H-, He-like recombination spectra – V. On the dependence of the simulated line intensities on the number of electronic levels of the atoms

ABSTRACT

This paper presents a study of the dependence of the simulated intensities of recombination lines from hydrogen and helium atoms on the number of |$n\ell$|-resolved principal quantum numbers included in the calculations. We simulate hydrogen and helium emitting astrophysical plasmas using the code cloudy and show that, if not enough |$n\ell$|-resolved levels are included, recombination line intensities can be predicted with significant errors than can be more than 30 per cent for H i IR lines and 10 per cent for He i optical lines (⁠|$\sim$|20 per cent for He i IR recombination lines) at densities |$\sim 1\text{cm}^{-3}$|⁠, comparable to interstellar medium. This can have consequences in several spectroscopic studies where high accuracy is required, such as primordial helium abundance determination. Our results indicate that the minimum number of resolved levels included in the simulated hydrogen and helium ions of our spectral emission models should be adjusted to the specific lines to be predicted, as well as to the temperature and density conditions of the simulated plasma.

1 INTRODUCTION

The expansion rate of the early universe might have increased with the presence of additional neutrino flavours, affecting the neutron-to-photon ratio, which in turn results in higher production of primordial He (Olive, Steigman & Walker 2000; Steigman 2012). Assessing this scenario needs the determination of the helium primordial abundances, |$Y_p$|⁠, to high precision (Izotov & Thuan 1998; Cyburt, Fields & Olive 2002).

|$Y_p$| is commonly obtained by fitting the Big Bang Nucleosynthesis models to the barion density from measurements of the Cosmic Microwave Background (CMB). Fields et al. (2020a, b) give a maximum likelihood determination of the number of neutrino families |$N_\nu =2.843\pm 0.154$|⁠, resulting from the Planck mission data. Alternatively, spectroscopic determinations of the helium primordial abundance focus on extragalactic low-metallicity H ii regions, where abundances are obtained from selected helium and hydrogen line ratios (Izotov & Thuan 1998; Aver, Olive & Skillman 2015; Valerdi et al. 2019). |$Y_p$| has also been obtained from observations of the intergalactic medium (Cooke & Fumagalli 2018). High statistics on the observed targets help to minimize instrumental and observational errors (Izotov, Stasińska & Guseva 2013), although the introduction of systematic error in the observations can be dominant (Aver, Olive & Skillman 2010, 2011; Peimbert, Peimbert & Luridiana 2016). Other significant uncertainties reside in the values of the probabilities of the electron collisional processes with the hydrogen and helium ions.

Comparisons with synthetic spectra is then essential to model helium abundances. Modelling has its own sources of uncertainty. For example, a complete description of each species’ infinite number of quantum levels is not available to any computer. For moderate densities (⁠|$\boldsymbol {n}=10^4\,\text{cm}^3$|⁠), high Rydberg levels can be considered in local thermodynamic equilibrium (LTE), where the collisions balance the electron populations statistically. As electron density decreases, more and more levels must be explicitly considered, eventually elevating the calculation times and computational memory to prohibitive numbers.

Several techniques have been used to bypass this difficulty. Seaton (1964) describes the equations that lead to high n population departures from LTE. Brocklehurst (1972) simplified the collisional-radiative (CR) matrix using a ‘condensation’ scheme, interpolating high-n departure coefficients from a number of representative levels. For optically thin plasmas, Burgess & Summers (1976) created CR coefficients to describe the fast ‘ordinary’ collisional radiative processes in function of the slower metastables, which are assumed to be in pseudo static equilibrium.

A related problem is the treatment of |$n\ell$| sub-levels that should be accounted for explicitly in order to correctly describe emission lines from electronic transitions between |$n\ell \rightarrow n^\prime \ell ^\prime$| sub-shells. Every n-shell has |$n^2$| sub-shells, which increases the computing operations in |$\sim n^4$| (Guzmán et al. 2019, hereafter Paper III). The standard approach consists in setting a minimum n for which proton collisions dominate over radiative transitions, and the populations of the n- shells will be |$\ell$|-mixed (Pengelly & Seaton 1964, hereafter PS64). Over this limit, density and |$\ell$|-changing collisional rates are assumed to be large enough for the |$\ell$|-shells to populate statistically with electrons so n-shells can be treated as unresolved in |$\ell$| (also known as collapsed; see Ferland et al. 2017). Under this limit, n-shells will be resolved. The number of resolved levels should be small enough to avoid long calculation times but high enough to account for cascade electrons that will contribute to line emission.

In this paper, we will use the self-consistent spectral code cloudy (Chatzikos et al. 2023) to analyse the influence of the number of |$n\ell$|-resolved states on the predicted intensities of the hydrogen and helium lines. We aim to signal a potential source of uncertainty in the models and provide a method to avoid it. We show that these corrections are essential in studies that require a great accuracy, such as the primordial helium determinations mentioned above. Moreover, they could also be important in calculating helium abundances in solar metallicity H ii regions with low to moderate electron density (see e.g. Méndez-Delgado et al. 2020).

This paper is the fifth in a series where we carefully review the theory and methods for predicting the recombination spectra of H- and He-like ions. Section 2 reviews the atomic data employed in the code cloudy. Section 3 analyses the models’ dependence on the number of resolved and unresolved levels included. Section 4 describes our benchmark simulation of hydrogen and helium plasmas. Section 5 presents our results, followed by a discussion in section  6.

2 ATOMIC DATA

The code cloudy uses a collisional-radiative approach for H-like and He-like ions. All other ions are treated using a two-level calculation, where ionization-recombination balance is obtained by only considering ionization from the ground state of the species under study and where electrons recombined to all excited states will eventually decay to the ground state. For these ions, emissions from low-lying states are assumed not to be affected by the ionization/recombination processes due to the largely different time-scales for ionization/ recombination and excitation (see Ferland et al. 2017, for a discussion on the limits of this approximation). In contrast, coupling of the H-like and He-like iso-electronic sequences’ excited levels with the continuum can make an impact in the optical and infrared emission lines.

The atomic data used in this work for H-like and He like iso-sequences are summarized below.

2.1 Bound-free radiative transitions

We calculate radiative recombination coefficients for H-like ions using the Milne relation (detailed balance) from the photoionization cross-sections  (Brocklehurst 1971).

Hummer & Storey (1998) provide radiative photoionization data for atomic helium for |$n\le 25$| and |$\ell \le 3$| or |$n>25$| and |$\ell \le 1$|⁠. For higher |$\ell$|’s, these authors scale the recombination coefficients to hydrogen.

Data for photoionization from the ground level of He-like ions are obtained from the fits of Verner et al. (1996), while excited levels up to |$n=10$| from lithium through calcium are obtained from the TopBase database at the Opacity project¹ (Cunto et al. 1993). Photoionization coefficients from higher n’s are roughly approximated to be hydrogenic.

2.2 Bound-bound radiative transitions

We calculate the Einstein radiative de-excitation coefficients as a function of the oscillator strengths |$f_{nl,n^\prime l^\prime }$| (see Martin & Wiese 2006, equation 10.17):

where |$m_e$|⁠, e, and c are the electron mass, the electron charge, and the speed of light, respectively, |$\epsilon _0$| is the permittivity of free space, |$\nu$| is the frequency of the transition, and |$g_{nl}$| is the statistical weight of the |$nl$| level. The oscillator strengths are calculated as a function of the radial integral as (Mansky 2006)

with |$\omega =2\pi \nu$| the angular frequency of the transition in s|$^{-1}$|⁠, and |$R_\infty$| the Rydberg constant for infinite nuclear mass. The radial integrals |$R_{nl}^{n^\prime l^\prime }$| are calculated using the recursion relation of the hyper-geometric functions (Hoang-Binh 1990), which are part of the exact solution provided by Gordon (1929).

In cloudy’s collisional-radiative approach, high Rydberg levels are unresolved on the angular momentum. Those levels are dubbed collapsed and are assumed to have statistical populations in the |$\ell$|-subshells (Ferland et al. 2017). For them, we use the formula of Johnson (1972):

where |$g_i(n)$| are the factors of a polynomial approximation (⁠|$g(n,x)=g_0(n) +g_1(n)x^{-1}+g_2(n)x^{-2}$|⁠) of the bound-free Gaunt factor (Menzel & Pekeris 1935).

2.3 |$\ell$|-changing collisions

|$\ell$|-changing or |$\ell$|-mixing collisional transitions change the angular momentum |$\ell$| of an electron of a given n-shell by proton Stark field mixing (Pengelly & Seaton 1964; Vrinceanu & Flannery 2001). The transition probabilities increase at lower temperatures due to a longer interaction of the proton electric field with the target atom shells (Guzmán et al. 2016, hereafter Paper I). |$\ell$|-changing collisions produced by slow charged heavy particles, such as protons or alpha particles, are dominant.

We use an improved version of the approach of PS64, which we named PS-M, developed by Guzmán et al. (2017, hereafter Paper II), and extended to non-degeneracy cases (such as hydrogen-like helium ions) and low-temperature/high-densities cases, where the classical PS64 produced nonphysical results, in Badnell et al. (2021, hereafter paper IV). We use the formulas recommended in the equation (9) and (12) of paper IV. Comparison of these results with the ones from PS64, used by HS87, are given in table 1 of paper IV for |$n=30$|⁠, showing a ratio between the two theories of |$\sim 30~{{\ \rm per\ cent}}$| for |$\ell =4 \rightarrow \ell ^\prime =3$| and |$\sim 4~{{\ \rm per\ cent}}$| for |$\ell =29 \rightarrow \ell ^\prime =28$|⁠.

2.4 n-changing collisions

For hydrogen atoms, energy changing electron-impact de-excitation effective rate coefficients up to |$n=5$| are taken from the R-Matrix with pseudo-states results in table 2 of Anderson et al. (2000, 2002). For other H-like ions, fits from Callaway (1994) and Zygelman & Dalgarno (1987) are used for |$n=1-2$| collisions. For helium atoms, effective rate coefficients for |$n\le 5$| are taking from the convergent close coupling (CCC) calculations from Bray et al. (2000). For He-like ions except Fe|$^{24+}$|⁠, |$n=1,2 \rightarrow n=2$| electron impact collisional transitions are taken from the fits and tabulations of Zhang et al. (1987) based on their own calculations. For He-like iron, we prefer the more accurate results of Si et al. (2017) using the independent process and isolated resonances approximation using distorted waves (IPIRDW).

For higher principal quantum numbers, straight trajectory Born approximation is used (equation 8.30 in Lebedev & Beigman 1998). These rates are angular momentum unresolved |$n\rightarrow n^\prime$|⁠. To obtain |$n\ell \rightarrow n^\prime \ell ^\prime$| resolved effective rates, we average over the statistical weight of the initial and final levels:

2.5 Collisional Ionization and three-body recombination

We use Vriens & Smeets (1980) formulae for collisional ionization. Three-body recombination values are obtained from detailed balance.

3 DEPENDENCE OF THE CALCULATIONS ON THE NUMBER OF ATOMIC LEVELS

The more energy levels included in the calculations, the more accurate the description of recombination electrons cascading down to the lower levels to produce recombination lines. Additionally, the separate treatment of |$\ell$|-subshells when they are not statistically populated, e.g. |$\ell$|-changing collisions do not dominate over the radiative probabilities, leads to a precise calculation of the line intensities. In modelling, choosing the correct number of resolved levels can be tricky. In this section, we briefly show how to calculate the limiting n values for which the non-inclusion of |$\ell$|-resolved levels in the spectral simulations can make an impact.

3.1 Continuum coupling

To account for the offset that the computation of a finite number of levels of a model ion can have in the final calculation we ‘top off’ the highest n-shell (Bauman et al. 2005) by increasing its ionization rate by one hundred times, forcing it to LTE with the continuum. The immediate lower levels will then receive the electron cascade from the continuum, coupling through radiative decay and mostly from n-changing collisions. However, this treatment is inaccurate for models with |$n{<}100$| at intermediate densities (see appendix in Paper III). At higher densities, continuum-lowering effects (Bautista & Kallman 2000) can reduce the top-off errors. As a result, high n-shells tend to approach LTE with the continuum and between themselves due to the high excitation rates (Paper III). We define departure coefficients |$b_n$| as the departure of the shell population from LTE:

where |$\boldsymbol {n}_+$|⁠, |$\boldsymbol {n}_e$|⁠, |$m_e$|⁠, and |$T_e$| are the parent ion density, electron density, electron mass, and electron temperature, respectively; h is the Planck constant; k is the Boltzmann constant; |$E_n$| is the n-shell binding energy; and |$g_n$| is the statistical weight of the shell n. When the departure coefficients are close to 1, the levels are at LTE with each other and with the continuum. In Fig. 1, we show departure coefficients for a slab of gas at density |$\boldsymbol {n}_\text{H} = 10^4 \,\text{cm}^{-3}$| under a monochromatic ionization source at 2 Ryd. In the Figure, the departure coefficients for high n smoothly approach unity. A calculation must include enough n-shells at LTE to describe the electron cascade from higher levels accurately. This method works well at intermediate hydrogen densities of |$\boldsymbol {n}_{\text{H}} \lesssim 10^5 \,\text{cm} ^{-3}$| when the principal quantum number of the last n-shell is |$n_{\text{last}}\gtrsim 100$| (Paper III). At higher densities, the effects of continuum lowering can reduce errors (Bautista & Kallman 2000).

More rigorous approaches have been used in the literature. Hummer & Storey (1987) CR matrix condensation (Burgess & Summers 1969; Brocklehurst 1970; Burgess & Summers 1976) interpolates and extrapolates high-n departure coefficients to a smooth function.

3.2 |$\ell$|-changing critical densities

The line emissivity for a transition from the upper level |$n\ell$| to the lower level |$n^\prime \ell ^\prime$| is given by

where |$\nu$| is the frequency of the transition, |$\boldsymbol {n}_{n\ell }$| is the density of the upper level, and |$A_{n\ell ,n^\prime \ell ^\prime }$| is the Einstein coefficient value of that transition. It is important to obtain precise equilibrium populations on the upper levels of the transitions to obtain accurate line emissivities. If these levels are not |$\ell$|-mixed, they must be resolved in their corresponding |$\ell$|-shells. |$\ell$|-changing collisions are more effective the higher the principal quantum numbers n are, the effective coefficients varying as |$q_{\ell \ell ^\prime } \sim n^4$| if |$\ell \ll n$| (PS64). At high n, |$\ell$|-changing is dominant over n-changing transitions, and the n-shells are statistically populated in all the |$n\ell$|-subshells,

where |$n_{n\ell }$| is the density of the |$n\ell$|-subshell. The competing process is spontaneous decay. The radiative lifetime is |$\tau _{n\ell } \sim n^5$|⁠, and so it can be compared with the collisional redistribution lifetime,

where |$n_\text{coll}$| is the collider density. We define the critical density at the point where |$\tau _\text{coll}=\tau _{n\ell }$| as

In astrophysical plasmas, such as H ii regions, we can assume |$\boldsymbol {n}_\text{coll} \sim \boldsymbol {n}_\text{H} \sim \boldsymbol {n}_e$| because hydrogen atoms are about 90 per cent of all particles, although Helium ions and alpha particles could contribute to |$\ell$|-changing collisions, decreasing critical densities in equation (9), as He abundance is usually around 10 per cent. In the rest of this paper, we will assume that |$\boldsymbol {n}_\text{coll} \sim \boldsymbol {n}_\text{H}$| unless otherwise specified.

The total radiative lifetime depends on the principal quantum number n as |$\tau _{n\ell } = A_{n\ell }^{-1} \propto n^{5}$|⁠, and, together with the dependence of the collisional rates on n, will make |$\boldsymbol {n}^\text{crit}_\text{coll} \propto n^{-9}$|⁠. In Fig. 2, we fitted the critical densities obtained for pure hydrogen gas at |$T = 10\,000$| K. Our fit, |$\boldsymbol {n}^\text{crit}_\text{coll}=An^\beta$| with |$A=7.6\times 10^{11}\text{cm}^{-3}$| and |$\beta = 8.68$|⁠, coincides with our estimations.

This simple fit produces a method to estimate the minimum principal quantum number to be |$\ell$|-resolved. However, it is necessary to account for the temperature dependence of the collisional rates |$q_{n\ell }$|⁠, which can significantly vary the critical densities. In Fig. 3, the critical densities range over two orders of magnitude between |$T=100$| K to |$T=10^8$| K. For example, at electron density |$\boldsymbol {n}_e=10^4\,\text{cm}^{-3}$|⁠, the level |$n=30$| can be |$\ell$|-mixed at |$T_e=100$| K, but it will not be so at |$T_e>10^5$| K. We have fitted the dependence of the critical density with the temperature to |$\mathbf {n}^\text{crit}_\text{coll} = B+CT^\gamma$|⁠, where |$\gamma \approx 0.44$|⁠, and B and C depend on the specific principal quantum number.

This situation is slightly more complicated for helium atoms, as |$\ell$|-changing collision probabilities depend on an extra cut-off of the probability at large impact parameters due the broken degeneracy of the |$\ell$|-shells (Paper II). The cut-off acts at low temperatures, increasing the critical density, lowering the rate coefficients, and making them have a more complicated temperature dependence than in the hydrogen case. This effect is mainly happening at low n’s, with a larger degeneracy. In Figs 4 and 5, critical densities are plotted as a function of the principal quantum number and temperature. The dependence of the |$\ell$|-changing rate coefficients on the electron temperature reflects on the critical densities. In Fig. 4, the critical densities of high Rydberg helium levels still depend on the principal quantum number as |$\boldsymbol {n}^\text{crit} _\text{coll} \sim n^{-9}$|⁠.

n-shells are at critical densities when outbound radiative decay and inbound |$\ell$|-changing collisions are equally competing. Thus, these levels should not be expected to be statistically populated in the |$\ell$|-subshells until collisions strongly dominate, making it necessary to resolve n-shells until the critical densities are well below the density of the simulated plasma. A rule of thumb is to keep resolved levels up to 10 + n, with n the principal quantum number of the level for which the critical density equals the density of the plasma. However, as seen in Fig. 2, critical densities are more spaced for lower levels. Therefore, this rule will work better at higher densities, where a relatively lower number of n-shells needs to be resolved. Specifically, for densities below |$\sim 1\text{cm}^{-3}$|⁠, levels well over 10+ n must be kept |$\ell$|-resolved.

4 HYDROGEN AND HELIUM IONIZED PLASMA SIMULATIONS

To quantify the error in the line intensities produced by an insufficient atom description, we carried simulations using the last development version of the spectral code cloudy (master 13108b32, Chatzikos et al. 2023), varying the number of resolved levels. We restrict ourselves to a slab of pure hydrogen gas or a mixture of hydrogen and helium to understand the effect of the size of atoms. We have illuminated our plasma with a monochromatic radiation of 2 Ryd, enough to ionize helium and hydrogen atoms. We set the ionization parameter as

where |$\phi _\nu$| is the photon flux² at the surface of the cloud in |$\text{ergs}\cdot \text{s}^{-1}\cdot \text{cm} ^{-2}$|⁠, |$\boldsymbol {n}_\text{H}$| is the hydrogen density, and c is the speed of light. This value will ensure enough photons to ionize our gas at all densities.³ We restrict the thickness of our slab to 1 cm to minimize optical depth and reduce the computational time. We assume case B (Baker & Menzel 1938) approximation, where Lyman emission lines (except Ly  |$\alpha$|⁠) are reconverted into higher series. Strict Case B approximation is accurate at densities |$\boldsymbol {n}_e\le 10^8\text{cm}^{-3}$| (Hummer & Storey 1987). We relax strict Case B approximation by explicitly accounting for collisions from |$n=2$| levels, which allows to carry this approximation up to higher densities.

Our simple models demonstrate the dependence of the electron populations on the number of levels of the modelled atoms. For the pure hydrogen model, we have tested the intensities of all recombination lines corresponding to the decay from |$n=10$| to |$n=2{\!-\!}8$|⁠, given in Table 1. For the mixture of hydrogen and helium we monitor recombination lines up to |$n=6$| in Table 2. Many of these lines are used to obtain primordial helium abundances (Porter & Ferland 2007; Porter et al. 2012, 2013).

We can expect similar behaviour from diffuse gas emission in clouds illuminated by multiwavelength spectral energy distribution sources where the gas is almost completely ionized, i.e. H ii regions, Broad Line regions, and Planetary Nebulae.

5 RESULTS: DEPENDENCE OF THE RECOMBINATION LINES INTENSITIES ON THE NUMBER OF LEVELS

To demonstrate how |$\ell$|-mixing influences electronic population, and, by extension, diffuse gas line emission, we increase the principal quantum number for which we computed resolved |$\ell$|-subshells until convergence in the line intensities. This will provide us with the number of resolved levels needed to be included in the calculations to obtain accurate line intensities. For our models, the minimum principal quantum number resolved in |$\ell$| is |$n=10$|⁠, which is the default for hydrogen in cloudy (see Ferland et al. 2017). Then, we increased the number of |$\ell$|-resolved n in steps of five up to |$n=70$|⁠. The rest of the levels remained collapsed up to |$n=200$|⁠. We then calculated the minimum resolved n necessary for converging the line intensity to 1 per cent and 5 per cent.

5.1 Pure hydrogen gas

Our first model used pure ionized hydrogen gas. We set the electron temperature of the plasma to 10 000K, and its hydrogen density varied between |$\boldsymbol {n}_H = 10^0{\!-\!}10^7\,\text{cm} ^{-3}$|⁠. The hydrogen density variation impacts the |$\ell$|-changing rates and can slightly affect critical densities in Fig. 2. To study the influence of the number of resolved levels in cloudy’s predictions, we have progressively increased the highest resolved principal quantum number in 5 units from |$n=10$| to 70. Meanwhile, we kept the total number of n-shells at |$n=200$|⁠. We have monitored the convergence of the intensities of all lines in Table 1.

In Fig. 6, we show the effect that insufficient resolved levels in the hydrogen atom produce on the H|$\alpha$| line. In the figure, we represent the predicted intensity as a function of the highest resolved principal quantum number considered in the calculation. For H |$\alpha$|⁠, the critical density of the upper level (⁠|$n=3$|⁠) is |$\boldsymbol {n}^\text{crit}\approx 10^9\text{cm}^{-3}$|⁠, well above the densities of most astrophysical nebulae, so cloudy keeps the level |$n=3$| resolved by default. However, electrons in upper n-shells influence the population of this level by decay. How the electron population of the |$\ell$|-subshells of these upper levels is distributed will depend mainly on the hydrogen density, as shown in Figs 2 and 3. It is important to include enough levels resolved in |$\ell$| to account for the correct electronic cascade towards these low levels. For example, the default calculation in cloudy includes |$\ell$|-resolved levels up to |$n=10$|⁠, which will have a statistical electron population on the |$\ell$|-subshells when the density is well above |$\mathbf {n}_\text{H} = 10^5 \text{cm}^{-3}$| for |$T=10^4$| K (see Fig. 2). In Fig. 6, this is only happening at the bottom right panel, for densities |$\boldsymbol {n}_\text{H} \ge 10^6\, \text{cm}^{-3}$|⁠. At |$\boldsymbol {n}_\text{H} = 10^4\, \text{cm}^{-3}$|⁠, we need to increase the resolved levels to |$n=20$| (⁠|$\boldsymbol {n}^\text{crit} > 100\, \text{cm}^{-3}$|⁠) to achieve convergence. In Fig. 6, the highest resolved level needed for a 1 per cent converged prediction of the intensity of H |$\alpha$| does not change at lower densities: the transition probabilities from levels immediately over |$n = 3$| dominate over the upper levels. As shown in Fig. 7, radiative and collisional transition probabilities to |$n=3$| decrease rapidly for |$\Delta n> 1$|⁠, and levels over |$n=15$| contribute only marginally to the electron population of the upper level. Even if levels with |$n \ge 15$| are not adequately resolved, their electron population will preferentially go to immediate lower levels and will not directly contribute to the H |$\alpha$| line. Note that the jump at |$n=5$| on the effective collisional rates in Fig. 7 comes from the change from the R-matrix approach to the Born approximation (Section 2.4) and is probably an artifice. However, suppose the coefficients for |$n>5$| would be one order of magnitude higher and smoothly join with |$n=5$|⁠. In that case, we can speculate that this would only slightly change Fig. 6, as the effective coefficients for |$n=15\rightarrow 3$| are already two orders of magnitude lower than |$n=6\rightarrow 3$|⁠. Better collisional data are indeed necessary to obtain more accurate results.

Table 1 shows the minimum highest resolved n for which the predicted line intensity converges with the highest resolved level to 5 per cent and 1 per cent for |$\boldsymbol {n}_\text{H} = 1 \,\text{cm}^{-3}$| and |$\boldsymbol {n}_\text{H} = 10^4 \,\text{cm}^{-3}$|⁠. We have also included the percentile difference of the line intensity prediction between our largest models (highest resolved level |$n=70$|⁠) and the default (highest resolved level |$n=10$|⁠) to indicate the sensitivity of the line to the number of resolved levels.

The variation in the number of resolved levels does not strongly influence the upper lines of the Balmer series. Their better convergence can be understood considering the dominance of |$\Delta n =1$| radiative transitions, making the electron flux to |$\Delta n >1$| residual and the convergence errors smaller. This effect is more noticeable for the Paschen series, where the highest error is on the P|$\alpha$| line (10.30 per cent for |$\mathbf {n}_\text{H}= 1 \text{cm}^{-3}$| and 7.61 per cent for |$\boldsymbol {n}_\text{H} = 10^4 \,\text{cm}^{-3}$|⁠). As seen in Fig. 7, the electron effective coefficients and the radiative transition rates to |$3s$| from different upper n’s dramatically decay as n increases. As we move towards the infrared to higher series, it takes more resolved levels to achieve convergence (Table 1) because the upper levels of the transitions are closer to the highest resolved levels and more affected by electrons cascading from unresolved levels. These two effects, the predominance of lower |$\Delta n$| decays and the larger deviation on the electronic populations of the upper levels closer to the resolved limit, combine to make the strongest lines from progressively higher series more sensitive to the number of resolved levels. That is shown in Fig. 8, where we represent the per cent differences in intensity plotted against the lowest level of the transition. In this figure, we have tagged each data point with the upper level of the corresponding transition. Larger lower n’s tend to have a larger error, while larger upper levels that decay to the same lower level have less error. At higher densities, the critical densities of a smaller number of levels will be over the plasma density, so a lower number of resolved levels is required. That can be seen in Fig. 9, where we represent the highest resolved level to achieve 1 per cent of convergence. The highest resolved n needed tends to decrease for higher densities. It is, however, still around |$n=35$| for NIR hydrogen transitions at electron densities of |$\boldsymbol {n}_e=10^4\, \text{cm}^{-3}$|⁠.

5.2 Helium recombination lines

A second set of models consisted of an ionized hydrogen and helium mixture gas where the helium abundance is 10 per cent with respect to hydrogen. The gas is illuminated with a monochromatic radiation of 2Ryd, enough to ionize hydrogen and also He atoms to He|$^+$| in typical conditions for an H ii region. Hydrogen densities were the same as in the pure hydrogen gas case models. The default highest resolved level for the helium atom in a cloudy calculation is |$n=6$| (Ferland et al. 2017), corresponding to a critical density of |$\boldsymbol {n}^\text{crit} \approx 1.7\times 10^8 \,\text{cm}^{-3}$|⁠. It is clear that in the range of densities considered in this study, where the highest hydrogen density |$\mathbf {n}_\text{H} = 10^7 \text{cm}^{-3}$|⁠, levels higher than |$n=6$| should be resolved. As we focus on the convergence of the emission lines of the recombined helium atoms, we have deemed it sufficient to start our models with the highest resolved level |$n^\text{res} = 10$|⁠. As with the pure hydrogen gas, we have successively increased the number of resolved levels until |$n=70$|⁠, where convergence was achieved for all the lines we considered. Note that this will not be necessarily the case for lines belonging to high-n transitions in the FIR or the radio regions of the spectrum. We have considered transitions corresponding to electron decays between helium levels with principal quantum numbers |$n=1$| to |$n=6$|⁠, listed in Table 2, covering a spectral range from the UV to the NIR. For simplicity, we kept only those lines that could be important for the determination of primordial helium abundances, as in Porter, Ferland & MacAdam (2007).

Table 2 lists the minimum resolved principal quantum number needed in our models to achieve a 5  per cent and 1  per cent convergence on the line intensity. Helium recombination lines in Table 2 present the same competing factors as in the case of pure hydrogen: the dominance of |$\Delta n = 1$| and the closeness of the high levels to the lowest non-resolved level. However, in this case, two spin systems, singlet and triplet, and the split of the degeneracy of triplet levels complicate the picture. Moreover, as stated in Section 3 and shown in Fig. 5, critical densities do not follow the same pattern for different n at different temperatures. As an illustration, in Table 3, we have taken the lines corresponding to spin-conserved transitions to levels |$1\, ^1S$| and |$2\, ^1S$|⁠. The differences between calculations with maximum resolved |$n^\text{res}=10$| and |$n^\text{res} =70$| increase up to |$n=5$|⁠, to decrease again from there. In Table 3, the electron collision transition rates from the upper-level |$n=6$| are at least an order of magnitude smaller than from smaller principal quantum numbers, revealing that |$n=6$| is not as strongly connected with |$n=2$| or |$n=1$| as the other levels with smaller n, mitigating the influence of the electron populations from a wrongly statistically populated level ahead of it. This influence is dominant for the transitions with |$n\le 5$|⁠, which have an increasing error as the level is closer to the maximum resolved level. This effect is possibly artificial because the separation of collisional rates between |$n=5$| and |$n=6$| coincides with different sets of collisional data obtained from different theoretical approaches (see Section  2.4). As in the case of hydrogen, better collisional data is necessary to assess this problem.

A final exception comes from the transitions from upper |$n\ell$|-shells with larger angular momentum. These transitions show a high per cent difference for different maximum resolved levels. This is explained by the more effective decay of the yrast levels through |$\Delta n =1$| transitions. Thus, they are more influenced by the forced statistical population of the upper non-resolved levels. This is the case of the lines with the more significant uncertainty on Table 2. For example, the transition (⁠|$2\, ^1P_1-3\, ^1D_2$|⁠) |$\lambda 6678.15$| Å has a maximum per cent difference of 7.42 per cent, while (⁠|$2\, ^1P_1-3\, ^1S$|⁠) |$\lambda 7281.35$| Å only amounts to 0.62 per cent. We obtain similar comparative ratios for other transitions with a high uncertainty, like (⁠|$2\, ^3P_J-3\, ^3D$|⁠) |$\lambda 5875$| Å (blended in J) compared to (⁠|$2\, ^3P_J-3\, ^3S$|⁠) |$\lambda 7065.25$| Å, or (⁠|$3\, ^3D-4\, ^3F$|⁠) |$\lambda 18685.1$| Å compared to |$3\, ^3D-4\, ^3P$||$\lambda 19543.1$| Å

While helium recombination lines of interest typically involve low levels, specifically in the optical range, their predicted intensities can still be significantly affected by the wrong choice of the number of resolved levels. This effect is more considerable when the hydrogen densities are smaller, so more levels fall under their critical densities. As shown in Fig. 10, convergence to 1 per cent needs at least a |$n^\text{res} = 15$| for most optical lines, but it can get to |$n^\text{res} = 35$| for infrared lines.

In Fig. 11, we show the convergence of the helium line intensities with the most significant per cent differences for densities of |$\boldsymbol {n}_\text{H} = 1 \,\text{cm}^{-3}$|⁠, corresponding to typical interstellar medium densities. Even when 1 per cent convergence is achieved fast at |$n^\text{res} = 15{\!-\!}20$|⁠, except for some extreme cases, most of the lines keep slowly varying their intensity up to |$n^\text{res} = 60{\!-\!}70$|⁠. As a rule of thumb, a safe value for the lines would be to set |$n^\text{res} \approx 40{\!-\!}50$|⁠, well over the minimum value required for the critical density shown in Fig. 4. This must be taken with care, and some lines might need a more conservative approach even at much higher densities. For example, (⁠|$3\, ^3D-4\, ^3F$|⁠) |$\lambda 18685.1$| Å has a 1 per cent convergence of the intensity at a maximum resolved level of |$n^\text{res} = 30$| at |$\boldsymbol {n}_\text{H} = 10^4 \,\text{cm}^{-3}$|⁠, while Fig. 4 suggests a |$n^\text{res} = 15{\!-\!}20$|⁠, well below that value.

6 CONCLUSIONS

The main aim of this study is to predict the number of |$n\ell$|-resolved levels needed to obtain accurate line intensities from hydrogen and helium in photoionization models, which applies to the determination of primordial helium abundances from line emission on metal-poor galaxies. While we do not think modelling should be added to the error budget of primordial helium determination in table 1 of Peimbert et al. (2007), it is crucial to consider atoms with an adequate number of levels to avoid the derived uncertainties on the line emissivities. This work will help quantify the number of levels to remove the uncertainties of high precision problems.

In this paper, we have shown the dependence of the calculated hydrogen and helium recombination line intensities on the number of resolved |$n\ell$|-shells included in the photoionization models. Our results show that, especially at low densities, a high number of |$n\ell$| resolved levels need to be included to get an accurate value of the line intensities. This error can be more significant for infrared lines with a high upper n. The temperature dependence on the critical densities emanating from the |$\ell$|-changing cross-sections can be more significant for low level transitions inside helium atoms (Fig. 5).

Our literature search found that different studies have varied ways of dealing with this problem. Storey & Hummer (1995) used the radiative to collisional excitations ratio to obtain the maximum resolved n for which the transition |$np\rightarrow n(n-1)$| has a 90 per cent chance to happen before a radiative transfer can occur. Using this method they find differences in 1 per cent in the departure coefficients of high |$nl$|-levels with respect to a fully resolved calculation. Some authors avoided the problem by accounting for a number of resolved levels in their photoionization models which is big enough to be over any considered critical densities, both in hydrogen and helium. For example, Porter et al. (2005) use |$n^\text{res} = 100$| for modelling He i emission in H ii regions. Porter et al. (2007, 2009) reduce the resolved levels to |$n^\text{res} = 40$| and |$n\le 100$|⁠, safely converging the helium lines of Table 2 for densities of |$\boldsymbol {n}_\text{H} =10^4\, \text{cm}^{-3}$|⁠. Porter et al. (2012, 2013) set |$n^\text{res}\le 100$| for a |$n\le 101$| model for helium. While a calculation of this size will give accurate results for visible and most infrared lines, we deem it unnecessary to resolve as many levels for the densities considered in their study, |$10^1\,\text{cm}^{-3} \le \boldsymbol {n}_e \le 10^{14} \,\text{cm}^{-3}$|⁠, that are well over the critical densities for |$n=100$|⁠. In their models for helium emission, Del Zanna & Storey (2022) use |$n\le 100$| and |$n^\text{res}\le 40$|⁠, which is suitable for all helium visible lines at most densities of nebular astrophysics. Del Zanna et al. (2020) use the same, |$n\le 100$| and |$n^\text{res}\le 40$|⁠, for their models of the solar corona at electron densities of |$\mathbf {n}_e = 10^8\text{cm}^{-3}$| and temperatures of |$T=10^6$|K, corresponding to the critical densities of |$n\approx 5$| (Figs 4 and 5). Therefore, we can consider their solar and nebular results well converged. Recently, Balser & Wenger (2024) included up to |$n^\text{res}=25$| for hydrogen and |$n^\text{res}=20$| for helium in their simulations of the H ii region’s emission lines for densities that ranged from |$\boldsymbol {n}_e=10^1\,\text{cm}^{-3}$| to |$\boldsymbol {n}_e=10^4\,\text{cm}^{-3}$|⁠. According to Tables 1 and 2, this provides convergence of most of the optical recombination lines of hydrogen and helium. Other authors have used these results: Aver et al. (2015) worked with the emissivities provided by Porter et al. (2012) on their study to obtain helium primordial abundances. Similarly Izotov, Thuan & Guseva (2014) use the emissivities from the models from Porter et al. (2009).

However, many studies that cite the latest papers of cloudy (Ferland et al. 2017; Chatzikos et al. 2023) omit the number of resolved levels accounted for in their models. It is important to note that if the commands that set the number of resolved and collapsed levels,

database h-like hydrogen resolved levels 80

database h-like hydrogen collapsed levels 70

database he-like helium resolved levels 50

database he-like helium collapsed levels 100,

are ignored in the inputs, the default number of resolved levels is |$n^\text{res}=10$| for hydrogen atoms and |$n^\text{res}=6$| for helium. These numbers are insufficient at the densities and temperatures of most astrophysical plasmas.

Supported by the results of this work, we strongly recommend selecting a maximum number of resolved levels that will provide converged emissivities for the lines under interest. The most straightforward method is using Figs 2, 3, 4, and 5 to select the level at which the density is over the critical density and add at least ten principal quantum numbers to determine the maximum resolved level. We provide ASCII files with copies of Tables 1 and 2 attached to this paper for all densities (⁠|$1\text{cm}^{-3} \le \mathbf {n}_\text{H} \le 10^7 \text{cm}^{-3}$|⁠) considered in this work. These will help to provide a guide for the required convergence of the emissivities of specific lines.

ACKNOWLEDGEMENTS

MC and GJF acknowledge support from NSF (1910687), and NASA (19-ATP19-0188, 22-ADAP22-0139). GJF acknowledges support from JWST through AR6428, AR6419, GO5354, and GO5018.

DATA AVAILABILITY

The files attached with this paper include a version of Tables 1 and 2 for all densities considered: |$1\,\text{cm}^{-3} \le \boldsymbol {n}_\text{H} \le 10^7$|⁠. These tables have been generated using the developed version of cloudy C23.01 with versions of the following input script:

set save prefix ''h40_4''

#

init file ''honly.ini''

#

laser 2 ryd

ionization parameter -2

#

hden 4

constant temperature 4 log

set dr 0

#

case B

#

database h-like hydrogen levels resolved 40

database h-like hydrogen levels collapsed 160

#

stop zone 1

iterate 2

#

print critical densities h-like

print lines column

#

save line list ''.list'' ''linesh.dat'' column

absolute last no hash

save line list ratios ''.rat'' ''linesh_ratio.dat''

column last no hash

Here, the last two lines of the script have been broken to fit in the column. Here, the number of resolved levels for hydrogen, density (hden), and temperature can be changed to a grid. Input scripts for helium are similar. The file linesh.dat contain the list of lines to be tested, while the file linesh_ratio.dat also contains the case B lines that are compared with them.

Footnotes

REFERENCES

APPENDIX A: Convergence at densities |$1\,\rm{cm}^{-3} \le \boldsymbol {n}_\rm{H} \le 10^7\,\rm{cm}^{-3}$|⁠.

Below, we reproduce Tables 1 and 2 for all other densities between |$1\,\text{cm}^{-3} \le \boldsymbol {n}_\text{H} \le 10^7\,\text{cm}^{-3}$|⁠, corresponding the density range investigated in this work. Tables A1, A2, A3, A4, and A5 can serve as a guide for setting the number of levels to obtain accurate emissivities for He i and H i recombination lines from UV to IR.