3 Kinematics

Fig. 2 shows the spatial distribution maps of the flux intensity, continuum, radial velocity and velocity dispersion of H$\alpha$ $\lambda$6563 and $[$N II$]$ $\lambda$6584 for Abell48. The white contour lines in the figures depict the distribution of the emission of H$\alpha$ obtained from the SHS (Parker et al., 2005), which can aid us in distinguishing the nebular borders from the outside or the inside. The observed velocity $v_{\rm obs}$ was transferred to the local standard of rest (LSR) radial velocity $v_{\rm LSR}$ by correcting for the radial velocities induced by the motions of the Earth and Sun at the time of our observation. The transformation from the measured velocity dispersion $\sigma_{\rm obs}$ to the true line-of-sight velocity dispersion $\sigma_{\rm true}$ was done by $\sigma_{\rm true}=\sqrt{\sigma^2_{\rm obs}-\sigma^2_{\rm ins}-\sigma^2_{\rm th}}$, i.e. correcting for the instrumental width (typically $\sigma_{\rm ins}\approx42$km/s for $R\sim 3000$ and $\sigma_{\rm ins}\approx18$km/s for $R\sim 7000$) and the thermal broadening ( $\sigma_{\rm th}^2=8.3\,T_{\rm e}[{\rm kK}]/Z$, where $Z$ is the atomic weight of the atom or ion).

Figure 3: (a) The SHAPE mesh model before rendering at the best-fitting inclination and corresponding rendered model. (b) The normalized synthetic intensity map and the radial velocity map at the inclination of $-35{}^{\circ }$ and the position angle of $135{}^{\circ }$, derived from the model ( $v_{\rm sys}=0$), which can be compared directly with Fig. 2.

(a) Morpho-kinematic mesh model

(b) Model results

We have used the three-dimensional morpho-kinematic modelling program SHAPE (version 4.5) to study the kinematic structure. The program described in detail by Steffen & López (2006) and Steffen et al. (2011), uses interactively moulded geometrical polygon meshes to generate the 3D structure of objects. The modelling procedure consists of defining the geometry, emissivity distribution and velocity law as a function of position. The program produces several outputs that can be directly compared with long slit or IFU observations, namely the position-velocity (P-V) diagram, the 2-D line-of-sight velocity map on the sky and the projected 3-D emissivity on the plane of the sky. The 2-D line-of-sight velocity map on the sky can be used to interpret the IFU velocity maps. For best comparison with the IFU maps, the inclination ($i$), the position angle `PA' in the plane of the sky, and the model parameters are modified in an iterative process until the qualitatively fitting 3D emission and velocity information are produced. We adopted a model, and then modified the geometry and inclination to conform to the observed H$\alpha$ and [N II] intensity and radial velocity maps. For this paper, the three-dimensional structure has then been transferred to a regular cell grid, together with the physical emission properties, including the velocity that, in our case, has been defined as radially outwards from the nebular centre with a linear function of magnitude, commonly known as a Hubble-type flow (see e.g. Steffen et al., 2009).

The morpho-kinematic model of Abell48 is shown in Fig. 3(a), which consists of a modified torus, the nebular shell, surrounded by a modified hollow cylinder and the faint outer halo. The shell has an inner radius of $10\hbox{$^{\prime\prime}$}$ and an outer radius of $23\hbox {$^{\prime \prime }$}$ and a height of $23\hbox {$^{\prime \prime }$}$. We found an expansion velocity of $v_{\rm exp}=35\pm 5$kms${}^{-1}$ and a LSR systemic velocity of $v_{\rm sys}=65 \pm 5$kms${}^{-1}$. Our value of the LSR systemic velocity is in good agreement with the heliocentric systemic velocity of $v_{\rm hel}=50.4\pm4.2$kms${}^{-1}$ found by Todt et al. (2013). Following Dopita et al. (1996), we estimated the nebula's age around 1.5 of the dynamical age, so the star left the top of the AGB around $8880$ years ago.

Fig. 3 shows the orientation of Abell48 on to the plane of the sky. The nebula has an inclination of $i=-35^{\circ}$ between the line of sight and the nebular symmetry axis. The symmetry axis has a position angle of ${\rm PA}=135^{\circ}$ projected on to the plane of the sky, measured from the north towards the east in the equatorial coordinate system (ECS). The PA in the ECS can be transferred into the Galactic position angle (GPA) in the Galactic coordinate system (GCS), measured from the north Galactic pole (NGP; ${\rm GPA}=0^{\circ}$) towards the Galactic east ( ${\rm GPA}=90^{\circ}$). Note that ${\rm GPA}=90^{\circ}$ describes an alignment with the Galactic plane, while ${\rm GPA}=0^{\circ}$ is perpendicular to the Galactic plane. As seen in Table 3, Abell48 has a GPA of $197$   $\mbox{$.\!\!^\circ$}$$8$, meaning that the symmetry axis is approximately perpendicular to the Galactic plane.

Based on the systemic velocity, Abell 48 must be located at less than 2kpc, since higher distances result in very high peculiar velocities ( $v_{\rm pec}>189$kms$^{-1}$; $v_{\rm pec}=170$kms$^{-1}$ found in few PNe in the Galactic halo by Maciel & Dutra, 1992). However, it cannot be less than 1.5kpc due to the large interstellar extinction. Using the infrared dust maps³ of Schlegel et al. (1998), we found a mean reddening value of $E(B-V)=11.39 \pm 0.64$ for an aperture of $10 \hbox{$^\prime$}$ in diameter in the Galactic latitudes and longitude of $(l,b)=(29.0,0.4)$, which is within a line-of-sight depth of $\lesssim20$kpc of the Galaxy. Therefore, Abell 48 with $E(B-V)\simeq2.14$ must have a distance of less than $3.3$ kpc. Considering the fact that the Galactic bulge absorbs photons overall 1.9 times more than the Galactic disc (Driver et al., 2007), the distance of Abell 48 should be around 2kpc, as it is located at the dusty Galactic disc.

Table: Kinematic results obtained for Abell48 based on the morpho-kinematic model matched to the observed 2-D radial velocity map.

Parameter	Value
$r_{\rm out}$ (arcsec)	$23 \pm 4$
$\delta r$ (arcsec)	$13 \pm 2$
$h$ (arcsec)	$23 \pm 4$
$i$	$-35^{\circ} \pm 2^{\circ}$
PA	$135^{\circ} \pm 2^{\circ}$
GPA	$197^{\circ}48\hbox{$^\prime$}\pm 2^{\circ}$
$v_{\rm sys}$(km/s)	$65 \pm 5$
$v_{\rm exp}$(km/s)	$35\pm 5$