## 5 Bremsstrahlung Emission Rates in Lorentzian Kappa Plasmas  
In most astrophysical and space plasmas, the external disturbances in thermal plasmas would produce the high-energy tail in the distribution of plasma electrons so that the deviations from the thermal Maxwellian distribution is expected due to the interaction between the plasma and the external perturbations. A pioneering work by [PERSON], [PERSON], and [PERSON] [17] showed that the nonthermal distribution due to the entropy generalization mechanism in the presence of external radiation field in astrophysical plasmas obeys the Lorentzian (kappa) velocity distribution function \(f_{L}(v)\)[17; 18; 24] in the form:  
\[f_{L}(v)\ =\ n_{e}\left(\frac{m}{2\pi\kappa E_{\kappa}}\right)^{3/2}\frac{ \Gamma(\kappa+1)}{\Gamma(\kappa-1/2)}\left(1+\frac{mv^{2}}{2\kappa E_{\kappa} }\right)^{-(x+1)}, \tag{21}\]  
where \(v\) and \(n_{e}\) are the velocity and the density of electron, \(\kappa(>3/2)\) is the spectral index in the Lorentzian distribution, \(E_{\kappa}[\equiv(\kappa-3/2)E_{M}/\kappa]\) is the effective energy of the Lorentzian electrons, \(E_{M}\equiv k_{B}T\), \(k_{B}\) is the Boltzmann constant, \(T\) is the electron temperature, and \(\Gamma(z)\) represents the gamma function with the argument \(z\). Then, the differential Lorentzian electron distribution function can be represented by \(dn_{L}(v)\ =\ 4\pi v^{2}f_{L}(v)dv\). It has been also shown that the radiation interaction modifies the conventional diffusion process in astrophysical plasmas so that the correction on the total diffusion coefficient can be represented by the factor \((1\,+\,\alpha_{R}v^{2})\), where \(\alpha_{R}\) is a constant related to the external radiation field intensity, since the non-Coulombic diffusion coefficient is found to be proportional to the square of the electron velocity \(v\) and can be also induced by the interaction with the field [17]. It is also interesting to note that the Lorentzian distribution acquiesces a simple power-law form at high energies, i.e., we have \(f_{L}(v)\,\propto\,\left(mv^{2}/2\kappa E_{\kappa}\right)^{-(\kappa\,+\,1)}\) when \(mv^{2}/2>>\kappa E_{\kappa}\). Moreover, the nonthermal astrophysical Lorentzian distribution with the infinity spectral index \(\kappa\), equivalent to the absence of the external interaction, turns out to be the thermal distribution for all velocities such as \(f_{L}(\kappa\rightarrow\infty)\,\propto\,\exp(\,-\,mv^{2}/2E_{\kappa\, \rightarrow\,\infty})\) owing to the mathematical limiting relation: \(\lim\limits_{t\rightarrow\infty}(1\,+\,x/t)^{t}\,=\,e^{x}\), where \(E_{\kappa\,\rightarrow\,\infty}\) corresponds to the thermal energy in the Maxwellian plasmas, i.e., \(k_{B}T(\,=\,E_{M})\)[17]. Hence, we have found that the Lorentzian distribution \(f_{L}(v)\) encompasses a wide range of plasma velocity distributions from the Maxwellian distribution to the inverse power law distribution. In addition, the effective Debye length \(\lambda_{\kappa}\) in nonthermal Lorentzian plasmas can be represented by \(\lambda_{\kappa}\,=\,\lambda_{D}\mu_{\kappa}\)[18] where \(\lambda_{D}\) is the conventional Debye length in Maxwellian plasmas and \(\mu_{\kappa}[\,\equiv\,\sqrt{(\kappa-3/2)/(\kappa-1/2)}]\) stands for the fractional measure of the nonthermal population in astrophysical Lorentzian plasmas. Hence, the bremsstrahlung emission rate \(P_{\varepsilon}\) for a given differential Lorentzian electron density distribution \(dn_{L}(v_{0})\) can be written by:  
\[P_{\varepsilon}\,=\,\frac{dE_{rad}}{dVdt\epsilon}\,=\,\int dn_{L}(v_{0})\,n_{a }\,v_{0}\,\varepsilon\left(\frac{d\sigma_{b}}{d\varepsilon}\right)^{MM}_{e-a} f_{CF}(\overline{E}_{0},\,\overline{\varepsilon},\,\overline{Z}_{eff})\,, \tag{22}\]  
where \(E_{rad}\) is the bremsstrahlung radiation energy and \(n_{a}\) is the atom density. Then, the scaled bremsstrahlung emission rate \(\overline{P}_{\varepsilon}(\,=\,P_{\varepsilon}/P_{0})\) in units of \(P_{0}[\,=\,\,(32/3\pi^{1/2})r_{0}^{2}cn_{e}n_{a}]\) is represented by:  
\[