Additional formulas, such as equation( 12), describe energy values beyond the upper boundary of interval 2, while equations( 5)-( 9) detail ionization losses for different intervals, both with and without boundary crossings.
This sequence of intervals and energy values expands up to 5 GeV for ACRs, and the bounds are included in the respective calculations.
( 12) E �. ��, � �h� � �E �
�. �� �
���. ���� � h� � 0.34Z � �. �E �� � � 0.15 �. �� �. ��
��
This model takes into consideration energy values above the upper boundary of interval 2, which are characteristic for the spectra presented in the table below( related to the figure).
It is valid for various combinations of evaluated parameters and for other mathematical expressions of the CORSIMA and CORIMIA models, which reflect the full energy range under consideration. The energy values and intervals for ACRs expand up to 5 GeV and beyond, and these bounds are incorporated into the respective calculations [ 7 ].
We now present the initial energy values for the interval boundaries that are critical for calculating the electron production rate by Multiply Charged ACRs( MC ACRs), as described in formula( 10). These values depend on the depth of penetration into the atmosphere. The initial energy for Emin = kT, when located in interval 2 *, is given by:
( 13) E �� �h� � �E �
�. �� �
���. ���� � h� � 0.32Z � �0.15 �. � � �kT� �. � � � 0.34Z � �E �
�. �� � 0.15 �. �� �� �. ��.
The initial energy situated in interval 1 is given by formula( 14):
( 14) E �� �h� � ��kT� �. � � ���� h �� �
�
The initial energy for the boundary Ea, before the spectrum penetrates into the atmosphere, when it is situated in interval 2, is given by formula( 15):
( 15) E �, � �h� � �E �
�. �� �
4. RESULTS
���. ���� �
�. �� h��
Figures 1-3 present the evolution of the spectra for the main ACR species— protons( Fig. 1), helium( Fig. 2), and oxygen( Fig. 3). These spectra are specifically calculated for a geomagnetic latitude of λm = 90. In the lower portion of the profiles, the proton and helium spectra are primarily influenced by atmospheric cut-offs. For protons, the spectrum drops to zero below 40 km( Fig. 1), while the helium spectra decrease below 40 km due to atmospheric cut-off effects( Fig. 2). Notably, the helium and oxygen spectra display the highest intensities among the ACR species.
Figure 1 illustrates a 3D model of the energy spectra for hydrogen( protons) H + H^ + H + within anomalous cosmic rays. The x-axis represents kinetic energy in Mega-electronvolts per nucleon( MeV / n), capturing the range of energies for ACR protons. The y-axis represents the altitude range from 40 km to 50 km in Earth ' s atmosphere, while the z-axis shows the differential flux, quantifying the intensity of protons at each energy level. The surface plot, marked with a color gradient, visualizes the expected intensities of proton flux at various energies and altitudes.
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