Optical and near-infrared recombination lines of oxygen ions from Cassiopeia A knotsTables 5 to 23 are only available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/

Optical and near-infrared recombination lines
of oxygen ions from Cassiopeia A knotsthanks: Tables 5 to 23 are only available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/

D. Docenko Max Planck Institute for Astrophysics, Karl-Schwarzschild-Str. 1, 85741 Garching, Germany Institute of Astronomy, University of Latvia, Raiņa bulvāris 19, Riga LV-1586, Latvia    R.A. Sunyaev Max Planck Institute for Astrophysics, Karl-Schwarzschild-Str. 1, 85741 Garching, Germany Space Research Institute, Russian Academy of Sciences, Profsoyuznaya 84/32, 117997 Moscow, Russia
Key Words.:
atomic processes - supernovae: individual: Cassiopeia A - infrared: ISM
Abstract

Context:Fast-moving knots (FMK) in the Galactic supernova remnant Cassiopeia A consist mainly of metals and allow to study element production in supernovae and shock physics in great detail.

Aims:We work out theoretically and suggest to observe previously unexplored class of spectral lines – metal recombination lines in optical and near-infrared bands – emitted by the cold ionized and cooling plasma in the fast-moving knots.

Methods:By tracing ion radiative and dielectronic recombination, collisional -redistribution and radiative cascade processes, we compute resulting oxygen, silicon and sulphur recombination line emissivities. It allows us to determine the oxygen recombination line fluxes, based on the fast-moving knot model of Sutherland and Dopita (1995b), that predicts existence of highly-ionized ions from moderate to very low plasma temperatures.

Results:The calculations predict oxygen ion recombination line fluxes detectable on modern optical telescopes in the wavelength range from 0.5 to 3 m. Line ratios to collisionally-excited lines will allow to probe in detail the process of rapid cloud cooling after passage of a shock front, to test high abundances of , and ions at low temperatures and measure them, to test existing theoretical models of a FMK and to build more precise ones.

Conclusions:

1 Introduction

The brightest source on the radio sky – a supernova remnant Cassiopeia A – is so close and so young that the best instruments are able to observe exceptionally fine details in its rich structure in different spectral bands. One of the most interesting phenomena are dense ejecta blobs of the supernova explosion, observed as numerous bright optical “fast-moving knots” (FMKs). These knots radiate mostly in [O iii] doublet near 5000 Å and other forbidden lines of oxygen, sulphur, silicon and argon.

Early observations using 200-inch telescope on Mount Palomar (Baade & Minkowski, 1954; Minkowski & Aller, 1954; Minkowski, 1957) demonstrated that the emitting knot plasma has temperatures about  K and electron densities about  cm. Very unusual spectra of the fast-moving knots have lead Shklovskii (1968) to suggestion, later confirmed by detailed measurements and analysis (Peimbert & van den Bergh, 1971; Chevalier & Kirshner, 1978, 1979), that they have extremely high abundances of some of heavy elements. Some of the knots consist of up to 90% of oxygen, others contain predominantly heavier elements, such as silicon, sulphur, argon and calcium (Chevalier & Kirshner, 1979; Hurford & Fesen, 1996).

It was also discovered that a typical lifetime of a FMK is of the order of 10-30 years. Existing knots disappear, but other bright knots appear on the maps of Cassiopeia A (hereafter Cas A).

Detailed Chandra observations (Hughes et al., 2000; Hwang et al., 2004) revealed presence of similar knots in X-rays, being very bright in spectral lines of hydrogen- and helium-like ions of silicon and sulphur and showing traces of heavily absorbed oxygen X-ray lines.

Similar bright optical knots are observed in other oxygen-rich supernova remnants (e.g., Puppis A, N132D, etc., Sutherland & Dopita (1995a)), but the case of Cas A is the most promising for further investigations.

The FMK optical spectra were first treated as the shock wave emission by Chevalier & Kirshner (1978, 1979). Subsequently it was understood that used shock models, that assumed solar abundances, are inappropriate for the case of the FMK plasma, as its high metal abundances result in a different shock structure.

There have been published several theoretical models describing the shock emission in pure oxygen plasmas (Itoh, 1981a, b; Borkowski & Shull, 1990) and oxygen-dominated plasmas (Sutherland & Dopita, 1995b, hereafter SD95). In these models, the reverse shock of the supernova remnant encounters a dense knot and decelerates entering its dense medium (Zel’dovich & Raizer, 1967; McKee & Cowie, 1975) to velocities of several hundred km/s. Shortly after the shock wave enters the cloud, the material ahead of the shock front is ionized by the radiation from the heated gas after the shock front. All the theoretical models assume a shock wave propagation with a constant velocity through a constant density cloud.

The optical emission is formed in two relatively thin layers: one immediately following the shock front, where plasma rapidly cools from X-ray emitting temperatures of about  K down to below  K; another at the photoionization front before the shock wave. The relatively cold layers having temperatures below  K contribute most strongly to the line emission, as approximate pressure equilibrium results in much higher densities and emission measures of these regions.

Important detail is that the recombination time is much longer than the plasma cooling time. Therefore recombining and cooling plasma simultaneously contains ions in vastly different ionization stages at all temperatures down to several hundred Kelvins – the plasma is in a non-equilibrium ionization state. It would be very important to find a way to confirm experimentally this prediction of the computational models.

It is interesting to note that predictions of various theoretical models differ in many ways. For example, as we show in an accompanying article (Docenko & Sunyaev, to be submitted), the model predictions of the fine-structure far-infrared line intensities differ by several orders of magnitude. We also show there that the models of SD95 and of Borkowski & Shull (1990) are the best of available ones in reproducing the far-infrared line emission of the fast-moving knots, although each of them is only precise up to a factor of several.

Nevertheless, these two theoretical models still have a lot of differences between them. For example, in the Borkowski & Shull (1990) models the and higher-ionized species are essentially absent, whereas in the SD95 model the is the dominant ion after the shock at temperatures above about 5000 K. Unfortunately, , and ions have no collisionally-excited lines in visible and infrared ranges, and their ultraviolet and soft X-ray lines are almost or completely undetectable due to high interstellar absorption on the way from the Cas A.

From large differences between the theoretical model predictions it is clear that they have lack of observational constraints and more diagnostic information in form of various line ratios is needed to pin down the true structure of fast-moving knots. Part of such information can be obtained from the far-infrared lines.


In this paper we argue that still more information may be obtained from metal recombination lines (RL) in optical and near-infrared spectral ranges. These lines arise in the transitions between highly-excited levels (principal quantum number ) populated by the processes of dielectronic and radiative recombination. The RLs are emitted by all ionic species, including, for example, and that have no other lines detectable from the Cas A.

Unfortunately, the collisional excitation process that is much more efficient at exciting lines corresponding to transitions between lowest excited states, is negligibly weak at excitation of high- levels. Therefore, emissivities of the RLs are normally several orders of magnitude weaker than ones of the collisionally-excited lines, but, as we show in this article, intensities of the recombination lines from the FMKs are reaching detectable levels.

For oxygen-dominated knots, using plasma parameters (electron temperature, density and emission measure of the gas at different distances from the shock wave) from the SD95 model with the 200 km/s shock speed we have computed the spectrum in the optical and near-infrared bands accessible by the modern ground-based telescopes. The cloud shock speed in the Cas A FMKs of about 200 km/s is determined by SD95 from the comparison of observed optical spectra with their model predictions. Although similar analysis based on the Borkowski & Shull (1990) model would also be valuable, the original article does not give enough information to allow us to compute the RL fluxes (e.g., the ionization state distributions as functions of temperature cannot be resolved from the plots in the regions significantly contributing to the recombination line emission).

The lines under consideration are predicted to be times weaker than e.g. the [O iii] line at 5007 Å. However, their fluxes are well above the sensitivity of the best present-day telescopes. These fluxes are only 2–5 times weaker than the detection limits of observations done in 1970’s and later on the 200-inch and smaller telescopes (Peimbert & van den Bergh, 1971; Chevalier & Kirshner, 1979; Hurford & Fesen, 1996). Observations of these lines do not demand high spectral resolution due to velocity spreads and turbulence in the shocked plasma, as well as due to intrinsic line splitting.

The recombination lines discussed below are radiated by all coexisting ionized oxygen species, from to . Study of these lines will permit to observe in details the process of the non-equilibrium cooling and recombination of the pre- and post-shock plasmas overabundant with oxygen or sulphur and silicon.

The discussed spectral lines allow to observe the coldest ionized pre-shock and post-shock regions in the FMKs, that are not possible to detect using the ground-based observations by any other means. The fine-structure far-infrared emission lines are also able to give such information, but demand an observatory to be located outside the atmosphere and cannot reach angular resolution sufficient to distinguish individual knots.

Observations of recombination lines of elements other than oxygen will allow to improve our knowledge on many yet unknown physical parameters characterizing the Cas A ejecta.

The metal recombination lines may be a promising tool also for studies of other oxygen-rich supernova remnants like Puppis A, N132D, G292+1.8, etc.


The paper structure is following. In the second Section we describe our method of the recombination line flux computation. In the Section 3 we apply the method to predict the optical and near-infrared line fluxes from the optical knots in Cassiopeia A and compare them with existing observational constraints. Description of the recombination line substructure allowing to identify the parent ion is given in the Section 4. In the Section 5 we discuss the ways to get information about physical parameters of the emitting region from the line ratios. Finally, in Section 6 we conclude the article.

2 Computation of recombination line fluxes

In the process of radiative recombination, especially at low temperatures (when is at least several times below the ionization potential), significant part of electrons recombine onto excited states. Dielectronic recombination for majority of heavy ions populate excited states even more efficiently than the radiative recombination.

In low-density plasmas such recombination onto excited states, characterized by quantum numbers , is followed by the electron radiative cascade to the ground state and emission of multiple photons in the course of the cascade. In case of recombining highly-charged ions, these photons are emitted in microwave, infrared, optical and ultraviolet spectral bands.

In this paper we describe the lines produced as a result of such radiative cascade, when both collisional and induced transitions of the type are unimportant. The only non-radiative process influencing electron cascade in our model are the collisional transitions, as they have much higher cross sections and significantly affect the level populations at relatively low . They also are the main reason of changes of the recombination line emissivities with electron density.

We show in Appendix A that the -changing transitions may be neglected for computations of optical recombination line fluxes, while still retaining reasonable accuracy of the results (better than 20%).

2.1 Elementary processes

In the current work we account for the following processes:

  • Radiative recombination (RR). Its level-specific rates were computed as described in Appendix B.1.

  • Dielectronic recombination (DR). Its level-specific rates were computed as described in Appendix B.2.

  • Radiative transitions. Their rates were computed using hydrogenic formulae with radial integral expressions from Gordon (1929). This may be not accurate for non-hydrogenic ions at , but electrons recombined to those low- states mostly transit to and do not contribute to the optical RL emission111 After all computations have been performed, the authors were informed about a more elegant and modern way of hydrogenic radial integral computation using associated Laguerre polynomials (Malik et al. (1991); note that there is a typographical error in one of their equations, which is rectified in the appendix of Heng & McCray (2007))..

  • Collisional -redistribution. Its rates were computed using sudden collision approximation expressions from Pengelly & Seaton (1964) and Summers (1977), in the region of their applicability. Outside it (where maximum cut-off parameter becomes less than the minimum one ), we approximated the Vrinceanu & Flannery (2001) classical expressions introducing cut-off at large impact parameters equal to . Details of this approximation are of minor importance, since the cross sections are proportional as the , being therefore relatively small. Energies of -levels and their differences determining the collisional rates in many cases, were computed as described in Appendix B.4.

Usage of hydrogenic approximation does not allow us to reliably compute emissivities and wavelengths of the lines corresponding to transitions involving levels . Therefore we do not provide the results concerning these levels.

Details of the atomic physics and approximations utilized for computations of the line emissivities are given in Appendix B.

2.2 Cascade and -redistribution equations

The equation describing radiative cascade of a highly-excited electron, accounting for the collisional -redistribution, is

(1)

where , and are the number densities of electrons, recombining ions and recombined ions with electron on the level .

Analytic solution of this coupled system of equations involves inversion of large matrices of sizes of up to several tens of thousands and is not feasible.

We solved the problem the following way. Starting from some maximum , relevant for the problem (defined in the Appendix A), we neglected its cascade population. Knowing the level-specific recombination and -redistribution rates, we computed resulting populations of levels by numerical solution of the system of linear equations (1) for .

Using level-specific radiative rates, we could compute -resolved cascade population of the levels for . Having computed total population rates, we solved the system (1) for this . In such a way, moving downwards in , populations of all levels were computed.

2.3 Recombination line emissivities

Having obtained level populations , we have computed recombination line emissivities , cm/s, defined as

(2)

where d is the number of transitions from level to per second in the unit volume.

As an example, predicted O v 8 line222As it is usual, by we denote spectral line formed by electronic transition from level to level , by – from to , etc. low-density emissivity with - and -components summed up is shown on Figure 1 as a function of temperature for different approximations.

Figure 1: Temperature dependence of the O v 8 recombination line emissivity in the low-density limit. Lower curve correspond to DR rate computations using quasi-classical autoionization rates, upper curve – to the ATOM rates (for details see Appendix B.2), middle curve does not account for the process of line emission without recombination (DCL), described in Appendix B.6. At low temperatures the emissivity increase is determined by the radiative recombination, but the peak around  K is arising due to the dielectronic recombination.

Emissivities of recombining optical and near-infrared recombination lines are given in Table 1 as functions of temperature in the low-density limit.

Approximate wavelengths333Here and everywhere below we present the vacuum wavelengths. of the brightest lines of all ions as functions of ionization stage are given in Table 2. They can be easily estimated using hydrogenic expressions for any recombination line arising in transition between levels with , as

(3)

where is the recombining ion charge.

Emissivity , cm/s, for electron temperature
Line , m  K  K  K  K
0.298
0.495
0.762
1.112
1.554
2.100
2.761
3.548
Table 1: Wavelengths and emissivities of several O v optical and near-infrared recombination lines in the low-density limit.
Line I II III IV V VI
4.0501 1.0125 0.4500 0.2531 0.1620 0.1125
7.4558 1.8640 0.8284 0.4660 0.2982 0.2071
12.365 3.0913 1.3739 0.7728 0.4946 0.3435
19.052 4.7629 2.1168 1.1907 0.7621 0.5292
27.788 6.9471 3.0876 1.7368 1.1115 0.7719
38.849 9.7122 4.3165 2.4281 1.5540 1.0791
52.506 13.127 5.8340 3.2817 2.1003 1.4585
69.035 17.259 7.6705 4.3147 2.7614 1.9176
88.707 22.177 9.8563 5.5442 3.5483 2.4641
Table 2: Hydrogenic vacuum wavelengths, m, of some optical and near-infrared recombination -lines for several ionization stages, denoted by the ion spectroscopic symbol after recombination.

2.4 Resulting line fluxes

Fluxes , erg/cm/s, of the lines were finally computed by integrating along the line of sight

(4)

where is the photon energy, is the distance from the observer to the emitting region and is the emitting region area.

The integral over distance in the plain-parallel approximation can be easily transformed into integral over temperature by substitution

where is the shock front speed, is a cooling function and is the total number density of all ions in plasma.

The parameters of this equation – cooling function, electron and ion densities as functions of temperature – were taken from the SD95 200 km/s shock model, described in more details in Section 3 below.

For purpose of qualitative analysis, we introduce below the oxygen differential emission measure per logarithmic temperature interval

where is the total oxygen ion number density. It shows contribution of a given logarithmic temperature interval to the total emission measure, showing where most of the line emission originates. Using this notion, we can express the line flux from Eq. (4) as

Here the fraction is abundance of a given ionic species and is also dependent on temperature.

In our computation of the post-shock recombination line emission, we artificially stop integrating expression (4) when plasma temperature drops below 1000 K. This results in some, possibly significant, underestimate of the cooling region line fluxes arising from recombinations of ions. Though, as we show below, the O ii recombination lines from the photoionized region are expected to be much brighter.

It should be remembered that, e.g., O v recombination lines arise in the radiative cascade in the ion triggered by the recombination of ion, so the line fluxes are proportional to the ionic abundance of , not .

3 Astrophysical application: FMKs in Cassiopeia A

Figure 2: The plain-parallel SD95 model schematic structure. The drawing is not to scale. Arrow shows the shock wave direction. Optical lines are emitted from the photoionization front and the post-shock cooling region at temperatures of several tens of thousands Kelvin. Recombination lines arise in the cold photoionized region and the post-shock cooling region.

The SD95 model describes fast-moving knot emission as arising in the interaction of the oxygen-dominated dense cloud with the external shock wave, entering the cloud and propagating through it (see Figure 2). The heated region just after the shock front produces high flux of the ionizing radiation that results in appearance of two photoionized regions: before and after the shock front.

The plasma after the shock front passage is rapidly cooling and, at temperatures of  K, emitting the lines observable in the visible and near-infrared spectra (Chevalier & Kirshner, 1978, 1979; Hurford & Fesen, 1996; Gerardy & Fesen, 2001). Thickness of this emitting layer is extremely small (about  cm), but due to the high density of the cooling material at  K it is able to produce bright emission lines of highly-ionized atomic species, such as [O iii]. This region also gives rise to the recombination line (RL) emission.

The SD95 model does not include the photoionized region after the shock front. Until now, even the presence of this region is disputed (Itoh, 1986) because, if present, it would produce too bright optical lines of neutral oxygen.

Between the photoionization front and the shock wave the plasma is predicted to be extremely cold ( K) and rather highly ionized (ionic abundances of and are about 60% and 30% in the SD95 200 km/s shock model). This cold ionized region, invisible in optical and X-ray bands, should be actively recombining and emitting bright recombination lines.

Resulting RL emission computed based on the SD95 200 km/s shock model is described separately in the following subsections for each of these two regions: after and before the shock front.

In this Section we provide results on the -summed recombination line fluxes, whereas in the Section 4 we discuss in details the individual line spectral substructure, that separates in some cases the recombination lines into several components, thus diminishing the individual component fluxes up to factors of 2–4.

3.1 Cooling region after the shock front

It is useful to make order-of-magnitude estimates of some timescales in the post-shock rapidly cooling plasma at temperatures between several hundred and  K, where the recombination line emission occurs (see below). As derived from SD95 model, the plasma cooling time is roughly equal to

where  K and  cm. It should be compared with the time needed to reach the collisional ionization equilibrium (CIE), that is in our caseis approximately equal to the recombination timescale . For example, for ion and the recombination time is

where  cm and is the total radiative recombination rate for this ion, taken from Pequignot et al. (1991). It is easy to see that the recombination timescale in this temperature interval is always longer than the cooling time for any abundant ion. Therefore the plasma is strongly overionized for its temperature, that results in high emissivities of the recombination lines.

The quantitative comparison of these timescales in the model is given on Figure 3, where ionization and recombination timescales for several ions are compared with the cooling time. At high temperatures the cooling timescale is longer than the ionization timescales and the material has enough time to converge to the CIE. At  K the cooling is much faster than the recombination down to temperatures of several hundred Kelvin and the high ionization state is stays “frozen” until very low temperatures are reached.

Figure 3: Oxygen ion and recombination and ionization timescales as compared to the plasma cooling time using the electron density dependence on temperature and the cooling function from the SD95 model with 200 km/s cloud shock speed.

Let us now compare the thermalization timescales and mean free paths to the quantities determined above. Following Spitzer (1956), the electron or ion self-collision timescale is equal to

where is the atomic weight (equal to 1/1836 for electrons), is the particle charge,  cm is the particle concentration and . In our case of the highly-charged plasma having the timescale of the ion-ion collisions is even shorter than for the electron-electron collisions.

Nevertheless, the mean free path even for electrons stays significantly below any other characteristic scale, including the typical temperature change length scale of more than  cm:

For illustration of the characteristic conditions after the passage of the shock front, on Figure 4 we show Li-like ion density as function of plasma temperature for this region444 We made our own computations of the post-shock plasma recombination and discovered that our oxygen ion distribution over ionization stages is rather similar to the one presented on the lower panel of Figure 3 of SD95, but only if the ion spectroscopic symbols on that figure are increased by unity. Therefore we assume that the Figure 3 of SD95 paper has a misprint and the ion spectroscopic symbols should be read, e.g., “O VII” instead of “O VI”, “O VI” instead of “O V”, etc. . The figure demonstrates incredibly high abundance of this ion in relatively dense plasma with  K after the shock front, much higher than in the CIE, where this ion exists only in a narrow temperature range  K (Mazzotta et al., 1998).

Figure 4: Differential oxygen emission measure per logarithmic temperature interval d/d in the post-shock cooling region of the SD95 FMK model. Ionic abundances () of the ion in the FMK model and in the collisional ionization equilibrium in low-density plasmas (Mazzotta et al., 1998) are also shown. Temperature range not taken into account in our calculations is shaded.

On the same figure we present also the oxygen emission measure distribution over temperature dd(log) in the SD95 model. This distribution together with the line emissivity dependence on temperature (e.g., Fig. 1) allows one to calculate the relative contributions to the total line flux from different temperature intervals (see Section 2.4 above).

Such line flux distributions for the prominent [O iii] line at 5007 Å and 8 lines of different oxygen ions are shown on Figure 5. Note that at low temperatures (below 4000 K), the [O iii] 5007 Å line emission is dominated by contribution from the recombinaiton cascades, not the collisional excitation. The cascade contribution to the post-shock intensity of this line reaches about 30%.

Because of high abundances at both low and high temperatures, the distribution of the O vi 8 line emission is much wider than that of the collisionally-excited optical [O iii] line and has 90% of its area within temperature range  K. Note that the dielectronic recombination is contributing less than 20% to the resulting line intensity for any of oxygen ions in the discussed SD95 model.

Figure 5: Differential luminosity distribution per logarithmic temperature interval according to the SD95 model for several spectral lines, including prominent 5007 Å [O iii] line and predicted O ii – O vi 8 lines. Curves are normalized so that the area enclosed under each of them equals one. Temperature range not taken into account in our calculations is shaded.

The O v and O vi lines are the brightest expected ones. Lines of O iii and O iv from this region are predicted to be dimmer by a factor of about three, and the O ii lines – by a factor of about ten.

On Figure 6 we show the predicted oxygen recombination line fluxes from the post-shock region of a model FMK. The brightest recombination lines around 1 m are about 300 times less intense than the reddened [O iii] 5007 Å line. Expected flux values are given also in Table 3.

Ion Line , m , erg/cm/s , erg/cm/s
O vi 7 0.5293 3.43 3.15 0.99
O vi 8 0.7720 1.83 1.07 0.99
O vi 9 1.0792 1.03 2.14 1.31
O vi 12 1.3740 2.19 7.42 0.99
O vi 10 1.4586 6.08 2.24 1.00
O vi 13 1.7189 1.49 6.65 1.00
O vi 11 1.9177 3.72 1.83 1.00
O vi 12 2.4642 2.35 1.38 1.00
O v 6 0.4943 4.07 2.56 0.62
O v 7 0.7618 1.99 1.10 0.81
O v 8 1.1114 1.06 2.36 0.89
O v 10 1.1929 2.91 7.66 0.92
O v 9 1.5538 5.92 2.36 1.21
O v 12 1.9784 1.24 6.26 0.96
O v 10 2.1002 3.47 1.83 0.94
O v 11 2.7613 2.12 1.32 0.97
O iv 7 1.1902 2.52 6.62 0.80
O iv 8 1.7363 1.32 5.97 0.74
O iii 5 0.8262 6.03 4.57 0.25
O iii 6 1.3727 2.65 8.95 0.25
O iii 7 2.1149 1.28 6.78 0.76
[O iii] 0.5007 9.32 6.30
[O iii] 0.5007 1.11 7.52
Including 11 line that is overlapping with 9
Including the pre-shock photoionized region contribution

Note. Assumed cloud area is  cm (size about 06). Reddening was applied using Schild (1977) reddening curve and  (Hurford & Fesen, 1996). The brightest knots are a factor of 3–10 more intense in the [O iii] 5007 Å line than the values given in the table. The values were determined from simulated spectra having 200 km/s Doppler line width at the half maximum level (FWHM), that is equal to the optical line widths observed in the Cas A knots (van den Bergh, 1971); see also Section 4.

Table 3: Brightest recombination line -summed emitted and reddened fluxes and from the post-shock cooling region computed following the SD95 model. The optical [O iii] 5007 Å line flux also given. The last column contains the flux part in the strongest unresolved line component . The wavelength corresponds to the strongest line component.

Figure 6: Reddened recombination line -summed fluxes from the post-shock cooling region. Details are as in note to Table 3. Here the lines are represented as “infinitely thin”; their fine structure is discussed in Section 4.

3.2 Cold photoionized region

The ions in the photoionized region before the shock front are rapidly recombining because of their low temperature. This results in rather bright recombination lines, with fluxes proportional to the distance between the ionization front and the shock wave.

The plain-parallel SD95 model does not determine this distance. Assumption of small optical depth of material between the shock wave and the ionization front results in restriction of  cm for relevant cloud models for preshock ion number density of 100 cm (illustrated on the Figure 12 of SD95). Below we take conservatively  cm, keeping in mind that the line fluxes scale linearly with it.

The cloud model also do not determine exact temperature in the photoionized region. We take  K as a reference value, but Dopita et al. (1984) mention that it may be as low as 100 K. If the real temperature in the region is less than our assumed value, recombination line emissivities are respectively higher (see Section 5.1 below for quantitative dependences).

Predicted oxygen line fluxes for the SD95 model OP200 with shock speed of 200 km/s and total preshock ion number density 100 cm are shown on Figure 7 and given in Table 4. It is seen that even with our quite conservative assumptions resulting line fluxes are still somewhat stronger than from the shock front. Indeed, the predicted line brightest components are on the level of almost half a percent of the [O iii] 5007 Å line.

Ion Line , m , erg/cm/s , erg/cm/s
O iii 5 0.8262 4.13 3.26 0.26
O iii 6 0.8327 1.05 8.48 0.31
O iii 7 1.2552 6.09 1.77 0.73
O iii 6 1.3727 1.95 6.61 0.27
O iii 8 1.7998 3.69 1.72 0.86
O iii 7 2.1149 1.00 5.32 0.82
O iii 9 2.4804 2.32 1.36 1.32
O iii 8 3.0871 5.50 3.59 0.91
O ii 5 1.8645 4.03 1.94 0.24
O ii 6 3.0922 1.90 1.24 0.59
Including 10 line that is overlapping with 9
Table 4: Brightest recombination line -summed emitted and reddened fluxes and from the photoionized region before the shock front. computed following the SD95 model. Details are as in note to Table 3. Photoionized region thickness is  cm. The last column contains the flux part in the strongest unresolved line component for FWHM of 200 km/s (see Note to Table 3). The wavelength corresponds to the strongest line component.

Figure 7: Recombination line -summed reddened fluxes expected from the photoionized region before the shock front computed following the SD95 ionization front model. Details are as in note to Table 3. Photoionized region thickness is  cm.

If they would be just a factor of several stronger, the lines would have been detected in spectroscopic observations of the fast-moving knots (see discussion on existing observation limits in Section 3.4 below).

Our estimate of the observational limit on the O iii 5 recombination line flux at 0.83 m based on the data by Hurford & Fesen (1996) of (5007 Å) may be transformed into a joint constraint on the thickness of the photoionized region , its temperature and total ion density :

assuming and ionic abundance after the ionization front of 0.6.

On this example it is easy to see that detection of the lines or even tight upper limits will allow to perform detailed observational tests of the SD95 and other FMK models.

3.3 Separating pre- and post-shock spectral lines

Observationally, the lines formed in the regions before and after the shock wave will be distinguishable because of two effects. Firstly, the lines arising in the photoionized region yet dynamically undisturbed by the shock wave should be intrinsically very narrow, but the lines arising in the post-shock region are expected to be Doppler-broadened by turbulence.

Secondly, the spectral lines arising in the post-shock region should be shifted with respect to the pre-shock lines. The relative Doppler shift between these two line families appears due to motion of the post-shock gas relative to the initial knot speed induced by the shock wave. In the strong-shock limit this velocity difference equals for specific heat ratio (Zel’dovich & Raizer, 1967), i.e. up to 150 km/s in the SD95 model, but the relative Doppler shift will also depend on the angle between the shock front and the direction towards observer.

Both collisionally-excited and recombination lines should display these features, but for the former group such differences should be easier to detect because of higher line intensities.

The observations with higher spectral resolution () could also result in detection of several narrow components forming a line. They could arise from the individual vortice or velocity component emission, if there is only one or a few of them dominating the plasma motions after the shock, like observed in laser experiments simulating cloud-shock interaction in supernova remnants (Klein et al., 2003). The resulting spectral line profile might be quite complicated (e.g., Inogamov & Sunyaev (2003)), but such observations will provide data on the post-shock dynamics of the fast-moving knot that is presently poorly constrained from observations.

3.4 Existing observational limits

As it is seen from Figures 6 and 7, the brightest recombination lines are expected to lie in the wavelength range between 0.7 and 3 m. Several detailed spectroscopic investigations have already been performed in this range and could have found these lines, provided that they had low enough detection limits. Below we discuss such existing optical and near-infrared limits on the line fluxes and compare them with the SD95 model predictions.

Optical spectra of fast-moving knots with wide spectral coverage were obtained, e.g., by Chevalier & Kirshner (1979) and Hurford & Fesen (1996). According to the authors, the detection limits around 0.75–0.85 m are about 300 times less than the [O iii] 5007 Å line flux for the brightest observed features.

Our results give flux ratios of the reddened 5007 Å line to the brightest components of the O vi 8, O v 7 and O iii 5 lines of about 300–500 (see Figures 6 and 7 and Tables 3 and 4). These values show that the model is on the boundary of consistency with the observational results. They also imply that the oxygen recombination line detection should be possible, provided that the physics in the post-shock and pre-shock regions corresponds to the one described by the SD95 model.

Recently, near-infrared (0.95–2.5 m) spectra of Cas A fast-moving knots, obtained at the 2.4 m Hiltner telescope, were published by Gerardy & Fesen (2001). It is more difficult to compare their detection limits with our predictions, as there is only one oxygen O i line blend present around 1.129 m. It arises in transitions between excited states of neutral oxygen and is also blended with [S i] line. If we assume that the overlapping [S i] line emission is negligible, our estimate of the observational detection limit corresponds to about 1/100 of the reddened optical [O iii] line flux. This value is a factor of several higher than the predicted fluxes of the O vi 9, O v 8 and O iii 6 lines, also showing feasibility of the line detection.

4 Individual line substructure

Figure 8: Fine structure of recombination lines. Line Doppler width at half maximum of 200 km/s is assumed. (a): O iii line component structure for . Sum of all components is set to be equal for each . Emissivities of the components are much weaker and structure of resulting lines is not shown. (b),(c): Spectral emissivity dd, cm/s/Å of the simulated O iii, Si iii and S iii and line structure for low-density case and temperature  K. Note that Si iii lines have been scaled down to fit in the same scale. (d): Spectral emissivity for the simulated O v and S v line structure in the same conditions. Absence of -splitting results in much simpler profiles.

In the non-relativistic hydrogenic approximation, level energies depend only on the ion charge and the principal quantum number of the level. Though, this approximation is not fully applicable for the description of the spectral lines corresponding to transitions between the levels with , as other effects start playing a role. They arise due to energy level shifts, described in Appendix B.4, resulting in the “fine” structure appearing in the spectral lines. As a result, these effects help to distinguish lines of different elements having equal ionization stages and initial and final ’s.

We calculated the line – and –substructure following the method outlined in Appendix B.5 ( is the quantum number used to characterize additional interaction of the highly-excited electron orbital momentum with the total angular momentum of other electrons, present in O ii, O iii, S ii and S iii lines). The results are shown on Figure 8 for several most interesting cases, assuming recombination lines having widths around 200 km/s (full width at half maximum level, FWHM), of the same order as the observed widths of forbidden optical lines (van den Bergh, 1971).

It is easily seen that high- levels have both smaller quantum defects and smaller -splitting. Complicated line structure in case of presence of the -splitting results in weaker individual line fluxes making them more difficult to detect.

One should also keep in mind that spectral lines arising in transitions between lower- levels have higher - and -splittings. Therefore the line components have larger separations and the amount of individual components increase even more with each of them being respectivly less intense. As a bottomline, the near-infrared lines are more promising for the first detection than the optical ones, as they have less substructure (compare panels (b) and (c) on Figure 8).

The substructure of the 6 recombination lines of Mg i, Al i and Si i have been observed in solar spectra near 12 m and explained using similar, although somewhat more elaborate, theoretical description (Brault & Noyes, 1983; Chang & Noyes, 1983; Chang, 1984). We have used these observations as the test cases for our line substructure computation code.

In Tables 5–15, available as an electronic supplement at the CDS, we give O ii – O vi, S ii – S v, Si ii and Si iii recombination line component vacuum wavelengths, emissivities and relative intensities in the resolution for several temperatures (lg3.0, 3.5, 4.0, 4.5 and 5.0) in the low-density limit. First seven columns characterize the quantum numbers of the transition (total recombining ion electronic angular momentum and highly-excited electron quantum numbers before and after the transition , , , , , ), Column 8 gives the line component wavelength in microns, next columns in pairs state line component emissivity in cm/s and intensity ratio of this component with respect to the - and -summed emissivity . Only the lines most likely to be detected are given in these tables, selected by the following parameters – only , and lines having wavelengths between 0.3 and 5.0 m.

Tables 16–23, also available at the CDS, contain similar information on predicted oxygen line component fluxes in the SD95 200 km/s shock model. The fluxes from the cooling and photoionized regions are given separately in Tables 16–20 (for O ii – O vi) and 21–23 (for O ii – O iv), respectively. First eight columns again contain the quantum numbers characterizing the transition and the line wavelength, Column 9 lists line component flux in erg/cm/s, Column 10 contains intensity ratio of this component with respect to the - and -summed emissivity.

As discussed in Appendix B.4, the input atomic data are precise to about 10% and we cannot expect better precision of the resulting line wavelength differences from the hydrogenic values.

Sample region from the total resulting model spectrum is shown on Figure 9. It shows variety of the spectral shapes as well as illustrates the diminishing of the peak intensity due to -splitting for O iii lines.

Figure 9: Model spectra near 0.8 and 2.1 m containing recombination lines of O ii–O vi computed based on the SD95 200 km/s shock model. Contributions of cooling and photoionized regions are summed up. It is seen that at longer wavelengths the substructure is less influencing the line profiles. Note that O ii line intensity is multiplied by 10 for illustration purposes. Line FWHM of 200 km/s is assumed.

The line substructure change with density and temperature because of changes in the relative populations of the -states. At low temperatures the main population mechanism is the radiative recombination, which is populating high- levels relatively efficient. When the electron density increases, -redistribution modifies high- populations for greater than about 20, but for the changes are generally smaller.

At higher temperatures, DR populates relatively lower states having higher quantum defects and low probabilities of -line emission. Therefore a recombination line is split into many components and its emissivity is relatively low. In this case, -redistribution mostly shift the recombined electrons to higher- states, simplifying the line profile and significantly increasing its emissivity.

On Figure 10 we show the change of the O iii line fine structure with temperature and density, illustrating the described effects.

Another effect arising at low temperatures and electron densities – lower populations of excited core states. For O ii, O iii, S ii and S iii lines this results in damping of the -splitting, as it arises only from recombination of excited ions. All the flux from components in this case will be redistributed into the central components and only the -splitting will remain.

Figure 10: Variation of the fine structure of the O iii line at 2.1 m with temperature and density. Line FWHM of 200 km/s is assumed. Note that the low-density curve for  K is multiplied by 20 to be visible on this scale.

5 Plasma diagnostics using recombination lines

5.1 Temperature diagnostics

Line emissivity dependence on temperature is shown for several bright optical and near-infrared oxygen recombination lines in the low-density limit on Figure 11. Corresponding line wavelengths are given in Table 2 above. This figure allows to compute the recombination line fluxes for other models of the multi-temperature plasma with emission measure distribution different from the discussed SD95 model.

Figure 11: Low-density -summed emissivities of oxygen ion optical and near-infrared recombination lines. Different colors represent different ions, plotted line thickness increase with the spectral line wavelength.

The curves on Figure 11 has two distinct regions. At low temperatures the emissivity is determined by the radiative recombination and changes smoothly from ion to ion. At temperatures higher than several tens of thousands Kelvin the dielectronic recombination starts to dominate the recombination rates (except for the O vi lines due to absence at low temperatures of dielectronic recombination channels in He-like ion ) and line emissivities of different ions start being determined by the lowest excited states of the recombining ion, as described in Appendix B.2. The temperature of the strong emissivity rise is proportional to the typical energy of such excited states. Amplitude of the rise is proportional to the lowest excited state decay rate.

For the discussed SD95 200 km/s shock model the high-temperature region where the DR dominates is important only for O v lines, but it is easy to imagine other ionic abundance distributions where it will affect also ions in lower ionization stages.

Dependence of individual O v recombination -line ratios on temperature is shown on Figure 12. Again, the two distinct regions are seen at low and high temperatures corresponding to RR- and DR-dominated recombination.

Figure 12: Low-density -summed O v recombination line emissivity ratios as functions of electron temperature.

It is easy to notice that in the intermediate temperatures, the lines corresponding to transitions at lowest ’s become relatively brighter. Reason for such dependence, especially clearly visible on the line ratio, is the following. As follows from Appendix B.2, at low temperatures the dielectronic recombination rate is damped by the factor . For the low -levels the doubly excited state energy may be significantly lower than the core excitation energy and at low temperatures this makes a difference and the dielectronic recombination populates mostly low- levels, increasing emissivities of low- recombination lines.

The described effect is most pronounced for recombination lines of O v, Si iii and S v, although in all cases the most temperature-sensitive line – the -line formed by the transition from the lowest level, onto which DR is possible – is situated around 2000–3000 Å. Though, as can be seen from Figure 12, also the next line situated for these ions around 4500–5000 Å, can also be a useful diagnostical tool, being relatively much brighter in a narrow temperature interval.

5.2 Recombination lines as density diagnostics

Recombination line flux ratios allow in principle to determine also density of the emitting region, as their dependences on density and temperature differently affect the relative line fluxes.

As an example, on Figure 13 we plot the O v line emissivity as function of the electron density for different temperatures between and  K. For better representation, emissivity ratios to their low-density values are shown. Clearly, the density dependences are weak and can probably be neglected in the first stage of the qualitative analysis, especially at low temperatures.

Two distinct regions are seen once again. At low temperatures where only radiative recombination determines the level populations, emissivities decrease with density, but at higher temperatures they increase. The difference is explained by different initial populations of the levels.

Dielectronic recombination predominantly populates states with (see, e.g., Appendix B.2). Collisions in this case mostly transfer recombined electrons to higher ’s, increasing the probability of transitions and, therefore, -line emissivities.

Radiative recombination, especially at low temperatures, populates high- states much more efficiently. In this case, populations of low- levels relatively increase as a result of the -changing collisions, and recombination line emissivities somewhat decrease.

On Figure 14 we show the O v recombination line emissivities relative to their low-density values as functions of electron density at temperature 30 000 K. Lines arising in transitions between higher levels increase relatively more, but in absolute terms the increase in emissivity is approximately constant, determined by the -redistribution on the high- levels.

Figure 13: Dependence of O v line emissivity on electron density for different electron temperatures . Values are normalized to the low-density emissivity.

Figure 14: Increase of O v -line emissivities with electron density at temperature  K.

5.3 Recombination lines of other elements

Our previous analysis mainly concerned the oxygen recombination line emission, as the SD95 model allows making quantitative predictions about their fluxes and line ratios to the observed collisionally-excited lines. For other elements, no model results for ionic abundance distribution are available. Thus to predict their line fluxes it is necessary to solve a separate problem of non-equilibrium plasma cooling and recombination after the shock front for dense clouds of different compositions. Such analysis is outside the scope of this paper.

Nevertheless, it seems valuable to provide data on the recombination line emissivities as functions of temperature for the most typical elements. Then, from measured line ratios, it will be possible to reconstruct characteristic conditions in the emitting regions. As two most typical examples for elements composing FMKs in Cas A, we concentrate on the recombination lines of silicon and sulphur. Approximate wavelengths of their recombination lines are given in Table 2 above.

Figure 15 presents low-density emissivities of brightest optical and near-infrared recombination lines of silicon and sulphur ions, expected to exist in the fast-moving knots. Note that Si iv and S vi lines are much weaker than of the other ions at  K due to weakness of dielectronic recombination channels at these temperatures. Note also that the line emissivities in Si ii and Si iii ions start to increase sharply already at about 12 000 K.

Thus these ions are the most sensitive tracers of plasma at temperatures between 15 000 and 30 000 K, when emissivities of O and S ions are yet relatively weak. This is also seen on Figure 8(b,c) above, where the emissivities of the lines of Si iii are a factor of 20-50 higher than that of O iii and S iii in the same conditions.

Resembling the case of oxygen recombination lines, the density dependences are not pronounced.

Figure 15: Low-density -summed emissivities of several optical and near-infrared silicon and sulphur recombination lines. Different colors represent different ions, plotted line thickness increase with spectral line wavelength.

5.4 Ratios to collisionally-excited lines

Comparison of metal recombination line fluxes with the “traditional” collisionally-excited lines may also be a useful tool for plasma diagnostics.

As these two types of lines have different origin, their emissivity dependences on temperature and density are rather different. For example, collisionally-excited line emissivities exponentially drop at temperatures below about , as thermal electron energies are not sufficient to excite the ion electronic transition. In contrast, recombination line emissivities at low temperatures increase with the temperature decreasing.

Also a useful property is that, even in the case when the recombination lines are not detected at all, from the limits on the ratios to the collisional lines it is possible to put constraints on the plasma parameters.

On the Figure 16 we give ratios to several brightest collisionally-excited optical lines. The exponential increase of ratios at low temperatures is clearly seen. Collisional line emissivities have been computed using Chianti atomic database (Dere et al., 1997; Landi et al., 2006)

Ratios to the fine-structure far-infrared line emissivities are given on Figure 17. As the Chianti database does not allow to compute line emissivities down to 100 K, we computed them by extrapolating fine-structure transition electronic excitation collision strengths to low temperatures by a constant, that should be reliable to within a factor of two. The collision strength values were adopted from calculations by Lennon & Burke (1994); Blum & Pradhan (1992); Tayal & Gupta (1999); Tayal (2000, 2006).

We note that observations of the far-infrared lines in the spectral range from 10 to 100 m are impossible from the ground. But even from space, observations of these very intense lines cannot be performed with angular resolution sufficient for resolving individual knots. They can result only in signal integrated over many individual emitting objects.

Figure 16: Low-density emissivity ratios of recombination lines to several brightest optical collisionally-excited lines.

Figure 17: Emissivity ratios of recombination lines to several far-infrared collisionally-excited lines. Upper and lower panels show emissivity ratios in the low-density limit applicable for the cold photoionized region (except for the 88.4 m line) and the case of  cm typical for the cooling region after the shock front.

Density dependences of recombination and collisionally-excited lines are also different. Forbidden line emissivity starts decreasing as at some critical density, whereas the recombination line emissivities may both decrease and increase depending on the plasma temperature (see above).

The best density indicators for relatively low temperatures and densities in the fast-moving knots are obviously ratios of different optical and near-infrared line to the far-infrared lines that have critical densities of the order of  cm. In the accompanying paper (Docenko & Sunyaev, to be submitted), we present such analysis based on existing experimental data and compare the results with our predictions based on the SD95 model.

6 Conclusions

We have offered and developed in detail a new method of rapidly recombining plasma diagnostics based on measurements of optical and near-infrared recombination lines of multiply-ionized metal atoms.

As a promising example, we have applied our method to the SD95 theoretical model of fast-moving knots in Cassiopeia A supernova remnant and computed expected oxygen line fluxes from a single FMK of average size and resulting recombination line ratios. It turned out that both cold photoionized region before the shock front and rapidly cooling region immediately after the shock front produce oxygen recombination lines strong enough to be observed on modern optical telescopes in the wavelength range between 0.5 and 3 m.

At shorter wavelengths, two reasons hamper the line observations: high absorption in the interstellar medium on the way from Cas A and the line splitting into widely separated multiple components with consequently lower intensities in each of them. The lines corresponding to transitions between levels are the most promising, as they are not so strongly split and have 50-90% of intensity in one or a few narrow components.

The precision of our RL flux estimates from the Cas A is not expected to be better than a factor of several, as inconsistencies of similar magnitude are observed between the SD95 model predictions and far-infrared oxygen line observations (Docenko & Sunyaev, to be submitted).

Nevertheless, when detected, the recombination lines will allow to determine the details of the photoionization and rapid cooling processes in the FMKs from the line intensities, intensity ratios to each other and to forbidden collisionally-excited lines and from the recombination line fine structure measurements. The measurements of the line structure demand higher signal-to-noise ratios that can nevertheless be achieved on modern telescopes in a reasonable integration time of few hours.

One very interesting result is that the predicted O v and O vi recombination lines, being brightest in the cooling region recombination line spectra, arise in the temperature range significantly below  K, where there is essentially no O and O ions in the collisional ionization equilibrium conditions.

It would be also very interesting to observe the recombination lines from the knots consisting mostly of silicon and sulphur. Existence of high-temperature shock-heated plasma having such chemical composition have been proved by X-ray spectral observations (e.g., Hughes et al. (2000); Hwang et al. (2004)), but detailed theoretical predictions of cooling and ionization structure of such clouds have not been developed yet.

One of the purposes of this article is to attract attention to the metal recombination lines radiated by plasma being far out of the collisional ionization equilibrium and having highly-charged ions at low temperatures. Besides the supernova remnants, such plasma producing detectable metal recombination lines may exist in planetary nebulae, as well as near quasars and active galactic nuclei.

Acknowledgements.
We are grateful to L.A. Vainshtein for valuable advices, providing results from the ATOM computer code and permission to use it. We are also thankful to the anonymous referee for remarks permitting to make the paper easier to understand and for a useful reference on computation of hydrogenic radiative transition rates.

Appendix A Determination of the upper cutoff

Traditionally, the highly-excited level populations are characterized by the so-called departure coefficients , defined as the ratio of actual level population to its thermodynamic equilibrium value . With increasing, at levels the collisional -redistribution processes establish equilibrium population over ’s, i.e., . The value of depends on the ion, as well as on electron temperature and number density.

For higher ’s, rates of collisional transitions involving change start rapidly increasing and from some determine the highly-excited level populations. Thus the curve, itself defined only at , has two distinct ranges as a function of : below it is determined by recombination and radiative processes and above it rapidly tends to unity because of the processes relating different level populations with each other and with the continuum: collisional -redistribution, three-body recombination and collisional ionization. As an example relevant for our study, ’s of Li-like ion at temperatures 1000 and 20 000 K and electron density 50 000 cm are shown on Figure 18, assuming .

Figure 18: Departure coefficients of recombining ions computed accounting for all processes (solid lines) and neglecting collisional -redistribution, three-body recombination and collisional ionization (dashed lines) at electron temperatures  K and  K and density  cm. At higher temperature dielectronic recombination dominates and significantly increases the level populations.

On this figure, we also show the departure coefficients, computed accounting only for radiative transitions. It is seen that such approximation results in population computation errors at levels higher than about .

Let us now show that the electrons that recombined to levels are contributing relatively weakly to the total recombination and it is safe to introduce an upper cutoff , neglecting population of higher levels. Two cases – and – should be discussed separately here.

In case of not introducing such cutoff and neglecting collisions may result in some cases in significant overestimate of the recombination rate – up to a factor of several. The easiest estimate of the cutoff position is the level where the total recombination rate onto computed without accounting for transitions equals to the true total recombination rate. It is easy to understand that in this case , as illustrated in the lower panel of Figure 19.

In case of the rates obtained neglecting collisional transitions between high levels will always underestimate true recombination rates. Though for temperatures above about  K taking upper cutoff value approximately equal to does not result in error more than about 20% for relevant ions, as illustrated on the upper panel of the Figure 19. Lower cutoff values will result in more significant underestimates of the total recombination rate and the line emissivities.

Figure 19: Recombination rates of for  cm and  K (upper panel) and  K (lower panel). Meaning of solid and dashed lines is as on Figure 18. Plot shows both rates for a given (magenta curves) and cumulative rates for all levels below (blue curves). Vertical line show upper cutoff used for this ion in our calculations.

In case of lower densities, when is very high, another natural limit arises from general dependence of the DR rate on the principal quantum number, see, e.g., Sobelman et al. (1981).

Appendix B Atomic physics for level population computations

b.1 Radiative recombination

There are two major recombination processes populating high- states in low-density plasmas: radiative and dielectronic recombination (RR and DR, respectively). In the radiative recombination process, a free electron is directly recombining to a bound state . A photon is simultaneously emitted, carrying away the released energy.

The radiative recombination level-specific cross sections were computed from hydrogenic photoionization cross sections by applying the detailed balance relation:

(5)

where is the electron energy prior to recombination, is the level binding energy, is the energy of emitted photon, is the recombining ion charge and is the fine-structure constant. The photoionization cross sections were calculated by the computer program of Storey & Hummer (1991).

The level-specific radiative recombination rates were then computed by numerical integration over the electron Maxwellian energy distribution for the temperature (here denote electron velocity).

Using expressions from e.g. Spitzer (1956), it may be shown that the electron-electron collision timescale is always shorter than the cooling time at temperatures below  K where the line emission occurs, and electrons have enough time to reach the Maxwellian distribution. Roughly,

where  K,  cm and .

b.2 Dielectronic recombination

The DR is a two-step process (for reviews see, e.g., Beigman et al. (1968); Shore (1969); Seaton & Storey (1976); Sobelman et al. (1981)), illustrated in the simplest case by the following diagram:

(6)

Free electron e is first resonantly captured by the ion ( denotes the ion charge) with simultaneous core electron excitation from the ground state to the state having excitation energy . This process is called dielectronic capture. Resulting ion having two excited electrons is unstable with respect to autoionization – the inverse process of the dielectronic capture.

The recombined ion then either autoionizes, or its excited core electron transits back to the ground state and emits a resonant line photon, thus making the ion stable against the autoionization.

Therefore the level-specific dielectronic recombination rate is a product of two factors: dielectronic capture rate and doubly-excited state stabilization probability. In the simple case described above it is expressed as

(7)

where and are recombining ion ground and excited states, and are their statistical weights, is the doubly-excited state autoionization rate with core electron after the autoionization moving to the ground state , is the core state decay rate to the ground state and is the electron energy prior to the recombination, .

We have determined the dielectronic recombination rates using two methods, which differ by the way they compute autoionization rates. In the first method, the autoionization rate is expressed in terms of the photoionization cross section using the dipole approximation for the inter-electronic interaction. Simple expression of this cross section and the autoionization rate in the limit is then obtained using quasi-classical (QC) approach (Bureyeva & Lisitsa, 2000).

It is known that the dipole approximation is not applicable for this purpose for states (Beigman et al., 1981). Though, this is not a serious limitation in the case, when the dielectronic capture occurs via excitation of the recombining ion core electron without change of its principal quantum number (). Such transitions are the most important in our case, as relevant plasma temperatures are much lower than the transition excitation energies.

The second method is based on the usage of the program ATOM (Shevelko & Vainshtein, 1993). It computes autoionization rates by extrapolating below the threshold the electronic excitation cross section of the recombining ion core transition. This extrapolation method is based on the correspondence principle and is thus precise in the limit of . Both in QC approximation and ATOM approach, the autoionization rates decrease as .

Comparison plots of the DR rates of forming with recombined electron populating all -levels and level -states are given on Figure 20. It is seen that in the quasiclassical approximation, electrons are populating lower ’s, but higher ’s. As can be inferred from Eq. (7), this corresponds to sharper -dependence of the autoionization rates. Comparison of scaled quasiclassical and quantum autoionization rates of doubly-excited oxygen ion is shown on Figure 21, indeed showing the inferred dependence.

It is clear from these data that the two models for the DR rates will result in significantly different line emissivities (see also Figure 1). For final results we decided to use the ATOM rates, where available, as the theoretical arguments (Shevelko & Vainshtein, 1993) show that they are more precise than the quasiclassical ones.

For ions , , , , , , and we used the ATOM rates. For other ions we implemented the quasiclassical expressions.

Figure 20: Comparison of -resolved (upper panel) and -summed (lower panel) ion dielectronic recombination rates computed by the two methods described in the text: ATOM program and quasiclassical (QC) approximation. The values are given for  K, but the curves scale identically with the temperature as long as  K, when core excitations start playing a role. The -resolved rates are given for .

Figure 21: Comparison of scaled autoionization rates of ion in state autoionizing to ion in state, , s, in the quasiclassical approximation and computed by the program ATOM.

Comparison of the dielectronic recombination rates with other atomic process rates is shown on Figure 22 for the case of ion with highly-excited electron on levels and 200. It is seen that the dielectronic recombination populates lower ’s, but is more efficient at high- levels. In case of sufficiently high density the dielectronic recombination to low- states is followed by rapid redistribution onto much higher ’s resulting in enhancement of the recombination line intensities.

Figure 22: Elementary process rates for ion recombining to form with highly-excited electron having (upper panel) and (lower panel) at electron temperature  K. Collisional rates (total for transitions to all ) are shown for electron density  cm and are negligibly low for case.

b.3 Highly-excited level energies

In hydrogen atom the energies of levels are independent on because of the shape of the Coulomb potential. In atoms with several electrons the changes in the potential energy curve introduce variations of the highly-excited level energies with . This dependence is usually parametrized by expression

where denotes quantum defect of the level and is the ion spectroscopic symbol. The quantum defects rapidly decrease with increasing and tend to constant with increasing (Seaton, 1983).

Their values are important in this study for two reasons. Firstly, level energies and determine transition energies and, therefore, spectral line wavelengths. Transitions having the same values of and have similar wavelengths. Small wavelength differences are in this case determined by the energy differences within and groups of states – by the quantum defects of the levels.

Secondly, the collisional transition rates are dependent on the energy level splitting , growing as this splitting decreases. Therefore the quantum defects in this case determine rates of -redistribution and recombination line flux dependence on electron density in plasma.

In case of the quantum defects for most ions are known and have been taken for our study from Ralchenko et al. (2007)555URL: http://physics.nist.gov/asd3. In the next subsection we describe our method of computation of level energy shift from hydrogenic values for that can be expressed via quantum defects as

where we have assumed that , i.e. that the quantum defect does not depend on and that it is small: .

The latter assumption is valid in our case of , when the states are “non-penetrating”, i.e., those in which highly-excited electron wave function significantly differs from zero only outside the atomic core region. This property also allows to use hydrogenic expressions for description of such highly-excited states.

b.4 Computation of the level shifts for high- states

Following the approach of Watson et al. (1980) and Dickinson (1981), we account for the non-penetrating state level shifts arising due to two effects: core polarizability and electrostatic quadrupole interaction with the core.

The energy shift induced by the core polarizability is proportional to the highly-excited electron radial integral and, expressing it with hydrogenic formula, equals (Seaton, 1983)

(8)