The electron energy loss rate due to radiative recombination ^{⋆}
^{1} SRON Netherlands Institute for Space Research, Sorbonnelaan 2, 3584 CA Utrecht, The Netherlands
email: J.Mao@sron.nl
^{2} Leiden Observatory, Leiden University, Niels Bohrweg 2, 2300 RA Leiden, The Netherlands
^{3} Department of Physics, University of Strathclyde, Glasgow G4 0NG, UK
Received: 13 September 2016
Accepted: 2 December 2016
Context. For photoionized plasmas, electron energy loss rates due to radiative recombination (RR) are required for thermal equilibrium calculations, which assume a local balance between the energy gain and loss. While many calculations of total and/or partial RR rates are available from the literature, specific calculations of associated RR electron energy loss rates are lacking.
Aims. Here we focus on electron energy loss rates due to radiative recombination of Hlike to Nelike ions for all the elements up to and including zinc (Z = 30), over a wide temperature range.
Methods. We used the AUTOSTRUCTURE code to calculate the levelresolved photoionization cross section and modify the ADASRR code so that we can simultaneously obtain levelresolved RR rate coefficients and associated RR electron energy loss rate coefficients. We compared the total RR rates and electron energy loss rates of H i and He i with those found in the literature. Furthermore, we utilized and parameterized the weighted electron energy loss factors (dimensionless) to characterize total electron energy loss rates due to RR.
Results. The RR electron energy loss data are archived according to the Atomic Data and Analysis Structure (ADAS) data class adf48. The RR electron energy loss data are also incorporated into the SPEX code for detailed modeling of photoionized plamsas.
Key words: atomic data / atomic processes
Full Tables 1 and 2 are available at the CDS via anonymous ftp to cdsarc.ustrasbg.fr (130.79.128.5) or via http://cdsarc.ustrasbg.fr/vizbin/qcat?J/A+A/599/A10
© ESO, 2017
1. Introduction
Astrophysical plasmas observed in the Xray band can roughly be divided into two subclasses: collisional ionized plasmas and photoionized plasmas. Typical collisional ionized plasmas include stellar coronae (in coronal/collisional ionization equilibrium), supernova remnants (SNRs, in nonequilibrium ionization) and the intracluster medium (ICM). In lowdensity, hightemperature collisional ionized plasma, for example, ICM, collisional processes play an important role (for a review see e.g., Kaastra et al. 2008). In contrast, in a photoionized plasma, photoionization, recombination and fluorescence processes are important in addition to collisional processes. Both the equations for the ionization balance (also required for a collisional ionized plasma) and the equations of the thermal equilibrium are used to determine the temperature of the photoionized plasma. Typical photoionized plasmas in the Xray band can be found in Xray binaries (XRBs) and active galactic nuclei (AGN).
For collisional ionized plasmas, various calculations of total radiative cooling rates are available in the literature, such as Cox & Daltabuit (1971), Raymond et al. (1976), Sutherland & Dopita (1993), Schure et al. (2009), Foster et al. (2012), and Lykins et al. (2013). These calculations take advantage of full plasma codes, such as SPEX (Kaastra et al. 1996) and APEC (Smith et al. 2001), and do not treat individual energy loss (cooling) processes separately. Total radiative cooling rates include the energy loss of both the line emission and continuum emission. The latter includes the energy loss due to radiative recombination (RR). Even more specifically, the energy loss due to RR can be separated into the electron energy loss and ion energy loss.
On the other hand, for photoionized plasmas, the electron energy loss rate due to RR is one of the fundamental parameters for thermal equilibrium calculations, which assume a local balance between the energy gain and loss. Energy can be gained via photoionization, Auger effect, Compton scattering, collisional ionization, collisional deexcitation and so forth. Energy loss can be due to, for example, radiative recombination, dielectronic recombination, threebody recombination, inverse Compton scattering, collisional excitation, and bremsstrahlung, as well as the line/continuum emission following these atomic processes. In fact, the energy loss and gain of all these individual processes need to be known. The calculations of electron energy loss rates due to RR in the Cloudy code (Ferland et al. 1998, 2013) are based on hydrogenic results (Ferland et al. 1992; LaMothe & Ferland 2001). In this manuscript, we focus on improved calculations of the electron energy loss due to radiative recombination, especially providing results for Helike to Nelike isoelectronic sequences.
While several calculations of RR rates, including total rates and/or detailed rate coefficients, for different isoelectronic sequences are available, for example, Gu (2003) and Badnell (2006), specific calculations of the associated electron energy loss rate due to RR are limited. The pioneering work was carried out by Seaton (1959) for hydrogenic ions using the asymptotic expansion of the Gaunt factor for photoionization cross sections (PICSs).
By using a modified semiclassical Kramers formula for radiative recombination cross sections (RRCSs), Kim & Pratt (1983) calculated the total RR electron energy loss rate for a few ions in a relatively narrow temperature range.
Ferland et al. (1992) used the nlresolved hydrogenic PICSs provided by Storey & Hummer (1991) to calculate both nresolved RR rates () and electron energy loss rates (). Contributions up to and including n = 1000 are taken into account.
Using the same nlresolved hydrogenic PICSs provided by Storey & Hummer (1991), Hummer (1994) calculated the RR electron energy loss rates for hydrogenic ions in a wide temperature range. In addition, Hummer & Storey (1998) calculated PICSs of He i (photoionizing ion) for n ≤ 25 with their closecoupling Rmatrix calculations. Together with hydrogenic (Storey & Hummer 1991) PICSs for n> 25 (up to n = 800 for low temperatures), the RR electronic energy loss rate coefficient of He i (recombined ion) was obtained.
Later, LaMothe & Ferland (2001) used the exact PICSs from the Opacity Project (Seaton et al. 1992) for n< 30 and PICSs of Verner & Ferland (1996) for n ≥ 30 to obtain nresolved RR electron energy loss rates for hydrogenic ions in a wide temperature range. The authors introduced the ratio of β/α (dimensionless), where β = L/kT and L is the RR electron energy loss rate. The authors also pointed out that β/α changes merely by 1 dex in a wide temperature range; meanwhile α and β change more than 12 dex.
In the past two decades, more detailed and accurate calculations of PICSs of many isoelectronic sequences have been carried out (e.g., Badnell 2006), which can be used specifically to calculate the electron energy loss rates due to RR.
Currently, in the SPEX code (Kaastra et al. 1996), the assumption that the mean kinetic energy of a recombining electron is 3kT/ 4 (Kallman & McCray 1982) is applied for calculating the electron energy loss rate due to RR. Based on the levelresolved PICSs provided by the AUTOSTRUCTURE^{1} code (v24.24.3; Badnell 1986), the electron energy loss rates due to RR are calculated in a wide temperature range for the Hlike to Nelike isoelectronic sequences for elements up to and including Zn (Z = 30). Subsequently, the electron energy loss rate coefficients (β = L/kT) are weighted with respect to the total RR rates (α_{t}), yielding the weighted electron energy loss factors (f = β/α_{t}, dimensionless). The weighted electron energy loss factors can be used, together with the total RR rates, to update the description of the electron energy loss due to RR in the SPEX code or other codes.
In Sect. 2, we describe the details of the numerical calculation from PICSs to the electron energy loss rate due to RR. Typical results are shown graphically in Sect. 3. Parameterization of the weighted electron energy loss factors is also illustrated in Sect. 3. The detailed RR electron energy loss data are archived according to the Atomic Data and Analysis Structure (ADAS) data class adf48. Full tabulated (unparameterized and parameterized) weighted electron energy loss factors are available in CDS. Comparison of the results for H i and He i can be found in Sect. 4.1. The scaling of the weighted electron energy loss factors with respect to the square of the ionic charge of the recombined ion can be found in Sect. 4.2. We also discuss the electron and ion energy loss due to RR (Sect. 4.3) and the total RR rates (Sect. 4.4).
Throughout this paper, we refer to the recombined ion when we speak of the radiative recombination of a certain ion, since the line emission following the radiative recombination comes from the recombined ion. Furthermore, only RR from the ground level of the recombining ion is discussed here.
2. Methods
2.1. Cross sections
The AUTOSTRUCTURE code is used for calculating levelresolved nonresonant PICSs under the intermediate coupling (IC) scheme (Badnell & Seaton 2003). The atomic and numerical details can be found in Badnell (2006); we briefly state the main points here. We use the Slatertypeorbital model potential to determine the radial functions. We calculated PICSs first at zero kinetic energy of the escaping electron. Subsequently, we calculated them on a zscaled logarithmic energy grid with three points per decade, ranging from ~ z^{2}10^{6} to z^{2}10^{2} ryd, where z is the ionic charge of the photoionizing ion/atom. PICSs at even higher energies are at least several orders of magnitude smaller compared to PICSs at zero kinetic energy of the escaping electron. Nonetheless, it still can be important, especially for the s and porbit, to derive the RR data at the high temperature end. We take advantage of the analytical hydrogenic PICSs (calculated via the dipole radial integral; Burgess 1965) and scale them to the PICS with the highest energy calculated by AUTOSTRUCTURE to obtain PICSs at very high energies. Fully nLSJresolved PICSs for those levels with n ≤ 15 and l ≤ 3 are calculated specifically. For the rest of the levels, we use the fast, accurate and recurrence hydrogenic approximation (Burgess 1965). Meanwhile, bundledn PICSs for n = 16, 20, 25, 35, 45, 55, 70, 100, 140, 200, 300, 450, 700, and 999 are also calculated specifically to derive the total RR and electron energy loss rates (interpolation and quadrature required as well).
The inverse process of dielectronic and radiative recombination is resonant and nonresonant photoionization, respectively. Therefore, radiative recombination cross sections (RRCSs) are obtained through the Milne relation under the principle of detailed balance (or microscopic reversibility) from nonresonant PICSs.
2.2. Rate coefficients
The RR rate coefficient is obtained by (1)where v is the velocity of the recombining electron, σ_{i} is the individual detailed (level/term/shellresolved) RRCS, f(v,T) is the probability density distribution of the velocity of the recombining electrons for the electron temperature T. The MaxwellBoltzmann distribution for the free electrons is adopted throughout the calculation, with the same quadrature approach as described in Badnell (2006). Accordingly, the total RR rate per ion/atom is (2)Total RR rates for all the isoelectronic sequences, taking contributions up to n = 10^{3} into account (see its necessity in Sect. 3).
The RR electron energy loss rate coefficient is defined as (e.g., Osterbrock 1989) (3)The total electron energy loss rate due to RR is obtained simply by adding all the contributions from individual captures, (4)which can be identically derived via (5)where (6)is defined as the weighted electron energy loss factor (dimensionless) hereafter.
The above calculation of the electron energy loss rates is realized by adding Eq. (3) into the archival postprocessor FORTRAN code ADASRR^{2} (v1.11). Both the levelresolved and bundledn/nl RR data and the RR electron energy loss data are obtained. The output files have the same format of adf48 with RR rates and electron energy loss rates in the units of cm^{3} s^{1} and ryd cm^{3} s^{1}, respectively. Ionization potentials of the ground level of the recombined ions from NIST^{3} (v5.3) are adopted to correct the conversion from PICSs to RRCSs at low kinetic energy for lowcharge ions. We should point out that the levelresolved and bundlednl/n RR data are, in fact, available on OPEN ADAS^{4}, given the fact that we use the latest version of the AUTOSTRUCTURE code and a modified version of the ADASRR code, here we recalculate the RR data, which are used together with the RR electron energy loss data to derive the weighted electron energy loss factor f_{t} for consistency. In general, our recalculate RR data are almost identical to those on OPEN ADAS, except for a few manyelectron ions at the the high temperature end, where our recalculated data differ by a few percent. Whereas, both RR data and electron energy loss data are a few orders of magnitude smaller compared to those at the lower temperature end, thus, the abovementioned difference has negligible impact on the accuracy of the weighted electron energy loss factor (see also in Sect. 4.4).
For all the isoelectronic sequences discussed here, the conventional ADAS 19point temperature grid z^{2}(10−10^{7}) K is used.
3. Results
For each individual capture due to radiative recombination, when kT ≪ I, where I is the ionization potential, the RR electron energy loss rate L_{i} is nearly identical to kTα_{i}, since the Maxwellian distribution drops exponentially for E_{k} ≳ kT, where E_{k} is the kinetic energy of the free electron before recombination. On the other hand, when kT ≫ I, the RR electron energy loss rate is negligible compared with kTα_{i}. As in an electronion collision, when the total energy in the incident channel nearly equals that of a closedchannel discrete state, the channel interaction may cause the incident electron to be captured in this state (Fano & Cooper 1968). That is to say, those electrons with E_{k} ≃ I are preferred to be captured, thus, L_{i} ~ Iα_{i}. Figure 1 shows the ratio of β_{i}/α_{i} = L_{i}/ (kTα_{i}) for representative nLSJresolved levels (with n ≤ 8) of Helike Mg xi .
Fig. 1
For Helike Mg xi, the ratio between levelresolved electron energy loss rates L_{i} and the corresponding radiative recombination rates times the temperature of the plasma, i.e. β_{i}/α_{i} (not be confused with β_{i}/α_{t}), where i refers to the nLSJresolved levels with n ≤ 8 (shown selectively in the plot). 

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In terms of capturing free electrons into individual shells (bundledn), owing to the rapid decline of the ionization potentials for those very highn shells, the ionization potentials can be comparable to kT, if not significantly less than kT, at the low temperature end. Therefore we see the significant difference between the top panel (lown shells) and middle panel (highn shells) of Fig. 2. In order to achieve adequate accuracy, contributions from highn shells (up to n ≤ 10^{3}) ought to be included. The middle panel of Fig. 2 shows clearly that even for n = 999 (the line at the bottom), at the low temperature end, the ratio between β_{n = 999} and α_{n = 999} does not drop to zero. Nevertheless, the bottom panel of Fig. 2 illustrates the advantage of weighting the electron energy loss rate coefficients with respect to the total RR rates, i.e. β_{i}/α_{t}, which approaches zero more quickly. At least, for the next few hundred shells following n = 999, their weighted electron energy loss factors should be no more than 10^{5}, thus, their contribution to the total electron energy loss rate should be less than 1%.
Fig. 2
Ratios of β_{i}/α_{i} for Belike Fe xxiii (upper and middle panel) and ratios of β_{i}/α_{t} (bottom panel), where i refers to the shell number. Low and highn shell results are shown selectively in the plot. The upper panel shows all the shells with n ≤ 8. The middle panel shows shells with n =100, 140, 200, 300, 450, 700, and 999. In the lower panel the shells are n =2, 8, 16, 49, 100, 300, and 999. 

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The bottom panels of Figs. 3 and 4 illustrate the weighted electron energy loss factors for Helike isoelectronic sequences (He, Si and Fe) and Fe isonuclear sequence (H, He, Be and Nlike), respectively. The deviation from (slightly below) unity at the lower temperature end is simply because the weighted electron energy loss factors of the very highn shells are no longer close to unity (Fig. 2, middle panel). The deviation from (slightly above) zero at the high temperature end occurs because the ionization potentials of the first few lown shells can still be comparable to kT, while sum of these nresolved RR rates are more or less a few tens of percent of the total RR rates.
Because of the nonhydrogenic screening of the wave function for lownl states in lowcharge manyelectron ions, the characteristic hightemperature bump is present in not only the RR rates (see Fig. 4 in Badnell 2006, for an example) but also in the electron energy loss rates. The feature is even enhanced in the weighted electron energy loss factor.
Fig. 3
Total RR rates α_{t} (top), electron energy loss rates L_{t} (middle) and weighted electron energy loss factors f_{t} (bottom) of Helike isoelectronic sequences for ions, including He i (black), Si xiii (red) and Fe xxv (orange). The temperature is downscaled by z^{2}, where z is the ionic charge of the recombined ion, to highlight the discrepancy between hydrogenic and nonhydrogenic. The captures to form the He i shows nonhydrogenic feature in the bottom panel. 

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Fig. 4
Top panel is total RR rates α_{t} of the Fe isonuclear sequence, including H (black), He (red), Be (orange) and Nlike (blue); middle panel is the RR electron energy loss rates L_{t}; and the bottom panel is the weighted electron energy loss factors f_{t}. The temperature of the plasma is downscaled by z^{2}, as in Fig. 3. 

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We parameterize the ion/atomresolved radiative recombination electron energy loss factors using the same fitting strategy described in Mao & Kaastra (2016) with the model function of (7)where the electron temperature T is in units of eV, a_{0} and b_{0} are primary fitting parameters, and c_{0},a_{1, 2}, and b_{1, 2} are additional fitting parameters. The additional parameters are frozen to zero if they are not used. Furthermore, we constrain b_{0−2} to be within –10.0 to 10.0 and c_{0} between 0.0 and 1.0. The initial values of the two primary fitting parameters a_{0} and b_{0} are set to unity together with the four additional fitting parameters a_{1, 2} and b_{1, 2} if they are thawed. Conversely, the initial value of c_{0}, if it is thawed, is set to either side of its boundary, i.e., c_{0} = 0.0 or c_{0} = 1.0 (both fits are performed).
In order to estimate the goodness of fit, the fits are performed with a set of artificial relative errors (r). We started with r = 0.625%, following with increasing the artificial relative error by a factor of two, up to and including 2.5%. The chisquared statistics adopted here are (8)where n_{i} is the ith numerical calculation result and m_{i} is the ith model prediction (Eq. (7)).
For the model selection, we first fit the data with the simplest model (i.e. all the five additional parameters are frozen to zero), following with fits with free additional parameters step by step. Thawing one additional parameter decreases the degrees of freedom by one. Thus, the more complicated model is only favored (at a 90% nominal confidence level) if the obtained statistics (χ^{2}) of this model improves by at least 2.71, 4.61, 6.26, 7.79, and 9.24 for one to five additional free parameter(s), respectively.
Parameterizations of the ion/atomresolved RR weighted electron energy loss factors for individual ions/atoms in Hlike to Nelike isoelectronic sequences were performed. A typical fit for nonhydrogenic systems is shown in Fig. 5 for Nlike iron (Fe xx). The fitting parameters can be found in Table 2. Again, the weighted energy loss factor per ion/atom is close to unity at low temperature end and drops toward zero rapidly at the high temperature end.
Fig. 5
Radiative recombination weighted electron energy loss factor for Nlike iron (Fe xx). The black dots in both panels (associated with artificial error bars of 2.5% in the upper panel) are the calculated weighted electron energy loss factor. The red solid line is the best fit. The lower panel shows the deviation (in percent) between the best fit and the original calculation. 

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In Fig. 6 we show the histogram of maximum deviation δ_{max} (in percent) between the fitted model and original calculation for all the ions considered here. In short, our fitting accuracy is within 4%, and is even accurate (≲2.5%) for the more important Hlike, Helike and Nelike isoelectronic sequences.
Fig. 6
Histogram of maximum deviation in percent (δ_{max}) for all the ions considered here, which reflects the overall goodness of our parameterization. The dashed histogram is the statistics of the more important Hlike, Helike and Nelike isoelectronic sequences, while the solid histogram is the statistics of all the isoelectronic sequences. 

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In addition, we also specifically fit for Case A (f_{A} = β_{t}/α_{t}) and Case B (Baker & Menzel 1938, f_{B} = β_{n ≥ 2}/α_{n ≥ 2}) the RR weighted electron energy loss factors of H i (Fig. 7) and He i (Fig. 8). Typical unparameterized factors (f_{A} and f_{B}) and fitting parameters can be found in Tables 1 and 2, respectively.
Unparameterized of RR weighted electron energy loss factors for H i, He i and Fe xx.
4. Discussions
4.1. Comparison with previous results for H I and He I
Fig. 7
Case A (solid line, filled circles) and Case B (dashed line, empty diamonds) RR weighted electron energy loss factor (f_{A / B}) for H i. The black dots in both panels (associated with artificial error bars in the upper one) are the calculated weighted electron energy loss factor. The red solid line is the best fit. The lower panel shows the deviation (in percent) between the best fit and the original calculation. 

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Fig. 8
Similar to Fig. 7 but for He i. 

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Fig. 9
Comparison of the RR data for H i among results from this work (black), Seaton (1959, blue), Ferland et al. (1992, orange), and Hummer (1994, red). Both results of case A (solid lines) and case B (dashed lines) are shown. The total RR rates () and electron energy loss rates () are shown in the top two panels. The RR weighted electron energy loss factors (f_{A / B}) are shown in the middle panel. The ratios of f_{A / B} from this work and previous works with respect to the fitting results (Eq. (7) and Table 2) of this work, i.e., , are shown in the bottom two panels. 

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Figure 9 shows a comparison of RR rates (), electron energy loss rates (), weighted electron energy loss factors () from this work, Seaton (1959, blue), Ferland et al. (1992, orange) and Hummer (1994, red). Since both Ferland et al. (1992) and Hummer (1994) use the same PICSs (Storey & Hummer 1991), as expected the two results are highly consistent. The Case A and Case B results of this work are also consistent within 1% at the low temperature end and increase to ~5% (underestimation). For the high temperature end (T ≳ 0.1 keV), since the ion fraction of H i is rather low (almost completely ionized), the present calculation is still acceptable. A similar issue at the high temperature end is also found in Case A results of Seaton (1959) with a relatively significant overestimation (≳5%) from the other three calculations.
Fitting parameters of RR weighted electron energy loss factors for H i, He i and Fe xx.
Fig. 10
Similar to Fig. 10 but for He i between this work (black) and Hummer & Storey (magenta 1998). The latter only provides data with T ≤ 10^{4.4} K. 

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Likewise, the comparison for He i between this work and Hummer & Storey (1998) is presented in Fig. 10. The Case A and Case B results from both calculations agree well (within 2%) at the low temperature end (T ≲ 2.0 eV). At higher temperatures with T ≳ 2 eV, the RR rate and electron energy loss rate for He i are not available in Hummer & Storey (1998).
4.2. Scaling with z^{2}
In previous studies of hydrogenic systems, Seaton (1959), Ferland et al. (1992), and Hummer & Storey (1998), all use z^{2} scaling for . That is to say, , where z is the ionic charge of the recombined ion X. The same z^{2} scaling also applies for (or ). LaMothe & Ferland (2001) also pointed out that the shellresolved ratio of (=) can also be scaled with z^{2}/n^{2}, i.e., where n refers to the principle quantum number.
In the following, we merely focus on the scaling for the ion/atomresolved data set. In the top panel of Fig. 11 we show the ratios of f_{t}/z^{2} for Hlike ions. Apparently, from the bottom panel of Fig. 11, the z^{2} scaling for the Hlike isoelectronic sequence is accurate within 2%. For the rest of the isoelectronic sequences, for instance, the Helike isoelectronic sequence shown in Fig. 12, the z^{2} scaling applies at the low temperature end, whereas, the accuracies are poorer toward the high temperature end. We also show the z^{2} scaling for the Fe isonuclear sequence in Fig. 13.
Fig. 11
z^{2} scaling for the Hlike isoelectronic sequence (Case A), including H i (black), O viii (red), Ar xviii (orange) and Ni xxviii (green). The top panel shows the ratios of f_{t}/z^{2} as a function of electron temperature (T). The bottom panel is the ratio of (f_{t}/z^{2})^{X} for ion X with respect to the ratio of (f_{t}/z^{2})^{H} for H. 

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Fig. 12
Similar to Fig. 11 but for the z^{2} scaling for the Helike isoelectronic sequences. 

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Fig. 13
z^{2} scaling for the Fe isonuclear sequence. The top panel shows the ratios of f_{t}/z^{2} as a function of electron temperature (T). The bottom panel is the ratio of (f_{t}/z^{2})^{X  like} for Xlike Fe with respect to the ratio of (f_{t}/z^{2})^{H  like} for Hlike Fe xxvi. 

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4.3. Radiative recombination continua
We restrict the discussion above to the RR energy loss of the electrons in the plasma. The ion energy loss of the ions due to RR can be estimated as P^{RR} ~ I_{i}α_{i}, where I_{i} is the ionization potential of the level/term into which the free electron is captured, and α_{i} is the corresponding RR rate coefficient. Whether to include the ionization potential energies as part of the total internal energy of the plasma is not critical as long as the entire computation of the net energy gain/loss is selfconsistent (see a discussion in Gnat & Ferland 2012). On the other hand, when interpreting the emergent spectrum due to RR, such as the radiative recombination continua (RRC) for a lowdensity plasma, the ion energy loss of the ion is essentially required. The RRC emissivity (Tucker & Gould 1966) can be obtained via (9)where n_{e} and n_{i} are the electron and (recombining) ion number density, respectively. Generally speaking, the ion energy loss of the ion dominates the electron energy loss of the electrons, since f_{t} is on the order of unity while kT ≲ I holds for those Xray photoionizing plasmas in XRBs (Liedahl & Paerels 1996), AGN (Kinkhabwala et al. 2002) and recombining plasmas in SNRs (Ozawa et al. 2009). Figure 14 shows the threshold temperature above which the electron energy loss via RR cannot be neglected compared to the ion energy loss. For hot plasmas with kT ≳ 2 keV, the electron energy loss is comparable to the ion energy loss for Z> 5. We emphasize that we refer to the electron temperature T of the plasma here, which is not necessarily identical to the ion temperature of the plasma, in particular, in the nonequilibrium ionization scenario.
Fig. 14
Threshold temperature above which the electron energy loss via RR cannot be neglected, compared to the ion energy loss, for Hlike (solid lines) and Helike ions (dashed lines). 

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4.4. Total radiative recombination rate
Various calculations of (total or shell/term/levelresolved) RR data are available from the literature. Historically, different approaches have been used for calculating the total RR rates, including the DiracHartreeSlater method (Verner et al. 1993) and the distortedwave approximation (Gu 2003; Badnell 2006). Additionally, Nahar and coworkers (e.g., Nahar 1999) obtained the total (unified DR + RR) recombination rate for various ions with their Rmatrix calculations. Different approaches can lead to different total RR rates (see a discussion in Badnell 2006) as well as the individual term/levelresolved RR rate coefficients, even among the most advanced Rmatrix calculations. Nevertheless, the bulk of the total RR rates for various ions agrees well among each other. As for the detailed RR rate coefficients, and consequently, the detailed RR electron energy loss rate, the final difference in the total weighted electron energy loss factors f_{t} are still within 1%, as long as the difference among different methods are within a few percent and given the fact that each individual RR is ≲10% of the total RR rate for a certain ion/atom. In other words, although we used the recalculated total RR rate (Sect. 2.2) to derive the weighted electron energy loss factors, we assume these factors can still be applied to other total RR rates.
Acknowledgments
J.M. acknowledges discussions and support from M. Mehdipour, A. Raassen, L. Gu, and M. O’Mullane. We thank the referee, G. Ferland, for valuable comments on the manuscript. SRON is supported financially by NWO, the Netherlands Organization for Scientific Research.
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All Tables
Unparameterized of RR weighted electron energy loss factors for H i, He i and Fe xx.
Fitting parameters of RR weighted electron energy loss factors for H i, He i and Fe xx.
All Figures
Fig. 1
For Helike Mg xi, the ratio between levelresolved electron energy loss rates L_{i} and the corresponding radiative recombination rates times the temperature of the plasma, i.e. β_{i}/α_{i} (not be confused with β_{i}/α_{t}), where i refers to the nLSJresolved levels with n ≤ 8 (shown selectively in the plot). 

Open with DEXTER  
In the text 
Fig. 2
Ratios of β_{i}/α_{i} for Belike Fe xxiii (upper and middle panel) and ratios of β_{i}/α_{t} (bottom panel), where i refers to the shell number. Low and highn shell results are shown selectively in the plot. The upper panel shows all the shells with n ≤ 8. The middle panel shows shells with n =100, 140, 200, 300, 450, 700, and 999. In the lower panel the shells are n =2, 8, 16, 49, 100, 300, and 999. 

Open with DEXTER  
In the text 
Fig. 3
Total RR rates α_{t} (top), electron energy loss rates L_{t} (middle) and weighted electron energy loss factors f_{t} (bottom) of Helike isoelectronic sequences for ions, including He i (black), Si xiii (red) and Fe xxv (orange). The temperature is downscaled by z^{2}, where z is the ionic charge of the recombined ion, to highlight the discrepancy between hydrogenic and nonhydrogenic. The captures to form the He i shows nonhydrogenic feature in the bottom panel. 

Open with DEXTER  
In the text 
Fig. 4
Top panel is total RR rates α_{t} of the Fe isonuclear sequence, including H (black), He (red), Be (orange) and Nlike (blue); middle panel is the RR electron energy loss rates L_{t}; and the bottom panel is the weighted electron energy loss factors f_{t}. The temperature of the plasma is downscaled by z^{2}, as in Fig. 3. 

Open with DEXTER  
In the text 
Fig. 5
Radiative recombination weighted electron energy loss factor for Nlike iron (Fe xx). The black dots in both panels (associated with artificial error bars of 2.5% in the upper panel) are the calculated weighted electron energy loss factor. The red solid line is the best fit. The lower panel shows the deviation (in percent) between the best fit and the original calculation. 

Open with DEXTER  
In the text 
Fig. 6
Histogram of maximum deviation in percent (δ_{max}) for all the ions considered here, which reflects the overall goodness of our parameterization. The dashed histogram is the statistics of the more important Hlike, Helike and Nelike isoelectronic sequences, while the solid histogram is the statistics of all the isoelectronic sequences. 

Open with DEXTER  
In the text 
Fig. 7
Case A (solid line, filled circles) and Case B (dashed line, empty diamonds) RR weighted electron energy loss factor (f_{A / B}) for H i. The black dots in both panels (associated with artificial error bars in the upper one) are the calculated weighted electron energy loss factor. The red solid line is the best fit. The lower panel shows the deviation (in percent) between the best fit and the original calculation. 

Open with DEXTER  
In the text 
Fig. 8
Similar to Fig. 7 but for He i. 

Open with DEXTER  
In the text 
Fig. 9
Comparison of the RR data for H i among results from this work (black), Seaton (1959, blue), Ferland et al. (1992, orange), and Hummer (1994, red). Both results of case A (solid lines) and case B (dashed lines) are shown. The total RR rates () and electron energy loss rates () are shown in the top two panels. The RR weighted electron energy loss factors (f_{A / B}) are shown in the middle panel. The ratios of f_{A / B} from this work and previous works with respect to the fitting results (Eq. (7) and Table 2) of this work, i.e., , are shown in the bottom two panels. 

Open with DEXTER  
In the text 
Fig. 10
Similar to Fig. 10 but for He i between this work (black) and Hummer & Storey (magenta 1998). The latter only provides data with T ≤ 10^{4.4} K. 

Open with DEXTER  
In the text 
Fig. 11
z^{2} scaling for the Hlike isoelectronic sequence (Case A), including H i (black), O viii (red), Ar xviii (orange) and Ni xxviii (green). The top panel shows the ratios of f_{t}/z^{2} as a function of electron temperature (T). The bottom panel is the ratio of (f_{t}/z^{2})^{X} for ion X with respect to the ratio of (f_{t}/z^{2})^{H} for H. 

Open with DEXTER  
In the text 
Fig. 12
Similar to Fig. 11 but for the z^{2} scaling for the Helike isoelectronic sequences. 

Open with DEXTER  
In the text 
Fig. 13
z^{2} scaling for the Fe isonuclear sequence. The top panel shows the ratios of f_{t}/z^{2} as a function of electron temperature (T). The bottom panel is the ratio of (f_{t}/z^{2})^{X  like} for Xlike Fe with respect to the ratio of (f_{t}/z^{2})^{H  like} for Hlike Fe xxvi. 

Open with DEXTER  
In the text 
Fig. 14
Threshold temperature above which the electron energy loss via RR cannot be neglected, compared to the ion energy loss, for Hlike (solid lines) and Helike ions (dashed lines). 

Open with DEXTER  
In the text 