The AstroStat Slog

Differential Emission Measures (DEMs) are a summary of the temperature structure of the outer atmospheres (aka coronae) of stars, and are usually derived from a select subset of line fluxes. They are notoriously difficult to estimate. Very few algorithms even bother to calculate error envelopes on them. They are also subject to numerous systematic uncertainties which can play havoc with proper interpretation. But they are nevertheless extremely useful since they allow changes in coronal structures to be easily discerned, and observations with one instrument can be used to derive these DEMs and these can then be used to predict what is observable with some other instrument.

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.