Improved Radiative Transfer Corrections in Helium Emission Lines

TL;DR

This work replaces the old radiative-transfer framework for He I lines with a modern, self-consistent collisional-recombination model built on up-to-date atomic data and an explicit treatment of optical-depth effects from the metastable $2^3S$ state. By extending the modeled level set to $n_{\max}=50$ and integrating collision strengths directly, the authors obtain more accurate line emissivities across a broad grid of $n_e$ and $T$, and derive a new optical-depth correction function that remains valid beyond the previous limits. The paper shows that, in the optically thin limit, emissivities agree with prior results to within ~1%, but non-zero optical depths lead to 1–2% corrections in the $f_\tau$ function and up to 10% deviations in the common product $E(\lambda,n_e,T)\times f_\tau(\lambda,n_e,T)$ for critical lines like $\lambda$3889 and $\lambda$7065, driven by non-linear $\tau$ behavior and inconsistent data usage in prior models. A practical 3-parameter-like fitting form $f_\tau(n_e,T,\tau)\approx (1+a(n_e,T)\tau)/(1+b\tau)$ with $a(n_e,T)$ expressed as polynomials in $\lg n_e$ and temperature-dependent coefficients yields accuracy $\lesssim 0.1\%$ over the target domain, enabling straightforward incorporation into primordial helium analyses and other spectroscopic studies of He I lines.

Abstract

We present a new detailed model of the He I collisional-recombination spectrum based on the most up-to-date atomic data. The model accounts for radiative transfer effects and the influence of a non-zero optical depth in He I lines arising from transitions to the metastable 2^3S state. The model reveals substantial deviations in the emissivities of the lambda3889 and lambda7065 lines in the case of a non-zero optical depth, with previous models systematically underestimating and overestimating them by 5 to 20 percent, respectively. In the optically thin case, however, our results show good agreement with previous studies. Using the new model, we compute optically thin emissivities for a wide set of UV, optical, and IR He I recombination lines over a fine grid of electron densities and temperatures typical for H II regions and planetary nebulae (1 <= ne <= 10^4 cm^-3, 8000 <= Te <= 22000 K). In addition, we present new fitting formulae for radiative transfer corrections for several He I lines relevant to optical and near-infrared observations, covering 0 <= tau_3889 <= 10 within the same density and temperature ranges. The accuracy of the obtained approximations is <= 0.1 percent within the specified parameter range. These results can be readily implemented in modern codes for determining the primordial 4He abundance and are also applicable to a broader range of spectroscopic analyses of He I emission lines.

Figures (9)

• Figure 1: Extrapolation of oscillator strength data from dm2007 using $n^1F \rightarrow 4^1D$ transition as an example. Lower values of $x$ correspond to higher excited states. Abbreviation DM2007 represents paper dm2007.
• Figure 2: A flowchart describing algorithm for calculation of A-values for dipole allowed transitions in Helium atom. Abbreviation DM2007 corresponds to the paper dm2007.
• Figure 3: Scaling factor $f(T, \gamma)$ for calculation of radiative recombination rate to levels $n^1P$ for $T = 10^4$, $1.5\times10^4$ and $2\times10^4$ K. Blue circles represent exact data on ratio $\alpha_{\gamma}/\alpha_{\rm H}$. Red line represents fit defined in eq. (\ref{['eq:rrr_scale_eq']}).
• Figure 4: A flowchart describing algorithm for calculation of collisional rates in Helium atom. Abbreviation LB1998 corresponds to collision rate from lebedev1998.
• Figure 5: Emissivities of He I $\lambda10830$ line in units $10^{-26}$ erg cm$^3$ sec$^{-1}$, calculated for different electron densities as a function of temperature. Orange line represents emissivity calculated using our model. Blue line represents emissivity from porter2012aver2013. Red dashed line represents $T = 10^{4.25} = 17783$ K corresponding to interpolation node on the grid of averaged collision strengths from bray2000 (see text).