Many papers mention emission measure, and some of them give the expression of EM, BUT there is no clear definition of EM.

My question is what EM is on earth? Its unit is supposed to be cm-3.

How to calculate EM? here(4B5) gives an expression, but it can not be derived from a light curve directly.

Given an X-ray light curve(not from the Sun), probably I need to devide the light curve into many bins and do spectral fitting.

Then what should I do? Rererence papers with clear definition and expression are welcome.

  • $\begingroup$ The very first google hit defines it as the integral of $N_2^2$ along the line of sight, where $N_e$ is the electron number density. That gives dimensions of pc cm$^{-6}$, though. $\endgroup$
    – pela
    Jan 30, 2016 at 13:49
  • $\begingroup$ @pela Ne is not a direct observational quantity. There are other expressions of EM, which can be derived by analysing a light curve. $\endgroup$ Jan 30, 2016 at 15:09
  • $\begingroup$ If you say that Many papers mention, ot would be nice if you added some example links to improve the quality of your question. $\endgroup$
    – user1569
    Jan 31, 2016 at 15:42

2 Answers 2


Emission measure is (usually) used in X-ray and EUV astronomy, though I suppose also in cases of optically thin radio emission. It is defined as the square of the number density of free electrons integrated over the volume of the plasma.

$${\rm EM} = \int n_e^2 \ dV$$

The flux of optically thin emission from a plasma (e.g. thermal bremsstrahlung) is then directly proportional to the emission measure of the plasma multiplied by a temperature dependent cooling loss law.

In other words, when you measure the flux of X-rays from an unresolved optically thin emitter, there is a degeneracy between the electron number density (squared) and the overall plasma volume.

When you fit an X-ray spectrum with an optically thin model, the emission measure (divided by $4\pi d^2$, where $d$ is the distance to the object), is a multiplicative free parameter.

Your question about calculation is extremely difficult to answer. Suppose I measure a count-rate of $N$ X-ray counts per second using some X-ray telescope (I can only assume that's what you mean by a "X-ray light curve".).

The count-rate received at the telescope depends on: the emission measure (as defined above) multiplied by a term that depends on the temperature (or temperatures) of the source, the chemical composition of the source and the adopted emission process (is it free-free thermal bremsstrahlung, a thermal plasma or something else). It is then attenuated by any intrinsic absorption in the source and any absorption between us and the source and by the distance to the source (assuming the radiation is isotropic). Finally what is detected is determined by the response of the X-ray detector to X-ray photons as a function of energy.

  • $\begingroup$ Your answer can be found in general papers, but they do not give a clear and direct definition about EM. There is no direct expression of EM either. Usually expressions of EM carry electron density, but Ne can not be derived from a light curve. $\endgroup$ Jan 31, 2016 at 15:15
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    $\begingroup$ @questionhang how much clearer can the definition be? $EM = \int n_e^2\ dV$. $\endgroup$
    – ProfRob
    Jan 31, 2016 at 15:36
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    $\begingroup$ What does this interation mean? Why do we integrate like this? You can not say E is $mc^2$ and that is all. We can not derive $Ne$ in a light curve directly. How to derive it then? I have specified my question clearly before your answer, right? $\endgroup$ Feb 1, 2016 at 2:12
  • $\begingroup$ I know how to calculate it now. Thank you. $\endgroup$ Feb 2, 2016 at 9:19

@tefalya Somebody told me "EM is the integral of electron density squared over the emitting volume. In collisional ionized plasmas, it gives the amount of material emitting at a given temperature. In the low density regime, temperature, emission measure, and abundance together with an atomic code completely specify the observed spectral energy distribution of coronal plasma.

The normalization parameter you get out of APEC, MEKAL, etc. relate to the emission measure. If you read the description of the APEC model in XSPEC, you will see that the normalization = 10^-14* n_e^2 dV/(4 !pi distance^2). The Emission Measure EM is n_e^2 dV, and has units of cm^-3. So the emission measure is an observable, and by assuming that the volume is not changing significantly during the flare, the changing emission measure is a proxy for the changing electron density."


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