4 Results and discussion

In Table 5 we give recombination coefficients for the 5g ($J\pi$)levels at temperatures of $10\,000$ K and $20\,000$ K, and a density of 10³ cm^-3, calculated from Eq. (1) and from the fitting formulae given in Sect. 3.1. These are provided as a benchmark for the use of the formulae. The comparison shows that the fitted coefficients $\alpha_\mathrm{fit}$ agree with the calculated ones $\alpha_\mathrm{calc}$within $1.5 \%$.The intensities of any of the lines in the 5g - 4f transition array may be obtained, within the range of validity of the fits ($5000\leq T_\mathrm{e}\mathrm{[K]}\leq 20\,000$ K and $10^2\leq N_\mathrm{e}\mathrm{[cm}^{-3}\mathrm{]}\leq10^6$), by combining the recombination coefficients calculated from the formulae of Sect. 3.1 with the branching ratios in Table 2.

  

Table 5: Calculated $\alpha_\mathrm{calc}$ and fitted $\alpha_\mathrm{fit}$ effective recombination coefficients (in $10^{-12}~\mathrm{cm}^3$ s^-1) for the 5g levels of O²⁺ at the electron temperatures $T_{\mathrm e} = 10\,000$ K and $T_{\mathrm e} = 20\,000$ K and the electron density $N_{\mathrm e} = 10^3$ cm^-3. Indices n₁ correspond to the energy level indices from Table 1

  

Figure 1: Synthetic recombination spectra for the 5g - 4f transition array at electron temperature $T_{\mathrm e} = 10\,000$ K and various electron densities: a) $N_{\mathrm e} = 10^2$,b) $N_{\mathrm e} = 10^3$,c) $N_{\mathrm e} = 10^4$,d) $N_{\mathrm e} = 10^5$,e) $N_{\mathrm e} = 10^6$ cm-3. The line profiles are Gaussian with $FWHM = 2\sqrt{\ln 2}$ Å

In Fig. 1 we show synthetic spectra of the entire 5g - 4f transition array calculated at $T_{\mathrm e} = 10\,000$ K and various electron densities. The lines are taken to have a FWHM of $$2\sqrt{\ln 2}$$ Å. Some of the lines show a pronounced variation in intensity with density at typical nebular densities, between 10² and 10⁴ cm^-3. The line at 4434.6 Å, for example, which is the only transition from the (²P$_{3/2}^\mathrm{o}$)5g ³H₆ level (level 21 in Table 1) increases in intensity by a factor of 5.3 between $N_{\mathrm e} = 10^2$ and 10⁴ cm^-3 as the population of the O^{3+ 2}P$_{3/2}^\mathrm{o}$ rises. The intensity of this line, which is one of the stronger lines in the 5g - 4f group, is 1.99 N(O³⁺)/N(H⁺) relative to H$_{\beta}$ at $T_{\mathrm e} = 10\,000$ K and $N_{\mathrm e} = 10^4$ cm^-3, where N(O³⁺) and N(H⁺) are the number densities of O³⁺ and H⁺ respectively.

The coefficients $g^\mathrm{casc}$ given in Table 4 show that the direct photorecombination and cascade contributions to each level are dominated by contributions from one or other of the O³⁺ parent states, ²P$_{J_0}^{\rm o}$, indicating that the 5g levels are approaching jj-coupling. We have investigated the effects of the interaction of the two series on the line intensities by carrying out a calculation of the coefficients in which there is no interaction. In this simpler approximation, we use hydrogenic recombination coefficients throughout. The recombination coefficients to the 5g $J\pi$ levels are directly proportional to the series parent population as follows:

 

$$\alpha\left[^2\mathrm{P}^\mathrm{o}_{J_0};\; 5\mathrm{g}\;(J... ...g}) \\ \times \frac{(2J+1)} {\omega(5\mathrm{g}) \; (2J_0+1)},$$ (10)

where $\alpha_\mathrm{eff}(5\mathrm{g})$ is the total hydrogenic recombination coefficient to the 5g level (Storey & Hummer 1995) and J₀ is the total angular momentum of the parent level. This coefficient includes both the direct radiative recombination to the 5g level and the cascading processes from the higher levels.

To demonstrate the difference between the two approximations, we compare synthetic spectra for the two models in Fig. 2 at the lowest electron density, $N_{\mathrm e} = 10^2$ cm^-3. At this density the fraction of the O³⁺ population in the ²P$_{3/2}^\mathrm{o}$ is very low (2%), so that in the approximation of Eq. (10), the Rydberg states (²P$_{3/2}^\mathrm{o}$)nl have very low populations and the resulting lines are weak, while in the more realistic approximation of Eq. (1), population can be transferred between the (²P$_{1/2}^\mathrm{o}$)nl and (²P$_{3/2}^\mathrm{o}$)nl series increasing the line intensities. It should be noted that the differences between the two approximations become smaller as the density increases and the parent state populations approach the Boltzmann distribution.

  

Figure 2: Synthetic recombination spectra for the 5g - 4f transition array at electron temperature $T_{\mathrm e} = 10\,000$ K and electron density $N_{\mathrm e} = 10^2$ cm-3. The dashed line shows the spectrum calculated using Eq. (10), the solid line using Eq. (1). The line profiles are Gaussian with $FWHM = 2\sqrt{\ln 2}$ Å