Features of the Generation of Ultralow-Frequency Electromagnetic Waves in the Earth’s Magnetosphere with Consideration of the Final Plasma Pressure of Hot Particles

Abstract

An analysis of the influence of the plasma pressure of hot anisotropic protons of the Earth’s radiation belt β on the development of cyclotron instability of ULF waves is presented. It is shown that the growth rate γ decreases significantly upon an increase in the Earth’s radiation belt β and that the generation of ULF waves may stop when the values of its damping are reached. The generation of ULF waves requires small values of the Earth’s radiation belt β, which are characteristic of low magnetic activity. This makes it possible to explain the observed fact that low magnetic activity is the most favorable for the appearance of Pc1 pulsations on the Earth’s surface. These features of the generation of ULF waves in the Earth’s magnetosphere, with consideration of the influence of the finite values of the plasma pressure of hot particles in the Earth’s radiation belt β, were not examined in previous works on a similar topic.

APPENDIX

The formulas given here for reference, though simple, can lead to confusion due to their number and interchangeability.

$$\begin{gathered} c_{{\text{A}}}^{2} = {{B_{0}^{2}} \mathord{\left/ {\vphantom {{B_{0}^{2}} {4\pi {{n}_{0}}m,}}} \right. \kern-0em} {4\pi {{n}_{0}}m,}}\,\,\,\,{{v}^{2}} = {{2T} \mathord{\left/ {\vphantom {{2T} {m,}}} \right. \kern-0em} {m,}}\,\,\,\,v_{\parallel }^{2} = {{2{{T}_{\parallel }}} \mathord{\left/ {\vphantom {{2{{T}_{\parallel }}} {m,}}} \right. \kern-0em} {m,}} \\ v_{ \bot }^{2} = {{2{{T}_{ \bot }}} \mathord{\left/ {\vphantom {{2{{T}_{ \bot }}} {m,}}} \right. \kern-0em} {m,}} \\ \beta = {{({{{{n}_{h}}} \mathord{\left/ {\vphantom {{{{n}_{h}}} {{{n}_{0}}}}} \right. \kern-0em} {{{n}_{0}}}}){{v}^{2}}} \mathord{\left/ {\vphantom {{({{{{n}_{h}}} \mathord{\left/ {\vphantom {{{{n}_{h}}} {{{n}_{0}}}}} \right. \kern-0em} {{{n}_{0}}}}){{v}^{2}}} {c_{{\text{A}}}^{2},\,}}} \right. \kern-0em} {c_{{\text{A}}}^{2},\,}}\,\,\,{{\beta }_{ \bot }} = {{({{{{n}_{h}}} \mathord{\left/ {\vphantom {{{{n}_{h}}} {{{n}_{0}}}}} \right. \kern-0em} {{{n}_{0}}}})v_{ \bot }^{2}} \mathord{\left/ {\vphantom {{({{{{n}_{h}}} \mathord{\left/ {\vphantom {{{{n}_{h}}} {{{n}_{0}}}}} \right. \kern-0em} {{{n}_{0}}}})v_{ \bot }^{2}} {c_{{\text{A}}}^{2},}}} \right. \kern-0em} {c_{{\text{A}}}^{2},}} \\ {{\beta }_{\parallel }} = {{({{{{n}_{h}}} \mathord{\left/ {\vphantom {{{{n}_{h}}} {{{n}_{0}}}}} \right. \kern-0em} {{{n}_{0}}}})v_{\parallel }^{2}} \mathord{\left/ {\vphantom {{({{{{n}_{h}}} \mathord{\left/ {\vphantom {{{{n}_{h}}} {{{n}_{0}}}}} \right. \kern-0em} {{{n}_{0}}}})v_{\parallel }^{2}} {c_{{\text{A}}}^{2}}}} \right. \kern-0em} {c_{{\text{A}}}^{2}}}, \\ {{c_{{\text{A}}}^{2}} \mathord{\left/ {\vphantom {{c_{{\text{A}}}^{2}} {{{v}^{2}}}}} \right. \kern-0em} {{{v}^{2}}}} = {{({{{{n}_{h}}} \mathord{\left/ {\vphantom {{{{n}_{h}}} {{{n}_{0}}}}} \right. \kern-0em} {{{n}_{0}}}})} \mathord{\left/ {\vphantom {{({{{{n}_{h}}} \mathord{\left/ {\vphantom {{{{n}_{h}}} {{{n}_{0}}}}} \right. \kern-0em} {{{n}_{0}}}})} {\beta ,}}} \right. \kern-0em} {\beta ,}}\,\,\,\,{{c_{{\text{A}}}^{2}} \mathord{\left/ {\vphantom {{c_{{\text{A}}}^{2}} {v_{ \bot }^{2}}}} \right. \kern-0em} {v_{ \bot }^{2}}} = {{({{{{n}_{h}}} \mathord{\left/ {\vphantom {{{{n}_{h}}} {{{n}_{0}}}}} \right. \kern-0em} {{{n}_{0}}}})} \mathord{\left/ {\vphantom {{({{{{n}_{h}}} \mathord{\left/ {\vphantom {{{{n}_{h}}} {{{n}_{0}}}}} \right. \kern-0em} {{{n}_{0}}}})} {{{\beta }_{ \bot }},}}} \right. \kern-0em} {{{\beta }_{ \bot }},}} \\ {{c_{{\text{A}}}^{2}} \mathord{\left/ {\vphantom {{c_{{\text{A}}}^{2}} {v_{\parallel }^{2}}}} \right. \kern-0em} {v_{\parallel }^{2}}} = ({{{{{{n}_{h}}} \mathord{\left/ {\vphantom {{{{n}_{h}}} {{{n}_{0}}}}} \right. \kern-0em} {{{n}_{0}}}})} \mathord{\left/ {\vphantom {{{{{{n}_{h}}} \mathord{\left/ {\vphantom {{{{n}_{h}}} {{{n}_{0}}}}} \right. \kern-0em} {{{n}_{0}}}})} {{{\beta }_{\parallel }},}}} \right. \kern-0em} {{{\beta }_{\parallel }},}} \\ F = 1 + \frac{1}{2}{{\beta }_{ \bot }}\frac{{(1 - {{{{T}_{\parallel }}} \mathord{\left/ {\vphantom {{{{T}_{\parallel }}} {{{T}_{ \bot }}}}} \right. \kern-0em} {{{T}_{ \bot }}}} - x)}}{{{{{(1 - x)}}^{3}}}} \equiv 1 + \frac{1}{2}{{\beta }_{ \bot }} \\ \times \,\,\frac{{(1 - {{{{\beta }_{\parallel }}} \mathord{\left/ {\vphantom {{{{\beta }_{\parallel }}} {{{\beta }_{ \bot }}}}} \right. \kern-0em} {{{\beta }_{ \bot }}}} - x)}}{{{{{(1 - x)}}^{3}}}}. \\ \end{gathered} $$

Using the dispersion equation for ion-cyclotron waves with allowance for the hot component ${{n}_{h}}$ (3) and substituting ${{k}_{\parallel }}$ in ${{\alpha }^{2}},$ we obtain

$$\begin{gathered} {{\alpha }^{2}} = \frac{{{{{(1 - x)}}^{3}}}}{{{{x}^{2}}}}\frac{{c_{{\text{A}}}^{2}}}{{v_{\parallel }^{2}}}F = \frac{{{{{(1 - x)}}^{3}}}}{{{{x}^{2}}}}\frac{{{{n}_{h}}}}{{{{n}_{0}}}}\frac{F}{{{{\beta }_{\parallel }}}}, \\ {{\alpha }^{2}} = \frac{{{{n}_{h}}}}{{{{n}_{0}}}}\frac{{{{{(1 - x)}}^{3}}}}{{{{x}^{2}}{{\beta }_{\parallel }}}}\left[ {1 + \frac{1}{2}{{\beta }_{ \bot }}\frac{{(1 - {{{{\beta }_{\parallel }}} \mathord{\left/ {\vphantom {{{{\beta }_{\parallel }}} {{{\beta }_{ \bot }}}}} \right. \kern-0em} {{{\beta }_{ \bot }}}} - x)}}{{{{{(1 - x)}}^{3}}}}} \right] \\ = \frac{{{{n}_{h}}}}{{{{n}_{0}}}}\left[ {\frac{{{{{(1 - x)}}^{3}}}}{{{{x}^{2}}{{\beta }_{\parallel }}}} + \frac{1}{{2{{x}^{2}}}}\frac{{{{\beta }_{ \bot }}}}{{{{\beta }_{\parallel }}}}\left( { - \frac{{{{\beta }_{\parallel }}}}{{{{\beta }_{ \bot }}}} + 1 - x} \right)} \right] \\ = \frac{{{{n}_{h}}}}{{{{n}_{0}}}}\left[ {\frac{{{{{(1 - x)}}^{3}}}}{{{{x}^{2}}{{\beta }_{\parallel }}}} + \frac{1}{{2{{x}^{2}}}}\left( {\frac{{{{\beta }_{ \bot }}}}{{{{\beta }_{\parallel }}}} - 1 - x} \right)} \right] \\ = _{{}}^{{}}\frac{{{{n}_{h}}}}{{{{n}_{0}}}}\frac{1}{{{{x}^{2}}{{\beta }_{\parallel }}}}\left[ {{{{(1 - x)}}^{3}} + \frac{1}{2}({{\beta }_{ \bot }} - {{\beta }_{\parallel }} - {{\beta }_{\parallel }}x)} \right], \\ {{\alpha }^{2}} = \frac{{{{n}_{h}}}}{{{{n}_{0}}}}\frac{1}{{{{x}^{2}}}}\left[ {\frac{{{{{(1 - x)}}^{3}}}}{{{{\beta }_{\parallel }}}} + \frac{1}{2}\left( {\frac{{{{\beta }_{ \bot }}}}{{{{\beta }_{\parallel }}}} - 1 - x} \right)} \right] \\ = \frac{{{{n}_{h}}}}{{{{n}_{0}}}}\frac{1}{{{{x}^{2}}}}\left[ {\frac{{{{{(1 - x)}}^{3}}B_{0}^{2}}}{{8\pi {{n}_{h}}{{T}_{\parallel }}}} + \frac{1}{2}\left( {\frac{{{{\beta }_{ \bot }}}}{{{{\beta }_{\parallel }}}} - 1 - x} \right)} \right] \\ = \frac{{{{n}_{h}}}}{{{{n}_{0}}}}\frac{1}{{{{x}^{2}}}}\left[ {\frac{{{{{(1 - x)}}^{3}}B_{0}^{2}}}{{4\pi {{n}_{0}}\frac{{{{n}_{h}}}}{{{{n}_{0}}}}mv_{\parallel }^{2}}} + \frac{1}{2}(A - x)} \right] \\ = \frac{{{{n}_{h}}}}{{{{n}_{0}}}}\frac{1}{{{{x}^{2}}}}\left[ {\frac{{{{{(1 - x)}}^{3}}c_{{\text{A}}}^{2}}}{{\frac{{{{n}_{h}}}}{{{{n}_{0}}}}v_{\parallel }^{2}}} + \frac{1}{2}(A - x)} \right], \\ \end{gathered} $$

(6)

$$\begin{gathered} \beta = \frac{{{{n}_{h}}}}{{{{n}_{0}}}}\frac{{{{v}^{2}}}}{{c_{{\text{A}}}^{2}}} = \frac{{{{n}_{h}}}}{{{{n}_{0}}}}\frac{{v_{ \bot }^{2} + v_{\parallel }^{2}}}{{c_{{\text{A}}}^{2}}} = \frac{{{{n}_{h}}}}{{{{n}_{0}}}}\frac{{v_{\parallel }^{2}}}{{c_{{\text{A}}}^{2}}}\left( {\frac{{{{T}_{ \bot }}}}{{{{T}_{\parallel }}}} + 1} \right) \\ = \frac{{{{n}_{h}}}}{{{{n}_{0}}}}\frac{{v_{\parallel }^{2}}}{{c_{{\text{A}}}^{2}}}(A + 2). \\ \end{gathered} $$

(7)

where $\frac{{c_{A}^{2}}}{{v_{\parallel }^{2}}} = \frac{{{{n}_{h}}}}{{{{n}_{0}}}}\frac{{(A + 2)}}{\beta }.$

Lastly, we get the expressions ${{\alpha }^{2}}$ across $\beta$ and through ${{c_{{\text{A}}}^{2}} \mathord{\left/ {\vphantom {{c_{{\text{A}}}^{2}} {v_{\parallel }^{2}}}} \right. \kern-0em} {v_{\parallel }^{2}}}{\text{:}}$

$${{\alpha }^{2}} = \frac{{{{n}_{h}}}}{{{{n}_{0}}}}\frac{1}{{{{x}^{2}}}}\left[ {\frac{{{{{(1 - x)}}^{3}}(A + 2)}}{\beta } + \frac{1}{2}(A - x)} \right],$$

(8)

$${{\alpha }^{2}} = \frac{{{{{(1 - x)}}^{3}}}}{{{{x}^{2}}}}\frac{{c_{{\text{A}}}^{2}}}{{v_{\parallel }^{2}}} + \frac{{{{n}_{h}}}}{{{{n}_{0}}}}\frac{1}{{2{{x}^{2}}}}(A - x).$$

(9)