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$\begin{figure} \par\includegraphics[width=13.5cm]{3853.f1}\end{figure}$ Figure 1: Relative differences between the two formulae (6) and (8) deriving the power emitted by one particle. The energy is scaled to the line peak energy. The harmonics 1 and 5 are shown at three particle energies $\beta = 0.1, 0.5, 0.9$ corresponding to the energy ranges explored in this work.
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In the text

$\begin{figure} \par\includegraphics[width=\columnwidth]{3853.f2}\end{figure}$ Figure 2: Relative differences between Eq. (6), the power emitted by one particle integrated over the solid angle and Eq. (4), the power radiated by one particle in the non-relativistic regime. Both emissivities have been calculated at a pitch-angle $\theta _{\rm p} = \pi /2$ . The plot is provided for the ten first harmonics and for two different particle energies $\beta =10^{-2} \ (\rm {triangles}) , \ \rm {and} \ 0.1 \ (\rm {crosses})$ . The emissivities have been normalised to the harmonic one. For scaling reasons, the errors have been divided by 50 for the $\beta = 0.1$ case.
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In the text

$\begin{figure} \par\includegraphics[width=\columnwidth]{3853.f3_1}\vspace*{3mm} ... ...2}\vspace*{3mm} \includegraphics[width=\columnwidth]{3853.f3_3}\par\end{figure}$ Figure 3: Thermal absorption coefficients (in cm^-1) at T = 5, 15, 25 keV. The photon angles are $\theta = 80\hbox {$^\circ $ }$ three dotted-dashed line, $60\hbox {$^\circ $ }$ long-dashed line, $30\hbox {$^\circ $ }$ dot-dashed line. The results can be compared with C89. The calculations have been performed for a magnetic field $B = 10 \ \rm {MG}$ .
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In the text

$\begin{figure} \par\includegraphics[width=7.5cm]{3853.f4}\end{figure}$ Figure 4: Relative differences between the ratio of the calculated emission and absorption coefficients $j_{\nu }/\alpha _{\nu }$ and $B_{\nu }(T)$ at non-relativistic temperatures $T = 1 \ \rm {(triangles)}, 10 \ \rm {(squares)}, 50 \ \rm {(stars)}$ keV, in the frequency range 1-100 $\nu /\nu _{\rm b}$ .
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In the text

$\begin{figure} \par\includegraphics[width=\columnwidth,clip]{3853.f5_1}\par\includegraphics[width=\columnwidth,clip]{3853.f5_2}\end{figure}$ Figure 5: Results for non-thermal lines at $\beta = 0.1$ . The first upper panel shows the spectra produced using the fitting functions (solid line). The flux obtained from numerical integration, indistinguishable at this scale, has been skipped for clarity. The ratio of the fitting function to the numerical results is plotted in the lower panel. The second panel shows the fits and the ratio for the absorption line. The emission and the absorption coefficients are normalised to $(2 \pi n_{\rm e} e B^3)/ (m_{\rm e} c^2)$ and $3/8 \ \pi \ N_{\rm e} \ (\sigma _{\rm T}/\alpha _{\rm f}) \ (B_{\rm cr}/B)$ respectively ( $\alpha _{\rm f} = 1/137$ is the fine structure constant and $B_{\rm cr}\sim 4.4 \times 10^{13} \ \rm {Gauss}$ is the critical magnetic field). The same normalisation is adopted for all figures. The formula (8) of GHS98 and (2.23) of GS91 in the non-relativistic limit has been used to plot the emission and absorption coefficients with a dotted line.
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In the text

$\begin{figure} \par\includegraphics[width=\columnwidth,clip]{3853.f6_1} \includegraphics[width=\columnwidth,clip]{3853.f6_2}\end{figure}$ Figure 6: Same as in Fig. 5 but for $\beta = 0.3$ . The formula (2.16) of GS91 in the relativistic limit for the emission coefficient is plotted in dotted line in the upper panel. The relativistic formula (2.17) of GS91 in the relativistic limit for the absorption cross-section has been used to derive the absorption coefficient plotted in dotted line in the lower panel.
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In the text

$\begin{figure} \par\includegraphics[width=\columnwidth,clip]{3853.f7_1} \includegraphics[width=\columnwidth,clip]{3853.f7_2}\end{figure}$ Figure 7: Same as in Fig. 6 but for $\beta = 0.7$ .
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In the text

$\begin{figure} \par\includegraphics[width=\columnwidth,clip]{3853.f8_1}\par\includegraphics[width=\columnwidth,clip]{3853.f8_2}\end{figure}$ Figure 8: Same as in Fig. 6 but for $\beta = 0.9$ .
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In the text

$\begin{figure} \par\includegraphics[width=\columnwidth,clip]{3853.f9_1}\par\incl... ...]{3853.f9_2}\par\includegraphics[width=\columnwidth,clip]{3853.f9_3}\end{figure}$ Figure 9: From top to bottom: Thermal synchrotron emission coefficients for $kT_{\rm e}=5.36\times 10^{-2}$ , 0.153 and 0.511 keV. Upper panels: the solid curves show the approximation using the fitting functions described in Sect. 6. The results from numerical integration are indiscernable. The dotted curves show the results from the formula of WZ00. Lower panels: ratio of the fitting functions to the numerical results.
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In the text

$\begin{figure} \par\includegraphics[width=\columnwidth,clip]{3853.f10_1}\par\inc... ...3853.f10_2}\par\includegraphics[width=\columnwidth,clip]{3853.f10_3}\end{figure}$ Figure 10: Thermal synchrotron emission coefficient for $kT_{\rm e}=1.53$ , 5.11 and 15.3 keV from top to bottom respectively. See caption of Fig. 9.
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In the text

$\begin{figure} \par\includegraphics[width=\columnwidth,clip]{3853.f11_1}\par\inc... ...3853.f11_2}\par\includegraphics[width=\columnwidth,clip]{3853.f11_3}\end{figure}$ Figure 11: Thermal synchrotron emission coefficient for $kT_{\rm e}=51.1$ , 153 and 511 keV from top to bottom respectively. See caption of Fig. 9.
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In the text

$\begin{figure} \par\includegraphics[width=13.5cm]{3853.f1}\end{figure}$	Figure 1: Relative differences between the two formulae (6) and (8) deriving the power emitted by one particle. The energy is scaled to the line peak energy. The harmonics 1 and 5 are shown at three particle energies $\beta = 0.1, 0.5, 0.9$ corresponding to the energy ranges explored in this work.
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$\begin{figure} \par\includegraphics[width=\columnwidth]{3853.f3_1}\vspace{3mm} ... ...2}\vspace{3mm} \includegraphics[width=\columnwidth]{3853.f3_3}\par\end{figure}$	Figure 3: Thermal absorption coefficients (in cm^-1) at T = 5, 15, 25 keV. The photon angles are $\theta = 80\hbox {$^\circ $ }$ three dotted-dashed line, $60\hbox {$^\circ $ }$ long-dashed line, $30\hbox {$^\circ $ }$ dot-dashed line. The results can be compared with C89. The calculations have been performed for a magnetic field $B = 10 \ \rm {MG}$ .
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$\begin{figure} \par\includegraphics[width=7.5cm]{3853.f4}\end{figure}$	Figure 4: Relative differences between the ratio of the calculated emission and absorption coefficients $j_{\nu }/\alpha _{\nu }$ and $B_{\nu }(T)$ at non-relativistic temperatures $T = 1 \ \rm {(triangles)}, 10 \ \rm {(squares)}, 50 \ \rm {(stars)}$ keV, in the frequency range 1-100 $\nu /\nu _{\rm b}$ .
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$\begin{figure} \par\includegraphics[width=\columnwidth,clip]{3853.f6_1} \includegraphics[width=\columnwidth,clip]{3853.f6_2}\end{figure}$	Figure 6: Same as in Fig. 5 but for $\beta = 0.3$ . The formula (2.16) of GS91 in the relativistic limit for the emission coefficient is plotted in dotted line in the upper panel. The relativistic formula (2.17) of GS91 in the relativistic limit for the absorption cross-section has been used to derive the absorption coefficient plotted in dotted line in the lower panel.
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$\begin{figure} \par\includegraphics[width=\columnwidth,clip]{3853.f7_1} \includegraphics[width=\columnwidth,clip]{3853.f7_2}\end{figure}$	Figure 7: Same as in Fig. 6 but for $\beta = 0.7$ .
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$\begin{figure} \par\includegraphics[width=\columnwidth,clip]{3853.f8_1}\par\includegraphics[width=\columnwidth,clip]{3853.f8_2}\end{figure}$	Figure 8: Same as in Fig. 6 but for $\beta = 0.9$ .
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$\begin{figure} \par\includegraphics[width=\columnwidth,clip]{3853.f10_1}\par\inc... ...3853.f10_2}\par\includegraphics[width=\columnwidth,clip]{3853.f10_3}\end{figure}$	Figure 10: Thermal synchrotron emission coefficient for $kT_{\rm e}=1.53$ , 5.11 and 15.3 keV from top to bottom respectively. See caption of Fig. 9.
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$\begin{figure} \par\includegraphics[width=\columnwidth,clip]{3853.f11_1}\par\inc... ...3853.f11_2}\par\includegraphics[width=\columnwidth,clip]{3853.f11_3}\end{figure}$	Figure 11: Thermal synchrotron emission coefficient for $kT_{\rm e}=51.1$ , 153 and 511 keV from top to bottom respectively. See caption of Fig. 9.
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