that the profiles obtained for casescorresponding to larger inclination and small launching angle haddouble-horned profiles and mentioned that this contradicts obser- We denote σ FWHM ( ϑ = x ◦ ) by σ ϑ = x ◦ . vational results. The analysis presented in this section shows thatsuch combinations are in fact ruled out. As mentioned above, Mor et al. (2009) adopted the more detailedexpression for the escape probability proposed by N08. In thatmodel, the torus is clumpy, consisting of optically thick clouds andthe quasar is obscured when one of such clouds is seen along theLOS. The torus is characterized by the inner radius of the clouddistribution (set to the dust sublimation radius, R d , that depends onthe grain properties and mixture) and six other parameters. Equa-tion (3) in Mor et al. (2009) provides the weighting factor that weused in our evaluation: P esc ( i ) = exp (cid:20) − N exp (cid:18) − ( π/ − i ) σ (cid:19)(cid:21) (55)where N is the mean number of clouds along a radial equatorialline and σ is the torus width parameter (analogous to its opening an-gle). Implicitly, it is assumed that the disc and the torus are aligned.In their Fig. 6, Mor et al. (2009) present the torus parameter distri-butions for their sample, and from there it is clear that the distri-bution of the two parameters we need to input in Eq. 55 (namely, N and σ ) are very broad. For completeness, we have reproducedin Table 2 the minimum, mean and maximum values of the two pa-rameters, taken from Mor et al. (2009). Note that within this model,we only have i max = π/ . The resulting distribution of the disper-sions with the launching angle are presented in Figure 11, whereeach line corresponds to a given combination of σ and N . For clar-ity, in the figure we excluded combinations such that at least one ofthe parameters takes its minimum value, as such combinations yieldlines farther away from the observed upper limit dispersion.For ϑ . ◦ , all cases depart significantly from the Fineat al. result. For ϑ & ◦ , most cases remain highly incom-patible with the Fine et al. results, but the curves are closer tothem when N = max( N Mor+090 ) , with σ = max( σ Mor+09 ) and σ = ¯ σ Mor+09 . That motivated us to look for combinations of theseparameters such that the results based on Mor et al. (2009) modelmatch the Fine et al. limit, at least for some launch angles. Adopting σ = ¯ σ Mor+09 and increasing N , one finds that for N ∼ (thelikely uppr limit, according to N08), all dispersions correspond-ing to ϑ & ◦ are below the Fine et al. boundary and the resthave decreased in a similar amount (the effect is that the curve hasalmost rigidly moved down). When N ∼ , only dispersionscorresponding to ϑ . ◦ are above the Fine et al. boundary. If,instead, σ = max( σ Mor+09 ) is adopted, for N ∼ . the dis-persion corresponding to ϑ ∼ ◦ is already below the Fine et al.limit, and for N ∼ . , all dispersions for the cases ϑ & ◦ arebelow that line. On the other hand, keeping N = max( N Mor+090 ) and increasing σ does not lead to an improvement.Note that Mor et al. (2009) found P ( i = 50 ◦ ) ≃ and P ( i = 70 ◦ ) < when N and σ were set to their mean val-ues, and, based on that, suggested that the inclination angle fortype-1 objects should lie in the range ◦ − ◦ . However, the au-thors found that for the case in which all the parameters but thetorus width are set to their mean values, the escape probability fallsrapidly if σ > ◦ . Our results indicate that the parameter N is important when considering the dispersions of the line widths.In effect, as mentioned above, the only curves that are relativelyclose to the Fine et al. (2010) constraints correspond to the case N = max ( N Mor+090 ) and to have a better match larger N val- c (cid:13) , 1–18 HD Disc Winds and Line Width Distributions Out[553]= i max @ ° D < l og H L i n e w i d t h (cid:144) k m s - L > i max @ ° D D i s p @ l og H L i n e w i d t h (cid:144) k m s - L D FWHMIPV
Model J = J = J = J = J = J = Figure 9.
Averages (top panesl) and dispersions (bottom panel) of log(
FWHM ) (solid lines) and log( IPV25 ) (dashed lines) evaluated for different launchingangles, using P ( i ) = 1 . The minimum viewing angle is i min = 2 . ◦ . Included is the plot (dotted curve) of the dispersion for a model of the form g ( i ) = FWHM ( i ) = i (1000 km s − ) (see text). The dashed horizontal line plotted together with the dispersions corresponds to the Fine et al. (2010) results,its meaning is commented below. Table 2.
Minimum, mean and maximum values of the N and σ torus parameters from the Mor et al. (2009) sample.Parameter Minimum Mean Maximum N σ [ ◦ ] 15 34 57c (cid:13) , 1–18 L. S. Chajet and P. B. Hall J @ ° D i M a x @ ° D FWHM J @ ° D i M a x @ ° D IPV25
Figure 10.
Contour plot of the standard deviation of the FWHM vs launching and inclination angles. Only the contours within the region that matches theFine et al. results are shown.
Out[1693]= æ æ æ æ æ æà à à à à à ì ì ì ì ì ì ò ò ò ò ò òô ô ô ô ô ôç ç ç ç ç ç J @ ° D D i s p @ l og H F W H M (cid:144) k m s - L D ç LE10 - Fully Twisted ô LE10: Tilt - only ò Mor + Σ=Σ
Max , N = N ì Mor + Σ=Σ
Max , N =< N > à Mor + Σ=<Σ> , N = N æ Mor + Σ=<Σ> , N =< N > Figure 11.
Dispersions of log(
FWHM ) using P ( i = π/ − β ) as given by Eq. 55. Here, i min = 2 . ◦ , i max = 84 ◦ . Each of the lines obtained usingthe Mor et al. (2009) prescription represents a different combination of σ and N . Also included is a line corresponding to dispersions calculated assuming atilt-only warped disc (see next section). The dashed line corresponds to the Fine et al. (2010) results. ues are needed. In that case, the σ values should still be close to orlarger than the mean from the Mor et al. (2009) sample.These results were obtained from a set of profiles correspond-ing to both mass and luminosity fixed, whereas Fine et al. (2010)and Mor et al. (2009) samples involve a range of masses and lumi-nosities. However, as already mentioned, Fine et al. (2010) foundthat the dispersion in C IV linewidths essentially does not de-pend on luminosity. As can be seen from their Figure 2, the IPV linewidth measurements are bound by . . M BH /M ⊙ . and . . L/L
Edd . . Based ion that, we then antici-pate that our results would not be strongly affected by consideringdifferent masses and/or luminosities. c (cid:13)000
Edd . . Based ion that, we then antici-pate that our results would not be strongly affected by consideringdifferent masses and/or luminosities. c (cid:13)000 , 1–18 HD Disc Winds and Line Width Distributions As mentioned in the Introduction, the model of Lawrence & Elvis(2010, LE10 hereafter) replaces the torus by a warped disc. In thissection we explore whether we can infer new constraints on theBLR or on the parameters of warped discs by applying a restrictedset of such warped disc models. Briefly, we evaluate the unob-scured solid angle distribution as a function of observer inclination i , dC ( i ) , calculated for arbitrary disc tilt angle θ . Then, we restrictour attention to the subset within our constraint i < π/ − β , usingthe calculated unobscured solid angle distribution to determine theprobability of the object being unobscured. Finally, we apply thatprobability to our emission line profiles, in a way analogous to thatemployed with the Fine et al. (2010) model.LE10 studied the fraction of type 2 AGN, f ≈ . among allAGN, and proposed a framework to account for it. They assumedthat randomly directed infalling material at large scales would pro-duce a warped disc at smaller scales, where it eventually aligns withthe inner disc. They analysed both fully twisted and tilt-only cases(explained in detail below) under that assumption and showed thatfully twisted discs can not reproduce the observed f . Models as-suming tilt-only discs, on the other hand, match the observed f .A warped disc can be analysed as a series of annuli, each char-acterized by its radius and the two angles θ ( r ) (the angle betweenthe spin axes of the annulus and the inner planar disc, i.e., the tilt)and φ ( r ) (the angle of the line of nodes measured with respect toa fixed axis on the equatorial plane, i.e., the twist). Thus, a fullytwisted disc corresponds to φ ( r ) = 0 , φ ( r + δr ) = 2 π , δr ≪ r anda tilt-only disc corresponds to the case of constant φ ( r ) . Each of thetwo warp modes can be associated to a covering factor C depend-ing on the misalignment, and the distribution of covering factorscan then be inferred from the probability distribution of the mis-alignment. Conversely, knowing the distribution of covering fac-tors, the probability distribution of misalignments can be evaluated.This is the approach we take below. Note that LE10 calculated theazimuthally integrated covering factor while we study the coveringfraction as a function of the azimuth.Consider first the tilt-only case. In Appendix A we derive theexpression for the differential covering factor dC ( i ) correspond-ing to this case. We estimated the unobscured fraction P ( i ) = dC ( i ) / π sin( i ) as a function of the tilt angle and found that con-sidering a random distribution in solid angle of θ up to θ max = π yields results incompatible with our line profiles. This is becausethat case corresponds to i max = π/ for the Fine et al. case inall our diagrams. In a random distribution of incoming orientationswith θ max = π , for every case of orientation θ there is a corre-sponding case with orientation π − θ (statistically speaking), whichmeans that this model is identical to the case i m ax = pi/ for ourpurposes. A variant of that model, with a random distribution insolid angle of θ up to θ max = π/ was also analysed. Using equa-tion A7 for the probability and equations 53 and 54 we performedthe same analysis applied in the Fine at al. case. Note that the onlyangle to be considered in this case is i max = π/ , so we also ruledout this model. We have included in Figure 11 the resulting dis-persions for this model. The dispersions are far from the Fine et al.(2010) observational results for any launch angle considered.An analogous analysis was performed for the full twist case,with θ π/ . The probability of being unobscured for ran-dom inclinations is given by P ( i ) = cos( i ) and, again, the regionin the i − ϑ plane is the same obtained using other prescriptions. In this paper, we combined an improved version of the MC97 discwind model (Hall et al. 2012, in preparation) with the MHD driv-ing of EBS92. We analysed how the resulting line profiles dependon different parameters of the model. In particular, we studied howthe observing angle i and the wind launch angle ϑ affect the emis-sion line profiles. We found that for fixed ϑ all profiles are slightlyasymmetric, with more asymmetric profiles for smaller inclination.For a given launching angle, less inclined objects have a larger frac-tion of their flux corresponding to motion towards the observer,therefore their profiles are less symmetric. For fixed i , the angle tobe considered is ϑ > ϑ , i.e the angle at which a wind that startedat ϑ intercepts the base of the emitting region. Two different casescan be found. If ϑ > i , wind velocity projections are mostly to-wards the observer, with red wings increasingly important for thecases ϑ i .Our main conclusion is that the shape of the line profiles, theirFWHMs and shift amounts (whether red or blue) with respect to thesystemic velocity depend not only on the viewing angle but also onthe angle (with respect to disc plane) at which the outflow starts.In fact, the relevant quantity is neither of the angles, but a combi-nation of them. This is a consequence of how the model has beenconstructed. In effect, the optical depth expression includes a de-pendance on the wind contribution, which ranges from v p sin( ϑ + i ) when φ = 0 , to v p sin( ϑ − i ) when φ = π . The launch angle param-eter, although included in the models, has been less explored in theliterature. Note, however, that MC97 have reported that their C IV line profiles do not strongly depend on ϑ ( λ in their notation).This difference could be due to our use of EBS92 streamlines in-stead of MC97 streamlines, or it could be due to our more rigorouscalculation of L ν as compared to MC97. Similar results oppositeto our findings are reported by Flohic et al. (2012), in their study ofBalmer emission lines.Using as a constraint the observational results obtained byFine et al. (2010) for the C IV lines in their sample, we found thatthe allowed region in the i − ϑ plane has an upper limit that de-pends on the torus half-opening angle, i max . For instance, a launchangle ϑ ∼ ◦ is only allowed for torus half-opening angle . ◦ ,while ϑ is . ◦ for i max ∼ ◦ . We found that the maximumtorus half-opening torus angle that is compatible with the obser-vations is about ◦ . Considering a model that replaces the toruswith a tilt-only warped disc, formed by the alignment at smallerdistances of material falling at large distances from random direc-tions, yields no difference in the resulting allowed region of theinclination-launch angle plane.These results were obtained for a single mass and luminosity,as opposed to the Fine et al. (2010) and Mor et al. (2009) resultswhich were obtained from datasets spanning an order of magnitudein both parameters. However, as mentioned in section 6, Figure 2of Fine et al. (2010) indicates a mild dependence of the dispersionof the linewidth on both these parameters. Thus, we expect thatsimulations for different masses and luminosities (currently beingundertaken) will yield similar results to those reported here. Futurework will also consider applying the model to other high ionizationlines, such as Si IV , as well as low ionization lines, such as Mg II .Other properties of the observed line profiles that could bemeasured and compared against the model include the line asym-metry (e.g. Whittle 1985, EBS92) and the cuspiness at x , C x , re-lated to the kurtosis and proposed by EBS92, where x is definedas a given fraction of the peak flux. In their analysis of Balmerlines, Flohic et al. (2012) have also studied other line profile mo- c (cid:13) , 1–18 L. S. Chajet and P. B. Hall
Figure 12.
Representation of a tilt-only disc (warped but not twisted). The outermost disc (shown here as an annulus) is tilted at an angle θ with respect tothe inner disc. That is also the angle between the spin axes of the annulus and the inner disc. The transition from outer disc to inner disc occurs over a rangeof radii in reality, but is shown here happening at a single radius for convenience. Adopting the azimuth of the ascending node as φ = 0 , at azimuth φ ′ theobscuration from the outer disc extends an angle θ ′ above the inner disc given by sin( θ ′ ) = sin( θ ) sin( φ ′ ) . The equivalent polar angle i ′ = π − θ ′ is givenby cos( i ′ ) = sin( θ ) sin( φ ) . At each azimuth, the light purple shading represents the obscuration due to the tilted disc (which takes the form of two wedges,each of maximum width θ ). The light shadow represents the obscuration due to the tilted disc at all azimuths. The region at azimuths φ φ ′ has a darkershadow to emphasize the angles intervening in the θ ′ calculation. ments. In addition to the FWHM, they considered the full width atquarter maximum (FWQM) as well as the asymmetry and kurto-sis indexes (A.I. and K.I. respectively), and centroid shift at quar-ter maximum ( v c (1 / ) as defined by Marziani et al. (1996). Thepresent work does not include the treatment of resonance scatter-ing of continuum photons or general relativistic (GR) effects. Intheir improvement of the MC97 and Chiang & Murray (1996) mod-els, Flohic et al. (2012) found that when relativistic effects are in-cluded, the line profiles become skewed to the red. For a givencombination of parameters, the line wings and centroids are in-creasingly redshifted with decreasing inclination. The amount ofredshift is also a decreasing function of the inner radius of theline-emitting region, that satisfies r min > r g , where r g is thegravitational radius. Although their results correspond to Balmer(i.e., low ionization) lines, we expect that high ionization linessuch as C IV may exhibit a similar or stronger response if theGR effects were to be considered, as C IV is expected to be emit-ted predominantly at smaller radii. However, our analysis and thatof Flohic et al. (2012) utilize different velocity fields, and there-fore a comparison of our results and theirs is perhaps not straight-forward. Giustini & Proga (2003) treat the absorption line depen-dence on wind geometry but we have not found in the literaturea similar study for emission lines. The relativistic MHD case hasbeen studied by several authors, often in the context of jet launch-ing and collimation and in relation to several different astrophys- ical environments, such as AGN, microquasars, young stellar ob-jects, pulsars and gamma-ray burst. The problem has been consid-ered both in the steady (e.g. Camenzind 1986; Chiueh et al. 1991;Li et al. 1992; Contopoulos 1994, 1995; Fendt & Greiner 2001;Heyvaerts & Norman 2003) and in the time-dependent regimes(e.g. Koide et al. 1999; Porth & Fendt 2010). In the relativis-tic MHD framework, the line formation problem has been con-sidered in the X-ray range in relation to the iron K-line (e.g.M¨uller & Camenzind04 2004). However, to the knowledge of theauthors, the combination with a relativistic version of MC97 hasnot been yet explored in the literature. We consider that as one ofthe possible future lines of work to pursue. ACKNOWLEDGMENTS
LSC and PBH acknowledge support from NSERC, and LSC fromthe Faculty of Graduate Studies at York University. PBH thanks theAspen Center for Physics (NSF Grant
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APPENDIX A: UNOBSCURED SIGHTLINEDISTRIBUTION FOR TILT-ONLY DISCS
Below we outline the evaluation of dC ( i ) for arbitrary disc tilt an-gle θ , to within the numerical factor required so that the total proba-bility in a given situation is unity. Recall that in the following anal-ysis the covering fraction is a function of the azimuth, while it isazimuthally integrated in the calculations of LE10.Define the line of nodes of the tilted outer disc relative to theinner disc to be at φ = 0 . Then at each φ , the obscuration fromthe outer disc extends an angle θ ′ above the inner disc given by sin θ ′ = sin( θ ) sin( φ ) above the inner disc (sine rule for sphericalright triangles); see Figure 12.The equivalent polar angle i ′ = π − θ ′ is given by cos( i ′ ) =sin( θ ) sin( φ ) . Solving the latter equation for the maximum unob-scured φ at a given i , φ max ( i, θ ) , yields φ max ( i, θ ) = arcsin[cos( i ) / sin θ ] , (A1)where < φ max ( i, θ ) < π .For < θ < π , there is an unobscured polar cap (at i < π − θ ) and a region where obscuration increases from 0% at i = π − θ to 50% at i = π . The differential solid angle in each region is: dC (cid:16) i < π − θ (cid:17) = 2 π sin( i ) di (A2) dC (cid:16) π − θ < i < π (cid:17) = sin( i ) di π + 2 Z φ max ( i,θ ) φ =0 dφ ! = sin( i ) di (cid:18) π + 2 arcsin (cid:20) cos( i )sin θ (cid:21)(cid:19) (A3)For π < θ < π , there is a polar cap of complete obscuration ( i < θ − π ) and a region where obscuration decreases from 100%at i = θ − π to 50% at i = π . The differential solid angle in thepartially unobscured region is: dC (cid:16) π θ < i < π (cid:17) = sin( i ) di π − Z φ max ( i,θ ) φ =0 dφ ! = sin( i ) di (cid:18) π − (cid:20) cos( i )sin θ (cid:21)(cid:19) (A4)For < θ < π , large i values are underrepresented, while small i values are underrepresented for π < θ < π . At θ = π , the halfof the hemisphere with < φ < π is obscured, which leads to auniform 50% reduction in the probability of observing the quasaralong every sightline as compared to the no-obscuration case. A1 Random orientations with θ π . Here we analyse in more detail a restricted variant of the modelwhere instead of a fixed tilt angle θ , a distribution of such anglesrandomly distributed in solid angle from θ π is considered.Combining equations A2 and A3 we obtain: dC ( i ) = sin( i ) di "Z π − iθ =0 π sin θdθ + Z π θ = π − i (cid:18) π + 2 arcsin (cid:20) cos( i )sin θ (cid:21)(cid:19) sin θdθ , (A5) c (cid:13) , 1–18 L. S. Chajet and P. B. Hall which becomes dC ( i ) = sin( i ) di " π − π cos (cid:16) π − i (cid:17) + 2 Z π θ = π − i arcsin (cid:20) cos( i )sin θ (cid:21) sin θdθ (A6)From dC ( i ) we define P ( i ) , the probability of being unob-scured, as P ( i ) = dC ( i )2 π sin( i ) . (A7)In the text, we comment on the results of combining the latterexpression with equations 53 and 54 to perform the same analysisapplied in the Fine at al. case.This paper has been typeset from a TEX/ L A TEX file prepared by theauthor. c (cid:13)000