At present, the HelMod modulated spectra framework does not include ultra-heavy galactic cosmic ray ions (UHGCRs, i.e., GCRs with Z>28) due to their expected negligible contribution to the radiation risk evaluation  (e.g., SEE rate) in space, particularly when compared to the impact of iron  (see Appendix  A in Boschini et al. 2024).

UHGCRs are GCRs that are heavier than nickel, i.e. with a nuclear charge larger than 28. The origin of UHGCRs is a long-standing and important question in the fields of astronomy and astrophysics (see recent reviews inWiedenbeck et al., 2007; Lingenfelter, 2019; Tatischeff et al., 2021, and references therein). Due to their rarity, measuring their abundance is extremely challenging, and their spectral shape as a function of energy is still unknown. Data mainly consists of ratios of elements or groups of elements in an unspecified energy band. This is due to the limitations of the detectors used to evaluate the abundance, which is typically measured in a range of energies around a few GeV/nucleon.

Fig. 1. Relative abundances of GCRs from iron (Z=26) to berkelium (Z=97). Open circles represent GCR relative abundances taken from Table 2 of the ISO-15390 model (ISO, 2004). SPENVIS spectra at 16.2 GeV/nucleon are also shown as open stars. These values have been compared with the relative abundance measurements of UHGCRs from various experiments collected in CRDB (Maurin et al., 2020). Different colors refer to abundances measured with respect to the abundance of the ion or the group of ions reported in the respective color legend label; multiplicative factors are used to rescale these values to the Fe abundance. [Figure from Boschini et al. (2024)]

Abundance measurements of UHGCR are available in the Cosmic-Ray DataBase (CRDB, Maurin et al., 2014, 2020). Figure 1 shows their values normalized to the iron abundance plotted against the atomic number Z. The most recent measurements in the Z=30-40 range are from the SuperTIGER experiment (Rauch et al., 2009; Murphy et al., 2016). However, experiments in the high charge range date back mainly to the late 70s to late 90s of the last century, such as HEAO3 (Binns et al., 1981, 1982, 1983, 1985, 1989), ARIEL6 (Fowler et al., 1987), UHCRE (Domingo et al., 1996; Dutta et al., 2003; Donnelly et al., 2012), and Trek (Westphal et al., 1998; Weaver & Westphal, 2002). Figure 1 also shows abundances relative to iron for "non-modulated" spectra, listed in Table~2 of the ISO-15390 report (ISO, 2004). This model is recognized by the ECSS guidelines (e.g., see ECSS, 2020) as the standard model to assess the space radiation environment. The Figure also reports ratios of UHGCR fluxes and the iron flux, obtained by the ISO-15390 model implemented in ESA’s SPace ENVironment Information System (SPENVIS, https://www.spenvis.oma.be/), at 16.2 GeV/nucleon —at this energy the solar modulation effects can be considered negligible.

Fig. 2. Relative abundances (black-filled squares) and relative abundances multiplied by Z2 (open circles) of GCRs at 16.2 GeV/nucleon, normalized to the iron abundance, as a function of the atomic number Z. The plot ranges from copper (Z=29) to uranium (Z=92). [Figure from Boschini et al. (2024)]

Figure 2 displays the ratio of GCR abundances from copper (Z=29) to uranium (Z=92) and the iron flux at 16.2 GeV/nucleon, obtained from SPENVIS, as a function of the atomic number Z. Additionally, the relative abundances are also shown multiplied by Z², considering that the collision energy-loss in matter at sufficiently high energies is proportional to Z². It is observed from the plot that the total amount of energy loss experienced by UHGCRs in the matter is less than 0.5% of the iron energy loss. Therefore, the contribution of UHGCRs to the radiation hazard in space, such as the SEE rate, can be considered negligible.

Bibliography

Boschini et al (2024), Fast and accurate evaluation of deep-space galactic cosmic ray fluxes with HelMod-4/CUDA, M. J. Boschini, G. Cavallotto, S. Della Torre, M. Gervasi, G. La Vacca, P. G. Rancoita, M. Tacconi, Advances in Space Research, April 2024, In Press

Binns, W. R., Fickle, R. K., Waddington, C. J. et al. (1981) Cosmic-ray abundances of elements with atomic number 26 less than or equal to 40 measured on HEAO 3. ApJ, 247, L115–L118. doi:10.1086/183602.

Binns, W. R., Israel, M. H., Klarmann, J. et al. (1982). The abundance of the actinides in the cosmic radiation as measured on HEAO 3. ApJ, 261, L117–L120. doi:10.1086/183899

Binns, W. R., Israel, M. H., Klarmann, J. et al. (1983). Cosmic-ray abundances of Sn, Te, Xe, and BA nuclei measured on HEAO 3. ApJ, 267, L93–L96. doi:10.1086/184010.

Binns, W. R., Israel, M. H., Brewster, N. R. et al. (1985). Lead, platinum, and other heavy elements in the primary cosmic radiation - HEAO 3 results. ApJ, 297, 111–118. doi:10.1086/163508. 

Binns, W. R., Garrard, T. L., Gibner, P. S. et al. (1989). Abundances of Ultra-heavy Elements in the Cosmic Radiation: Results from HEAO 3. ApJ, 346, 997. doi:10.1086/168082.

Domingo, C., Font, J., Baixeras, C. et al. (1996). Source abundances of ultraheavy elements derived from UHCRE measurements. Radiation Measurements, 26(6), 825–832. doi:10.1016/S1350-4487(96)00090-X.

Donnelly, J., Thompson, A., O’Sullivan, D. et al. (2012). Actinide and Ultra-Heavy Abundances in the Local Galactic Cosmic Rays: An Analysis of the Results from the LDEF Ultra-Heavy Cosmic-Ray Experiment. ApJ, 747(1),40. doi:10.1088/0004-637X/747/1/40.

Dutta, A., Batra, V., & Biswas, S. (2003). Abundance of actinides in cosmic radiation. Radiation Measurements, 36(1-6), 287–290. doi:10.1016/S1350-4487(03)00137-9. 

ECSS (2020). Space Environment. Technical Report ECSS-E-ST-10-04C Rev.1 European Cooperation for Space Standardization, ESA. URL: https://ecss.nl/standard/ecss-e-st-10-04c-rev-1-space-environment-15-june-2020/

Fowler, P. H., Walker, R. N. F., Masheder, M. R. W. et al. (1987). Ariel Measurements of the Fluxes of Ultra-heavy Cosmic Rays. ApJ, 314, 739. doi:10.1086/165101

ISO (2004). Space Environment (Natural and Artificial) - Galactic Cosmic Ray Model. Standard International Organization for Standardization Geneva, CH. URL: https://www.iso.org/standard/37095.html.

Lingenfelter, R. E. (2019). The Origin of Cosmic Rays: How Their Composition Defines Their Sources and Sites and the Processes of Their Mixing,Injection, and Acceleration. ApJS, 245(2), 30. doi:10.3847/1538-4365/ab4b58.

Maurin, D., Dembinski, H. P., Gonzalez, J. et al. (2020). Cosmic-Ray DatabaseUpdate: Ultra-High Energy, Ultra-Heavy, and Antinuclei Cosmic-Ray Data(CRDB v4.0). Universe, 6(8), 102. doi:10.3390/universe6080102.

Maurin, D., Melot, F., & Taillet, R. (2014). A database of chargedcosmic rays. A&A, 569, A32. doi:10.1051/0004-6361/201321344.

Murphy, R. P., Sasaki, M., Binns, W. R. et al. (2016). Galactic Cosmic RayOrigins and OB Associations: Evidence from SuperTIGER Observations ofElements 26Fe through 40Zr. ApJ, 831(2), 148. doi:10.3847/0004-637X/831/2/148.

Rauch, B. F., Link, J. T., Lodders, K. et al. (2009). Cosmic Ray origin in OB Associations and Preferential Acceleration of Refractory Elements: Evidence from Abundances of Elements 26Fe through 34Se. ApJ, 697(2), 2083–2088. doi:10.1088/0004-637X/697/2/2083.

Tatischeff, V., Raymond, J. C., Duprat, J. et al. (2021). The origin of Galacticcosmic rays as revealed by their composition. MNRAS, 508(1), 1321–1345.doi:10.1093/mnras/stab2533.

Weaver, B. A., & Westphal, A. J. (2002). Extended Analysis of the Trek Ultra-heavy Collector. ApJ, 569(1), 493–500. doi:10.1086/339265.

Westphal, A. J., Price, P. B., Weaver, B. A. et al. (1998). Evidence againststellar chromospheric origin of Galactic cosmic rays. Nature, 396(6706),50–52. doi:10.1038/23887

Wiedenbeck, M. E., Binns, W. R., Cummings, A. C. et al. (2007). An Overviewof the Origin of Galactic Cosmic Rays as Inferred from Observations ofHeavy Ion Composition and Spectra. Space Sci. Rev., 130(1-4), 415–429.doi:10.1007/s11214-007-9198-y

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GALPROP obtained LIS using modulated spectra from HelMod

The local interstellar spectra (LIS) are input cosmic ray intensities for any modulation models. Fluxes are assumed isotropically distributed at heliosphere boundary, in a steady-state configuration. The LIS is usually computed using numerical model accounting for the propagation in galaxy. The heliosphere modulation is the added using, e.g.,the modulation module available here for on line use or download.

Recently,  Boschini et al 2017, 2018a, 2018b, 2020a, 2020b deduced LIS's for protons, helium nuclei and antiprotons using the most recent experimental results combined with GALPROP model for propagation in galaxy and HelMod for the propagation in heliosphere.

Successive work extend the available sets of LIS's. Boschini et al 2018b, using the same procedure, deduced LIS's  for Carbon- and Oxygen-nuclei, while Boschini et al 2018a deduced LIS for galactic electrons. 

HelMod and GALPROP were combined to provide a single framework and run to reproduce a comprehensive set of observations of CR species collected in different time periods, from 1997 up to 2015.

The numerical table of computed LIS is available as supplementary material of  Boschini et al 2017, Boschini et al. 2018a,  and  Boschini et al 2020b.

Antiproton LIS

Boschini et al 2017 provides analytical fits to the derived LIS. The fit to the antiproton LIS provides an accuracy of 2-3% for 3 GV \(<R<1000\) GV and 10% for 1.5 GV \(<R<3\) GV:

\begin{equation}
F(R)\times R^{2.7} = 1.34-4.99\times 10^{-4}R+\frac{44.3}{12.6+R}-\frac{168}{31.1+R^{2}} +\frac{13600}{22200+R^{2}}, \quad R>1.5\ {\rm GV},
\end{equation}
while the average accuracy of AMS-02 data is about 10%-20%.

Electron LIS

In Boschini et al 2018a., it was provided a parameterization \(F(T)\) of the electron GALPROP LIS from 2 MeV up to 90 GeV as a function of kinetic energy in GeV:

\begin{align}\label{EQ::ele}
&F(T) = \\
&\left\{
\begin{array}{ll}
\frac{1.181\!\!\times\!\! 10^{11} T^{-12.061}}{1+4.307\!\!\times\!\! 10^{8} T^{-9.269}+3.125\!\!\times\!\! 10^{8} T^{-10.697}} & T<6.88 GeV\,\\ \nonumber
995.598\, T^{-3.505}+4.423\, T^{-2.620}, & T\ge 6.88 GeV, \nonumber
\end{array}
\right.\nonumber
\end{align}

where the units are (m\(^2\) s sr GeV)\(^{-1}\). This fit reproduces the \galprop{} electron LIS with an accuracy better than 5\% for the whole quoted energy range.

The electron LIS that results from the model calculations is in a good agreement with data (See discussion in  Boschini et al 2018a.). Meanwhile, it may harbor an additional electron component from an unknown source of the same nature as that of the excess positrons. If charge-sign symmetry is assumed, i.e. that the electron and positron components coming from an unknown source have identical spectra, then the spectral shape of such an additional electron component can be derived from AMS-02 positron measurements. The spectrum of an additional component, ``the signal,'' \(S(T)\) can be parametrized as a function of kinetic energy as:
\begin{equation}
S(T)= 4.5\times10^{-3}\ T^{-1.53} e^{-\frac{T}{400\ {\rm GeV}}}\ ({\rm m}^2\ {\rm s\ sr\ GeV})^{-1}
\end{equation}
This involves a re-tuning of the electron injection spectrum above the break at 95 GV (See discussion in  Boschini et al 2018a.). This parameterization also takes into account the standard astrophysical background of secondary positrons evaluated to be \(\le\)6\% at 30 GeV.

LIS for Proton and Nuclei up to Z<=28

Boschini et al 2020b provide analytical parameterizations of the LIS for each species, with parameters summarized inTable C1. The presented formulas  parametrize numerical LISs obtained from the join effor of GALPROP and HelMod with an accuracy below 3% from 0.1 GV up to 10⁵ GV and complete the numerical tables.

Bibliography

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By exploiting experimental results, the combined effort of the physicists involved with the Galprop  model for propagation in galaxy and HelMod for the propagation in heliosphere, the local interstellar spectra (LIS) for Galactic Cosmic Rays species  up to Z=28 (Nickel) were derived. These spectra are available and accessible from the current webpage.

Some of the currently available LIS's were derived accounting for AMS-02 data published up to TV rigidity region. The exploitation of AMS-02 data allowed one to approach the procedure with high statistic data of unprecedent accuracy. Currently, the observation data at Earth on cosmic rays species from HEAO3-C2 (from october 1979 to June 1980) and AMS-02 were employed for absolute scale normalization of fluxes (see Sects. 3-3.2 in Boschini et al. 2020).

The GALPROP LIS for all CR species (dashed lines) are compared to the Voyager 1 data (filled circles, Cummings et al 2016). We also show updated Voyager 1 data for H and He (open circles) taken from September 1, 2012 to November 13, 2019. The elements are sorted by approximate amount of primary contribution: first group is mostly primary, second – with significant primary contribution, and third – mostly secondary. 

Galprop-HelMod join effort

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• LIS
• Ions
• GALPROP
• Nuclei

The following text is based on / Extracted from Boschini et al 2019  and regards HelMod version 4 and beyond.

The boundary of the heliosphere, called heliopause (HP), is a contact discontinuity separating the solar cavity -- in which the SW plasma is flowing -- from the interstellar space. It also represents the extreme limit beyond which solar modulation does not affect CR flux. Thus, outside HP the truly pristine LIS of GCR spectra could be observed. After being accelerated in the solar corona (Parker, 1958) SW adiabatically expands in radial direction with supersonic speed. In its journey towards the external regions of the heliosphere SW flow changes its supersonic regime, through the formation of a physical boundary called Termination Shock (TS) which, in practice, separates the inner  part of the heliosphere¹ from the outer region², also known as heliosheath (HS).

The heliosphere boundaries (i.e., TS and HP) were extensively discussed in the literature (e.g., Parker, 1961,1963; Axford, 1972; Holzer, 1989; Zank, 1999,2015  and references therein). Recently, fundamental advancements in the knowledge of the outer heliosphere were achieved by means of both Voyager probes, which provided on-site observations of TS and HP positions, SW plasma and magnetic field properties (e.g., see Richardson et al, 2008; Richardson, 2013,Nasa-Voyager,Burlaga and Ness, 2016,Nasa-OMNIWeb). In Parker's model of the heliosphere (Parker, 1961, 1963), such pieces of information can be exploited for allowing us to estimate the time dependence of both TS and HP positions -- i.e., those currently used in HelMod --, as discussed in Heliospheric boundaries section. For instance, the predicted TS values are in good agreement with those observed: for Voyager 1 (Voyager 2) the detected TS position is 93.8 AU (83.6 AU) and the predicted is 91.8 AU (86.3 AU), i.e. within 3 AU; and, using the HP position observed by Voyager 1, the predicted value is 120.7 AU at the time of the Voyager 2 HP crossing which occurred at 119 AU. Furthermore, the interstellar magnetic field strength measured by Voyager 1 is \((0.48\pm0.04)\)nT (Burlaga e Ness, 2016). In the context of the Parker model, at the TS positions observed by Voyager 1 and Voyager 2, such a magnitude requires that the value of the dimensionless stagnation pressure (Parker, 1963) approaches its maximum value, i.e., the one allowed for a spherical diamagnetic solar cavity (see discussion in Heliospheric boundaries section). It should be remarked that, from the measurements of energetic neutral atoms by IBEX (McComas et al, 2009) and Cassini (Krimigis, 2009), Dialynas et al. (2017) strongly suggested a diamagnetic bubble-like heliosphere³ with negligibly small tail-like features (see also Drake, 2015,Opher et al 2015, 2017).

In HelMod model, so far the inner region of the heliosphere was described as an effective heliosphere (Bobik et al 2012) with a radius -- i.e., an effective TS distance -- of 100AU. Solar modulation was, then, treated by subdividing it in 15 radially equally-spaced regions. Each i-th region traveled by CR particles is characterized by heliospheric parameters evaluated at i-Carrington rotations back-in-time, corresponding to the time needed by SW for reaching it (Bobik et al 2012, Boschini et al 2018). 
The actual dimensions of the heliosphere are accounted for by scaling the position of TS by the time dependent values obtained as described in Heliospheric boundaries section. The HS is included as a single additional region on top of the 15 inner regions with its actual size as calculated in Heliospheric boundaries section. As discussed in Heliospheric boundaries section the TS and HP latitudinal profile in the nose region  was determined taking into account the latitudinal variation of the ISM ram pressure component of the total pressure on the HP surface. In the anti-nose region only the magnetic and plasma pressure were accounted for. As a consequence, an asymmetry in the direction of the ISM flow is introduced (see fig. below). Finally a cylindrical symmetry with respect to the axis along the sun-nose direction was applied to have full three-dimensional heliospheric boundaries.

¹The inner part of the heliosphere -- corresponding to the space region from the Sun up to the TS -- will be indicated as inner heliosphere in the following.
²The outer part of the heliosphere - corresponding to the space region from the TS up to the HP - will be indicated as outer heliosphere in the following.
³Under the condition that there is no interstellar magnetic field, Parker, (1961; 1963)  discussed the case of a steady subsonic interstellar wind leading to a comet-like shape (see, e.g.,  Axford, 1972; Holzer, 1989, Zank 1999, 2015 and references therein).

Bibliography

	Bobik, P., Boella, G., Boschini, M.~J., Consolandi, C., Della Torre, S., Gervasi, M., Grandi, D., Kudela, K., Pensotti, S., Rancoita P.G. and Tacconi, M. (2012). Systematic Investigation of Solar Modulation of Galactic Protons for Solar Cycle 23 Using a Monte Carlo Approach with Particle Drift Effects and Latitudinal Dependence. Astrophys. J., 745:132 .
	Boschini, M.J., Della Torre, S., Gervasi, M., La Vacca, G. and Rancoita, P. G. (2018a) Propagation of Cosmic Rays in Heliosphere: the HelMod Model  Adv. Space Res. 62(10):2859 - 2879, 2018
	Boschini, M.J., Della Torre, S., Gervasi, M., La Vacca, G. and Rancoita, P. G. (2019) The HelMod Model in the Works for Inner and Outer Heliosphere: from AMS to Voyager Probes Observations  Advances in Space Research 64(12):2459 - 2476, 2019.
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	Burlaga, L.F., 1984. Space Sci. Rev. 39, 255–316.
	Burlaga, L.F., Ness, N.F., 2016. Astrophys. J. 829, 134.
	Dialynas, K., et al., 2017. Nature  Astronomy 1, 0115.
	Drake, J.F., Swisdak, M., Opher, M., 2015. Astrophys. J. Letter 808, L44.
	Holzer, T.E., 1989.  Annu. Rev. Astron. Astrophys. 27, 199 – 234.
	Krimigis, S.M., 2009. Science 326, 971.
	McComas, D.J., et al. 2009. Science 326, 959
	NASA-OMNIweb, 2018. online database http://omniweb.gsfc.nasa.gov/form/dx1.html .
	NASA-Voyager, 2018. online database https://voyager.gsfc.nasa.gov/flux.html .
	Opher, M., Drake, J.F., Swisdak, M., Zieger, B., Toth, G., 2017. Astrophys. J. Letter 839, L12
	Opher, M., Drake, J.F., Zieger, B., Gombosi, T.I., 2015. Astrophys. J. Letter 800, L28.
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	Parker, E.N., 1963. Interplanetary dynamical processes.
	Richardson, J.D., Kasper, J.C., Wang, C., Belcher, J.W., Lazarus, A.J., 2008. Nature 454, 63–66
	Richardson, J.D., 2013. Journal of AdvancedResearch 4, 229–233.
	Zank, G., 2015. Annual Review of Astronomy and Astrophysics 53, 449–500.
	Zank, G.P., 1999. Space Sci. Rev. 89, 413–688.

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