Deep survey of millimeter radio recombination lines toward the planetary nebulae IC 418 and NGC 7027

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Table 2

Summary of initial parameters and outputs from Co³RaL.

Parameters		IC 418	NGC 7027
Input parameters
Distance (pc)	d	1 200 ^(a)	860 ^(b)
Inclination angle	i	–20° ^(c)	–55° ^(d)
Ellipsoid semi-axis	A	7500	4000
	B	9750	6000
	C	7500	4000
Radial structure (AU)	R_in	10	10
	R₀	500	300
	R₁	5800	2800
	R₂	6500	3400
	R_out	9750	6000
Electron density (cm^–3)	N_e(r) ∈ [R_in, R₀]^(e)	27 240	8000
	N_e(r) ∈ [R₀, R₁]^(e)	$27 240 {(\frac{r}{R_{0}})}^{- 0.9}$ $27\,240{\left( {{r \over {{R_0}}}} \right)^{ - 0.9}}$	8000
	N_e(r) ∈ [R₁, R₂]^(e)	23 000	150 000
Electron temperature (K)	T_e(r) ∈ [R_in, R₀]^(e)	11 000	25 000
	T_e(r) ∈ [R₀, R₁]^(e)	11 000	$25 000 - 3000 (\frac{r - R_{0}}{R_{1} - R_{0}})$ $25\,000 - 3000\left( {{{r - {R_0}} \over {{R_1} - {R_0}}}} \right)$
	T_e(r) ∈ [R₁, R₂]^(e)	11 000	22 000
Expansion velocity (km s^–1)	υ_exp,0	1.0	5.0
	υ_∞	13.0	25.0
	υ_exp(r) ∈ [R_in, R₂]^(e)	υ_exp,0 – (υ_∞ – υ_exp,0)	$(\frac{r - R_{in}}{R_{2} - R_{in}}) (f)$ $\left( {{{r - {R_{{\rm{in}}}}} \over {{R_2} - {R_{{\rm{in}}}}}}} \right){\,^{\left( f \right)}}$
Turbulent velocity^(g) (km s^–1)	υ_turb	8.0	10.0
Abundances	H	0.90	0.90
	He I	0.09^(h)	0.064⁽ⁱ⁾
	He II	0	0.042⁽ⁱ⁾
Derived parameters
Ionized mass ( M_⊙)		5.12 · 10^–2	2.92 · 10^–2
Scalar momentum (g cm s^–1 )		9.97 · 10³⁷	1.19 · 10³⁸
Kinetic energy (erg)		5.07 · 10⁴³	1.21 · 10⁴⁴
Mech. luminosity (L_⊙)		0.143	1.203
Kinematic timescale (a)	Aver	2930	828
	Min	317	90
	Max	3565	1141
Mass-loss rate (M_⊙ a^–1)		1.75 · 10^–5	3.53 · 10^–5

Notes.^(a) Intermediate value from Guzmán et al. (2009) and Ali et al. (2015), also similar to González-Santamaría et al. (2021).^(b) Ali et al. (2015).^(c) Combination of observations at different wavelengths from Garay et al. (1989); Guzmán et al. (2009); Ramos-Larios et al. (2012).^(d) Rodríguez et al. (2009).^(e) These parameters are defined between two ellipsoidal shells labeled using their characteristic radius.^(f) Same expression for both sources.^(g) This parameter is defined as the σ of a Voigt profile (see Eqs. (B.9), (B.22) of Sánchez Contreras et al. 2024) ^(h) Pottasch et al. (2004), Table 19. ⁽ⁱ⁾ Bernard Salas et al. (2001) Table 6.

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