4 Kinematic Modeling

To disentangle the three-dimensional gaseous structure of Th2-A, we used the morpho-kinematic modeling tool SHAPE (Version 5.0; Steffen et al., 2011; Steffen & López, 2006). This program has been used for many objects, such as Hb 5 and K 3-17 (López et al., 2012), NGC 2392 (García-Díaz et al., 2012), Hen 3-1333 and Hen 2-113 (Danehkar & Parker, 2015), and NGC 3242 (Gómez-Muñoz et al., 2015). It uses interactively molded geometrical polygon meshes to reconstruct three-dimensional structures based on kinematic and spatial observations. It constructs a cell grid, each cell representing a volume, and uses a ray-casting algorithm to perform radiative transfer through these cells. It produces several outputs that can be directly compared with observations, namely synthetic images and position-velocity (P-V) diagrams, and velocity channels. However, synthetic images do not include explicit photo-ionization process, so under such conditions the emissivity distribution for each spectral line is modeled ad-hoc based on the observations of the corresponding emission line. To determine a best-fitting model, geometrical and kinematic parameters are iteratively modified until acceptable solutions are obtained.

To model Th2-A, we used the H$ \alpha $ velocity slices presented in Fig.3 since all the kinematic components are included in H$ \alpha $ emission rather [NII] emission due to the very high excitation feature of this nebula. The velocity channel maps of the morpho-kinematic model are closely compared with the observed channel maps. All these geometrical structures were slightly modified until the model outputs reasonably match the observational maps. As a starting point, a circular shell was assumed for the ring of Th2-A. The velocity is defined as radially outward from the nebular center with a linear function of magnitude, commonly known as a Hubble-type flow (Steffen et al., 2009). The inclination angle of this shell was then manipulated until synthetic images at different velocity channels match the observational maps (see the channels between $ -51$ and $ 51$kms$ ^{-1}$). Assuming that the observed circular ring is a toroidal shell, the inclination of major axis is found to be $ -10^{\circ} \pm 5^{\circ}$ with respect to the line of sight ($ 0^{\circ}$ being pole-on, $ 90^{\circ}$ being edge-on). The best-fitting model describes a toroidal shell with an expansion velocity of $ 40\pm10$kms$ {}^{-1}$.

Table: Parameters of the Best-fitting SHAPE Model of Th2-A
Parameter Value
Inclination of major axis, $ i$ $ -10^{\circ} \pm 5^{\circ}$
Position angle of major axis, P.A. $ -45^{\circ} \pm 5^{\circ}$
Outer radius of the ring, $ r_{\rm out}$ $ 14\pm2$ arcsec
Thickness of the ring, $ \delta r$ $ 9\pm2$ arcsec
Waist expansion velocity $ 40\pm10$kms$ {}^{-1}$
Polar expansion velocity $ 90\pm20$kms$ {}^{-1}$
Systemic velocity (LSR), $ V_{\rm sys}$ $ -52\pm5$kms$ {}^{-1}$

Figure: Synthetic images at different velocity channels obtained from the best-fitting SHAPE model. Each channel is 20.5kms$ ^{-1}$ wide.

Figure: Top panels: P-V arrays of Th2-A in H$ \alpha $ emission for slits oriented with P.A. $ =-45^{\circ }$ and $ 45^{\circ }$ passing through the central star. The velocity axis is with respect to to the systemic velocity of the central star, in kms$ {}^{-1}$ unit. The angular offset at 0 arcsec defines the central star position. Bottom panels: the corresponding synthetic P-V diagrams obtained from the best-fitting SHAPE model.
\includegraphics[width=2.3in]{figures/fig6_6563_pv_m45.eps} \includegraphics[width=2.3in]{figures/fig6_6563_pv_p45.eps}

We further modeled the bipolar collimated outflows using a prolate ellipsoid with a density lower than that of the toroidal shell. It explains the high velocity components seen in the $ -92$, $ -72$, $ 72$ and $ 93$kms$ ^{-1}$ channels (see Fig.3). However, the length of the bipolar outflows cannot precisely be constrained, as they are slightly oriented, relative to the line of sight (inclination $ -10^{\circ}$). Assuming a homologous outflow, the distance of the bipolar outflow from the nebular center is nearly twice larger than the polar radius of the toroidal shell. From the model, the collimated bipolar outflows are found to have a polar expansion velocity of $ 90\pm20$kms$ ^{-1}$.

In Figure 4, we present a 3D representation of the final best-fitting model viewed from different orientations (from $ 0^{\circ}$, $ 90^{\circ}$), followed by the resultant mesh model (inclination $ -10^{\circ}$), before rendering, and the final rendered model, respectively. As seen in Fig.4, the obtained synthetic image fairly resembles the observation (see Fig.2). The parameters of the best-fitting model are summarized in Table 1.

The velocity channel maps of the resultant kinematic model are given in Fig.5, which can be subsequently compared with the observational maps presented in Fig.3. The close match between them suggests that our morpho-kinematic model is able to reproduce the observed kinematic structure of this object.

Fig. 6 (top panels) shows the H$ \alpha $ emission line P-V arrays of Th2-A extracted from the IFU datacube for two slits oriented with P.A. $ =-45^{\circ }$ and $ 45^{\circ }$, which pass through the the central star. We present these slits because the best-fitting model has a symmetric axis with P.A. $ =-45^{\circ }$. The velocity axis on all plots is relative to the LSR systemic velocity of the central star ( $ v_{\rm sys}=-52$kms$ ^{-1}$). The stellar continuum from the CSPN has also been subtracted. There are two separate velocity components reaching $ \pm 110$kms$ ^{-1}$. The two bright knots represent the dense torus expanding with a velocity of $ 40\pm10$kms$ ^{-1}$. The synthetic P-V diagrams derived from the model are shown in Fig. 6 (bottom panels) under the observed ones. It is seen that they reasonably match the observed diagrams. Note that the south part of Th2-A was not completely covered by the IFU field-of-view, which is visible in the lower parts of the P-V arrays.

Ashkbiz Danehkar