4.1 Plasma diagnostics

The derived electron temperatures ($T_{\rm e}$) and densities ($N_{\rm e}$) are listed in Table 5, together with the ionization potential required to create the emitting ions. We obtained $T_{\rm e}$ and $N_{\rm e}$ from temperature-sensitive and density-sensitive emission lines by solving the equilibrium equations of level populations for a multilevel atomic model using EQUIB code (Howarth & Adams, 1981). The atomic data sets used for our plasma diagnostics from collisionally excited lines (CELs), as well as for abundances derived from CELs, are given in Table 4. The diagnostics procedure to determine temperatures and densities from CELs is as follows: we assume a representative initial electron temperature of 10000K in order to derive $N_{\rm e}$ from $[$S II$]$ line ratio; then $T_{\rm e}$ is derived from $[$N II$]$ line ratio in conjunction with the mean density derived from the previous step. The calculations are iterated to give self-consistent results for $N_{\rm e}$ and $T_{\rm e}$. The correct choice of electron density and temperature is important for the abundance determination.

We see that the PN Abell 48 has a mean temperature of $T_{\rm e}([$N II $])=6980 \pm 930$ K, and a mean electron density of $N_{\rm e}([$S II $])=750 \pm 200$ cm${}^{-3}$, which are in reasonable agreement with $T_{\rm e}([$N II $])=7\,200 \pm 750$ K and $N_{\rm e}([$S II $])=1000 \pm 130$ cm${}^{-3}$ found by Todt et al. (2013). The uncertainty on $T_{\rm e}([$N II$])$ is order of $40$ percent or more, due to the weak flux intensity of [N II] $\lambda$5755, the recombination contribution, and high interstellar extinction. Therefore, we adopted the mean electron temperature from our photoionization model for our CEL abundance analysis.

Table 4: References for atomic data.

Ion	Transition probabilities	Collision strengths
N${}^{+}$	Bell et al. (1995)	Stafford et al. (1994)
O${}^{+}$	Zeippen (1987)	Pradhan et al. (2006)
O${}^{2+}$	Storey & Zeippen (2000)	Lennon & Burke (1994)
Ne${}^{2+}$	Landi & Bhatia (2005)	McLaughlin & Bell (2000)
S${}^{+}$	Mendoza & Zeippen (1982)	Ramsbottom et al. (1996)
S${}^{2+}$	Mendoza & Zeippen (1982)	Tayal & Gupta (1999)
	Huang (1985)
Ar${}^{2+}$	Biémont & Hansen (1986)	Galavis et al. (1995)
Ion	Recombination coefficient	Case
H${}^{+}$	Storey & Hummer (1995)	B
He${}^{+}$	Porter et al. (2013)	B
C${}^{2+}$	Davey et al. (2000)	B

Table 5 also lists the derived HeI temperatures, which are lower than the CEL temperatures, known as the ORL-CEL temperature discrepancy problem in PNe (see e.g. Liu et al., 2000; Liu et al., 2004b). To determine the electron temperature from the HeI $\lambda\lambda$5876, 6678 and 7281 lines, we used the emissivities of He I lines by Smits (1996), which also include the temperature range of $T_{\rm e} < 5000$K. We derived electron temperatures of $T_{\rm e}({\rm He~I})=5110$K and $T_{\rm e}({\rm He~I})=4360$K from the flux ratio HeI $\lambda\lambda$7281/5876 and $\lambda\lambda$7281/6678, respectively. Similarly, we got $T_{\rm e}({\rm He~I})=6960$K for HeI $\lambda\lambda$7281/5876 and $T_{\rm e}({\rm He~I})=7510$K for $\lambda\lambda$7281/6678 from the measured nebular spectrum by Todt et al. (2013).

Table 5: Diagnostics for the electron temperature, $T_{\rm e}$ and the electron density, $N_{\rm e}$. References: D13 - this work; T13 - Todt et al. (2013).

Ion	Diagnostic	I.P.(eV)	$T_{\rm e}({\rm K})$	Ref.
$[$N II$]$	$\frac{\lambda6548+\lambda6584}{\lambda5755}$	14.53	$6980 \pm 930$	D13
			$7200 \pm 750$	T13
$[$O III$]$	$\frac{\lambda4959+\lambda5007}{\lambda4363}$	35.12	$11870 \pm 1640$	T13
HeI	$\frac{\lambda7281}{\lambda5876}$	24.59	$5110 \pm 2320$	D13
			$6960 \pm 450$	T13
HeI	$\frac{\lambda7281}{\lambda6678}$	24.59	$4360 \pm 1820$	D13
			$7510 \pm 4800$	T13
			$N_{\rm e}({\rm cm}^{-3})$
$[$S II$]$	$\frac{\lambda6717}{\lambda6731}$	10.36	$750 \pm 200$	D13
			$1000 \pm 130$	T13