3 Morpho-kinematic model

We have used the morpho-kinematic modeling program SHAPE (version 5.0) described in detail by Steffen & López (2006) and Steffen et al. (2011). This program has been used for modeling many PNe, such as NGC 2392 (García-Díaz et al., 2012), NGC 3242 (Gómez-Muñoz et al., 2015), Hen 2-113 and Hen 3-1333 (Danehkar & Parker, 2015b). It uses interactively molded geometrical polygon meshes to generate three-dimensional structures of gaseous nebulae. The program produces several outputs that can be directly compared with observations, namely position-velocity diagrams, velocity channels and synthetic images. The modeling procedure consists of defining the geometry, assigning a density distribution and defining a velocity law. Geometrical and kinematic parameters are modified in a manual interactive process until a satisfactorily fitting model has been constructed.

Figure 4 (a) shows the morpho-kinematic model before rendering at two different orientations (inclination: 0$^{\circ }$ and 90$^{\circ }$), and their best-fitting inclination, together with the result of the rendered model. The morpho-kinematic model consists of an equatorial dense torus (main shell) and a pair of asymmetric bipolar outflows. The values of the parameters of the final model are summarized in Table 4. For the velocity field, we assume a Hubble-type flow (Steffen et al., 2009).

The velocity-channel maps of the final model are shown in Figure 4 (b), where they can be directly compared with the observed velocity-resolved channel maps presented in Figure 3. The model maps are a good match to the observational maps. The model successfully produces two kinematic components of the jets moving in opposite directions on both sides of the torus. From the morpho-kinematic model, we derived an inclination of $i=-82^{\circ}\pm 4^{\circ}$ with respect to the line of sight. Taking the inclination derived by the best-fitting model, we estimated a “jet” expansion velocity of $120 \pm 40$ kms$^{-1}$ with respect to the central star.

Figure: Top panels: SHAPE mesh model of M2-42 before rendering at two different orientations (inclination: 0$^{\circ }$ and 90$^{\circ }$), the best-fitting inclination, and the corresponding rendered image, respectively. Bottom panels: Synthetic images at different velocity channels obtained from the best-fitting SHAPE model.

(a) SHAPE model

(b) Velocity channels

Table: Parameters of the morpho-kinematic model of M2-42.

Parameter	Value
Inclination of major axis, $i$	$-82^{\circ} \pm 4^{\circ}$
Position angle of major axis, PA	$50^{\circ} \pm 5^{\circ}$
Galactic position angle of major axis, GPA	$112^{\circ}24\hbox{$^\prime$}\pm 5^{\circ}$
Outer radius of the main shell	$3\pm1$ arcsec
NE Jet distance from the center	$12\pm2$ arcsec
SW Jet distance from the center	$9\pm2$ arcsec
Jet velocity from the center	$120 \pm 40$ kms${}^{-1}$

As seen in Table 4, the symmetric axis of the bipolar outflows has a position angle (PA) of $50^{\circ} \pm 5^{\circ}$ measured from the north toward the east in the equatorial coordinate system (ECS). This leads to a Galactic position angle (GPA) of $112$   $\mbox{$.\!\!^\circ$}$$4$. The GPA is the position angle of the nebular symmetric axis projected on to the sky plane, measured from the North Galactic Pole toward the Galactic east. Note that ${\rm GPA}=90^{\circ}$ describes an alignment with the Galactic plane, whereas ${\rm GPA}=0^{\circ}$ is perpendicular to the Galactic plane. Therefore, the symmetric axis of M2-42 is roughly aligned with the Galactic plane. This alignment could have some implications for other studies of GBPNe (see e.g. Falceta-Gonçalves & Monteiro, 2014; Rees & Zijlstra, 2013; Danehkar & Parker, 2015a).