Emission and recombination coefficients for hydrogen with κ-distributed electron energies

Abstract

We provide a data set of emission and recombination coefficients of hydrogen using a κ-distribution of electron energies rather than the more traditional Maxwell–Boltzmann (MB) distribution. The data are mainly relevant to thin and relatively cold plasma found in planetary nebulae and H ii regions. The data set extends the previous data sets provided by Storey and Hummer which were computed using an MB distribution. The data set, which is placed in the public domain, is structured as a function of electron number density, temperature and κ. Interactive fortran 77 and c++ data servers are also provided as an accessory to probe the data and obtain Lagrange-interpolated values for any choice of all three variables between the explicitly computed values.

atomic data, atomic processes, radiation mechanisms: general, radiation mechanisms: non-thermal, ISM: abundances, planetary nebulae: general

1 INTRODUCTION

There are many studies related to the recombination of hydrogen and hydrogenic systems, the most comprehensive being those of Hummer & Storey (1987); Storey & Hummer (1988) and Storey & Hummer (1995). However, all the past work was based on a Maxwell–Boltzmann (MB) distribution of electron energies. There has been, and still is a general consensus that this is the appropriate distribution for thin nebular plasmas. However, this was disputed in the past (Hagihara 1944) where considerable deviations from the thermodynamic equilibrium on which the MB relies were claimed although this claim was later discounted (Bohm & Aller 1947).

There has been a recent revival (Nicholls, Dopita & Sutherland 2012) of the proposal of a non-thermal electron energy distribution in planetary nebulae and H ii regions in the light of the long standing problem in nebular physics of the contradiction between the results for elemental abundance and electron temperature as obtained from the optical recombination lines (ORL) and those obtained from the collisionally excited lines (CEL). According to the recent proposal, the ORL-CEL discrepancy problem can be resolved by assuming a non-MB electron distribution, specifically the κ-distribution. There have been a few recent attempts to assess the merit of this suggestion (Sochi 2012; Storey & Sochi 2013, 2014; Zhang, Liu & Zhang 2014) using spectroscopic means to directly sample the free electron energy distribution. With the exception of Storey & Sochi (2014) they are all inconclusive, in the sense that the data do not differentiate between a single κ-distribution and other models, such as one with two MB components at different temperatures. Storey & Sochi (2014) do, however argue that, with a high degree of certainty, the Balmer line and continuum spectrum of the extreme planetary nebula Hf 2-2 cannot be modelled with a single κ-distributed electron energy distribution, while it can be with a model comprising two MB distributions. It should be noted however that Zhang et al. (2014) analyse the same spectra of the same object and conclude that either model can adequately model the spectrum. We return to this apparent contradiction below.

Both Storey & Sochi (2014) and Zhang et al. (2014) model the Balmer line and continuum spectrum with MB and κ-distributions. Typically, the continuum intensity is modelled relative to one of the high Balmer lines, chosen to be close in wavelength to the Balmer edge and apparently unblended. The continuum spectrum is relatively easy to model with an arbitrary electron energy distribution but the modelling of the high Balmer line intensities requires a full treatment of the collisional–radiative recombination process as a function of κ as well as the usual density and temperature variables. Storey & Sochi (2014) made such a calculation in Case B of Baker & Menzel (1938) and presented some results for the line which they used to normalize intensities, H11. Here, we publish the full results from those calculations. We note that Zhang et al. (2014) also use H11 for normalization but they rely on an approximate treatment taken from Nicholls et al. (2012) which applies a κ-dependent scaling function to the H11 emission coefficient calculated with an MB distribution. Storey & Sochi (2014) show that this approximation is poor for the very low values of κ that are required to model the Hf 2-2 spectrum, which may explain why Zhang et al. (2014) reach a different conclusion to Storey & Sochi (2014) in the case of Hf 2-2.

The results are provided in two text files where the emission and recombination coefficients are given as a function of electron number density N_e, electron temperature T_e and κ. We also provide interactive data servers, in the form of fortran 77 and c++ codes, for mining the data and obtaining interpolated values in the three variables between the explicitly computed values. In Section 2, we give a brief theoretical background about the atomic computational model used to generate the data; while in Section 3, we explain the structure of the data set and provide general instructions and clarifications about how it should be probed and used. Section 4 contains general conclusions and discussions.

2 ATOMIC MODEL

The fundamental quantity required for the calculation of the hydrogen recombination line intensities is the coefficient for recombination to a state nl of H. For a free electron energy distribution f(E), this is given by (e.g. Storey & Sochi 2014)

\begin{eqnarray} \alpha (nl) &=& \frac{R^{{5}/{2}}}{\sqrt{2}c^2 m_{\rm e}^{{3}/{2}}} \frac{\omega (nl)}{\omega ^+}\nonumber\\ &&\int _0^\infty \left( \frac{h\nu }{R} \right)^2 \left( \frac{R}{E} \right)^{{1}/{2}} \sigma (\nu ,nl)\ f(E)\ {\rm d}\left(\frac{E}{R} \right) , \end{eqnarray}

(1)

where R is the Rydberg energy constant, ω⁺ and ω(nl) are the statistical weights of the initial and final states, respectively, ν is the frequency of the emitted photon, E is the energy of the free electron, σ(ν, nl) is the cross-section for photoionization which is the inverse process to recombination, and the other symbols have their standard meanings. The calculation of the recombination line emission coefficients in a full collisional–radiative treatment has been described by Hummer & Storey (1987) and Storey & Hummer (1995) and we use the same methods here.

Traditionally, f(E) is the MB distribution function. Here, we use instead the κ distribution which is given by (Vasyliunas 1968; Summers & Thorne 1991)

\begin{eqnarray} &&{f_{\kappa }\left(E, T_{\kappa }\right)=}\nonumber\\ &&{\qquad\frac{2\sqrt{E}}{\sqrt{\pi (kT_{\kappa })^3}}\frac{\Gamma \left(\kappa +1\right)}{(\kappa -\frac{3}{2})^{{3}/{2}}\Gamma \left(\kappa -\frac{1}{2}\right)}\left(1+\frac{E}{(\kappa -\frac{3}{2}) kT_{\kappa }}\right)^{-\left(\kappa +1\right)} ,} \end{eqnarray}

(2)

where κ is a parameter defining the distribution, Γ is the gamma function of the given arguments, and T_κ is a temperature characteristic to the particular distribution. For sufficiently large κ, f_κ(E, T_κ) tends to the MB distribution.

For energetically low-lying states of H, the dominant processes are recombination and radiative decay. For higher states, collisional processes become important, with l-changing collisions being the most frequent. In our calculations the nl states of hydrogen are assumed degenerate with respect to l and in this case the dominant processes that change l are collisions not with electrons but with H⁺, He⁺ and He⁺⁺ ions. In the results described here, we retain an MB distribution for these heavier particles. We also retain an MB distribution for the processes that change energy and n, which are dominated by collisions with electrons. Consequently the emissivities that we calculate should be treated with caution for the high-n states for which l- and n-changing collisional processes become important. The boundary of this region is primarily a function of the electron density, being at approximately n = 100, 75, 50, 30 and 20 for densities of 10², 10³, 10⁴, 10⁵ and 10⁶ cm⁻³, respectively.

The rate coefficients for l-changing collisions used to obtain the above boundary n values were calculated using the theory described by Pengelly & Seaton (1964). Vrinceanu and co-workers have published a series of papers (Vrinceanu 2005; Vrinceanu, Onofrio & Sadeghpour 2012, 2014) on electron and proton induced collisions with Rydberg states of hydrogen. In Vrinceanu et al. (2012), they state that the rate coefficients for proton induced Δl = 1 transitions are overestimated by the theory of Pengelly & Seaton (1964) by about an order of magnitude. The calculation of the rate coefficient depends upon an integration of the probability for an l-changing transition over the impact parameter of the incident particle, assumed to travel on a straight-line trajectory. It is well known that this integral is divergent for Δl = 1 transitions in a quantum mechanical treatment. The approximate treatment of the transition probability by Pengelly & Seaton (1964) converges on the quantum mechanical result at large impact parameter as illustrated in fig. 1 of Vrinceanu et al. (2012). Pengelly & Seaton (1964) introduce a cut-off at large impact parameter to remove the divergence based on collective effects in the plasma or the finite lifetime of the Rydberg state. The semiclassical approach of Vrinceanu et al. (2012) does not correctly replicate the quantum behaviour at large impact parameter with the probability instead falling discontinuously to zero at a finite and relatively small value of the impact parameter. The missing contribution from large impact parameter is the origin of the order of magnitude difference they report between their results and those of Pengelly & Seaton (1964). We see no reason to prefer their semiclassical result over the quantum treatment at large impact parameters and therefore consider the Pengelly & Seaton (1964) results to be more reliable.

3 DATA

The κ-dependent emission coefficients, ϵ(N_e, T_e, κ) are provided in a single text file called ‘e1bk.d’. The energy emitted per unit volume per unit time is then N_eN₊ϵ(N_e, T_e, κ) where N₊ is the H⁺ number density and where all quantities are in cgs units. The structure of this file is explained in the following points.

• The first row of this file contains (in the following order) the number of N_e values, the number of T_e values and the number of κ values for which data are provided.

• The 9 values of N_e are specified by log₁₀N_e = 2.0(0.5)6.0.

• The 16 values of T_e are specified by log₁₀T_e = 2.0(0.2)3.8, 3.9(0.1)4.4.

• The 44 values of κ are specified by log₁₀κ = 0.20(0.01)0.30, 0.35(0.05)1.0, 1.1(0.1)2.0, 2.2(0.2)3.0, 3.5, 4.0, 5.0, 6.0.

• The data therefore consist of 6336 blocks (= 9 × 16 × 44).

• The first row of each block contains information about the block which consists of the following:

  Z    log₁₀N_e    log₁₀T_e    log₁₀κ    B    n_min    n_max,

  where Z = 1 is the atomic number of hydrogen, ‘B’ refers to Case B and n_min and n_max are the minimum and maximum upper state principal quantum numbers for which emission coefficients are tabulated. Each block therefore contains 4850 (⁠|$=\frac{1}{2}n_m(n_m-1)-1$|⁠) entries. These 4850 entries are arranged in 607 rows and hence the total number of rows in each block is 608.

• Of the three variables, N_eT_e and κ, the fastest varying is N_e followed by T_e followed by κ, and hence the ordinal number of a block is given by

  \begin{equation} O_{B}=O_{N}+(O_{T}-1)9+(O_{\kappa }-1)144, \end{equation}

  (3)

  where O_B, O_N, O_T and O_κ are the ordinal numbers of block, N_e value, T_e value and κ value, respectively. For example the ordinal number of the block for log₁₀N_e = 4 (O_N = 5), log₁₀T_e = 2.6 (O_T = 4) and log₁₀κ = 0.27 (O_κ = 8) is

  O_B = 5 + (4 − 1)9 + (8 − 1)144 = 1040

  and hence it starts on row 631714 (= 608(O_B − 1) + 2) and ends on row 632321 (= 608O_B + 1).

• The 4850 values of emission coefficients in each block are arranged for transitions from upper levels n_u to lower levels n_l with n_u in descending order from n_m, and n_l in ascending order from 1 to (n_u − 1), and hence the ordinal number for a transition tr(n_u, n_l) is given by

  \begin{equation} O_{{\rm tr}}=n_{\mathbf l}+\frac{1}{2}(n_m-n_{\mathbf u})(n_m+n_{\mathbf u}-1). \end{equation}

  (4)

  Emission coefficients to n = 1 are not calculated for Case B and hence are set to zero.

A second file named ‘t1bk.d’ contains the hydrogen total recombination coefficients in Case B, α(N_e, T_e, κ), and the total recombination coefficients to the 2s state of hydrogen, α_2s(N_e, T_e, κ), such that the number of recombinations per unit volume per unit time is N_eN₊α in cgs units. This file contains 12 672 entries arranged in 1584 rows. The first half (6336) of these entries are the total recombination coefficients of hydrogen while the second half are the total recombination coefficients to the 2s state. The entries in each one of these two blocks correspond to the 6336 values of physical conditions (i.e. various combinations of N_e, T_e and κ) positioned according to equation (3).

4 CONCLUSIONS AND DISCUSSIONS

In this paper, we computed atomic emission and recombination coefficient data for hydrogen with electron energies described by a κ-distribution. The atomic model used in the computation of these data uses the techniques described by Hummer & Storey (1987) and Storey & Hummer (1995). The data, which are placed in the public domain, span ranges of electron number density, temperature and κ useful for modelling and analysing plasmas such as those found in planetary nebulae and H ii regions. Interactive data servers provide easy access to the data with Lagrange-interpolated values in all three variables.

The work of PJS was supported in part by STFC (grant ST/J000892/1). We would like to thank Professor Gary Ferland for his constructive review and useful comments.

Statement

The data described in this paper, with the interactive data servers, can be obtained in computer readable format from the Centre de Données astronomiques de Strasbourg data base catalog VI/142. The fortran 77 data server is compiled and tested thoroughly using gfortran, f77 and Intel fortran compilers on Ubuntu 12.04 and Scientific Linux platforms, while the c++ server is compiled and tested thoroughly using g++ compiler on Ubuntu 12.04 and Dev-C++ 5.7.1 and Microsoft Visual Studio 6.0 compilers on Windows XP and Windows 8 64-bit versions. Representative sample results from all these compilers and on all those platforms are compared and found to be identical within the reported numerical accuracy.

SUPPORTING INFORMATION

Additional Supporting Information may be found in the online version of this article:

intrat3D.d

Screenshot

t1bk.d

ReadMe.dat

intrat3D.f

intrat3D.cpp

e1bk.d (Supplementary Data).

Please note: Oxford University Press are not responsible for the content or functionality of any supporting materials supplied by the authors. Any queries (other than missing material) should be directed to the corresponding author for the article.