For Model A, which assumes the very broad component around 4450 Å as coming from the Balmer emission regions, the following model functions are included. A narrow plus a broad Gaussian function (second moment smaller or larger than 500 km s−1) G
Hβ, N
([λ
0, β
N
,  σ

β
N
,  f

β
N
]) (parameter λ
0 and σ as central wavelength and second moment in units of angstrom) and G
Hβ, B
([λ
0, β
B
,  σ

β
B
,  f

β
B
]) are applied to describe the Hβ. Narrow and broad Gaussian functions G
Hγ,N
([λ
0, γ
N
,  σ

γ
N
,  f

γ
N
]) and G
Hγ, B
([λ
0, γ
B
,  σ

γ
B
,  f

γ
B
]) are applied to describe the Hγ. Two narrow Gaussian functions G

c1([λ
0, c1,  σ

c1,  f

c1])and G

c2([λ
0, c2,  σ

c2,  f

c2]) are applied to describe the core components of the [O iii]λ4959, 5007 Å doublet. Two broad Gaussian functions G

e1([λ
0, e1,  σ

e1,  f

e1]) and G

e2([λ
0, e2,  σ

e2,  f

e2]) are applied to describe the extended components of the [O iii] doublet (Greene & Ho 2005a; Shen et al. 2011; Zhang 2021a), considering probably spatially extended emission regions for the outflow-related extended [O iii] components as discussed in Zakamska et al. (2016). A broad and a narrow Gaussian functions G

c3([λ
0, c3,  σ

c3,  f

c3]) and G

e3([λ
0, e3,  σ

e3,  f

e3]) are applied to describe the core and the extended components of the [O iii]λ4363 Å. A power-law component 



A×λ4100ÅB

 is applied to describe the continuum emissions underneath the emission lines. When the model functions are applied, the following restrictions are accepted. First, the line flux and second moment of each Gaussian component are not smaller than zero. Second, ratios of the central wavelengths (in units of angstrom) and second moments (in units of angstrom) of G
Hβ, N
 to G
Hγ, N
 are fixed to be
1



λ0,γN4341.68Å=λ0,βN4862.68Åσ0,γN4341.68Å=σ0,βN4862.68Å,

leading the narrow Balmer lines to have the same redshift and line widths. Third, ratios of the central wavelengths, second moments, and fluxes of G

c1 to G

c2 to G

c3 are fixed to be
2



λ0,c15008.24Å=λ0,c24960.295Å=λ0,c34364.436Åσ0,c15008.24Å=σ0,c24960.295Å=σ0,c34364.436Åfc1=3×fc2(fc3free).

Fourth, the ratios of the central wavelengths, second moments, and fluxes of G

e1 to G

e2 to G

e3 are fixed to be
3



λ0,e15008.24Å=λ0,e24960.295Å=λ0,e34364.436Åσ0,e15008.24Å=σ0,e24960.295Å=σ0,e34364.436Åfe1=3×fe2(fe3free).

The model parameters of G
Hγ, B
 and G
Hβ, B
 are free to model parameters, which will provide further clues to the origin of the very broad components around 4450 Å from Balmer emissions or from extended [O iii] emissions or other physical origins. Then, through the Levenberg–Marquardt least-squares minimization technique, the best-fitting results to the emission lines obtained using Model A can be well determined, and are shown in Figure 2 with χ
2/dof ∼ 0.88 (where χ
2 and dof are the summed squared residuals for the best-fitting results and the degrees of freedom, respectively). Here, in order to show clearer results, the left panel of Figure 2 shows the best-fitting results to the emission lines around Hβ with a rest wavelength range from 4600–5400 Å, and the right panel of Figure 2 shows the best-fitting results for the emission lines around Hγ with the rest wavelength range from 4200–4600 Å.
Citation: Zhang 2021a
Citation function: Uses