Revisiting the BE99 method for the study of outflowing gas in protostellar jets

Context. An established method for measuring the hydrogen ionisation fraction in shock-excited gas is the BE99 method, which utilises six bright forbidden emission lines: [S II] λλ 6716, 6731, [N II] λλ 6548, 6583, and [O I] λλ 6300, 6363. The main assumptions of this technique are that the gas is in a low-excitation state ( x e < 0.3) and that the equilibrium of the underlying ionisation network is reached. Aims. Our aim is to extend the BE99 method by including more emission lines in the blue and near-infrared part of the spectrum ( λ = 3500 − 11 000 Å), and by considering higher hydrogen ionisation fractions ( x e > 0.3). In addition, we investigate how a non-equilibrium state of the gas and the presence of extinction influence the BE99 technique. Methods. We numerically solved a network of ionisation reactions in the time domain, which lead to the BE99 equilibrium. We applied the BE99 method on synthetic spectra also in the non-equilibrium state. We extented the BE99 technique to higher ionisation fractions by considering additional reactions, which involve higher ionisation states of oxygen, nitrogen, and sulphur. We tested our concepts on the low-excitation outflow of Par Lup 3−4 and the high-excitation 244−440 Proplyd in Orion. Results. Many additional emission line ratios can in principle be exploited as extended curves (or stripes) in the ( x e , T e ) diagram. If the BE99 equilibrium is reached and extinction is corrected for, all stripes overlap in one location in the ( x e , T e ) diagram indicating the existing gas parameters. We find that the BE99 equilibrium is reached faster than the hydrogen recombination time. The application to the Par Lup 3−4 outflow shows that the classical BE99 lines together with the [N I] λλ 5198+5200 lines do not meet in one location in the ( x e , T e ) diagram. This indicates that the gas parameters derived from the classical BE99 method are not fully consistent with other observed line ratios. A multi-line approach is necessary to determine the gas parameters. From our analysis we derive n e ∼ 45000 cm −3 − 53000 cm −3 , T e = 7600 K − 8000 K, and x e ∼ 0.027 − 0.036 for the Par Lup 3−4 outflow. For the 244−440 Proplyd we were able to use the line ratios of [S II] λλ 6716+6731, [O I] λλ 6300+6363, and [OII] λλ 7320, 7330 in the BE99 diagram to estimate the ionisation fraction at knot E3 ( x e = 0.58 ± 0.05). Conclusions. The BE99 method can be extended by utilising more emission line ratios in the ( x e , T e ) diagram and considering higher ionisation states. Exploiting new line ratios reveals more insights on the state of the gas. Our analysis indicates, however, that a multi-line approach is more robust in deriving gas parameters, especially for high-density gas.