Fig. 1. Modeled altitude-dependent differential spectra of hydrogen protons( H +) in anomalous cosmic rays( ACRs), showing flux variation with kinetic energy and altitude in Earth ' s stratosphere.
This model is based on initial spectra extracted from references [ 9 ], [ 10 ], providing insight into the distribution and penetration of ACR protons into the stratosphere.
Figure 2 presents a 3D model of the energy spectra for ACR helium nuclei He ++. The x-, y-, and z-axes correspond to those in Fig. 1. The data in Fig. 2 are related to Table 1 [ 7 ], which shows eventsubtracted spectra for helium observed by two telescopes, LET1( red) and HET B( blue), from the ISOIS instrument onboard the Parker Solar Probe( PSP). Data were taken during the periods 2018.7 – 2019.9 and 2018.7 – 2021.2, respectively. Particles in the 1-3 MeV / n range represent solar cosmic rays( SCRs) or solar energetic particles( SEPs); 3-60 MeV / n are ACRs, and above 60 MeV / n are Galactic Cosmic Rays( GCRs). GCRs at 1 AU were simulated using the HelMod-4( Heliospheric Modulation Model, version 5.1) online calculator( www. helmod. org).
The initial MC ACR helium spectrum before entering the atmosphere is described by the mathematical expression( 15), modified by the particle ' s path during penetration toward Earth. The spectrum shapes at heights above 40 km are shown in Fig. 2.
( 16) D( E) = 3,34 × 10-7 × E 2 + 9,19 × 10-9 × E 3 + 1,49 × 10-3 × e-E.
This MC ACR helium spectrum is calculated based on numerical values from Table 1, which is derived from observations by the LET1 and HET B telescopes onboard the PSP [ 7 ].
Figure 3 illustrates modeled altitude-dependent differential spectra of multi-charged oxygen nuclei in ACRs, showing flux variation with kinetic energy and altitude in Earth ' s stratosphere. This figure provides a theoretical visualization of the differential spectra based on computational modeling, rather than experimental data. The x-axis signifies the kinetic energy of ACRs( in MeV / n), while the y- axis represents altitudinal levels in the stratosphere( from 40 km to 50 km). The z-axis measures the differential flux, indicating the density of cosmic ray particles at each energy level. This model is based on initial spectra for different charge states of oxygen nuclei presented in reference [ 12 ].
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