I know that the line strength ratio tells us how hot the electron plasma in a nebula is, and also give information about the electron densities in the nebula. But how do you compute the line ratios? Say we consider two emission lines (both forbidden) decaying to the ground state. If I have the Einstein A coefficients and the statistical weights, how do I compute the line ratios? Any explanation (preferably with examples) will be highly appreciated.
You need a model for the excitation of the line. But since you say you have two forbidden lines, you can try assuming the density is well below the "critical density" (which is the density at which point the upper levels of the transitions get quenched by collisions instead of waiting as long as it takes for the "forbidden" radiative transition to occur). Then if you are talking about two transitions from the same upper level, the line ratio is simply the ratio of the Einstein A values for the two transitions (note that the Einstein A value normally already includes the statistical weights of the lower levels). If you are talking about two different upper levels, then you need a model for how those upper levels get populated, and the line ratios will be the ratios of the upper level populations times the Einstein A values. It's not always easy to know how the upper levels get populated, so that's why it's hard to give you an answer to your question, it is very context dependent. But the main simplification of using forbidden lines is that you don't have to worry about the photons getting reabsorbed before they get out. (A detail is that line ratios are often given in terms of energy flux ratios, rather than photon ratios, so you'd need to multiply the Einstein A values by the photon energies if the transitions are at very different energies.)