Kinematic analysis of nuclear spirals

# Kinematic analysis of nuclear spirals: feeding the black hole in NGC 1097

Glenn van de Ven11affiliation: Max Planck Institute for Astronomy, Königstuhl 17, 69117 Heidelberg, Germany: glenn@mpia.de 22affiliation: Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA; Hubble Fellow , Kambiz Fathi33affiliation: Stockholm Observatory, Department of Astronomy, Stockholm University, AlbaNova, 106 91 Stockholm, Sweden; kambiz@astro.su.se 44affiliation: Oskar Klein Centre for Cosmoparticle Physics, Stockholm University, 106 91 Stockholm, Sweden
###### Abstract

We present a harmonic expansion of the observed line-of-sight velocity field as a method to recover and investigate spiral structures in the nuclear regions of galaxies. We apply it to the emission-line velocity field within the circumnuclear star forming ring of NGC 1097, obtained with the GMOS-IFU spectrograph. The radial variation of the third harmonic terms are well described by a logarithmic spiral, from which we interpret that the gravitational potential is weakly perturbed by a two-arm spiral density wave with inferred pitch angle of . This interpretation predicts a two-arm spiral distortion in the surface brightness, as hinted by the dust structures in central images of NGC1̃097, and predicts a combined one-arm and three-arm spiral structure in the velocity field, as revealed in the non-circular motions of the ionised gas. Next, we use a simple spiral perturbation model to constrain the fraction of the measured non-circular motions that is due to radial inflow. We combine the resulting inflow velocity with the gas density in the spiral arms, inferred from emission line ratios, to estimate the mass inflow rate as a function of radius, which reaches about  M yr at a distance of  pc from the center. This value corresponds to a fraction of about of the Eddington mass accretion rate onto the central black hole in this LINER/Seyfert1 galaxy. We conclude that the line-of-sight velocity not only can provide a cleaner view of nuclear spirals than the associated dust, but that the presented method also allows the quantitative study of these possibly important links in fueling the centers of galaxies, including providing a handle on the mass inflow rate as a function of radius.

galaxies: active — galaxies: kinematics and dynamics — galaxies: nuclei — galaxies: structure — galaxies: individual: NGC 1097

## 1. Introduction

Gas transport to the centers of galaxies is still mainly an unsolved problem (e.g. Martini, 2004). Since most of this gas is residing in a rotating disk well beyond the center, it is essentially a problem of angular momentum transport. Proposed transport mechanisms range from galactic interactions and bars, to nuclear bars and spirals, to stellar mass loss and disruption near the central black hole (BH). This range in scales also closely represents a range in decreasing mass inflow rates, which in turn might be correlated with activity in the galactic nucleus (AGN), ranging from quasar, Seyfert, LINER to quiescent galaxies. However, not only is it challenging to (observationally) establish fueling mechanisms down to a few parsec from the center, time delays between changes in the mass inflow rate and the onset of nuclear activity further complicate linking them.

It is also likely that multiple fueling mechanisms are important and act together. Large-scale bars are efficient at transporting gas inward (e.g. Athanassoula, 1992), but the presence of an inner Lindblad resonance (ILR) will cause the gas to pile up in a nuclear ring, often clearly visible due to the intense star formation (see e.g. Figure 1). The gas might continue further inward through nested bars (e.g. Shlosman et al., 1989; Englmaier & Shlosman, 2004), although dynamical constraints on a double-barred system may prohibit inflow down to the center (Maciejewski et al., 2002). Moreover, observational evidence of nuclear bars is scarce (e.g. Regan & Mulchaey, 1999; Martini & Pogge, 1999), but should be interpreted with care since the prominent dust lanes in the main bar might be absent in secondary bars (e.g. Shlosman & Heller, 2002).

On the other hand, nuclear spirals seem to be commonly observed, both in active and quiescent galaxies (e.g. Laine et al., 1999; Pogge & Martini, 2002), ranging from flocculent to grand-design nuclear spirals (e.g. Martini et al., 2003). Whereas the former are suggested to form by acoustic instabilities (e.g. Elmegreen et al., 1998), the grand-design nuclear spirals are thought to be the result of gas density waves (e.g. Englmaier & Shlosman, 2000) or shocks (e.g. Maciejewski, 2004a, b) induced by the non-axisymmetric gravitational potential of a large-scale bar. The latter two might be connected in the sense that the bar-driven spiral shocks trigger the gas density waves throughout the disk (Ann & Thakur, 2005), which in turn seem to be necessary for the nuclear spiral to be long-lived (Englmaier & Shlosman, 2000). The inward extended gas inflow through a nuclear spiral not only depends on the torque of the large-scale bar, but also the gas having high enough sound speed so as to loose angular momentum (e.g. Englmaier & Gerhard, 1997; Patsis & Athanassoula, 2000), as well as the presence of a central mass concentration such as a super-massive BH to overcome a closer-in (inner) ILR (e.g. Fukuda et al., 2000; Ann & Thakur, 2005).

The deviations in the gas density due to nuclear spirals are typically only a few per cent (Englmaier & Shlosman, 2000), which makes direct imaging very difficult. Instead, the obscuration due to dust thought to be associated with the gas overdensities is often employed, but when the extinction is small or the dust not well-mixed with the gas, it can lead to unclear or even missed detections of nuclear spirals. At the same time, nuclear spirals induce non-circular motions in the gas, with resulting deviations in the observed velocity field that can be a significant fraction of the underlying circular velocity. If the gas is ionized, the kinematics might be inferred from emission lines, which in general can be detected more easily and at higher spatial resolution in the optical, than can be tracers of the molecular or atomic gas such as CO and Hi at radio wavelengths. Moreover, while imaging typically only yields a detection of the nuclear spiral, the observed non-circular motions in combination with the gas (over)density inferred from the simultaneously measured (line) fluxes might be used to derive an estimate of the gas mass in/out-flow rate.

An alternative approach to get a handle on the gas flow rates is to compute the gravitational torques from the observed surface brightness using a mass-to-light ratio conversion calibrated against (circular) velocity measurements (García-Burillo et al., 2005), or against stellar population models fitted to color measurements (Quillen et al., 1995). In these cases, various assumptions are made, most importantly that stellar light is a clean (once corrected for dust obscuration) and direct (mass follows light) tracer of the underlying gravitational potential. Even so, this approach shows that gravitational torques due to non-axisymmetric structures are very efficient at transporting gas inward while overcoming dynamical barriers such as the corotation resonance. However, when the resulting predicted mass inflow rates as functions of radius are compared with the non-circular motions in the observed gas velocity fields the correlation between both is often not evident (Haan et al., 2009).

The main difficulty with the non-circular motions is to identify the fraction that is due to pure radial flow. Gas on closed elliptic orbits not only contributes to non-circular motions (or elliptic streaming) in the azimuthal direction but also in the radial direction (Wong et al., 2004). However, since closed elliptic orbits are applicable strictly only to collisionless (stellar) orbits, we expect that as a result of shocks and other dissipational effects the angular momentum of the gas will change, leading to in/out-flow. While the measured non-circular motions are often taken to be directly representative of the radial flow velocities (e.g. Storchi-Bergmann, 2007), only a fraction of them are expected to be truly radial in/out-flows.

The goal of this paper is two-fold: (i) to show that harmonic expansion of gas velocity fields provides a clean way to detect nuclear spirals and to estimate their pitch angles; (ii) to use simple perturbation models to constrain the fraction of measured non-circular motions that is due to radial flow, and to estimate the corresponding mass inflow rate. In Section 2 we present the harmonic analysis, and in Section 3 we apply it to the observed emission-line velocity field within the circumnuclear star forming ring of NGC 1097. We fit the third harmonic terms as a three-arm logarithmic spiral structure consistent with a weak two-arm spiral perturbation of the gravitational potential. We then use the perturbation model derived in Appendix A to constrain the inflow rate as function of distance from the center of NGC 1097. In Section 4 we discuss the corresponding spiral distortion in the surface brightness, and possible additional non-circular motion contributions to the velocity field. Finally, we link the estimated mass inflow rate to the accretion onto the central BH of NGC 109. We summarize and draw our conclusions in Section 5.

## 2. Harmonic analysis

Maps of the surface brightness (SB), line-of-sight velocity (), velocity dispersion (), and higher order velocity moments (often expressed in term of Gauss-Hermite moments , , …) of nearby galaxies generally show organized, periodic features, which can be suitably studied by means of a harmonic expansion (e.g. Schoenmakers et al., 1997; Wong et al., 2004; Fathi et al., 2005; Krajnović et al., 2006; Spekkens & Sellwood, 2007).

### 2.1. Harmonic expansion

We start by dividing a map into a number of elliptic annuli, each with a different semi-major axis length , but we assume all have the same flattening and share the same angle and center , so that in terms of Cartesian coordinates on the map (i.e., on the plane of the sky)

 x′=x′0+Rcosψcosψ0−qRsinψsinψ0, y′=y′0+Rcosψsinψ0+qRsinψcosψ0. (1)

If the map is aligned such that is pointing North and pointing West, than , where is the common (observational) definition of the position angle of the major axis of the galaxy, measured from North through East. Next, we extract the profiles along each of the annuli and describe them by a finite number of harmonic terms ,

 P=c0(R)+n∑m=1cm(R)cosmψ+sm(R)sinmψ. (2)

There are various ways to obtain the set of best-fit ellipses, but given the highly non-linear nature of the optimization for the best-fit parameters, this is commonly done in a stepwise, iterative process. We follow a similar approach as described in Krajnović et al. (2006) — based on the usually adapted procedure in photometry (Jedrzejewski, 1987) — which we briefly summarize.

For (early-type) galaxies, the maps of SB, and even higher-order velocity moments (, , …) each are to first order well approximated by a function that is constant along (concentric, equally flattened and orientated) ellipses, and only varies as function of the semi-major axis length of these ellipses. Similarly, and odd higher-order velocity moments (, , …) each are well described by a similar function, but one that has an additional cosine variation along the ellipses. In these cases, the harmonic expansion in equation (2) truncates after the third-order term (). The parameters of the best-sampling ellipses for even velocity moments, can then be obtained by minimizing , and without for odd velocity moments. We first perform the minimization for a grid of (fixed) and values, where we might use external constraints such as the inclination and measured position angle together with an initial estimate of the center . Next, starting from the best-fit grid pair, we optimize for all four parameters.

We sample the semi-major axis lengths as for , with the initial depending on the spatial resolution; for the geometric increase factor we take . We achieve this sampling, together with uniform sampling in , by bilinear interpolation of the observed map. When fitting the observed line-of-sight velocity field shown in Figure 2, we assume all motions are within the equatorial plane. We divide all harmonic terms by to take into account the projection effect. For further details, including error estimates, see Fathi et al. (2005).

### 2.2. Spiral structure

Once we have obtained the set of best-fitting ellipses, we fit the harmonic expansion of equation (2) to each of the corresponding profiles to obtain the harmonic terms and (up to higher order ), as functions of . The difference with the above expansion up to and including , reveals the deviations from the latter smooth model: for example, boxiness and diskiness in the SB map, and non-circular motion in the residual map.

In the case of a spiral structure as observed in NGC 1097, both through the dust in the SB and in the residual map (Fathi et al., 2006), the deviations are more naturally described as a (radially varying) offset in the angle , than through a complex combination of variations in the amplitudes and . Henceforth, we rewrite the harmonic expansion of equation (2) (in a mathematically equivalent expression) as

 P=K0(R)+n∑m=1Km(R)cos(m[ψ−ψm(R)]), (3)

where the amplitudes and phase shifts are related to the coefficients and () by

 K2m=c2m+s2mandtan(mψm)=smcm. (4)

For example, the map of NGC 1097 in Figure 2 reveals, after subtracting the best-fit circular motion (with , since we set ), what seems to be a three-arm spiral structure, . Here, traces the spiral arms as function of radius .

### 2.3. Weakly perturbed gravitational potential

An axisymmetric gravitational potential in a frame that co-rotates with a weak perturbation of harmonic number can be written in polar coordinates as

 Φ(R,ϕ)=Φ0(R)+Φm(R)cos(m[ϕ−ϕm(R)]), (5)

where is the phase of the perturbation. Through Poisson’s equation it follows that the corresponding surface mass density exhibits a harmonic distortion.

To derive the line-of-sight velocity, we follow Schoenmakers et al. (1997). As described in Appendix A, we extend their collisionless analysis by including radial damping in the equations of motion to take into account the dissipative nature of gas. We assume that the gas moves on closed loop orbits in the equatorial plane, which we observe at an inclination away from its normal and at an (azimuthal) angle . Given a point in the equatorial plane, the projection of the azimuthal and radial velocity onto the corresponding line-of-sight yields

 Vlos=sini[vϕ(R,ψ)cosψ+vR(R,ψ)sinψ], (6)

where is zero on the line of nodes (see also Fig. 1 of Schoenmakers et al., 1997). To first order the solutions of the equations of motion in the perturbed gravitational potential of equation (5) yield

 vR(R,ψ) = vc(R)[cRcosmψ+sRsinmψ], (7) vϕ(R,ψ) = vc(R)[1+cϕcosmψ+sϕsinmψ], (8)

where is the circular velocity and , , and are functions of given in Appendix A. Substituting these solutions into equation (6), we obtain

 Vlos = V⋆cosψ (9) + cm−1cos(m−1)ψ+sm−1sin(m−1)ψ + cm+1cos(m+1)ψ+sm+1sin(m+1)ψ,

with and , and the circular velocity in projection. We thus find, as concluded before by Schoenmakers et al. (1997) and already qualitatively inferred by Canzian (1993) that, if the gravitational potential has a perturbation of harmonic number , the line-of-sight velocity field contains an and an harmonic term.

### 2.4. Pitch angle

How loosely or tightly wound a spiral is can be quantified via its pitch angle, , which, at a given radius, , measures the angle between the tangent of the spiral arm and a circle with radius in the plane of the disk. Inverting the phase-shift, , of a spiral, we can parameterize an arm of the spiral as

 x=R(ϕ)cosϕandy=R(ϕ)sinϕ. (10)

One can then show that the pitch angle is given by

 cotζ=dϕdlnR, (11)

which is positive (negative) if the spiral curves anti-clockwise (clockwise) with increasing radius. The smaller the pitch angle, the more tightly the spiral is wound, with a circle, while corresponds to a straight line. A specific case that is often encountered in nature is that of a logarithmic spiral

 ϕ(R)=1b0lnRa0↔R(ϕ)=a0exp(b0ϕ), (12)

with constants and . From equation (11), we find the well-known property that the logarithmic spiral has a constant pitch angle .

In the case of an -spiral in the line-of-sight velocity field, the corresponding pitch angle, , follows directly from the phase-shift, , in the harmonic expansion in equation (4). The pitch angles and generally take different values and bracket the pitch angle of the -spiral perturbation in the gravitational potential that caused them. To show this, we start from equations (A12) and (A13) in Appendix A and rewrite the coefficients and as

 cm±1 = Km±1cos(mφm−θm±1), sm±1 = Km±1sin(mφm−θm±1), (13)

so that after substitution into equation (4) we obtain

 mφm−θm±1=(m±1)ψm±1, (14)

where . This links the phase-shift of the -spiral perturbation in the gravitational potential with the phase-shifts of the -spirals in the line-of-sight velocity field. The corresponding pitch angles are related as

 mcotζm=(m±1)cotζm±1−dθm±1/dlnR. (15)

In general, and depend in a rather complex way on the gravitational potential, but as we show in Appendix A they possess some generic properties.

First, the amplitude is larger (smaller) than the amplitude outside (inside) the corotation radius , and equal to it at . This implies a transition in the line-of-sight velocity field at , going from the -spiral dominating outside to the -spiral dominating inside (see also Figure 5). This is also concluded by Schoenmakers et al. (1997) for the collisionless case and earlier by Canzian (1993) for the less general case of a tightly wound spiral in the linear density-wave theory (see also Canzian & Allen, 1997). However, the dissipational nature of the gas, which we model in Appendix A via radial damping (cf. Wada, 1994), can alter the relative amplitudes of the harmonic terms (see also Section 3.2 below).

Second, typically varies much less with radius than the spiral phase-shifts, so that in equation (15) is relatively small. As a result, we can estimate the pitch angle of the -spiral perturbation in the gravitational potential from the pitch angles and/or of the -spiral and -spiral in the observed line-of-sight velocity field, without constructing a full dynamical model (see also the last two panels of Figure 5).

Nuclear spirals in principle provide a mechanism to transport gas from kpc scales, where it often stalls inside a nuclear ring, into the center of the galaxy. Still, as mentioned in Section 1, if a nuclear spiral (or a nuclear bar) is due to gas moving on closed elliptic orbits this results in non-circular motions, also referred to as elliptic streaming, but not necessarily in net inflow towards and/or outflow away from the center. However, unlike stars, gas is not collisionless and its orbits interact and ex-change angular momentum leading to net radial flows (e.g. Wada, 1994).

Henceforth, in the analytic models in Appendix A, we assume a weak perturbation in the gravitational potential causing gas to deviate from circular onto elliptic orbits, while taking into account its dissipative nature via radial damping. These analytic spiral models are an extension of the analytic bar models introduced by (Wada, 1994), who showed that they describe well the gas behavior seen in hydrodynamical simulations. The radial damping causes the gas to lose/gain angular momentum inside/outside the corotation radius, which nicely matches the angular momentum transfer due to the torque from the bar potential in numerical simulations. The amount of radial damping, controlled through the dimensionless parameter111Note that our is the same as the dimensionless parameter in Wada (1994), while we define ., , thus provides a handle on the amount of net radial flow that is needed to explain the observed non-circular motions in addition to elliptic streaming.

In this way, a rather straightforward estimate of the net radial flow velocity, , can be obtained by comparing the radial velocity (equation A5) of the analytic model that includes radial damping () with the analytic model without radial damping (). At a given radius , the maximum radial velocity is given by

 vR,max=m(Ω−Ωp)R(A2+B2Δ2+Λ2)1/2. (16)

Here, and only depend on the weak perturbation as given in equation (A2), while

 Δ=κ2−m2(Ω−Ωp)2,Λ=2λκm(Ω−Ωp), (17)

are functions of the angular frequency and the epicycle frequency of the axisymmetric gravitational potential, as well as the harmonic number and pattern speed of the weak perturbation. Since without radial damping () all radial motion is due to elliptic streaming, we subscribe a fraction of the radial velocity to elliptic streaming, leaving as an estimate of the radial flow velocity

 vflow=(Λ2Δ2+Λ2)1/2vR. (18)

Even though the analytic models neglect possible non-linear effects, they can capture most features of observed non-circular motions (as we show next in the case of NGC 1097), and at the same time provide an estimate of the fraction of the observed non-circular motions that is due to net radial flow in addition to elliptic streaming. Note that radial flow here does not mean that the gas is following pure radial orbits with zero angular momentum, which would contribute only to the harmonic term (e.g. Wong et al., 2004). Instead, the gas is expected to gradually spiral inward/outward as it has both azimuthal and radial velocity components. In case the angular momentum loss/gain is driven by a weak gravitational potential perturbation with harmonic number , this results in a contribution to and .

## 3. Nuclear spiral in NGC 1097

We apply the above harmonic analysis to the observed emission-line velocity field within the circumnuclear star forming ring of NGC 1097. We recover in the non-circular motions a spiral structure and infer its pitch angle directly from the harmonic components. Next, we use a spiral perturbation model to constrain the radial inflow velocity and combine this with the gas density in the nuclear spiral to estimate the mass inflow rate as function of distance from the center of NGC 1097.

### 3.1. Non-circular motions

NGC 1097 (ESO 325–58) is a nearby (distance 14.5 Mpc, so  pc) LINER/Seyfert 1 host with a strong,  kpc long, bar and a  kpc in radius circumnuclear star forming ring (Figure 1). In this Sb galaxy dust can be traced within the large-scale spiral arms out to a (outer Lindblad resonance, OLR) radius of  kpc, in prominent lanes along the bar, and continuing within the nuclear ring as a spiral structure down to  pc from the center (Lou et al., 2001; Prieto et al., 2005). Non-circular motions associated with this nuclear spiral structure (Fathi et al., 2006; Davies et al., 2009), indicate a possible mechanism to drive gas from kpc scales down to a few pc from the center, where a double-peaked broad H emission profile (Storchi-Bergmann et al., 1993) indicates the presence of a super-massive BH.

Fathi et al. (2006) describe in detail the observations and reduction of the two-dimensional spectroscopy of NGC 1097 obtained with the GMOS-IFU on the Gemini South Telescope (GS-2004B-Q-25, PI: Storchi-Bergmann). Three pointings within the nuclear ring region (Figure 1) provided 1500 individual spectra covering 5600–7000 Å at a velocity resolution of 85 km s and with a spatial sampling of 01.

In the first panel of Figure 2, we present the [Nii] emission-line velocity field covering the inner  kpc. A systemic velocity of  km s (equivalent to a constant term) has been subtracted. For an inclination  (flattening ; Fathi et al., 2006), and position angle of , the other panels show the results of applying the harmonic expansion of Section 2.1. From left to right: the best-fit circular motion , the remaining non-circular motion, and the harmonic reconstruction from the sum of the latter two.

In Figure 3, the first two panels show the adopted position angle and systemic velocity. The next four panels show the coefficients and (in km s) for the first three harmonic terms, as function of the (deprojected) radius . The bottom three panels, show the phase shifts (in degrees) for the alternative formulation of the harmonic expansion given in equation (3), as functions of the (natural) logarithm of .

We interpret the non-circular residual motion (third panel of Figure 2) as a three-arm spiral structure. This interpretation is supported by the significant amplitudes of and , and in particular by the the smooth variation of the corresponding phase-shift with radius. Excluding the uncertain measurements within (vertical dotted line) and the single measurement at the edge of the map, the remaining measurements indicated by solid circles show a linear relation between and . This is consistent with a logarithmic spiral defined in equation (12), with fitted ″ and . The slope provides a direct and robust measurement of the pitch angle of .

In the third panel of Figure 2, this three-arm logarithmic spiral with pitch angle  is plotted with magenta circles on top of the non-circular motions. If this three-arm spiral in the velocity field is due to a two-arm spiral perturbation in the gravitational potential as we argue below, it should also give rise to a one-arm logarithmic spiral in the velocity field. Combining equation (12) and the right-hand-side of equation (14), we find (and equal ). This means a pitch angle  for the one-arm logarithmic spiral, resulting in the magenta triangles in the third panel of Figure 2. The contribution of the one-arm spiral to the non-circular motions is partly absorbed into the circular motions since the coefficient is indistinguishable from the circular velocity contribution. Moreover, as can be seen from Fig. 3, the remaining coefficient contributes in a different way than the combined and terms, so that it is not surprising that neither the one-arm nor the three-arm spiral alone traces the spiral structure in the non-circular motions. This is even aside from possible additional contributions, or contaminations in this case, from other (even) harmonic terms, as discussed below in Section 4.2.

The positive sign of the pitch angle indicates the nuclear spiral is curved anti-clockwise with increasing radius, equivalent to the spiral arms extending outwards of the large-scale bar, as can be seen in the left panel of Figure 1. Since the global rotation in NGC 1097 is clockwise from the observed velocity field, and the North-East is the “far side” from being more obscured, it follows that both the large-scale and nuclear spirals are trailing.

### 3.2. Two-arm spiral perturbation

The above three-arm spiral structure in the non-circular motions is consistent with a perturbation in the gravitational potential due to a harmonic spiral (see also Section 2.3). From equation (15) it then follows that the pitch angle of this two-arm spiral perturbation in the gravitational potential follows from the measured pitch angle of the three-arm spiral in the velocity field as . We show below that, as expected, the later term is small, so that for a measured , we calculate .

As illustrated in Figure 5 of Appendix A, the true value of might differ slightly depending on the details of the gravitational potential perturbation. We construct a model for the nuclear spiral in NGC 1097 to get a handle on the latter difference as well as to constrain the fraction of the observed non-circular motions that is due to net radial inflow in addition to elliptical streaming. We use the analytic solutions of Appendix A for gaseous orbits in an axisymmetric gravitational potential, , that is weakly perturbed by a logarithmic spiral with pitch angle . These analytic spiral models are based on linearized equations of motion under the epicycle approximation, but are not restricted to tightly wound spirals (e.g. Lin et al., 1969; Canzian & Allen, 1997). In this way, we show below that the observed loosely wound nuclear spiral in NGC 1097 is still consistent with a density wave, and not necessarily driven by shocks as suggested by Davies et al. (2009). Note that these simple models assume the existence of a weak perturbation in the gravitational potential without specifying neither how the perturbation arises nor how it is maintained. To include potential driving mechanism such as the large-scale bar requires more sophisticated models, which is beyond the goals and scope of this paper.

We adopt the power-law model (Evans & de Zeeuw, 1994) with axisymmetric gravitational potential

 Φ0(R)=⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩v202β/2β[1−(1+R2R2c)−β/2]β≠0,v2012[1−ln(1+R2R2c)]β=0, (19)

and corresponding circular velocity

 vc(R)=v02β/4RRc(1+R2R2c)−(1/2+β/4), (20)

so that at the core radius . The parameter controls the logarithmic gradient of the rotation curve at large radii: rising, flat, and falling. The three gravitational potential parameters , and are set by comparing the corresponding circular velocity with the measured radial profile of , taking into account the non-circular contribution due to the spiral perturbation. In the third panel of Figure 3, the black dashed curve shows for  km s, ″, and . Except for the three values within ″, this simple power-law model is a good representation of the measurements, once the non-circular contribution derived below is added, as indicated by the blue dashed curve. The advantage of such a simple analytic representation of the gravitational potential is that it makes all subsequent calculations concerning the perturbation very convenient.

For the amplitude of the gravitational potential perturbation we assume , with constant strength , while the phase-shift is given by . Here, ″ from the above fit to the harmonic terms in the line-of-sight velocity field, while the corresponding approximation , is taken as the initial value for the pitch angle. The three additional free parameters are the pattern speed, , of the spiral perturbation, the azimuthal viewing angle, , and finally the amount of radial damping, . Equations (A12) and (A13) then provide predictions for the non-circular motion contribution in terms of the harmonic coefficients , , , and , which we compare with the corresponding measured radial profiles for NGC 1097 in Figure 3.

The flattening of the Hi rotation curve in the outer parts of NGC 1097 (Sofue et al., 1999) implies a nearly constant circular velocity  km s. With the angular and epicycle frequencies approximately given by , the OLR being at  kpc yields a pattern speed  km s kpc. The latter places the corotation radius,  kpc, or % beyond the extent of the large-scale bar, consistent with numerical simulations of “fast bars” (e.g. Athanassoula, 1992; Debattista & Sellwood, 2000), and measured pattern speeds in similar galaxies (e.g. Aguerri et al., 2003; Gerssen et al., 2003; Rautiainen et al., 2008). Assuming that the two-arm nuclear spiral as a gas density wave is being driven by the large-scale bar (Englmaier & Shlosman, 2000), we adopt the same value for the pattern speed of the perturbation. Since we consider the harmonic coefficients well within the corotation radius, they are not sensitive to, and hence do not constrain .

On the contrary, a small change in the azimuthal viewing angle already causes a significant radial offset in the harmonic coefficients, so that we need . Next, too little radial damping results in too small amplitudes of the coefficients and with respect to and the non-circular contribution to . Specifically, since and are of similar amplitude and shape over most of the radial range, significant radial damping with is needed. As described in Section 2.5 and discussed below in Section 3.3, this implies that net radial flow makes up most of the intrinsic radial velocity. Matching the amplitudes of the harmonics terms yields a strength of the gravitational potential perturbation of . Finally, values for the pitch angle of the spiral perturbation that are in the range of , approximated above from the pitch angle of the three-arm spiral in the velocity field, yield predictions for the harmonic coefficients that are consistent with the measured harmonic coefficients. The effect of the additional term is indeed small, so that provides a robust measurement of .

This is further illustrated in Figure 3, where in addition to the measured harmonic coefficients, we show with dashed curves the predictions of the above two-arm spiral perturbation model with pitch angle , azimuthal viewing angle , and radial damping parameter . Both and , shown as dot-dashed curves in respectively the bottom-left and bottom-right panel, indeed vary only little with radius. Similarly, the dashed curve in the bottom-right panel shows the predicted relation between and while taking into account , which is nearly indistinguishable from the fitted solid line. The predicted harmonic terms match well the and measurements in the middle-right panel, and rather well the measurements in the middle-left panel, except for those at larger radii. However, in particular the last three points in radius are less certain because the corresponding ellipses to extract these measurements are not fully covered (see also second panel of Fig. 2), and they might well be disturbed by the nuclear ring. The blue dashed curve in the top-right panel is the spiral model prediction for , which apart from the innermost measurements, nicely traces the measured rotation curve.

Due to the significant non-circular motion contribution the latter is different from the circular velocity of the (power-law) axisymmetric gravitational potential shown as the black dashed curve. The corresponding black dot-dashed curve is still well above the pattern speed  km s kpc (solid horizontal line) at radius ″, which places the ILR well beyond the nuclear ring radius of  kpc. If indeed the radius of nuclear rings is set by the location of the ILR (e.g. Buta & Combes, 1996), this indicates that the nuclear ring in NGC 1097 has migrated inward. Inward migration has also been suggested for the nuclear ring in NGC 4314 (Benedict et al., 2002), has been seen in hydrodynamic simulations (e.g. Fukuda et al., 2000; Regan & Teuben, 2003), and might be a consequence of shepherding of the gas ring by star clusters that formed in it (van de Ven & Chang, 2009).

### 3.3. Mass inflow rate

We found that a spiral perturbation model with only elliptic streaming, in which gas is moving on closed elliptic orbits without radial damping, cannot explain the observed non-circular motions in NGC 1097. In Section 2.5, we showed that the amount of radial damping required provides an estimate of the radial flow velocity in terms of the radial velocity of the spiral model which matches the observed harmonic terms. Since in the case of NGC 1097 a large amount of radial damping is needed, the ratio in equation (18) is close to unity. In other words, the contribution to the radial velocity in the nuclear spiral is predominantly due to net radial flow, which is directed inward to the center of NGC 1097, as we are well within the corotation radius. For the same spiral model as shown in Figure 3 with the dashed curves, the corresponding radial inflow velocity is shown in the top-left panel of Figure 4 with a solid curve. Even though nearly identical to the radial velocity , it still is only a fraction of the total non-circular motion (dashed curve), which also includes the azimuthal velocity.

To turn this radial inflow velocity into a constraint on the mass inflow rate, we furthermore need to know the gas density in the nuclear spiral, as well as the geometry or the area through which the gas is flowing. The flux ratio of the [Sii]6716.4,6730.8 emission-line doublet included in the spectral range of the GMOS-IFU observations allows us to constrain the mean electron density over the observed field. Taking all values together yields an average flux ratio of , which, adopting the prescription by Shaw & Dufour (1994) and using the atomic parameters compiled by Mendoza (1983) and Osterbrock (1989), corresponds to  cm assuming a mean electron temperature of  K. Even though the flux ratio map might indicate a spiral structure similar to that in the non-circular motions, the signal-to-noise is not high enough to quantify the gas (over)density in the spiral arms from it. Instead, we apply the same elliptic annuli used to extract the harmonic coefficients to the flux ratio map. From the distribution within each annulus, we derive as a function of radius, resulting in the dashed curve in the top-right panel of Figure 4 (divided by 10 for illustrative purposes). Next, we assume that the width of the distribution, indicated by the solid curve, is driven by the density wave contrast, and hence yields an estimate of the electron (over)density within the spiral arms. Assuming a % lower (higher) electron temperate of  K, leads to an increase (decrease) in the electron densities by a factor of about 20%. In all cases, the density contrast is of the order of %, in agreement with hydrodynamic simulations (e.g. Englmaier & Shlosman, 2000; Maciejewski, 2004b) and -band imaging presented by Davies et al. (2009). Using the proton mass and a factor 1.36 to account for the presence of Helium, we convert to a gas mass overdensity .

For the geometry we assume in agreement with the spiral perturbation model that both the radial flow velocity and gas mass overdensity, at each radius in a disk with scale height , vary as a sinusoidal function in azimuthal angle . Integrating over the values for which the radial flow velocity is positive, the mass inflow rate then reduces to

 ˙M=mvinflowΔρgasπR2hR14m, (21)

for gas of density flowing in the spiral arms towards the center at a velocity . To estimate the scale height , we start from a marginally stable disk with Toomre’s (1964) , and substitute in a constant sound speed  km s, the epicycle frequency of the power-law axisymmetric gravitational potential, and , with inferred from the average electron density . The dashed curve in the bottom-left panel of Figure 4 shows as function of radius. Clearly, towards the center this leads to an unrealistically high scale height, so that we constrain to be not larger than the relative width of a spiral arm, resulting in the solid curve. Finally, in the bottom-right panel of Figure 4, we present the mass inflow rate (in M yr) as function of radius.

## 4. Discussion

We have argued for a two-arm spiral perturbation in the gravitational potential as the source of the three-arm spiral structure in the velocity field of NGC 1097. We verify the implied two-arm spiral distortion in the surface brightness, and discuss possible causes for contributions from additional harmonic terms. Finally, we link the mass inflow rate to the accretion onto the central BH.

### 4.1. Two-arm spiral perturbation?

An spiral perturbation in the gravitational potential that would explain the three-arm spiral structure in the velocity field of NGC 1097, implies an spiral distortion in the surface mass density and hence a two-arm spiral structure in the surface brightness.

Even if the latter distortion is too weak to measure directly, it might show itself through obscuration by correlated dust. Indeed, structure maps of NGC 1097 (e.g. Pogge & Martini, 2002; Martini et al., 2003; Fathi et al., 2006) show spiral-like features, but they are not evidently a two-arm nuclear spiral. The structure map in the right panel of Figure 1 is based on the Richardson-Lucy image restoration technique (Snyder et al., 1993), using a multi-step convolution of the HST/ACS high resolution camera FR656N image with a two-dimensional PSF model constructed using Tiny Tim (Krist & Hook, 1997). We have overplotted the structure map with the wavelet map from (Lou et al., 2001, their Fig. 1), which does seem to be consistent with a two-arm spiral structure, but additional spiral features cannot be ruled out.

The solid magenta curves show the two-arm spiral with pitch angle , which we inferred from the logarithmic spiral fitted to the harmonic terms (Section 3.2). The two open arms trace well the spiral structures in both the wavelet and structure map, except closer to the nuclear ring when, at least in the northern part, a more tightly wound spiral seems needed. The dashed magenta curves indicate the two additional arms in case of an spiral perturbation with the same pitch angle. Such higher-order even harmonic terms can result from non-linear coupling of modes, and —although smaller in amplitude then the spiral perturbation— might give rise to possibly additional spiral features. Furthermore, a spiral perturbation driven by the large-scale bar is not necessarily restricted to an harmonic term222For example, the axisymmetric power-law potential in equation (19), being perturbed by replacing the radius by with , creates besides also higher-order even harmonic terms.. Still, as long as the perturbation is bi-symmetric all resulting harmonic terms are even.

Nonetheless, Prieto et al. (2005) note in high-resolution VLT/NACO infrared images a central spiral with a three-arm symmetry, though one of the three arms does not seem to continue towards the nuclear ring, but instead splits into a number of spiral filaments. Davies et al. (2009) show the inner of the NACO -band residual image from Prieto et al. (2005) together with their SINFONI -band residual image (their Fig. 1). While the third arm is already weaker in the -band (in particular when taking into account the narrow intensity scaling that is saturating the two strong arms), it nearly disappears in the -band. Also, the residual flux distribution of (warm) H, as traced by the 2.12 m 1–0 S(1) line (their Fig. 3), reveals two strong arms. Nevertheless, Davies et al. (2009) claim a (weak) third arm in the stellar and gas density to give rise to the two-arm spiral structure they argue to see in the residual H velocity field (their Fig. 5). However, when the inner of our [Nii] non-circular motions (third panel of Fig. 2) are overlayed by their H residual velocity field (as in their Fig. 6), we believe the (observed and predicted) three-arm spiral structure is traced inwards by the H kinematics. This apparent agreement is not obvious as their near-infrared observations are less affected by dust than our optical measurements, but also because it is not evident that (warm) H emission and ionized emission trace the gas kinematics in the same way. Even so, we find that the data presented by Davies et al. (2009) do not contradict our interpretation.

As indicated by Davies et al. (2009), the kinematics of the stars do not show any significant deviations from axisymmetry (see their Fig. 2). Indeed, it is expected that the bulge stars are dominating the stellar kinematics, and that the perturbation is only visible in the intensity due to dust extinction in the equatorial plane. Since the stability of the gas in the equatorial plane inhibits self-amplification (see also Davies et al., 2009), the observed spiral pattern in the gas is most likely due to a weak spiral perturbation in the total, stellar-dominated, gravitational potential. The most natural driver of this perturbation is the large-scale bar, which might induce spiral shocks in the gas (e.g. Maciejewski, 2004a, b). Davies et al. (2009) argue for these shocks to be present in the inner region of NGC 1097, based on a large amplitude of the radial motions with respect to the velocity dispersion of the gas. However, as discussed above (see also top-left panel of Fig. 4), only part of the observed non-circular motions might be radial motions, and shocks might just be the trigger to create long-lived gas density waves (Englmaier & Shlosman, 2000; Ann & Thakur, 2005). Whether spiral shocks and/or bulge stars ionize the perturbed gas, the spiral perturbation can be traced through non-circular motions in the observed ionized gas kinematics.

In this way, we find a three-arm structure in the [Nii] non-circular motions, which we believe is consistent with the residual H velocity field, as well as with two spiral arms visible in the residual H flux, -band image, and -band image, presented by Davies et al. (2009). All this leads to our interpretation of a weak two-arm spiral perturbation in the gravitational potential driven by the large-scale bar; though further modeling is needed to find out if for example the spiral pattern in the gas are induced by shocks or are long-lived density waves. Instead, the explanation by Davies et al. (2009) of a three-arm spiral perturbation in the gravitational potential, requires quite a special driving mechanism, such as non-linear interactions between the large-scale bar and a nuclear bar or a (dark) massive orbiting compact object, which so far have not been demonstrated in models. In addition, observed structure in addition to that expected from a two-arm spiral perturbation in the gravitational potential, including a possible weak third arm, might be the result of asymmetric dust obscuration, as we discuss next.

The coefficients and , that provide the phase-shift , may be “contaminated” by additional contributions to the non-circular motion, but in general they are not expected to result in a smooth variation of the phase-shift with radius, as seen for a spiral structure. An exception is the harmonic term, since the measured coefficient also incorporates the circular velocity as , which in general dominates over the non-circular motions and . This results in a phase-shift that is everywhere close to zero, and due to the degeneracy in there is little hope of constraining a two-arm spiral perturbation from its contribution to the harmonic term in the velocity field. However, we can predict its pitch angle and phase-shift from the (logarithmic) spiral inferred from the contribution to the harmonic term: and . As a result, we might use as an estimate of the non-circular contribution to . The effect is shown in the top-right panel of Figure 3, where the black dashed curve is of the (power-law) axisymmetric potential, which after taking into account yields the blue dashed curve that matches the measured coefficients indicated by the blue diamonds. This provides a novel way to correct for non-circular motions, which otherwise might, for example, lead to an underestimation of the (inner) slope of the mass distribution (e.g. Hayashi & Navarro, 2006).

Figure 3 shows that, besides and harmonic terms in the velocity field expected from a perturbation in the gravitational potential, the non-circular motions also seem to contain an harmonic term. There are several effects that might (partly) cause this additional contribution.

As shown by Schoenmakers et al. (1997, their equation 7), an error in the (kinematic) center results in a spurious and contribution to the line-of-sight velocity as

 δVlos = V⋆[(1+α)δx′2R (22) −(1−α)(δx′2Rcos2ψ+δy′2Rsin2ψ)],

We see that the effect on and vanishes if , i.e., when the circular velocity curve increases linearly with radius, . As in most galaxies, the latter is also the case in the inner region of NGC 1097, but still is significantly positive within the central . Not surprisingly, varying the kinematic center (in the process of finding the best-fit set of ellipses as described in Section 2.1) does not remove the harmonic contribution, but in contrast makes the overall fit worse. This makes a significant effect due to an incorrect center unlikely.

Based on the analysis in Section 2.3, we expect a similar contribution of both and harmonic terms to the velocity field from an distortion of an axisymmetric distribution. It is unlikely that the gravitational potential itself is lopsided since the stars that dominate in mass do not show any such signature. Also, the nuclear spiral itself is expected to be bi-symmetric if it is indeed a gas density wave driven by the large-scale bar. Still, the phase-shift in the bottom-middle panel of Figure 3 seems to vary smoothly, and even close to linearly as a function of within the central where prominent dust features are present. Moreover, the amplitude of the slope is similar to the linear relation of versus in the bottom-right panel, but with negative instead of positive sign. This is consistent with a logarithmic spiral with the same pitch angle but orientated clockwise instead of anti-clockwise, i.e., leading instead of trailing. In principle, both leading and trailing nuclear spirals can exist, as shown by Wada (1994), but in their simulations the smaller leading spiral dissolves before the long-lived larger trailing spiral fully emerges. Alternatively, the spiral-like contribution to the harmonic term, appearing like a “negative image”, might result from an asymmetric dust obscuration, mimicking a lopsided distortion.

The often prominent dust lanes along the leading edges of bars in galaxies are associated with shocks in the gas streaming along the length of the bar (e.g. Athanassoula, 1992), which in turn lead to velocity jumps across the dust lanes (e.g. Mundell & Shone, 1999). Moreover, numerical models of dust lanes (e.g. Gerssen & Debattista, 2007) as well as analytical models of diffuse disks (e.g. Valotto & Giovanelli, 2004) show that dust extinction can have a significant effect on the velocity along the line-of-sight. Henceforth, we expect the dust and possible shocks associated with the spiral features to distort the velocity field, but the modeling required to understand the specific effects on the non-circular motion is beyond the scope of this paper.

### 4.3. Feeding the central black hole?

The Eddington accretion rate onto a central black hole

 ˙MEdd=2.2M⊙yr−1(ϵ0.1)−1(MBH108M⊙), (23)

adopting for the radiative efficiency, and a mass  M for the central black hole in NGC 1097 —based on the measured central stellar velocity dispersion of  km s (Lewis & Eracleous, 2006) substituted in the relation (Tremaine et al., 2002)— yields  M yr. This value is indicated in the bottom-right panel of Figure 4 by the upper dashed horizontal line. The lower dotted horizontal line at is the approximate transition from a Seyfert1 to a LINER active galactic nucleus (see for a review Ho, 2005), and thought to be the rate below which mass accretion becomes radiatively inefficient (see for reviews Quataert, 2001; Narayan, 2005).

Whereas NGC 1097 is typically classified as a LINER galaxy, monitoring of the nucleus reveals evolution in its activity up into the Seyfert1 regime (Storchi-Bergmann et al., 2003). Nemmen et al. (2006) find that the observed optical to X-ray spectral energy distribution in the nucleus of NGC 1097 is consistent with an inner radiatively inefficient accretion flow plus outer standard thin disk, with a mass accretion rate of . We find a mass inflow rate, down to  M yr() at a distance ( pc) from the center, where the gas kinematics are still accurately measured and well described by the two-arm spiral perturbation model. These constraints are obtained at very different scales (tenths versus tens of pc), and the mass inflow rate and the onset of nuclear activity are not necessarily linked in time. Even so, it is encouraging that we obtain comparable values.

Our mass inflow rate is significantly lower than  M yr estimated by Storchi-Bergmann (2007) at a distance  pc. In the latter estimate, a non-circular motion of  km s is adopted for the inflow velocity, whereas we find from our spiral model that even though the contribution of the radial inflow dominates over elliptic streaming, it is still only a fraction ( km s at  pc) of the non-circular motions. Moreover, the estimated electron density of  cm is similar to the average density we derive directly from emission line ratios, but more than an order of magnitude higher than the expected (over)density in the spiral arms ( cm at  pc). Finally, instead of assuming a fixed opening angle of , we incorporate the geometry from the spiral model, and allow for the scale height to vary, in order for the disk to remain marginally stable. In this way, we obtain an estimate of the mass inflow rate of similar value as the expected mass accretion rate onto the central BH in NGC 1097, without having to invoke an unclear filling factor (e.g. Storchi-Bergmann et al., 2007). Even so, the mass inflow rate might be even further constrained, in particular by measuring emission line ratios at higher signal-to-noise to obtain a more accurate estimate of the (over)density, as well as using other emission lines to break the degeneracy with temperature.

## 5. Summary and conclusions

We presented harmonic expansion of the line-of-sight velocity field as a suitable method to identify and quantify possible structures in the non-circular motions, including nuclear spirals. We confirmed earlier findings (Canzian, 1993; Schoenmakers et al., 1997) that a weak perturbation in the gravitational potential of harmonic number , causes the surface brightness to also exhibit an distortion, but leads to and harmonic terms in the velocity field. In the case of a -arm spiral perturbation in the gravitational potential, we found that the corresponding -arm and -arm spirals in the velocity field are respectively less and more tightly wound, with pitch angles approximately related as . In Appendix A, we derived an analytic perturbation model, which allows for a simple estimate of the fraction of the measured non-circular motions that is due to radial flow.

We applied this method to the emission-line velocity field within the circumnuclear star forming ring of NGC 1097, obtained with the GMOS-IFU spectrograph. The radial variation of the resulting harmonic terms can be fitted with a logarithmic spiral with a pitch angle . We linked these harmonic terms in the velocity field to a weak perturbation of the gravitational potential due to a two-arm nuclear spiral with an inferred pitch angle . This predicts a two-arm spiral distortion in the surface brightness, as hinted by the dust structure in central images of NGC 1097, although additional spiral structure might be present as a result of higher-order even harmonic terms. Furthermore, this two-arm spiral perturbation of the gravitational potential adds a combined and spiral structure to the velocity field, as revealed in the non-circular motions of the ionised gas in the center of this galaxy. We argued that it is unlikely that the presence of also harmonic terms in the velocity field is due to lopsidedness in the gravitational potential nor in the nuclear spiral itself, which we expect to be due to long-lived density waves in the gas driven by the bi-symmetric large-scale bar. Instead, we postulated that an asymmetric dust obscuration mimics a lopsided distortion, and gives rise to the additional even harmonic terms in the velocity field.

To match the measured odd harmonic terms in the velocity field of NGC 1097, a spiral perturbation model with a large amount of radial damping is required. This indicates both strong dissipation and, that, in addition to elliptic streaming, a significant fraction of the non-circular motions is due to radial inflow. We combined the inferred radial inflow velocity with the gas density in, and the geometry of, the spiral arms to estimate the mass inflow rate as function of radius. We calculated the gas density from the variation in the flux ratio of the [Sii]6716.4,6730.8 emission-line doublet, and we incorporated in the geometry the scale-height assuming a marginally stable disk. The resulting mass inflow rate decreases to  M yr at a distance of  pc from the center, down to where our measurements are still reliable. We showed that this corresponds to , where the latter Eddington accretion rate is onto a black hole in NGC 1097 with a mass  M based on the central stellar velocity dispersion. This mass inflow rate is consistent with the active galactic nucleus in NGC 1097 varying between LINER and Seyfert1, but comparison with mass accretion models are hampered by the (still) very different physical scales as well as possible time delay between mass inflow and the onset of nuclear activity.

We conclude that the line-of-sight velocity can provide not only a cleaner view on nuclear spirals than does the associated dust, but that the presented method also allows one to quantitatively study these possibly important links in fueling the centers of galaxies, including a handle on the mass inflow rate as a function of radius.

## Acknowledgments

It is a pleasure to thank Eva Schinnerer for helpful suggestions, and Paul Wiita for discussions and a critical reading of the manuscript. We thank the referee, Eric Emsellem, for constructive comments on this work. GvdV acknowledges support provided by NASA through Hubble Fellowship grant HST-HF-01202.01-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS 5-26555. KF is supported by the Swedish Research Council (Vetenskapsrådet), and gratefully acknowledges the hospitality of the Institute for Advanced Study, to which a visit contributed greatly in completing this project. In Figure 1, the VLT/VIMOS images (left panel) have been taken and pre-processed by European Southern Observatory (ESO) Paranal Science Operation astronomers, with additional image processing by Henri Boffin (ESO), and the VLT/NACO images (middle panel) is based on research published in the October issue of Astronomical Journal, vol. 130, p. 1472.

## Appendix A Gaseous orbits in a weakly perturbed gravitational potential

We derive analytic solutions of gaseous orbits in a weakly perturbed gravitational potential (for modeling of a strong perturbation see e.g. Spekkens & Sellwood, 2007). We follow the treatment of Binney & Tremaine (1987, p. 146-148) for a weak harmonic perturbation. Like Schoenmakers et al. (1997), we include an additional phase-shift as in equation (5) to accommodate for a spiral perturbation. Since gas at the same spatial location has the same velocity, we look for closed loop orbits that do not intersect themselves. However, to take into account the dissipative nature of gas, we include, like Wada (1994), radial damping, so that the first-order equation of motion becomes

 ¨R1+2λκ0˙R1+κ20R1=−R0(Acosη+Bsinη),η=m[ϕ0(t)−ϕm(R0)], (A1)

where controls the amount of radial damping. The subscript zero refers to zeroth-order with constant and , with the pattern speed of the perturbation. We have introduced the epicycle frequency, , and angular frequency, , with the logarithmic slope of the circular velocity, , of the axisymmetric part of the gravitational potential. Furthermore,

 A=ΦmR2[21−ω+dlnΦmdlnR],B=ΦmR2mcotζm, (A2)

where we have introduced , and is the pitch angle as defined in equation (11). Solving equation (A1) we find that the solution for closed loop orbits is

 R = R0[1−(acosη+bsinη)], (A3) ϕ = ϕ0+[2(asinη−bcosη)−ξsinη]/[m(1−ω)] (A4) vR = vc{m(1−ω)[asinη−bcosη]}, (A5) vϕ = vc{1+(1+α)[(a−ξ)cosη+bsinη)}, (A6)

with appearing in the last line because of a first-order conversion from guiding center to a point in the observed velocity field. Moreover,

 a=AΔ−BΛΔ2+Λ2,b=BΔ+AΛΔ2+Λ2,ξ=1κ2ΦmR221−ω, (A7)

where we have defined and . The orbit solutions have a singularity at the corotation radius where (), because the adopted epicycle approximation breaks down. Without radial damping (), the (collisionless) orbit solutions also have singularities when , i.e., at the Lindblad resonances given by .

Next, we assume to first order and replace it by which is zero along the line of nodes. We also define , so that . This allows us to recast the above expressions for and in multiple angles of as in equations (7) and (8), with

 cR = −m(1−ω)[asin(mφm)+bcos(mφm)], (A8) sR = m(1−ω)[acos(mφm)−bsin(mφm)], (A9) cϕ = (1+α)(a−ξ)cos(mφm)−(1+α)bsin(mφm). (A10) sϕ = (1+α)(a−ξ)sin(mφm)+(1+α)bcos(mφm), (A11)

Substitution in equation (6) results in the expression for the line-of-sight velocity in equation (9), where the harmonic coefficients are given by

 cm±1 = A±cos(mφm)−B±sin(mφm) (A12) sm±1 = A±sin(mφm)+B±cos(mφm), (A13)

with

 A±=12V⋆{[(1+α)∓m(1−ω)]a−(1+α)ξ},B±=12V⋆[(1+α)∓m(1−ω)]b. (A14)

Substituting and , with

 K2m±1=A2±+B2±,tanθm±1=B±/A±, (A15)

we arrive at the form given in equation (2.4).

In Figure 5, we present an example of a weakly perturbed axisymmetric logarithmic potential defined in equation (19), with  km s,  kpc and . The amplitude of the perturbation is a factor times the axisymmetric logarithmic potential, while the angular dependence is due to a logarithmic spiral defined in equation (12), with pitch angle . This results in non-circular motions which contribute a 1-arm and 3-arm spiral structure to the observed line-of-sight velocity field as shown in Figure 5 for an adopted inclination of .

In general, and in equation (A15) above depend in a rather complicated way on the gravitational potential. However, in case of the linear spiral density-wave theory (Shu et al., 1973; Canzian & Allen, 1997) the expressions reduce to

 Km±1=12vmsini[κ2Ω∓m(Ω−Ωp)κ],tanθm±1=±cotζm, (A16)

where the constant amplitude measures the strength of a tightly wound spiral without radial damping. In this case, and , so that , and in equation (A7), and hence and . The expression for in equation (A14) is proportional to the reduced expression for in equation (A16), since by substituting and , the term in square brackets in equation (A16) is proportional to in equation (A14).

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