4 The models

Table 3: The key parameters of the best-fitting kinematic models.
Hen3-1333 Hen2-113

Size (arcsec) PA($ ^{\circ}$) GPA($ ^{\circ}$) $ i$($ ^{\circ}$) Size (arcsec) PA($ ^{\circ}$) GPA($ ^{\circ}$) $ i$($ ^{\circ}$)
$ a-a'$ $ 6.2 \times 3.7$ $ -15\pm5$ $ 39.6\pm5$ $ -30\pm15$ $ 3.7 \times 2.4$ $ 65\pm5$ $ 93.3\pm5$ $ 40\pm15$

$ b-b'$
$ 6.8 \times 2.9$ $ 60\pm5$ $ 114.6\pm5$ - $ 4.8 \times 2.6$ $ 120\pm5$ $ 148.3\pm5$ -

The 3D kinematic modelling program SHAPE (Steffen & López, 2006; Steffen et al., 2011) was used to identify the spatial distribution of the outer tenuous lobes around the compact cores of Hen3-1333 and Hen2-113. The modelling procedure consists of defining a geometry, assigning emissivity distribution and defining a velocity law as a function of position. The program produces several outputs that can be directly compared with the observations, namely position-velocity diagram and appearance of the object on the sky.

For this study, we adopted a bipolar geometric model (marked as $ a-a'$ in Fig.3, left-hand panel). To replicate the HST images, another bipolar component, which is identical to the other component, is introduced with different orientations (marked as $ b-b'$ in Fig.3). We adjusted the inclination ($ i$) and position angle `PA' in an iterative process until the qualitatively fitting solution is produced. The velocity IFU maps derived from the H$ \alpha $ emission line, combined with the HST images, were used to constrain the primary bipolar lobes ($ a-a'$). The HST observations were also used to reconstruct the secondary bipolar lobes ($ b-b'$). Fig.3 (second panel) also shows the HST images with higher contrast, which reveal details not accessible at the previous contrast level.

Fig.3 (third panel) shows a 3D representation of the model at the best-fitting inclination of each object. From the best-fitting kinematic models, the primary lobes ($ a-a'$) of Hen3-1333 and Hen2-113 were found to have PA  $ =-15^{\circ}\pm5^{\circ}$ and $ 65^{\circ}\pm5^{\circ}$, and inclination angles of $ i=-30^{\circ}\pm15^{\circ}$ and $ 40^{\circ}\pm15^{\circ}$, respectively. The secondary lobes ($ b-b'$) of Hen3-1333 and Hen2-113, which are only noticeable in the HST images, were found to have PA  $ =60^{\circ}\pm5^{\circ}$ and $ 120^{\circ}\pm5^{\circ}$. Fig.3 (right-hand panel) shows the synthetic rendered images. As seen, the rendered images perfectly resembles the HST observations shown in Fig.3 (left-hand panel).

Table 3 lists the key parameters of the best-fitting morpho-kinematic models: the sizes, the PA, the Galactic position angle (GPA), and the inclination ($ i$) of the lobes, respectively. The PA is the position angle of the bipolar lobes projected on to the plane of the sky, and measured from the north towards the east in the equatorial coordinate system (ECS). The GPA is the position angle projected on to the sky plane, measured from the North Galactic Pole (NGP) towards the Galactic east. The inclination is measured between the line of sight and the nebular symmetry axis ( $ i=0^{\circ}$ being pole-on).

Ashkbiz Danehkar