Misestimation of temperature when applying Maxwellian distributions to space plasmas described by kappa distributions

Abstract

This paper presents the misestimation of temperature when observations from a kappa distributed plasma are analyzed as a Maxwellian. One common method to calculate the space plasma parameters is by fitting the observed distributions using known analytical forms. More often, the distribution function is included in a forward model of the instrument’s response, which is used to reproduce the observed energy spectrograms for a given set of plasma parameters. In both cases, the modeled plasma distribution fits the measurements to estimate the plasma parameters. The distribution function is often considered to be Maxwellian even though in many cases the plasma is better described by a kappa distribution. In this work we show that if the plasma is described by a kappa distribution, the derived temperature assuming Maxwell distribution can be significantly off. More specifically, we derive the plasma temperature by fitting a Maxwell distribution to pseudo-data produced by a kappa distribution, and then examine the difference of the derived temperature as a function of the kappa index. We further consider the concept of using a forward model of a typical plasma instrument to fit its observations. We find that the relative error of the derived temperature is highly depended on the kappa index and occasionally on the instrument’s field of view and response.

Appendix A: Energy at the spectrogram’s maximum

In the simplified case, where the instrument’s FOV is very narrow and its response is not a function of energy and direction, then Eq. (7) becomes

$$ C(E) \propto E^{2}f(E;\varOmega ). $$

(A.1)

In order to find the energy at the spectrogram’s maximum, we calculate the energy for which the first derivative of (A.1) is zero, $\partial C(E)/\partial E = 0$, leading to

$$ \frac{\partial \ln f ( E;\varOmega )}{\partial \ln E} = - 2. $$

(A.2)

If the observed plasma is described by the kappa distribution, we substitute Eq. (8) and use Eq. (5) in Eq. (A.1) that gives the observed counts maximum, which now becomes the quadratic equation

$$\begin{aligned} &(\kappa - 1)E_{\max} - (\kappa - 3)\cos \omega \sqrt{E_{0}} \sqrt{E_{\max}} \\ &\quad{}- 2\biggl[\biggl(\kappa - \frac{3}{2} \biggr)k_{\mathrm{B}}T - E_{0}\biggr] = 0. \end{aligned}$$

(A.3)

For small bulk energies, i.e., $E_{0} \ll (\kappa - \frac{3}{2})k_{\mathrm{B}}T$, $E_{0} \ll E_{\max}$, the maximum is given by $E_{\max} \cong 2k_{\mathrm{B}}T(\kappa - \frac{3}{2})/(\kappa - 1)$. If we model the instrument’s response assuming a Maxwell distribution, we obtain the above again Eq. (A.2) but for $\kappa \rightarrow \infty$, i.e.,

$$ E_{\max} - \cos \omega \sqrt{E_{0}} \sqrt{E_{\max}} - 2k_{\mathrm{B}}T = 0 $$

(A.4)

and for small bulk energies, the maximum is given by $E_{\max} \cong 2k_{\mathrm{B}}T$. Even in this simplified case which we demonstrate here, the location of the spectrogram’s peak varies as a function of all the plasma parameters.