9.1  The isothermal case

Once the isothermal assumption is made, it is straightforward to deduce from a line ratio a temperature T_*, since the intensity ratio I₁/I₂ is directly equal to the ratio of the contribution functions:

I1 I2 =  C1(T*, Ne) C2(T*, Ne)

(27)

Any ratios of lines of ions of close ionization stage can be used for this purpose, as long as the lines are not density-sensitive, otherwise the derived temperature T_* becomes density-dependent.

Different line ratios are expected to give different temperatures, if the plasma distribution is not isothermal.

Aside from the cited possible errors due to density and element abundance uncertainties, which can easily be avoided by careful choice of lines, there is one substantial source of error that surprisingly is normally neglected in the literature. This is the validity of the adopted ionization equilibrium.

The use of different ionization equilibrium calculations can produce significantly different results as shown in Figure 24.

Figure 24: Contribution functions of two iron lines and their ratio, for different ionization equilibria. Note how the ratio is different.