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Revista Mexicana de Astronomıa y Astrofısica, 45, 143–154 (2009)

THE 3D VELOCITY STRUCTURE OF THE PLANETARY

NEBULA NGC 7009

W. Steffen,1 M. Espındola,2 S. Martınez,3,4 and N. Koning5

Received 2008 October 15; accepted 2009 May 13

RESUMEN

En busca de desviaciones de una expansion homologa en nebulosas plane-tarias presentamos un modelo 3D morfo-cinematico de NGC 7009. El modelo hasido construido con Shape basado en diagramas posicion-velocidad de la literaturae imagenes del HST. Encontramos que los datos son congruentes con un perfil develocidad radial con un gradiente mayor a latitudes comparado con el de la regionecuatorial (Modelo 1). En un segundo modelo supusimos una componente de ve-locidad radial que aumenta de manera lineal con una componente poloidal adicionaldel orden de 10 km s−1 a latitudes alrededor de 70◦. El verdadero campo de ve-locidad probablemente es intermedio entre estos casos lımites. Encontramos que laexpansion de los ansae no es radial con respecto a la estrella central. Su campo develocidad parece apuntar en una direccion cercana al punto de salida de la cascaraprincipal. Predecimos un patron para los movimientos propios en el Modelo 2.

ABSTRACT

In search for deviations from homologous expansion in planetary nebulaewe present a 3D morphokinematical model of NGC 7009. The model has beenconstructed with Shape based on PV diagrams from the literature and HST images.We find that the data are consistent with a radial velocity field with increasedgradient at high latitudes compared to the equatorial region (Model 1). In a secondmodel we assume a linearly increasing radial velocity component with an addedpoloidal component of order 10 km s−1 at latitudes around 70◦. The true velocityfield is likely to be in between these two limiting cases. We also find that theexpansion of the ansae is non-radial with reference to the central star. Their velocityfield is focused near the apparent exit points from the main shell. We predict theproper motion pattern for the model with a non-zero poloidal velocity component.

Key Words: ISM: jets and outflows — planetary nebulae: individual (NGC 7009)— stars: AGB and post-AGB — stars: mass loss

1. INTRODUCTION

In recent years it has become clear that a com-plex three-dimensional (3D) structure of a planetarynebula (PN) poses a major challenge to the interpre-tation of its formation process. As has been shownfor NGC 7009 by Goncalves et al. (2006), eventhe interpretation of spectroscopic measurements for

1Instituto de Astronomıa, Universidad Nacional

Autonoma de Mexico, Mexico.2Instituto Politecnico Nacional, Mexico.3Instituto Nacional de Astrofısica, Optica y Electronica,

Tonanzintla, Mexico.4Benemerita Universidad Autonoma de Puebla, Mexico.5Department of Physics and Astronomy, University of Cal-

gary, Canada.

abundance determinations may be seriously affectedby the 3D structure of an object. Furthermore ex-pansion parallax determinations of badly needed im-proved distances to PNe are dependent on the ac-curacy of 3D models of their structure and velocityfield (e.g. Terzian 1997; O’Dell, Henney, & Sabbadin2009). It is well known that orientation effects area problem for any structural classification scheme orstatistical studies of planetary nebulae (e.g. Balick& Frank 2002). It is therefore crucial to obtain 3Dmorphokinematical information and reconstructions.

NGC 7009 is a planetary nebula with severalmorphological and kinematical sub-systems withmultiple shells, a halo, jet-like streams, ansae and

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144 STEFFEN ET AL.

small-scale filaments and knots. Sabbadin et al.(2004; STC04, for short) have produced a tomo-graphic 3D-reconstruction of NGC 7009 based onlong-slit observations at the ESO NTT telescopewith the EMMI instrument. Their reconstructionis based on the assumption of a linear and radial,i.e. homologous, velocity field.

Homologous expansion is evidenced in axisym-metric objects that show the same structure in di-rect images and in their echellograms (or position-velocity, P-V, diagram). The P-V diagrams encodethe velocity along the line of sight due to Doppler-shift of the spectral line. In homologous expansionthe Doppler-shift corresponds directly to the posi-tion along the line of sight, except for a constantfactor. In axisymmetric objects this factor can bedetermined by fitting an axisymmetric morphokine-matic model simultaneously to the image and PVdiagrams. If such a constant factor can not be foundto fit the whole image and PV diagrams, then de-viations from a homologous expansion are present.These might be in the form of a non-linear depen-dence of the radial velocity (radially outward) as afunction of distance from the central star or a non-zero poloidal velocity component. Deviations froma linear correspondence between image and PV di-agram may, however, also be due to structural de-viations from axisymmetry. In such a case, if thevelocity field is known and monotonous in its com-ponents, the 3D emission structure can be inferredunambiguously from the image and P-V diagrams.However, if the velocity field is not well known andthe object is intrinsically 3D, then ambiguities arisein the determination of the true 3D structure.

Hydrodynamical simulations show that in wind-driven ellipsoidal and bipolar nebulae, deviationsfrom a homologous expansion may introduce sub-stantial distortions in reconstructions of the 3Dstructure that assume homologous expansion (Stef-fen, Garcıa-Segura, & Koning 2009). We are there-fore undertaking a study of well-resolved planetarynebulae with detailed imaging and internal velocitymeasurements (using Doppler-effect and/or internalproper motion) in order obtain 3D reconstructionsthat take into account such deviations. In this pa-per we analyze the well-observed nebula NGC 7009in search of morphokinematical solutions that mightprovide information about the dynamical state of thenebula.

STC04 find that the large scale structure of themain shell is best described as a triaxial ellipsoid.Guerrero, Gruendl, & Chu (2002) find extended X-ray emission which precisely fills the main shell.

Since the shocked fast stellar wind is expected toproduce such x-ray emission, this observation sup-ports the existence of a wind or at least a high pres-sure bubble that drives the expansion. Steffen et al.(2009) showed that morphokinematical reconstruc-tions of such nebulae, when based on a homologousexpansion law, lead to an overestimation of the bub-ble cross section along the line of sight at mid to highlatitudes. In the reconstructions by STC04, such adifference in the cross sections along the line of sightand in the plane of the sky is clearly present.

In this paper we therefore estimate the deviationsfrom homologous expansion that are still consistentwith the current imaging and kinematical data whilemaintaining approximately the same cross section ofthe main shell in the plane of the sky and along theline of sight. We construct a model with purely ra-dial velocities but increasing velocity gradient withdistance (Model 1) and a linearly increasing radialcomponent with an additional poloidal component.With these boundary conditions the models shouldprovide upper limits for these velocity components.In order to allow testing of these results we predictinternal proper motion patterns to be compared withfuture observations.

Fernandez, Monteiro, & Schwarz (2004) providedproper motion measurements of the eastern ansa inNGC 7009 that revised the values obtained by Liller(1965). They also detected expansion of the mainshell, but did not quantify it, since they suggestedthat it might be a moving ionization front, whichdoes not provide the velocity of the bulk motion ofthe gas. However, STC04 find that the nebula isfully ionized, except possibly in the ansae and thecaps, such that the concern about a misinterpreta-tion of the measured velocities might not be justified.Rodrıguez & Gomez (2007) have recently used radioobservations with the Very Large Array (VLA) todetermine the tangential velocity of ansae and findthat the western ansa might be faster than the east-ern one by a factor of 1.5. However, the error boundsof the two values overlap, and hence a similar veloc-ity for both is not excluded (especially consideringthe fact that their distance to the central star differsby less than ten percent). Based on our models wepredict proper motion patterns of the main shell andthe ansae, which are crucial to eliminate or reduceambiguities in future models.

The tomographic reconstruction by STC04 usesground-based position-velocity diagrams at regularintervals of 12 position angles. Although these areof excellent quality, the coverage of the object is notcomplete and there are gaps between slits which had

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THE 3D VELOCITY STRUCTURE OF THE PLANETARY NEBULA NGC 7009 145

Fig. 1. The first and second image of NGC 7009 areHST images in Hα and [NII] adapted from Sabbadin etal. (2004). The third and fourth image are our Shape

model based on [NII] position-velocity diagrams from thesame paper and HST images (see Figures 3 and 4). Thebottom image has been rendered with red and blue colorcoding according to the Doppler-shift. The spatial reso-lution in the model images is 0.5 arcsec.

to be interpolated. Furthermore, the spatial reso-lution is seeing-limited. In this work we thereforeintend to improve the spatial resolution and fill ingaps by taking into account high-resolution imaging

Fig. 2. The mesh representation of our 3D-model ofNGC 7009. These meshes enclose the volumes that cor-respond to the various features of the Shape-model. Thevolumes are sampled with randomly distributed parti-cles. Highly tunable 3D-functions are then used to quan-tify the relative emission and velocity fields.

information from the Hubble Space Telescope in anew, manual, reconstruction with our morphokine-matic modeling package Shape.

The layout of our paper is as follows. In § 2 wedescribe the modeling procedures and § 3 containsthe results and discussions for the main structuresof NGC 7009. In § 4 we summarize our conclusions.

2. MORPHOKINEMATIC MODELING WITHSHAPE

We have modeled the morphokinematical struc-ture of NGC 7009 with the Shape software (version2.7) which is a rather new approach to reconstructingthe 3D structure of astrophysical objects from imageand Doppler-data (Steffen & Lopez 2006; Steffen etal. 2009). The model consists of a mesh-structurefor each significant large-scale and small-scale struc-ture that has been identified in the image data (Fig-ure 1). The meshes have been constructed with themodeling tools that Shape provides for this purpose.Figure 2 shows the resulting 3D mesh with the ori-entation that corresponds to Figure 1.

Note that the reconstruction of the 3D structurebased on 2D images and PV diagrams is not a uniqueprocess and relies on assumptions that allow us tomap the Doppler-velocity to a spatial position alongthe line of sight. The assumptions can involve thestructure of the object, the velocity field or both.Often some kind of symmetry is assumed to reduceambiguities. Magnor et al. (2005) have developedan algorithm to reconstruct nebulae with a cylindri-cal symmetry that relies on 2D images only. Theassumption of a homologous expansion, i.e. a veloc-ity vector that is proportional to the position vector,allows limited reconstruction of arbitrary spatial dis-tributions based on Doppler velocity measurementsto within a constant of proportionality along the lineof sight. If the extent of the object along the line of

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146 STEFFEN ET AL.

sight can be inferred from other sources, e.g. largescale symmetry, then this reconstruction is unique.The reconstruction of NGC 7009 by STC04 is of thistype.

Many planetary nebulae appear to show a ho-mologous expansion as evidenced by the similarity ofstructures in their images and PV diagrams. How-ever, we would like to point out that caution hasto be exercised when relating the homologous ex-pansion of a highly non-spherical shell-like structurewith velocity fields in hydrodynamical simulations.The high density shells represent the same hydrody-namical or ionization feature in different directionsas seen from the central star. Such a feature could bean ionization front or the cooling shock region causedby the fast stellar wind impacting on the slow AGB-wind. In a spherically symmetric object we then seea spherical shell. This shell is of course not emptyinside and is surrounded by external material. Alongeach direction from the central star we then have avelocity profile as a function of distance. As shownin detail by hydrodynamical simulations, the veloc-ity profile along such a “ray” may be very complexand usually is far from linear (e.g. Schonberner etal. 2007), i.e. along a single ray, the expansion isnon-homologous.

However, the story is very different when we con-sider the same hydrodynamical feature in differentdirections. Usually, due to varying ambient densi-ties in different directions, the expansion speed ofthe same hydrodynamical feature changes with di-rection. This is the main reason for the non-sphericalshapes of planetary nebulae in the generalized inter-acting stellar wind model (Kahn & West 1985; Bal-ick 1987). In a first approximation, the expansionalong each direction is independent from the otherdirections and features are “sorted” in a similar fash-ion as in a ballistic expansion: the faster, the fur-ther away, such that a near-homologous velocity fieldarises for each hydrodynamical or ionization feature.This means that flow problem admits a similarity so-lution (e.g. Kahn & West 1985). However, the pres-sure is not exactly constant along surfaces of thesestratified shells. Therefore non-radial velocity com-ponents and deviations from the homologous radialexpansion may arise. Two- or three-dimensional nu-merical hydrodynamical simulations show such devi-ations from radial motion (e.g. Mellema, Eulderink,& Icke 1991; Steffen et al. 2009). The simulationsexhibit deviations from a linear variation of the ra-dial velocity component with distance as well as non-zero poloidal velocity components. Fortunately, ex-cept for local small-scale deviations, the radial com-

ponent increases monotonically with distance andthe poloidal component varies rather smoothly withpoloidal angle. This behavior reduces considerablythe ambiguities inherent in the general problem ofmapping the Doppler-velocity to a position along theline of sight.

The classification and characterization of the ve-locity field of observed planetary nebulae is usuallydone based on such hydrodynamical or ionizationshells. In order to judge with any degree of reli-ability, whether the velocity field is homologous ornot, only such coherent structures can be used tocompare their structure in images and PV diagramsof a single emission line.

In contrast, to obtain a velocity field that can becompared with one-dimensional simulations, the ve-locity field across such shells has to be determined.To some extend this can be achieved consideringdifferent ionic species which trace different regionsalong any given direction. This type of analysishas been done, for example, by Wilson (1950) andrecently by Sabbadin et al. (2004, and referencestherein).

An important fact to take into account here isthat the velocity as a function of distance (measuredin different directions) of a single non-spherical shell-like feature is expected to differ from that measuredalong a single radial direction (which crosses all hy-drodynamical structures). Hence, the 3D velocityfields for different shells may also be very differentfrom each other. If different ionic species occupydistinct spatial regions, their velocity fields with dis-tance and poloidal angle can not be assumed to bethe same. For a reliable 3D reconstruction of an ob-ject it is therefore very important to distinguish thesedifferent shells and reconstruct or model them inde-pendently. Shape is excellently suited for this kind ofanalysis, since it allows the user to judge which fea-tures to reconstruct independently, even in a singleline observation.

Our main aim is to obtain information on thedeviations of the velocity field from homologous ex-pansion within these coherent shell structures.

Instead of making a strong assumption about thevelocity field, we make an assumption about thestructure along the line of sight. The main assump-tion is that the cross section along the line of sightshould be the same or very similar to the cross sec-tion in the plane of the sky at the same position alongthe axis of the main shell. A secondary, but usefulconstraint is that the streams should touch the tipsof the main shell, both in the plane of the sky as wellas along the line of sight.

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THE 3D VELOCITY STRUCTURE OF THE PLANETARY NEBULA NGC 7009 147

Fig. 3. The middle column shows [NII] position-velocitydiagrams of NGC 7009 at 6 different position angles sep-arated by 15◦ from each other (adapted from Sabbadinet al. 2004). The corresponding model PV diagrams areshown to the left (Model 1) and right (Model 2). Notethat the grayscale of the observations is logarithmic andthat of the observations is square-root (see main text).The spatial resolution in the PV diagrams is 0.5 arcsecand the velocity resolution is 5 km s−1.

In our model, velocity components are separable,i.e. the poloidal velocity is only dependent on the“latitude” and the radial velocity is only dependent

Fig. 4. Same as Figure 3, but for six additional positionangles.

on distance. This introduces a further constraintby reducing the degrees of freedom but relaxes theconstraint of a homologous expansion.

The kinematics of the outer shell indicate a sys-tematic blue-shift by approximately 2–3 km s−1 withrespect to the main shell. The origin of such a shiftcan be either structural, i.e. it is closer to the ob-server. It could be truly slower, due interaction witha density gradient or proper motion in the ambient

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148 STEFFEN ET AL.

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Fig. 5. The relative emissivity distribution as a functionof azimuthal angle θ around the axis of the object forthe north-western (solid line) and south-eastern (dashedline) ring and the oblique equatorial ring (dotted line).Emissivities are normalized to their maximum value foreach component.

medium. The effect is small and has been neglectedin our model, which is concerned mostly with thekinematics of the main shell. Any effects on the mainshell would be unnoticeable.

We present two solutions that are qualitativelydifferent in terms of their velocity field. In Model 1we have a radial velocity field with a monotonousnon-linearly increasing velocity magnitude. InModel 2 a linearly increasing radial velocity compo-nent and a non-zero poloidal component are applied.The azimuthal component is always assumed to bezero. These assumptions are based on the resultsfrom our recent study of the velocity fields in hydro-dynamical simulations of typical ellipsoidal and bipo-lar planetary nebulae (Steffen et al. 2009), wherea single dense hydrodynamical shell has been ex-tracted and analyzed. The simulations always showboth an increasing gradient for the radial compo-nent and a non-zero poloidal component at mid lati-tudes. Cylindrical and mirror symmetry in the sim-ulations demand that the poloidal velocity be zeroat the equator and at the axis.

We start with an ellipsoidal structure along theline of sight that is the same as that of the image out-line in the plane of the sky. This structure and thevelocity field are then iteratively adjusted to conformto the observed PV diagrams. While this proceduredoes not provide a unique solution, it is expected toprovide estimates for the deviations from a homolo-gous expansion. Based on the results we then makepredictions for the expansion in the plane of the sky,which can be verified with future proper motion mea-surements of nebular features.

Fig. 6. Images of the Shape-model as seen from selectedviewpoints. In the top row the observer’s point of view isrotated around the horizontal axis by the labeled angles.In the bottom row the viewpoint is rotated around thevertical axis by the specified angle. The spatial resolutionin these images is 0.5 arcsec.

Our modeling of NGC 7009 is based on im-ages from the Hubble Space Telescope (HST) andposition-velocity (PV) diagrams from STC04. Pleaserefer to this paper for details on the observations.Figure 1 shows the HST [NII] image and our corre-sponding model image. Although, for consistency,the images and PV diagrams of additional spectrallines have been consulted, we concentrate on themodeling of the [NII] emission. [NII] shows the high-est contrast between features and is more likely tobe concentrated along surfaces or in discrete knottystructures. These properties provide a less ambigu-ous mapping between imaging and spectral features.Note that the figures in STC04 are displayed with alogarithmic grayscale, i.e. the contrast between fea-tures is greatly reduced. In our work we focus on therelative positions and velocities of the structures inNGC 7009, rather than their relative brightness, sothat the key information is in the outlines and posi-tions of the features. Currently, Shape cannot repro-duce these high contrast structures and therefore therelative brightness should be taken to be only qual-itative, intended for identification of the individualfeatures and for visualization purposes.

The general modeling strategy is as follows. Fromthe overall characteristics of the images (Figure 1)and PV diagrams (Figures 3 and 4, adapted fromSTC04), as well as previous models and reconstruc-tions, a first approximation of the model is cre-

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THE 3D VELOCITY STRUCTURE OF THE PLANETARY NEBULA NGC 7009 149

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Fig. 7. The radial velocity component is shown as afunction of distance from the central star. The contin-uous line is for the main shell in Model 1. The dashedline is that of Model 2, which has a poloidal velocitycomponent as show in Figure 8. The same radial veloc-ity law (dashed line) has been adopted for the streamsand ansae (4.1 km s−1 arcsec−1). The dot-dashed lineis the velocity law adopted for the envelope, caps andequatorial rings (3.6 km s−1 arcsec−1).

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Fig. 8. The poloidal velocity component of the main shellis shown as a function of latitude (with the equator at90◦. The eastern tip of the main shell is at 0◦ and thewestern tip is at 180◦.

ated by interactively constructing the mesh struc-ture. This mesh structure is filled with a randomdistribution of particles (see below). Initially, a ho-mologous expansion is assumed for the velocity field,according to the results from STC04. Then the gen-eral orientation of these structures is changed to fitthe tilt of the PV diagrams. For the main and theouter shell we first assume that the cross-sectionperpendicular to the axis is circular. We then ap-ply large scale first order corrections to the shape

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Fig. 9. The model’s relative emissivity of the main shellis plotted as a function of the distance along the axis z

(top) and as a function of azimuthal angle θ (bottom).The distribution in angle is different for the northern(continuous line) and the southern half (dashed line) withpeaks at nearly opposite angles. Negative z-values in thetop graph are toward the East.

and velocity field. These corrections include triaxialstructure, bending and rotation around the axis. Forthe velocity field we then include a deviation from alinear radial velocity component (Model 1) or a non-zero poloidal component according to Steffen et al.(2009) (Model 2). We then make second-order cor-rections for the structure of the main shell to fit theoutline of the HST images. After these, similar ad-justments are made to the structure along the lineof sight and in the velocity field in order to fit thesmall scale structure of the P-V diagrams. Adjust-ments are then made to small-scale features in thebrightness distribution. This process is iterated untila satisfactory model is obtained.

A random distribution of particles is located ei-ther on the surface or in the volume of the mesh.They are the points where 3D space is sampled.

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150 STEFFEN ET AL.

The local average density of the particles is con-stant on the surface or in the volume distribution.The stochastic character of particle distribution in-troduces local noise in the brightness distributionwhich can be controlled by the number of parti-cles used. With the appropriate particle density,this noise simulates the small-scale emissivity vari-ations. Thus, the precise location of these variationsis modeled only statistically. Some small-scale fea-tures have been included adding or deleting particleslocally with a special tool for that purpose. The dis-tribution of sampling points is the first spatial selec-tion of the sub-structure that constitutes the wholeobject. A second selection is obtained via the as-signment of a relative emissivity, which may be zeroto exclude emission from certain mesh regions. Theemissivity is given as a function of spatial coordi-nates. This function can either be analytic or set byan interactive graph. Similarly, the velocity vectoris assigned as an analytical function of coordinatesor an interactive graph in each coordinate (Steffenet al. 2009).

The interactive graphical representation of thesequantities allows for very detailed and complexstructures to be reproduced. For instance, Figure 5shows the relative emissivity distribution eθ(θ) onthe equatorial ring structure as a function of az-imuthal angle θ in spherical coordinates. The distri-bution in each coordinate is then multiplied to thedistribution in other coordinates er(r) and eφ(φ), i.e.e(r, φ, θ) = e0 · eθ(θ) · er(r) · eφ(φ) (i.e. the distribu-tions are assumed to be separable in their coordi-nates). e0 is a total scaling factor.

After the observational parameters have beenset (like orientation, spectrograph slit position andwidth, seeing and velocity resolution) the emissiv-ity distribution is integrated along the line of sightand rendered as an image and PV diagram. In thismodel, we assume that the object is optically thin.Currently, the dynamic range of brightness model-ing in Shape does not reach that of the features inNGC 7009, which in some cases are only clearly vis-ible on a logarithmic scale. Since the precise relativebrightness of large and small-scale features is not themain concern of this work, rather than the locationand kinematics of structure in 3D space, this limita-tion is not important for our modeling. The model-ing of relative brightness between features has beenadjusted using only visual inspection. The imagesand PV diagrams have been displayed in a squareroot grayscale.

The comparison between model and observationis done by superimposing them with variable trans-

parency directly in Shape. This allows a very accu-rate visual comparison of positions and outlines inthe images and PV diagrams.

For NGC 7009, all structures, except the mainshell, have been modeled as volume distributions.The main shell is thought to be the thin, swept-upshell around the fast wind and is much thinner thanits size. The outer shell is approximated as a filledtriaxial ellipsoid that is limited on the inside by themain shell; this shell has been sampled between theouter ellipsoidal structure and the inner main shell.Other structures have been sampled throughout thevolume that is set by their mesh. Note that this doesnot necessarily mean that emission comes from thefull sampled region, because the relative emissivitydistribution may limit the emission to only part ofthis volume.

3. RESULTS AND DISCUSSION

The full model structure of NGC 7009 is shownin Figure 6, where the viewpoint has been rotatedaround the horizontal axis (top row) and the verticalaxis (bottom row). Model 1 and 2 have the same 3Dstructure, but different velocity fields, and only oneset of images has been shown. The position-velocitydiagrams for both models can be compared to theobserved data in Figures 3 and 4.

3.1. Main shell

We find that the data used in this work are con-sistent with a bent structure of the main shell, bothin the plane of the sky and along the line of sight(Figure 6).

Model 1 has a purely radial velocity field with anoutward increasing velocity gradient (Figure 7, solidline). Model 2 has a linear radial velocity component(Figure 7, dashed line) and a poloidal component asshown in Figure 8.

Since the purely radial velocity field is sphericallysymmetric, any differences between the western andeastern hemispheres in the PV diagrams have to beaccounted for by structure along the line of sight.Although the increasing velocity gradient in Model1 reduces the difference in the cross section of themain shell along the line of sight compared to that inthe plane of the sky (at high latitudes), the westerntip is still much wider along the line of sight thanin the plane of the sky. This is due to the fact thatin the PV diagrams the western and the eastern tiphave a similar thickness, but there is a considerabledifference between the eastern and western tip in theimage (while the radial velocity field is symmetric).

We have explored a model with the same radialvelocity structure as Model 1, but with an unbent

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THE 3D VELOCITY STRUCTURE OF THE PLANETARY NEBULA NGC 7009 151

structure, adjusting the cross section of the mainshell along the line of sight. Excellent agreementwith the PV diagram along the main axis can beachieved. However, this solution cannot be broughtinto agreement with PV diagrams from intermediateposition angles and has therefore been omitted. Thisimplies that the bending is at an angle to the planeof the sky and the plane marked by the main axis ofthe object and the line of sight.

The fact that X-ray emission has been observedthat is constrained by the main shell (Guerrero etal. 2002), makes it likely that it is still driven by thestellar wind or at least an over-pressured hot bub-ble. Following Steffen et al. (2009), in Model 2 wehave fitted a non-zero poloidal velocity field. Fig-ure 8 shows the poloidal velocity as a function oflatitude (with 90o being the equator). In additionto the features in the PV diagrams, the main con-straint for this velocity distribution was to have amain shell with a cross-section that was as circularas possible. As can be seen in Figure 6, top row, thecross section is non-circular. This is required by thestructure of slits that are approximately perpendic-ular to the main axis of NGC 7009. The adoptedrelative emissivity distribution of the main shell isplotted in Figure 9.

In Figures 3 and 4 we compare Models 1 and 2with the observed [NII] PV diagrams from STC04.Overall, the agreement with the data is very good.However, there are subtle deviations in Model 1which cannot be eliminated unless traded for moreserious deficiencies in other regions. In our opinion,these differences make Model 2 (which has a poloidalvelocity component adding degrees of freedom) su-perior to Model 1, with a purely radial velocity field.A comparison of the observations and model PV di-agrams at mid to high latitudes (PA = 49◦ – 109◦)shows better agreement of Model 2 with the data,except for the lower (East-Southeast, E-SE) sectionat PA = 94◦. It has proved impossible to obtain asatisfactory agreement simultaneously between thisE-SE region and the PV diagram along the main axisat PA = 79◦.

Regarding the accuracy of the velocity field, weestimate that, within our modeling framework andour basic assumptions about the cross section of themain shell, the graph for the upper limit of the ra-dial velocity is accurate to about 10− 15%, whereasthat of the poloidal velocity component is probablyaccurate to about 15 − 25%. This was estimated byvarying the parameters manually until the model be-came clearly inconsistent with the observations. Themain uncertainty is in the cross section of the object.

For the combination of velocity structure and crosssection there is no unique solution based on imagingand Doppler velocity measurements.

Considering the uncertainties, the differences inagreement with the observations are not sufficient todefinitely favor one model over the other. Consid-ering the results from hydrodynamical simulations(Steffen et al. 2009), the likely solution is some-where in between Models 1 and 2. This is consistentwith simulations that show a velocity field with botha radial component with an increasing gradient anda poloidal velocity component. Further restrictionwill have to wait for detailed internal proper mo-tion measurements of individual features in the mainshell. In Figure 10 we therefore make a predictionof the proper motion pattern expected for Model 2.A statistical analysis of such a proper motion pat-tern in an observed object will provide the observedequivalent of the graph in Figure 8. We expect allmeasured poloidal velocity values to fall below thoseof Model 2, with significant values beginning at a lat-itude of about 40◦. The outline of the highest valuesin the observed graph will directly provide the corre-sponding model component. This will then allow anobservational derivation of the radial component as afunction of distance from the central star (Figure 7).

Although we have not made a very detailed anal-ysis of relative brightness of features in NGC 7009,we still find some interesting results for the mainshell which are related to its structure. The relativemodel emissivity distributions of the main shell as afunction of distance along the symmetry axis and asa function of the angle around the axis are shown inFigure 9 (note that the ratios between peaks is sup-pressed in our model, due to the reduced dynamicalrange as compared to NGC 7009). As a function ofdistance along the axis the two halves are very sim-ilar. But we find that the western and eastern halfare brightest at approximately opposite sides almostalong the line of sight (Figure 9, bottom, θ = 0 isin the plane of the sky). This brightness differencemight well be related to the bent structure of themain shell and the brightness of the caps at similarangles around the axis (see Figure 6).

Considering these results, the bent and point-symmetric structure together with the increased ve-locity deviating from homologous expansion (eitherradial or poloidal) at high latitudes suggest the exis-tence of two separate mass-loss events that were atleast partially collimated, but in different directions.

Because of the presence of a hot bubble, we haveinvestigated the physically possible case of a velocitythat is locally perpendicular to the main shell with

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152 STEFFEN ET AL.

Fig. 10. The projected velocity vectors for a randomly selected subset of particles is shown. They represent the propermotion pattern predicted with the Shape-model. The color coding is according to the Doppler-shift of the particle. Inorder to show the off-center convergence of the proper motion vectors at mid-latitudes and the ansae, extended guidinglines have been drawn which extend some of the proper motion vectors backwards. To assist in locating the positions ofthe vectors in the image of NGC 7009, the model mesh of the main shell has been included. The position of the centralstar has been marked with a filled circle.

a magnitude increasing with distance from the star.We found no plausible structure that would fit theobservations at high latitudes. Such a model couldonly be fitted to the data at latitudes of less thanapproximately 60◦ from the equator.

3.2. Ansae

The main question about the ansae is their for-mation process. Did they originate near the cen-tral star, or are they the result of a global collima-tion process similar to a hydrodynamic Canto flow(Canto, Tenorio-Tagle, & Rozyczka 1988) or mag-netic focusing (Garcıa-Segura & Lopez 2000). Ve-locity vectors that point directly to the center wouldspeak in favor of an origin near the central star,whereas vectors that converge near the exit pointfrom the main shell might be regarded as evidencefor global focussing.

Rodrıguez & Gomez (2007) use Very Large Arrayradio observations to determine the proper motion ofthe ansae. They find 23 ± 6 and 34 ± 10 mas yr−1

for the eastern and the western ansae, respectively,i.e. the western ansa is almost 1.5 times faster thanthe eastern one. However, within the uncertainty,they could also be the same. We have thereforetested both options. The results shown in all fig-ures refer to the case of equal velocities using the4.1 km s−1 arcsec−1 velocity law shown in Figure 7.

Regarding the origin of the ansae, the measureddifference in velocity might be quite important. Thewidth of the ansae in Doppler velocity (see Figure 3,PV diagram at PA = 79◦) as compared to the widthof the main shell is larger than the ratio of widthsin the images, especially for the eastern ansa. This

may be, of course, simply because the ansae are ac-tually more extended along the line of sight (see thereconstruction in STC04). Let us assume that thesize of the ansae is approximately the same alongthe line of sight as in the plane of the sky; then theresult is quite different. If, in addition, the velocitiesof both ansae are at the lower end of the measuredvalues, then their Doppler velocity width requiresthat the velocity vectors focus near the tips of themain shell (see Figure 10). If the western ansa ac-tually has a velocity that is 1.5 times higher thanthe eastern counterpart, then the spread in the PVdiagram is consistent with velocity vectors emergingfrom the central star. However, this is true only forthe western ansa, whereas the eastern ansa definitelyrequires velocity vectors that expand from a positionnear the tip of the main shell. In order to further testthe origin of the ansae, high-resolution observationsof the proper motion of individual knots in the ansaewill be needed. In the [NII] the knots are so well de-fined that such a study might be possible within afew years using the HST.

3.3. Caps and rings

For completeness we have included caps and ringsin our model, following the same homologous expan-sion law for all features (3.6 km s−1 arcsec−1, seeFigure 7). We confirm the existence of at least one,somewhat oblique, equatorial ring. Additional ringsegments at different inclination angles to the equa-tor have been included to model some of the small-scale features in the PV diagrams and the emission.The relative emissivity of the main equatorial ringas a function of angle around the main axis has been

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THE 3D VELOCITY STRUCTURE OF THE PLANETARY NEBULA NGC 7009 153

plotted in Figure 5. The same figure also showsthe relative brightness distribution used to modelthe caps as regions in the rings at some distancefrom the equator (solid and dashed line; see also themesh in Figure 2). Note that the angular positionof the bright caps is similar to that of the brightregions in the main shell (see Figure 9). The capshave been modeled with few particles in order to re-produce their small scale structure. However, sincethe particles have been distributed randomly, knotsin the observations have no particular correspondingcounterpart in the model.

3.4. Proper motion pattern

Figure 10 shows the proper motion pattern pre-dicted from Model 2. The projected velocity vectorshave been drawn for a small random subset of par-ticles in the model. Since there are several velocitysubsystems, observation of the proper motion pat-tern requires local methods, rather than global ex-pansion measurements that might be described by asingle expansion factor (Fernandez et al. 2004). It ispreferable to apply methods based on the motion ofindividual features like those used by Li, Harrington,& Borkowski (2002) for the young planetary nebulaBD+30◦3639. Similar methods have been employedon NGC 6302 by Meaburn et al. (2005), and byO’Dell et al. (2009) on NGC 6720, the Ring Nebula.For clarity, we have included a few guiding lines thatemphasize the non-radial directions of proper motionin the main shell and the ansae. In an observationaltest, the poloidal velocity component could be de-rived from the proper motion vectors at the edge ofthe nebula. Plotted as a function of latitude, suchresults could be directly compared with the upperlimits for the poloidal velocity components predictedin Figure 8.

In order to measure the expansion of NGC 7009,Fernandez et al. (2004) have shown difference imagesbased on [NII] HST images from 1996 and 2001. Un-fortunately, one of the HST images did not includethe western ansa. They also had some problems toappropriately register the two images, so that theremight be some systematic errors in their results. It istherefore of interest to predict future results of suchmeasurements based on our 3D model. Fernandezet al. (2004) have not analyzed the expansion of themain shell, arguing that it might be a moving ioniza-tion front, which does not provide the velocity of thebulk motion of the gas. However, Sabbadin et al.(2004) find that the nebula is fully ionized, exceptpossibly in the ansae and the caps. Therefore, what-ever causes the observed expansion, a study of this

Fig. 11. The difference image of the model for [NII] emis-sion for time 1996 and 50 years later is shown. Zero emis-sion was normalized to 128 on a grey-scale of 255 shades.The dark regions show the residuals of the older image,whereas the bright residuals are due to the future image.

expansion might bring new insights. We thereforepredict the expected expansion pattern for Model 2,based on the assumption that the expansion is dueto bulk motion.

Shape has the option to use the model velocityfield to advance the position of the individual parti-cles in time, assuming a velocity that does not changewith time. With this feature we produced a differ-ence image for various time-spans. Figure 11 showsthe expected difference between [NII] images taken50 years apart assuming a distance of 1.6 kpc (Sab-badin et al. 2004). Note that the distance valuesgiven for this object vary approximately between 0.8and 1.6 kpc. At half the distance the same expansionwould be obtained in 25 years. Smaller time-scalesprovide similar, but noisier results. Since the spatialresolution of the HST images is better than that ofour model by about a factor of three, a similar differ-ence image with HST resolution could be obtainedwith a baseline of 15–20 years. A new observationalanalysis of this type is of interest, since there is a sys-tematic difference between the results of our modeland those of Fernandez et al. (2004). In their im-ages, the eastern tip of the main shell clearly showsstronger expansion north of the symmetry axis. Inour model, there is no such systematic difference be-tween North and South.

4. CONCLUSIONS

Based on detailed position-velocity diagramsfrom Sabbadin et al. (2004) and Hubble Space Tele-scope imagery, we have constructed a high-resolution3D morpho-kinematical model for the planetary neb-

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ula NGC 7009. We largely confirm the results bySabbadin et al. (2004), but obtain differences for themain shell due to the differences in the assumptionson the velocity fields. We have estimated the upperlimits for deviations from a homologous expansion.We find that the data are consistent with a radial ve-locity field with a stronger gradient at high latitudes(Model 1) as compared to the region near the equa-tor. In a second model we have assumed a linearlyincreasing radial velocity component with an addedpoloidal component of order 10 km s−1 at latitudesof about 70◦. Based on hydrodynamical simulationsof wind-driven PNs by Steffen et al. (2009), we ex-pect these velocity fields to represent limiting casesand that the real nebula has a velocity field havinga combination of increasing gradient in the radialvelocity and a poloidal component, but each withsmaller values. We exclude the possibility that thevelocity field of the main shell is locally perpendicu-lar to its surface at least at high latitudes. We alsofind that the expansion of the ansae is non-radial ifthe central star is used as reference. Their velocityfield is focused near the apparent exit points from themain shell. This points towards an origin as a Cantoflow, large scale magnetic focusing or direct inter-action with the ambient medium. High-resolutionproper motion measurements of individual featuresin the main shell and the ansae could yield a test ofthe non-radial expansion patterns by now or withina few years. We predict the proper motion patternfor the model with a non-zero poloidal velocity com-ponent.

We acknowledge support from Conacyt grant49447 and Universidad Nacional Autonoma deMexico DGAPA-PAPIIT IN108506-2. M.E. andS.M. acknowledge support by the Instituto deAstronomıa, Universidad Nacional Autonoma deMexico. N.K. acknowledges support by the Nat-ural Sciences and Engineering Research Council ofCanada (NSERC) and the Killam Trusts. We thank

Wolfgang Steffen: Instituto de Astronomıa, Universidad Nacional Autonoma de Mexico, Apdo. Postal 106,22800 Ensenada, B. C., Mexico (wsteffen@astrosen.unam.mx).

Moises Espındola: Escuela Superior de Fısica y Matematicas, Instituto Politecnico Nacional, Av. Politecnico,Edificio No. 9 U.P.A.L.M. 07738 Mexico D. F., Mexico (mogaoga@gmail.com).

Sergio Martınez: Instituto Nacional de Astrofısica, Optica y Electronica, Tonantzintla, Puebla and Facultad deCiencias Fısico Matematicas, Benemerita Universidad Autonoma de Puebla, Av. San Claudio y Rıo Verde,Col. San Manuel, C.P. 72570; Apdo. Postal 1152, Puebla, Puebla, Mexico (sergiomtz@inaoep.mx).

Nico Koning: Department of Physics and Astronomy, University of Calgary, Science B 504A, Canada (nkon-ing@iras.ucalgary.ca).

Franco Sabbadin and collaborators for kindly grant-ing permission to adapt their images and PV dia-grams of NGC 7009 for this work.

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