The flux at Earth due to an atomic transition u –> l from a volume element δV at a location ɼ,

I_ul(ɼ) = (1/4 π) (1/d(ɼ)²) A(Z,ɼ) G_ul(n_e(ɼ),T_e(ɼ)) n_e(ɼ)² δV(ɼ) ,

where n_e is the electron density and T_e is the temperature of the plasma, A(Z,ɼ) are the abundance of element Z, G_ul(n_e,T_e) is the atomic emissivity for the transition, and d is the distance to the source.

We can combine the flux from all the points in the field of view that arise from plasma at the same temperature,

I_ul(T_e) = (1/4 π) ∑_ɼ|Te (1/d(ɼ)²) A(Z,ɼ) G_ul(n_e(ɼ),T_e) n_e²δV(ɼ) .

Assuming that A(Z,ɼ), n_e(ɼ) do not vary over the points in the summation,

I_ul(T_e) ≈ (1 / 4 π d²) G_ul(n_e,T_e) A(Z) n_e² (ΔV / Δlog T_e) Δlog T_e ,

and hence the total line flux due to emission at all temperatures,

I_ul = ∑_Te (1 / 4 π d²) A(Z) G_ul(n_e,T_e) DEM(T_e) ΔlogT_e

The quantity

DEM(T_e) = n_e² (ΔV / Δlog T_e)

is called the Differential Emission Measure and is a very useful summary of the temperature structure of stellar coronae. It is typically reported in units of [cm^-3] (or [cm^-5] if ΔV is written out as area*Δh). Sometimes it is defined as n_e²(ΔV/ΔT) and has units [cm^-3K^-1].

The expression for the line flux is an instance of Fredholm’s Equation of the First Kind and the DEM(T_e) solution is thus unstable and subject to high-frequency oscillations. There is a whole industry that has grown up trying to derive DEMs from often highly unreliable datasets.