Differential Emission Measure [Eqn]

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,

Iul(ɼ) = (1/4 π) (1/d(ɼ)2) A(Z,ɼ) Gul(ne(ɼ),Te(ɼ)) ne(ɼ)2 δV(ɼ) ,

where ne is the electron density and Te is the temperature of the plasma, A(Z,ɼ) are the abundance of element Z, Gul(ne,Te) 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,

Iul(Te) = (1/4 π) ∑ɼ|Te (1/d(ɼ)2) A(Z,ɼ) Gul(ne(ɼ),Te) ne2δV(ɼ) .

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

Iul(Te) ≈ (1 / 4 π d2) Gul(ne,Te) A(Z) ne2 (ΔV / Δlog Te) Δlog Te ,

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

Iul = ∑Te (1 / 4 π d2) A(Z) Gul(ne,Te) DEM(Te) ΔlogTe

The quantity

DEM(Te) = ne2 (ΔV / Δlog Te)

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 ne2(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(Te) 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.

  1. aneta:

    I like your Eq. postings. However, they would be more “readable” with a better editor for the equations, so they could look nicer. You should immediately see what’s going on in terms of letters and symbols. Now it is hard to read them. I typically need to write them down on the paper to see what is going on. In your “I Like Eq.” posting you say a lot about writing and presentation of equations. Good editor is really helpful.

    08-14-2008, 10:52 pm
  2. vlk:

    I know, Aneta.. the problem is that MathML is not universal enough that we can assume it would be visible to everyone, and using inline jpegs makes it impossible to match with the text’s font size and makes it very difficult to correct errors. So my compromise thus far has been to use ISO-8859-1 unicode. Suggestions for alternatives would be much appreciated!

    08-14-2008, 11:44 pm
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