Spiral arms in broad-line regions of active galactic nuclei II. Loosely wound cases: Reverberation properties

There is growing evidence that broad-line regions (BLRs) in active galactic nuclei (AGNs) have regular substructures, such as spiral arms. This is supported by the fact that the radii of BLRs measured by reverberation mapping (RM) observations are generally consistent with the self-gravitating regions of accretion disks. We showed in Paper I that the spiral arms excited by the gravitational instabilities in these regions may exist in some disk-like BLRs. Here, in the second paper of the series, we investigate the loosely wound spiral arms excited by gravitational instabilities in disk-like BLRs and present their observational characteristics. We solve the governing integro-di ﬀ erential equation by a matrix scheme. The emission-line proﬁles, velocity-delay maps, and velocity-resolved lags of the BLR spiral arms are calculated. We ﬁnd that the spiral arms can explain some of the phenomena seen in observations: (1) di ﬀ erent asymmetries in the emission-line proﬁles in the mean and rms spectra; (2) complex subfeatures (incomplete ellipse) in some velocity-delay maps, for example that of NGC5548; and (3) the short timescales of the asymmetry changes in emission-line proﬁles (rms spectra). These features are attractive for modeling the observed line proﬁles and the properties of reverberation, and for revealing the details of the BLR geometry and kinematics.


Introduction
As the prominent features in the UV/optical spectra of active galactic nuclei (AGNs), the broad emission lines with velocity widths of ∼1000 -20000 km s −1 originate from the broadline regions (BLRs) photoionized by the continuum radiation from the central accretion disks around supermassive black holes (SMBHs).The physics of BLRs (e.g., the geometry, kinematics, mass distributions, and photoionization properties), which determines the profiles of broad emission lines, is not only related with the origin and evolution of the materials in the central regions of AGNs, but also closely connected with the measurement of BH masses in reverberation mapping (RM, e.g., Blandford & McKee 1982;Peterson 1993).It makes BLRs one of the core topics in AGN researches.
RM is a technique to probe the geometry and kinematics of BLRs and to measure the masses of SMBHs in AGNs.It has been successfully applied to more than 100 objects in the past decades (e.g., Peterson et al. 1998;Kaspi et al. 2000;Bentz et al. 2009;Denney et al. 2009;Barth et al. 2011;Rafter et al. 2011;Du et al. 2014Du et al. , 2018a;;Fausnaugh et al. 2017;Grier et al. 2017a;De Rosa et al. 2018;Rakshit et al. 2019;Hu et al. 2021;Yu et al. 2021;Bao et al. 2022).RM measures the delayed response of broad emission lines with respect to the varying continuum emission.Due to the limits of spectral resolution and flux calibration precision, most RM campaigns in the early days focused on the average time lags (τ Hβ ) of Hβ emission line (e.g., Peterson et al. 1998;Kaspi et al. 2000).In combination with the velocity widths (V Hβ ) of Hβ lines, the masses of SMBHs can be formulated with Hβ R Hβ /G, where R Hβ = cτ Hβ is the emissivity-weighted radius of BLR, c is the speed of light, G is the gravitational constant, and f BLR is a parameter called "virial factor", which is controlled by the BLR geometry and kinematics.Therefore, the accuracy of BH mass measurement is directly related with the understanding of BLR physics.Furthermore, with the improvement of flux calibration and spectral resolution in recent years, velocity-resolved RM, rather than only measuring an average time lag, has been gradually performed to more and more objects.It aims to measure the time lag as a function of velocity (e.g., Bentz et al. 2008Bentz et al. , 2010a;;Denney et al. 2010;Du et al. 2016bDu et al. , 2018b;;Pei et al. 2017;De Rosa et al. 2018;Hu et al. 2020a,b;Brotherton et al. 2020;Lu et al. 2021;U et al. 2022;Bao et al. 2022) or, more importantly, to reconstruct the "velocity-delay maps" (also known as transfer functions) of BLRs by model-independent methods such as maximum entropy method (e.g., Bentz et al. 2010b;Grier et al. 2013;Skielboe et al. 2015;Xiao et al. 2018a,b;Brotherton et al. 2020;Horne et al. 2021) or to constrain the BLR parameters by the Bayesian modeling through Markov Chain Monte Carlo (MCMC, e.g., Pancoast et al. 2012Pancoast et al. , 2014;;Grier et al. 2017b;Williams et al. 2018;Li et al. 2018;Villafaña et al. 2022).The general geometry and kinematics of the BLRs (e.g., disk-like, inflow, or outflow) in dozens of AGNs have been successfully revealed (see the aforementioned references).
Systematic researches on the inhomogeneity and substructures in BLRs are relatively scarce, however, their signs in observations have appeared gradually.Three pieces of evidence imply the existence of the inhomogeneity and sub-structures in BLRs.(1) Many AGNs show complex emission-line profiles, even with multiple wiggles or small peaks, rather than symmetric profiles or simply asymmetric profiles with a little stronger red or blue wings in their emission lines (e.g., the line profiles of Mrk 6, Mrk 715, or NGC 2617 in the Appendix of Du et al. 2018b).It indicates that the BLR gas distributions in those objects should be more complex than previously thought.(2) There is a well-known phenomenon that the line profiles in the mean and rms spectra of RM are commonly different for a same object (e.g., Peterson et al. 1998;Bentz et al. 2009;Denney et al. 2009; Barth et al. 2013;Fausnaugh et al. 2017;Grier et al. 2012;Du et al. 2018b;De Rosa et al. 2018;Brotherton et al. 2020).The profiles of the emission lines in rms spectra represent the geometry and kinematics of the gas that has response to the continuum variations and only is a portion of the total BLR gas.The differences between the mean and rms spectra suggest the gas inhomogeneity in BLRs.(3) More importantly, the velocity-delay maps of some objects (e.g., NGC 5548) have shown complex features (e.g., incomplete ellipse, bright strips) in comparison with the simple disk, inflow, or outflow models.They are probably the evidence of the BLR inhomogeneity and sub-structures (e.g., Xiao et al. 2018b;Horne et al. 2021).
The radii of BLRs measured by RM mostly span from 10 3 R g to 10 5 R g for different objects, where R g = 1.5×10 13 M 8 cm is the gravitational radius and M 8 = M • /10 8 M is the SMBH mass in unit of 10 8 solar mass (Du et al. 2016a).Such a range of radius is consistent with the self-gravitating region of accretion disk (e.g., Paczynski 1978;Shlosman & Begelman 1987;Bertin & Lodato 1999;Goodman 2003;Sirko & Goodman 2003).Besides, a number of objects (e.g., Arp 151,3C 120,NGC 5548) show clear RM signatures of Keplerian disks (Bentz et al. 2010b;Grier et al. 2013;Xiao et al. 2018b;Horne et al. 2021).The heuristic idea that the origin of BLRs is related with the self-gravitating regions of accretion disks was initially discussed by Shore & White (1982), and was further theoretically studied in the subsequent works (e.g., Collin-Souffrin 1987;Collin-Souffrin & Dumont 1990;Dumont & Collin-Souffrin 1990a,b).Although the detailed physics in the self-gravitating region is still far from fully understood, the existence of spiral arms may be a natural consequence resulted from the gravitational instabilities in this region (e.g., Lodato 2007).
On the other hand, the mass ratio of standard accretion disk (Shakura & Sunyaev 1973) to SMBH can be expressed as 0.04α −4/5 0.1 M 6/5 8 Ṁ 7/10 r 5/4 4 (or 0.7α −4/5 0.1 M 6/5 8 Ṁ 7/10 r 5/4 5 depending on the typical radius) if the disk extends to the scale size of BLR, where Ṁ = Ṁ• c 2 /L Edd is the dimensionless accretion rate, Ṁ• is the mass accretion rate, L Edd = 1.5 × 10 46 M 8 erg s −1 is the Eddington luminosity of solar composition gas, α 0.1 = α/0.1 is the viscosity parameter, and r 4 = R out /10 4 R g (or r 5 = R out /10 5 R g ) is the outer radius.This ratio is generally similar to the diskto-star mass ratios in protoplanetary systems, which commonly possess spiral arm structures (e.g., Andrews et al. 2013;Dong et al. 2018).This also leads to the possibility that BLRs can host spiral arms.
Therefore, it is important to investigate the spiral arms in BLRs and their potential characteristics in observations.Horne et al. ( 2004) calculated the velocity-delay map of a photoionized disk with two spiral arms mathematically without introducing any precise physics (through "twisting" the elliptical orbits).Gilbert et al. (1999), Storchi-Bergmann et al. (2003), Schimoia et al. (2012), andStorchi-Bergmann et al. (2017) assumed an analytical form of the spiral arms and explained the double-peaked profiles of the broad emission lines in AGNs, but similarly do not include any dynamical physics.As the first paper of this series, Wang et al. (2022) introduced the density wave theory of spiral galaxies (e.g., Lin & Shu 1964, 1966;Lin et al. 1969), which applies to self-gravitating disks (Goldreich & Tremaine 1979), into the research of BLRs for the first time (hereafter Paper I). Paper I explores the possibility of density waves in BLRs through discussing their physical conditions, and focuses on the simplest cases of tight-winding arms with short wavelengths and small pitch angles (adopting the formalism of tight-winding approximation).However, the loosely wound spiral arms have more significant features in line profiles or RM signals relative to the tightly wound cases (see more details in Paper I or in the following sections of the present paper).Hence, it is crucial to investigate the loosely wound spiral arms in BLRs and their characteristics in observations.
As the second paper of this series, here we calculate the surface density distributions of loosely wound spiral arms in a numerical manner without the tight-winding approximation, and their corresponding emission-line profiles, velocity-delay maps, and velocity-resolved lags.Comparing with Paper I, we adopt more general radial distributions of the BLR surface density and sound speed, which are assumed as power laws with free indexes.This is a natural extension of Paper I. The paper is organized as follows.In Section 2, we briefly introduce the density wave model and the numerical method.Some fiducial modes (arm patterns) and their observational signals (in emission-line profiles, velocity-delay maps, and velocity-resolved lags) for different azimuthal angles of the line of sight (LOS) are provided in Sections 3 and 4. We discuss and compare the models with the observations in Section 5. A brief summary is given in Section 6.

Theoretical Formulation
We adopt the density wave formalism in Lin & Lau (1979) and the numerical method in Adams et al. (1989) to calculate the spiral arms.The perturbation equations and numerical method in Adams et al. (1989) apply to both tightly and loosely wound arms, and can also derive one-armed density wave (azimuthal wave number m = 1).Details of the formula deduction and numerical procedures can be found in these papers and the references therein.For completeness, we briefly describe the key points in this section.The model in the present paper assume the general geometry of BLR is disk-like.It may apply to the objects which show clear features of Keplerian disks in their RM signals (e.g., Arp 151, 3C 120, NGC 5548 in Bentz et al. 2010b;Grier et al. 2013;Xiao et al. 2018b;Horne et al. 2021).

Perturbation Equations
Here we adopt the linear normal-mode formalism in Adams et al. (1989) (also refer to the more recent work of Chen et al. 2021).We use the cylindrical coordinates (R, ϕ, z).In a thin disk, the continuity equation and the motion equations in radial and azimuthal directions read and respectively, where u(R, ϕ, t) and υ(R, ϕ, t) are the radial and azimuthal components of velocity, σ(R, ϕ, t) is the surface density, V 0 is the gravitational potential of SMBH, ψ is the gravitational potential of disk, h is the enthalpy defined by dh = a 2 dσ/σ (governed by the thermodynamic property of gas), and a is the sound speed.It should be noted that the viscosity is neglected here.The m-fold linear perturbations of the equilibrium state are considered.The variables (u, υ, σ, ψ, h) can be expressed as −mϕ) , where F is u, υ, σ, ψ, or h.The subscript 0 represents the variables in the equilibrium state, and 1 represents the perturbation components.ω = mΩ p − iγ is the complex eigenfrequency.Its real part represents the pattern speed Ω p of the rotating arms, and the imaginary part gives the exponential growth rate γ of the density waves.Then, the linearized equations can be formulated as and where Ω(r) is the rotation curve and κ is the epicyclic frequency.
The perturbation ψ 1 of the gravitational potential can be given by the integral of the surface density where R in and R out are the inner and outer radius of the disk.
Combining the above equations, we can obtain the integrodifferential equation of ψ 1 and σ 1 where and ν = (ω − mΩ)/κ is the dimensionless frequency.Eqn ( 8) is the governing integro-differential equation of the density wave.Solving this equation numerically, if given the boundary conditions, can provide the perturbation of the surface density σ 1 .

Rotation Curve
For the SMBH and BLR disk system, the rotation curve has three components (see Adams et al. 1989) which come from the central SMBH, the unperturbed disk, and the pressure respectively.The disk component can be expressed as Given the rotation curve, the epicyclic frequency can be written as As is well known, the elliptic integral in the calculation of disk potential has singularity (e.g., Adams et al. 1989;Laughlin et al. 1997;Huré 2005).Some methods can handle this singularity in specific cases, e.g., the splitting method in Huré et al. (2007).
Here we follow Adams et al. (1989) and use the softened gravity method to calculate the disk potential.A softening term of η 2 R 2 is added into the square root of the denominator at the singular points.We adopt η = 0.1 in the calculation of rotation curve, and have checked that the deduced Ω(R) is similar to that obtained by the splitting method in Huré et al. (2007).For Eqn (7), we use a smaller value of η = 0.01 similar to Chen et al. (2021).We have also checked that the detailed values of the softening parameter η do not significantly change the spiral arms, emissionline profiles, or velocity-delay maps in the following sections1 .However, it should be noted that the softening parameter η may influence the growth rate of density wave (e.g., Laughlin et al. 1997), though it may not significantly change the spiral pattern (particularly, away from the corotation or Lindblad resonances, where ν = 0 or ±1).We mainly focus on the spiral pattern and the corresponding RM characteristics in the present paper.The influence of η to the growth rate will be discussed in future.

Boundary Conditions
The origin of BLRs is still under debate (e.g., Czerny & Hryniewicz 2011;Wang et al. 2017).Although the emissivityaveraged radii of BLRs (R BLR ) have been measured in more than 100 AGNs by RM campaigns (see, e.g., Bentz et al. 2013, Du et al. 2015, Du & Wang 2019, Grier et al. 2017a), the inner/outer radii of BLRs and their corresponding boundary conditions have large uncertainties so far.However, the radii of dusty tori in some AGNs have been successfully measured, which give us strong constraints to the outer radii of their BLRs.Infrared RM campaigns found a relation between the radius for the innermost dusty torus and the optical luminosity, which is written as R torus ≈ 0.1L 0.5 43.7 pc (e.g., Minezaki et al. 2019).L 43.7 is the V-band luminosity in units of 10 43.7 erg s −1 .We adopt a typical bolometric correction factor of 10 (from bolometric to V-band luminosity).We set the outer radius of BLR at the inner edge of dusty torus in our calculation (R out = R torus ).Considering R torus /R BLR ≈ 3 ∼ 7 (Du et al. 2015;Minezaki et al. 2019), we adopt R out /R in = 20, 50, 100 in the following calculations in order to ensure that the radial range of our calculation is wide enough, and to check the influence of different R out /R in to the spiral arms.
We adopt the same boundary conditions as in Adams et al. (1989) for simplicity, but keep in mind that the detailed BLR boundary conditions are still unknown.At outer boundary, the Lagrangian pressure perturbation is required to vanish, which means the confining pressure from the external medium (probably the gas in torus) is a constant.At the inner boundary, we assume the velocity perturbation u 1 = 0, so that the radial component of the velocity perturbation vanishes at the inner boundary.Inner and outer boundary conditions can be verified by comparing the arm patterns and the corresponding emission profiles, velocity-resolved lags and velocity delay maps with the RM observations in future.
2.4.Indirect Potential for One-armed Density Wave Adams et al. (1989) considered the influence that the one-armed perturbation makes the center of star be displaced from the center of mass of the protoplanetary system for the first time.We also take this effect into account in our calculation by the same method that incorporate an indirect potential component in Eqn (8) as in Adams et al. (1989).The indirect potential can be expressed as where M disk is the mass of BLR disk.
In the present paper, we adopt the matrix scheme in Adams et al. (1989) for searching the eigenvalues of ω and solving the governing integro-differential equation.The details of the matrix scheme can be found in Adams et al. (1989).We only briefly describe the general idea and some key points here.The integral and differential operators in Eqn (8) can be expressed into matrixes.By introducing the dimensionless surface density perturbation S (R) defined by σ 1 (R) = σ 0 (R)S (R) and dividing the radial axis to N grid in logarithmic space, the integro-differential equation can be reduced to the form of where i, k = 1, ..., N are the indices of the radial grid.The repeated subscript implies summation over its range as the convention in matrix manipulation.The first and last row of the matrix W ik (ω) are determined by the inner and outer boundary conditions.Eqn ( 17) is a homogeneous system with N equations and N unknowns, and has non-zero solutions only if the matrix W ik (ω) has a vanishing determinant which can yield the eigenvalue of ω.The matrix W ik (ω) is a 5th-order function of ω.
To find all of the eigenvalues simultaneously, Eqn ( 17) is rewritten into a 5N × 5N matrix equation where n, l = 1, ..., 5N are indices, W 1 nl and W 2 nl are two matrixes regrouped from W ik (ω) in light of the coefficients of ω with different orders, and S * l is a rearrangement of S k (see its detailed form in Adams et al. 1989 and Appendix B).We can obtain the eigenvalues ω and eigenvectors S by solving this generalized eigenvalue problem.Eqn (18) has 5N eigenvalues, which is corresponding to 5N modes.Most of modes have zero growth rate (imaginary part, see Section 2.1 and Appendix A) and are not physically relevant.We select the lowest order mode with significant growth rate which will be the most global in extent and can be self-excited to become significant.For the calculation efficiency, we use N = 500 in the present paper.

Fiducial Models
Before solving the governing equation, the equilibrium state of the BLR is required.The emissivity distributions of BLRs have been preliminarily reconstructed through BLR modeling in several objects (e.g., Pancoast et al. 2012Pancoast et al. , 2014;;Grier et al. 2017b;Williams et al. 2018;Li et al. 2018), however, the real surface density distributions are still unclear because the reprocessing coefficient distributions are not known.In Paper I, we adopt the polytropic relation as the prescription of the disk.Here, we generalize and assume that the distributions of the surface density and sound speed are power laws, which follow and We use q/2 rather than q as the index of a 0 in order to keep the same manner as Adams et al. (1989).
The stability of a disk can be quantified by the parameter Q = κa 0 /πGσ 0 (Toomre 1964).The disk is stable if Q 1, and very unstable if Q is far smaller than unity.Here we consider a quasi-stable BLR disk with the average value of Q parameter, defined by close to unity.We set Q as a free parameter in the following sections.
In total, the model used here has 7 parameters: the mass of SMBH M • , the mass ratio between disk and SMBH M disk /M • , the dimensionless accretion rate Ṁ , the power law indices p and q, parameter Q, and the ratio of outer and inner radii R out /R in .Among them, M • and Ṁ determine the outer radius, and the other 5 parameters control the pattern of spiral arms (see Adams et al. 1989).Changing M disk /M • is equivalent to adjusting σ 0 .The value of Q determines a 0 if M disk /M • (equivalently σ 0 ) is fixed2 .Our purpose is not to explore the entire parameter space but to demonstrate the observational characteristics for some typical cases of the BLR arms.Comparing with the standard accretion disks (Shakura & Sunyaev 1973), the surface density distributions in self-gravitating accretion disks are proposed to be steeper and p ≈ 1 ∼ 3/2 are always adopted in theoretical works (e.g., Lin & Pringle 1987;Goodman 2003).In addition, the sound speed distributions of self-gravitating disks are probably flatter (q = 0 ∼ 3/4, see, e.g., Goodman 2003;Sirko & Goodman 2003;Rice et al. 2005).We adopt (p = 3/4, q = 3/4) and (p = 3/2, q = 1/2) as two fiducial configurations, which are corresponding to the distributions in standard accretion disk and self-gravitating disk, respectively.We call them Models A and B hereafter (see Table 1).We fix M • = 10 8 M and Ṁ = 1.0, and leave the other parameters (M disk /M • , Q, and R out /R in ) as free parameters.M • and Ṁ determine the outer radius R out .After R out is determined, the parameter R out /R in controls the inner radius.

Spiral Arms with m = 1
Self-regulation (e.g., compression or shocks induced by the gravitational instabilities, see Bertin & Lodato 1999;Lodato & Rice 2004;Lodato 2007) has been proposed to maintain Toomre parameter Q so that it is not far smaller than unity.In the present paper, we do not aim to investigate the detailed self-regulation mechanisms, but simply assume that Q is a little larger than unity (see, e.g., Lodato & Rice 2004).It means the disk is quasi-stable but the instabilities can still be self-excited ( Q = 1.5, 2.0, 2.5).
It is intuitive that the one-armed density perturbation can produce the most significantly asymmetric emission-line profiles and velocity-delay maps.We first calculate the spiral arms of Model A with m = 1.For each set of parameters, there are more than one eigenvalues and solutions (modes).We adopt the mode with the lowest order and significant growth rate because it will be the most global and can grow in a relatively rapid rate (see the eigenvalues in Appendix A).For M disk /M • , it is still difficult to observationally determine its exact values in AGNs, especially for the self-gravitating regions where the BLRs may reside.But as mentioned in Section 1, it is possible to give an rough estimate of M disk /M • from standard accretion disk model (Shakura & Sunyaev 1973), that M disk /M • is in the range of ∼ 0.04 − 0.7 (corresponding to R out from 10 4 R g to 10 5 R g ).Similarly, from the marginally self-gravitating disk model of Sirko & Goodman (2003), M disk /M • in quasars can be as high as a few tenths (see Figure 2 in Sirko & Goodman 2003).Here we select M disk /M • = 0.2 and 0.8 as representatives in the present paper.It should be noted that the disks for Model A and B are both Notes.All of the parameters are in units of R BLR = 33 × (L 5100 /10 44 erg s −1 ) 0.5 lt−days.For the typical SMBH mass M • = 10 8 M and accretion rate Ṁ = 1.0 adopted in the present paper, R BLR = 40.4lt−days.For each case of Model A or B, we calculate the line profiles for two sets of parameters in order to simulate the mean and rms spectra with different widths (see Section 4.1).For the velocitydelay maps, we adopt the same parameters for comparison.
relatively thin with  1).Through comparing the cases with different disk-to-SMBH mass ratios, it is obvious that more massive disks have more loosely wound spiral arms (see more discussions in Section 5.5).In addition, the arms in more massive disks tend to locate in more outer radii.For the cases with the same disk-to-SMBH mass ratio, the arms are wound more loosely if Q are larger (see more discussions in Section 5.5).The influence of R out /R in looks very weak.
We also present the spiral arms of Model B in Figure 2 (the corresponding eigenvalues are also provided in Appendix A).In general, the spiral arms of Model B are more loosely wound than those in Model A. Moreover, similarly, the arms in more massive disks are more loosely wound.If Q is smaller, the spiral arms wind more tightly.The influence of R out /R in is still weak to the primary arms in the outer part of the disk, but the inner part of the disks with larger R out /R in show some weak small arms in the less massive disks.More importantly, in comparison with Model A, the spiral arms of Model B are more "banana"-like (see Adams et al. 1989).From the inside out, the arms in Model B do not extend continuously but show several gaps and wiggles.In contrast, this phenomenon is weaker in Model A. The arms in Model A extend outward more continuously.
Our goal is to investigate the observational characteristics of the loosely wound spiral arms.We focus on the cases with ( Q, M disk /M • , R out /R in ) = (2.5, 0.8, 100) and calculate their profiles of emission lines, the velocity-delay maps, and the velocityresolved lags in the following Sections 4.1, 4.2 and 4.3.

Spiral Arms with m = 2
We also calculate the two-armed density waves (m = 2).The m = 2 spiral arms of Models A and B with M disk /M • = 0.8 and R out /R in = 100 are shown in Figure 3. Similar to m = 1 modes, the m = 2 modes wind more loosely if Q is larger.Comparing with the m = 1 modes, the arms in the m = 2 modes can extend inward to smaller radii.The outer parts of the disks tend to be loosely wound, while the inner parts wind more tightly.In comparison with Model A, the pitch angles of the arms in Model B are larger and the "banana" shape of the arms is more significant.In the following Section 4.2, we also present the velocity-delay maps of the m = 2 spiral arms for the cases of ( Q, M disk /M • , R out /R in ) = (2.5, 0.8, 100).

Emission-Line Profiles
In our models, the surface density distributions are assumed to be power laws (see Section 3.1).However, the emissivities of broad emission lines do not necessarily follow the same rules.The locally optimally emitting clouds (LOC) scenario (e.g., Baldwin et al. 1995;Korista et al. 1997) has been successfully applied to investigate and reproduce the observed flux ratios of the prominent broad emission lines (e.g., Korista & Goad 2000;Leighly 2004;Nagao et al. 2006;Marziani et al. 2010;Negrete et al. 2012;Panda et al. 2018).Its main idea is that, although the BLR gas covers a wide range of physical conditions (e.g., density, ionization parameter), emission lines always tend to emit from their own optimal places (e.g., Baldwin et al. 1995;Korista et al. 1997).Following Paper I, we simply assume that the emissionline emissivity Ξ is a Gaussian function of ionization parameter U of the BLR gas with the form of where of the BLR gas is defined as where Q H is the number of hydrogen-ionizing photons, n H = ρ/m H is the number density, ρ = (σ 0 +σ 1 )/2H = (σ 0 +σ 1 )Ω/2a 0 is the hydrogen density, and m H is the mass of hydrogen.The line profile can be expressed as where λ 0 is the central wavelength of the emission line, υ(R, ϕ) is the velocity of the BLR gas, g(R, υ) is the velocity distribution at R, and n obs is the unit vector pointing from the observer to the source (LOS).
Many RM campaigns have demonstrated that their rms spectra have line widths different (narrower or broader) from the mean spectra (e.g., Peterson et al. 1998;Bentz et al. 2009;Denney et al. 2009;Barth et al. 2013;Fausnaugh et al. 2017;Grier et al. 2012;Du et al. 2018b;De Rosa et al. 2018;Brotherton et al. 2020), which means the responsivity (varying part) of BLR is different from its mean emissivity.For simplicity, to simulate this phenomenon, we simply assume the responsivity has the same form as Eqn ( 22) but with a different set of (µ U , σU ) rather than taking into account the real photoionization processes in our calculation (hereafter we use Ξ to denote both of emissivity and responsivity).Here we investigate two combinations of (µ U , σU ) corresponding to the typical cases that rms spectra are narrower or broader (called Cases I and II).The values of µ U , σU , and the maximum dimensionless surface density S max are listed in Table 2.We select these parameters because on one hand they can demonstrate the line profiles (or velocity-delay maps in the following Section 4.2) at different radii, and on the other hand it is easy for us to simulate the mean and rms spectra with different line widths.We set the maximum value of dimensionless surface density S max to 0.1 and 0.2 for Models A and B, respectively (also in the following Sections 4.2 and 4.3).It should be noted that the actual situations may probably be larger or smaller than these values.More detailed calculations including photoionization models will be carried out in a separate paper in future.

Line Profiles with m = 1
We present the emission-line profiles of single-epoch/mean spectra and rms spectra for the spiral arms of Models A with m = 1, for different azimuthal angles (ϕ los ) of LOS, in Figure 4.The disks are rotating counter-clockwise.The LOS inclination angle only changes the widths of emission lines, we fix the inclination angle to θ los = 30 • in our calculation (θ los = 0 • refers to looking at the disks from the face-on direction).The contribution of the sound speed a 0 is also taken into account by adding a macro-turbulence speed in the direction perpendicular to the disk.For each of Cases I and II, the mock mean and rms spectra are provided as two rows in Figure 4.As expected, the mean spectra are broader than the rms spectra in Case I, and are relatively narrower in Case II.It is obvious that the line profiles are generally double-peaked because the most efficient emitting region resemble a ring-like shape (determined by Eqn ( 22)).The stonger emissivities/responsivities of the spiral arms results in an obvious asymmetry in the line profiles (see Figure 4).Along with the azimuthal angle ϕ los increases from 0 • , the asymmetry of the profiles change between symmetric, blueward, and redward periodically.For some cases, the weaker peaks almost disappear (e.g., ϕ los = 90 • in the first row of Case II).In Case II, the asymmetries caused by the spiral arms are more significant because the µ U parameters are relatively larger and the σU are smaller.More importantly, the asymmetries of the mock mean and rms spectra can be totally different (blueward or redward) even if the LOS are exactly the same (see, e.g., ϕ los = 180 • in Case II).It implies that the spiral arms can naturally produce differentlyasymmetric mean and rms spectra without any further special assumptions.
In Model B, its emissivity/responsivity tends to be distributed in more outer radius (because U ∝ R 3/4 approximately).The emissivities/responsivities of the spiral arms and the corresponding emission-line profiles for Model B in Cases I and II are shown in Figure 5.The "banana"-like distributions of the spiral arms in Model B (see Section 3.2 and Figure 2) still make the emission-line profiles significantly asymmetric.Compared with Model A, Model B has relatively less asymmetric line profiles.Some of the mock line profiles in Figures 4 and 5 are very similar to the observations.We will provide a simple comparison between the models and observations in the following Section 5.1.

Line Profiles with m = 2
For the spiral arms with m = 2, the profiles of their corresponding emission lines are symmetric and double-peaked.The perturbation σ 1 is identical if ϕ increases every 180 • (m-fold axissymmetric), so the emissivities on the left and right sides of the LOS (blueshifted and redshifted) are exactly the same.Therefore, the line profiles of the arms with m = 2 have no asymmetry.The readers can refer to the dashed lines in Figures 4 and 5.

Velocity-delay Maps
RM can be approximated as a linear model that where Ψ(υ, τ) is the so-called "velocity-delay map" (or transfer function), ∆L c (t) is the continuum light curve, and ∆L (υ, t) is the variation of emission-line profile at different epochs (e.g., Blandford & McKee 1982).The velocity-delay map describes how the line profile responses to the varying continuum flux, and is determined by the geometry, kinematics and emissivity of the gas in BLR.The velocity-delay map of a simple Keplerian disk is symmetric, and has been calculated numerically and demonstrated in many works (Welsh & Horne 1991;Perez et al. 1992;Horne et al. 2004;Grier et al. 2013, or see Appendix D in Paper I).
The velocity-delay map can be calculated from In the calculation of emission-line profiles, we adopted two sets of parameters (µ U , σU ) for each case in Models A and B in order to simulate the mean and rms spectra (Ξ represents emissivity and responsivity, respectively).Strictly speaking, in the calculation of velocity-delay maps, we ought to employ the "responsivity" implication of Ξ, however, we do not distinguish responsivity and emissivity here because we simply assumed that they have the same form mathematically (Gaussian distributions, see Section 4.1).The only difference between them is that their (µ U ,  2, and use the nomenclature Ξ in the following discussions.The LOS inclination angle is fixed to θ los = 30 • .A smaller or larger angle will cause the velocity-delay maps narrow or broader in their velocity axes.

Velocity-delay Maps with m = 1
Similar to the line profiles, we calculate the velocity-delay maps of Models A and B for different LOS azimuthal angles.The results for both of Cases I and II are provided (see Figures 6 and  7).The sound speed has also been taken into account, so the corresponding velocity-delay maps look moderately smooth.The general morphologies of the velocity-delay maps are similar to the traditional "bell"-like envelope with a bright "elliptical ring" of a simple Keplerian disk (e.g., Welsh & Horne 1991; Perez et al. 1992;Horne et al. 2004;Grier et al. 2013).However, they are significantly asymmetric and show remarkable sub-features of bright arcs/strips (indicating strong responses from the arms).The asymmetries of the responses in the velocity-delay maps are consistent with the asymmetries of the line profiles in Figures 4  and 5.
In Model A, the contributions from the strong responsivities of the spiral arms look significant (see Figure 6).Along with the azimuthal angle increases from 0 • to 270 • , the asymmetry and the locations of the arcs/strips in the maps caused by the strong arm responsivities change correspondingly.
In Case II, the spiral arm patterns are more significant in the Ξ distributions if the strong-response regions are mainly located in larger radii.The bright arcs/strips (the strongest responses) in Compared with the cases of m = 1, the m = 2 arms can extend to more inner radii, thus their contributions in the velocitydelay maps are more significant.In addition, Ξ tends to be more "banana"-like in Model B, which is similar to the cases with m = 1.
It is obvious that the velocity-delay maps of the spiral arms with m = 2 are asymmetric and different from the velocitydelay map of a simple Keplerian disk.The distributions of the strongest responses (bright arcs/strips in Figures 8 and 9) in the maps change along with the LOS azimuthal angle.For example, for the velocity-delay map of µ U = 2.00 and σU = 0.05 in Model A, the strongest responses tend to be in the lower right corner if ϕ los = 0 • and rotates to the lowest place if ϕ los = 90 • .
For Model B, the arms in the central parts also contribute strong signals in the maps (see Figure 9).The maps look inhomogeneous and have many sub-features.The lower parts of the maps have multiple layers (similar to lasagna) in Case I of both Model A and B. This is a typical feature in velocity-delay maps if there are a number of arms in the inner radius of the Ξ-map.

Velocity-resolved Lags
Because of the high requirement of the data quality, it is not always easy to obtain velocity-delay maps.As a compromise, the velocity-resolved lag analysis is also useful for the probe of BLR geometry and kinematics, and has been applied in many RM campaigns (e.g., Bentz et al. 2008Bentz et al. , 2009;;Denney et al. 2009Denney et al. , 2010;;Grier et al. 2013;Du et al. 2016bDu et al. , 2018b;;De Rosa et al. 2018;Brotherton et al. 2020;Hu et al. 2021;Lu et al. 2021;U et al. 2022;Bao et al. 2022).We present the velocity-resolved lags for Models A and B in Cases I and II with m = 1 by averaging the velocity-delay maps (Figures 6 and 7) along their time axes.The results are shown in Figure 10.The blue lines are the velocity-resolved lags of Model A, and the orange lines are the the asymmetries of the emission-line profiles in AGNs.Capriotti et al. (1979) and Capriotti et al. (1981) suggested that the optically-thick clouds with inflow or outflow velocities in BLRs can produce asymmetric broad emission lines.Ferland et al. (1979) also proposed that a stronger red wing can be explained by the self absorption of the line radiation in an expanding BLR with optically-thick clouds.Raine & Smith (1981) established a disk BLR model illuminated by the scattered radiation from the wind, which can yield slight asymmetric line profiles.The double-peaked, asymmetric line profiles can be explained by a relativistic Keplerian disk (Chen et al. 1989).Eracleous et al. (1995) suggested that an elliptical BLR disk can interpret the double-peaked profiles whose red peak is stronger than the blue one, which is contrary to the prediction of a relativistic disk.More recently, Storchi-Bergmann et al. (2003, 2017); Schimoia et al. (2012) proposed that the spiral arms can explain the doublepeaked, asymmetric line profiles and their variations, but based on the mathematical models which presume the analytical forms of the perturbation rather than a physical model such as in the present paper.In addition, the asymmetries of the line profiles can also be attributed to supermassive binary black holes (e.g., Shen & Loeb 2010;Bon et al. 2012;Li et al. 2016;Ji et al. 2021).The physical model of density waves in this paper can produce the double-peaked and asymmetric line profiles as those in Fig- ures 4 and 5.
More importantly, if the emissivity distributions of the mean and rms spectra are different (it's always this case in observations), the line profiles of the mean and rms spectra in the BLR spiral-arm models of the present papers can naturally produce very different asymmetries.For instance, the mean spectrum has a blue asymmetry but the rms spectrum has a red asymmetry, or one is generally symmetric but the other is significantly asymmetric (see Figures 4 and 5).In observations, the mean and rms spectra in many objects have very different line asymmetries (e.g., Mrk 202, Mrk 704, 3C 120, NGC 2617, NGC 3227, NGC 3516, NGC 4151, NGC 4593, NGC 5548, NGC 6814,SBS 1518+693 in Peterson et al. 1998;Bentz et al. 2009;Denney et al. 2009;Grier et al. 2012;Barth et al. 2013;Fausnaugh et  NGC 4593 (rms) Fig. 11.Some examples of the comparisons between the emission-line profiles generated from the models and observed in RM campaigns.The upper panels are the models, and the lower are the observed rms spectra scanned and digitized from the references marked in the lower left corners.The models, the parameters (µ U , σU , and S max ), and the LOS azimuthal angles are marked in the lower left and upper right corners in the upper panels.The names of the objects are provided in the lower panels.al. 2017;Du et al. 2018b;De Rosa et al. 2018;Brotherton et al. 2020).The BLR model with spiral arms is a very promising mechanism that can easily explain the differences of the line profiles in the mean and rms spectra of RM campaigns.
Fitting the observed mean or rms line profiles with the present model is beyond the purpose of this paper.We simply select some line profiles from our Models A and B (without any fine-tuning), and then discover that it is easy to find some observed rms spectra that have almost the same profiles as these models.Some simple comparisons between the profiles of models and observations are provided in Figure 11.
The vertical radiation pressure may drive some gas flow from the disk surface (e.g., Wang et al. 2012;Czerny et al. 2017;Elvis 2017).This potential gas flow may contribute some velocity broadening or extra blueshift asymmetry to the line profiles (may also influence the velocity-resolved lags and velocity-delay maps).This effect will be considered in more details in the future.

Velocity-delay Map of NGC 5548 and Implications to BLR Spiral Arms
The high-quality velocity-delay maps of the Hβ emitting region in NGC 5548 have been successfully reconstructed by the maximum entropy method in two RM campaigns in 2014 and 2015, and are presented in Horne et al. (2021) and Xiao et al. (2018b), respectively.The two maps in 2014 and 2015 are very similar, and both of them show traditional "bell"-like envelopes with a bright "elliptical rings" which is the typical signature of a simple Keplerian disk.However, the responses at the red velocities (∼ 2000 km s −1 ) and long time lags (∼ 30 days) are relatively weaker than the other parts in both of the two maps (Horne et al. 2021 calls it an incomplete ellipse).Xiao et al. (2018b) sug-gested that this weak response is due to the inhomogeneity of the outer part of the BLR in NGC 5548.In addition, Horne et al. (2021) presents a helical "barber-pole" pattern in the C iv line of NGC 5548, which also implies the potential existence of some azimuthal structures in the BLR.
The spiral arms stimulated from the self-gravity instabilities are probably a physical origin of the weak response (incomplete ellipse) in the velocity-delay map of NGC 5548.The velocitydelay map produced by Model A with µ U = 0.60, σU = 0.10 (or µ U = 1.00, σU = 0.10), and ϕ los = 90 • (shown in Figure 6) has a similar weak response at red velocities and long time lags (incomplete ellipse).We will carry out detailed fitting to the velocity-delay map of NGC 5548 with the spiral-arm model in a separate paper in future.

Changes of Emission-line Profiles and Velocity-resolved
Lags: Arm Rotation, Changes of Emissivity/Responsivity, or Instabilities of Spiral Arms The real part of eigenvalues ω represents the rotation speed of the spiral arms, and depend on M disk /M • , Q, R out /R in , and the inner/outer radius.We provide the values of ω in Figures 1, 2, and 3.The timescale 2π/ω, that the arms rotate 360 • , spans from ∼ 70 years to ∼ 110 years for the cases with M disk /M • = 0.8 in the present paper.However, as shown in Figures 4, 5, and 10, the emission-line profiles and the velocity-resolved lags (or even velocity-delay maps) can vary significantly if ϕ los changes 90 • .Thus, observers will discover that the emission-line profiles and the velocity-resolved lags (or even velocity-delay maps) change significantly in ∼ 20 − 30 years if the BLR has similar parameters we adopted here (M • = 10 8 M and Ṁ = 1.0).Even if the parameters are different and the spiral arms prefer different mode (see Appendix A), the timescale can decrease further (even smaller than ∼ 10 years).From Appendix A, the real part of ω is generally on the order of (GM • /R 3 out ) 1/2 (or larger than (GM • /R 3 out ) 1/2 by factors of a few), where (GM • /R 3 out ) 1/2 is the Keplerian rotation frequency at the outer radius of the disk.The rotation timescale may be roughly Therefore, the rotation timescale may be smaller if the accretion rate and BH mass are smaller.
In observations, the emission-line profiles (mean or rms) and the velocity-resolved time lags have shown significant changes between two campaigns several to ten years apart.For instance, the line profile in the rms spectrum of NGC 3227 was symmetric and double-peaked in 2007 (Denney et al. 2009), but became asymmetric and single-peaked (the peak is redshifted) with strong blue wing in 2017 (Brotherton et al. 2020).Its velocityresolved lags changed from shorter in blue and longer in red velocities to inverse from 2007 to 2017 (Denney et al. 2009;Brotherton et al. 2020).The velocity-resolved lags of NGC 3516 changed from longer in blue and shorter in red to inverse to some extent from 2007 to 2012 (Denney et al. 2009;De Rosa et al. 2018), and changed back in 2018-2019 (Feng et al. 2021).Considering their smaller black hole masses, the timescales of these changes are generally consistent with the rotation timescale of the density waves.The spiral arms in BLR is probably a very natural explanation for such quick changes.In particular, some of the periodic variations in the line profiles (or in the velocityresolved lags or velocity-delay maps in future observations) can probably be explained by the spiral arms.Future detailed modeling will reveal the surface densities and azimuthal angles of the spiral arms in those objects.
Furthermore, if the continuum luminosities vary, the emissivity/responsivity distributions may change accordingly because of the photoionization physics (e.g., µ U , σU may be different).In this case, the line profiles, velocity-resolved lags, and velocitydelay maps can show significant changes within even shorter time scales (light-traveling time scale).Therefore, it must be crucial to monitor an object (especially the ones with large variations, or even changing-look AGNs) repeatedly in different luminosity states.
Finally, the instabilities of spiral arms can also be a mechanism for the short timescales of the changes in the emission-line profiles (single-epoch, mean, or rms) and the velocity-resolved lags.The growth rates can be comparable to the Keplerian timescales at outer radii, especially for Model A (for Model B, the growth timescale is longer than the Keplerian timescale by factors of a few to ten, see Figures A.1-A.3 in Appendix A), which means that the timescales of the instabilities for spiral arms can be relatively short.As mentioned above, the line profiles and the velocity-resolved lags can change within a period as short as 10 years (e.g., NGC 3227, NGC 3516).In addition, the line profiles (single-epoch, mean, or rms spectra) of some objects (e.g., Mrk 6 in Doroshenko et al. 2012and Du et al. 2018b, 3C 390.3 in Sergeev 2020and Du et al. 2018b) also showed obvious changes, but in longer timescales of ∼ 20 − 30 years.The instabilities of spiral arms can also be a possible explanation for those changes.But it should be noted that the growth timescale is still significantly longer than the rotation timescale (see Figures A.1-A.3), thus the changes caused by the instabilities of arms may be slower than those caused by the rotation.Moreover, the changes caused by the instabilities should be more chaotic, but the those caused by the rotation should be ordered and probably periodic.

Observational Tests in Future
As shown above, directly searching the spiral-arm signatures from the velocity-delay maps and emission-line profiles in RM campaigns is a very promising way to identify the spiral arms in BLRs.Recently, a trend with RM campaigns is to focus on a specific subclass of AGNs in order to investigate their unique properties, e.g., "Monitoring AGNs with Hβ Asymmetry" (MAHA) project targets to the AGNs with asymmetric Hβ emission lines (Du et al. 2018b;Brotherton et al. 2020;Bao et al. 2022).We may identify some BLRs with spiral arms from the velocitydelay maps or emission-line profiles in MAHA project in the future.In addition, it is also promising to search candidates of spiral-arm BLRs from some spectroscopic samples of the AGNs with asymmetry emission-line profiles (e.g., Eracleous et al. 2012).
Furthermore, RM to some AGNs with very large flux variations may be helpful.The velocity-delay maps of a same object at high and low states can probe different radii of its BLR (high state for larger radius and low state for smaller radius), and will provide a better constraints to spiral-arm pattern.

Roles of Parameters
In Section 3, we found that the spiral arms wind more loosely if the Toomre parameter Q and the mass ratio M disk /M • are larger.This phenomenon is easy to understand.The dispersion relation of the gravitational instabilities can be expressed, in lowest approximation, as (ω − mΩ) 2 = κ 2 + (ka 0 ) 2 − 2πGσ 0 |k|, where k is the wave number (Lin & Lau 1979).The waves are trailing if k < 0. The solution of the dispersion relation is where k 0 = κ 2 /πGσ 0 Q 2 .Considering that M disk /M • is proportional to σ 0 , the wave number |k| decreases and the wavelength increases (arms wind more loosely) if Q and M disk /M • are larger.5.6.Linear Analysis and σ 1 /σ 0 As a first step, we adopted the linear analysis to describe the density wave in disk-like BLRs and neglect the viscosity in the present paper for simplicity.The absolute amplitude of σ 1 cannot be directly deduced from Eqn (8) and is freely scalable (the solution of Eqn 18 can be S * l or CS * l with an arbitrary constant C).In more realistic calculations, the dissipation processes such as shocks, nonlinear growth of perturbations, or viscous stress should be taken into account.On one hand, the dissipation can lead to a deposit of the angular momentum carried by density wave to the disk, which may also induce changes in the surface density of the disk.On the other hand, the absolute amplitude of σ 1 may be determined if the growth of perturbation becomes saturated by the dissipation processes (e.g., Laughlin & Rozyczka 1996;Laughlin et al. 1997).These effects are not included in current equations of motion (Eqn 5 and 6) and the normal mode matrix equation (Eqn 18), and will be considered further in future.

Accretion Driven by Spiral Arms
The dimensionless accretion rate Ṁ is only used to determine the continuum luminosity and further the inner and outer radii, as well as the appropriate reference parameters for Ξ in Table 2.We mainly focus the spiral arms in BLRs which typically span from 10 3 R g to 10 5 R g .The UV/optical continuum luminosity comes from the more inner accretion disk ( 10 3 R g ), which could be in Shakura & Sunyaev regime (Shakura & Sunyaev 1973).Discussing the angular momentum transfer in details is beyond the scope of this paper.However, we can roughly evaluate if the accretion rate driven by the spiral structures in these regions is enough for the accretion in the inner disk.
In a viscous thin disk with quasi-Keplerian rotation, the radial velocity induced by a viscosity ν vis (Lynden-Bell & Pringle 1974) can be expressed as where ν vis = αa 0 H is an effective "alpha"-type viscosity, α is viscosity parameter, and H is the thickness of the disk.The mass accretion rate can be obtained with Ṁ• = 2πRuσ 0 .The global spiral arms may redistribute the disk material and be described in terms of a diffusive process with an effective viscosity α eff (Laughlin & Rozyczka 1996).α eff is on the order of 0.01 or so (especially in nonlinear regime, e.g., Laughlin & Bodenheimer 1994;Laughlin & Rozyczka 1996;Lodato & Rice 2005).We have checked that, with such a α eff , the disk properties assumed in the present paper (σ 0 , a 0 , H, and Ω) can very easily support the accretion with Ṁ ∼ 1.

Vertical Structures and Possible Influences
Given the sound speed a 0 and rotation curve Ω, the thickness of the disk is H/R ∼ R 1/8 and H/R ∼ R 1/4 for Model A and B, respectively.It means that the geometry of the disk is "bowlshaped" (concave, see Starkey et al. 2022).Such geometry can enable the disk to be illuminated by the ionizing photons from the inner region.
With surface density (σ 1 ) variations, the disk thickness is also likely to modulate.The wave crest of the arm may be more strongly irradiated by the ionizing photons because it protrudes from the disk surface.On the contrary, the wave trough may be more weakly irradiated.Therefore, the asymmetries of the line profiles and velocity-resolved lags, and the sub-features in the velocity-delay maps may be more stronger.A sophisticated treatment of the vertical structures and the corresponding influences to the observation are needed in the future.

Boundary Conditions
In this paper, we adopted the same boundary conditions as in Adams et al. (1989) for simplicity.Noh et al. (1991) and Chen et al. (2021) investigated the influence of boundary conditions on the pitch angles, pattern speeds, and growth rates of spiral arms in protoplanetary disks.They tried reflecting and transmitting boundaries besides the boundary conditions of Adams et al. (1989), and found that the boundary conditions mainly influence the growth rates but have little effect on the pitch angles and pattern speeds of the arms (the differences are 10% for different boundary conditions).Their works indicate that adopting the boundary conditions of Adams et al. (1989) is enough for exhibiting the general reverberation properties of the BLR arms in observations.In the future, the boundary conditions may be revised by comparing the models with the real observations.

Summary
In recent years, there are growing evidences that some of BLRs are inhomogeneous and have substructures.The radii of BLRs measured by RM are consistent with the self-gravitating regions of accretion disks, which implies that the spiral arms excited by the gravitational instabilities may exist in, at least, the disk-like BLRs.In this paper, we calculate the surface densities of the spiral arms in BLRs, for two typical configurations (called Models A and B) with different parameters, by using the density wave theory.We find that more massive disks (larger disk-to-SMBH mass ratios) with larger Toomre parameters tend to have more loosely wound arms (more significant in observations).In comparison with Model A, the spiral arms of Model B are more "banana"-like.
We present the emission-line profiles, velocity-delay maps, and velocity-resolved lags for the cases of loosely wound spiral arms (in more massive BLR disks).For m = 1 spiral arms, the emission-line profiles, velocity-resolved lags have significant asymmetries, and the velocity-delay maps are asymmetric and have complex substructures (bright arcs/strips).For m = 2 spiral arms, the emission-line profiles and velocity-resolved lags are symmetric, on the contrary, the velocity-delay maps are asymmetric and show complex substructures.The spiral arms in BLRs can easily explain some phenomena in observations: -For a same object, the mean and rms spectra in RM observations can have very different asymmetries.The rms spectra always have different widths compared to the mean spectra in RM campaigns, which implies that the emissivities/responsivities of the invariable and variable parts in BLRs are different.Considering the different emissivities/responsivities, the calculations in the present paper show that the spiral arms in BLRs can naturally produce differently-asymmetric line profiles in the mean and rms spectra of a same object without any further special assumptions.-Our models can generate emission-line profiles almost the same as the observations (rms spectra).-The spiral arms in the disk-like BLRs can produce complex features such as bright arcs/strips, and are probably a physical origin for the relatively-weak response region (incomplete ellipse) in the velocity-delay map of NGC 5548.-The timescale that the spiral arms rotate ϕ los ∼ 90 • (which can significantly changes the line profiles or velocityresolved lags) can be as short as 10 years.The rotation of the spiral arms can explain the quick changes of the asymmetries in the emission-line profiles, the velocity-resolved lags, or even velocity-delay maps between RM campaigns several to ten years apart.Futhermore, some of the periodic variations in the line profiles (or in the velocity-resolved lags or velocity-delay maps in future observations) can probably be explained by the rotation of the BLR spiral arms.-The line profiles, velocity-resolved lags, and velocity-delay maps can show significant changes within short time scales (light-traveling time scale) if the continuum vary significantly.
Sophisticated fitting to the observations by the spiral-arm models will reveal the detailed geometry and kinematics of BLRs in the future.

−−−−−−−−−−−−Fig. 1 .
Fig. 1.Dimensionless surface density of spiral arms (m = 1) for Model A. The 6 panels in upper left corner are the spiral arms for more massive disks (M disk /M • = 0.8), and the 6 panels in lower right corner are those for less massive disks (M disk /M • = 0.2).The values of Q, M disk /M • , and R out /R in are marked on the top of each panels.In general, more massive disks have more loosely wound spiral arms (see more details in Section 3.2).The eigenvalues (real and imaginary parts) of ω are also provided in each of the panels.

−−−−−−−−−−−−Fig. 2 .
Fig. 2. Dimensionless surface density of spiral arms (m = 1) for Model B. Similar to Figure 1, the 6 panels in upper left corner are the spiral arms for more massive disks (M disk /M • = 0.8), and the 6 panels in lower right corner are those for less massive disks (M disk /M • = 0.2).The values of Q, M disk /M • , and R out /R in are marked on the top of each panels.The eigenvalues of ω are also provided in each of the panels.

−−−−−−Fig. 3 .
Fig. 3. Dimensionless surface density of the spiral arms with m = 2 for Models A and B. The upper 3 panels are the arm patterns of Model A, and the lower 3 panels are those of Model B. We only plot the spiral arms with R out /R in = 100 as examples.

Pu
Fig. 4. Emission-line profiles of Model A in Cases I and II.The left panel in each row is the Ξ image.The values of (µ U , σU ) and S max are marked on the top of the Ξ images.The red dotted lines mark the LOS azimuthal angles ϕ los .The four panels on the right in each row are the line profiles (blue solid lines) corresponding to different ϕ los .The grey dashed lines are the profiles without spiral arms.The line profiles (mock mean and rms) of the spiral arms for Case I are provided in the upper two rows, and the profiles of Case II are shown in the lower.
Fig. 5. Emission-line profiles of Model B in Cases I and II.The meaning of panels and different lines (solid and dashed) are the same as in Figure 4.

Pu
Fig. 6.Velocity-delay maps of Model A (m = 1) in Cases I and II.The left panel in each row is the Ξ image.The red dotted lines mark the LOS azimuthal angles ϕ los .The four panels on the right in each row are the velocity-delay maps corresponding to different ϕ los Fig. 8. Velocity-delay maps of Model A (m = 2) in Cases I and II.The meaning of panels are the same as in Figure 6.

Fig. 10 .
Fig. 10.Velocity-resolved lags.The blue and orange lines are corresponding to Models A and B (m = 1), respectively.The reprocessing coefficients and LOS azimuthal angles are marked in each panel.

Table 1 .
Parameters of Models A and B