The origin of high energy cosmic rays (CR) remains uncertain, nevertheless supernova remnants are being promoted as the most promising candidates for providing the CR flux up to PeV energies measured at Earth . Due to shock front acceleration in the surrounding dust cloud, charged particles are able to reach energies up to that order. Here a sudden break in the cosmic ray spectrum occurs, referred to as the knee.

The experiments H.E.S.S., MAGIC, VERITAS and MILAGRO have proven gamma rays from SNRs up to energies as high as ∼ 100 TeV. By additionally using X-ray and radio data of those sources it can be identified whether hadronic or leptonic processes caused the very high energy (VHE) gamma rays. This differentiation has been conducted in .

This approach allows the derivation of jp(E), which gives information about a potential population of CR protons inside the SNRs. Since CR become deflected due to their electric charge by magnetic fields, and energy loss processes happen in interactions with Galactic gas clouds, it is impossible to directly extract information of those CRs regarding original direction and energy release by their SNR of origin. Hence, this raises the importance of the indirectly determined parameter jp(E).

On the contrary, gamma rays travel through the Galaxy nearly without being deflected, and their energy budget remains untouched by the Galactic environment.

In terms of propagating CRs through the Galactic disk to Earth we make use of the publicly available GALPROP code . By applying certain modifications to this code we are able to inject CR nuclei of SNRs with spectra as derived in .

However, the CR flux up to the knee will be underestimated when considering only today's observable SNRs as possible sources. This becomes clear, when comparing the average time of CRs travelling through the Galaxy to be about 106–107 years to the lifetime of one SNR, which is about 104 years, and thus significantly less.

We therefore take the spectra of all observable SNRs as a representative subsample of SNR spectra and luminosities. The time averaged sample of SNRs is obtained by simulating multiple SNRs at random positions in the Galactic disk, following the SNR distribution of , which has an increased density near the Galactic center. Although this is the distribution we use in our analysis, there are other distribution that are considered to be more realistic, see e.g. .

The primary objective of this proceeding is to describe the actual status of our analysis with special emphasis put on the overall normalization . We present an simulation that focuses on reproducing the slope of the proton flux measured at Earth. We do not present any optimization of model parameters, and all shown results are preliminarily ranked.

In the electromagnetic spectrum of 24 SNRs has been analyzed and the interaction type for generating VHE gamma rays, either leptonic or hadronic, has been identified. Sources with a very hard gamma ray spectra, e.g. 100 TeV and above, are necessary to explain the knee in the CR spectrum. Young SNRs provide the best candidates. It is shown that 21 SNRs are causing VHE gamma rays through π0-decay, hence characterized as a hadronic process. For further details see Table 3 in .

In our analysis we choose these 21 SNRs as the source spectra jp(E) jp(E)=apE2+2Empc2E02+2E0mpc2-αp2E+mpc2E2+2Empc2×tanh⁡EEmin,pexp⁡-EEmax,p, with E as the kinetic energy, ap as the normalization in units of MeV-1, the proton mass mp, the normalization energy E0 = 1 TeV as well as minimum and maximum energy Emin and Emax, respectively. The minimum energy is fixed to a value of Emin = 10 MeV while the maximum energy depends on the actual SNR. See Table 3 in for detailed information on that energy.

Equation () represents a power law in momentum space with low- and high energy cut-offs. Since we analyze cosmic ray energies in the range from ∼ GeV to ∼ PeV the low energy cutoff is negligible.

The Greens catalogue contains a total number of 274 SNRs, whereas only about ten percent have been observed in gamma rays. Hence, by only taking those SNRs into account, we expect a too low prediction of the CR flux.

The source spectra jp(E) as parametrized in Eq. () is propagated through the Galactic disk with the open source C++ tool GALPROP . In order to perform this, certain modifications have to be applied to the code. Secondly, a function has been implemented that selects a random spatial position of the SNRs, following the SNR distribution as presented in .

In this section we show our preliminary results. We have used N = 100 sets of 21 SNRs distributed randomly in the Galactic grid. Throughout this proceeding the notation < … > stands for the average of N = 100 simulations so that the simulation is comparable to experimental observations. Moreover, we refer to the kinetic energy as used in above equations with the letter T.

Generally we have estimated the lifetime of one SNR, where its CR radiation is relevant, to TSNR = 104 years, and in each run the SNRs are all lit up all at once. Figure  shows a selection of the simulated N = 100 GALPROP runs with 21 SNRs each, along with its average < dF/dT >. Furthermore, it has been normalized to CREAM data at 3.25 TeV in order to better compare its slope to the data. Also, the solar potential has a value of 550 MV. In this plot, the yellow band represents the statistic variance over all simulations. That is, the mean value of all runs has been allowed to have an 1σ uncertainty.

As shown in Fig.  the spectral behavior is described very well for energies higher than E = 1 TeV. PAMELA data, on the contrary shows a different slope than predicted by our simulation.

Since we derive the CR spectrum from gamma ray measurements we expect a best match of data in the range of 1–100 GeV for Fermi and 0.1–10 TeV for IACTs. Our assumption of η = 10 % of CR energy is converted in photons, be expect reasonably good agreement in a range of 10 GeV to 100 TeV. Extrapolation effects do not cause this discrepancy.

The Boron over Carbon ratio serves as a cross check and is shown in Fig. . It can be seen that the simulation with heavier nuclei includes reproduces AMS-01 data well. More importantly, this cross check proves that the number of SNRs indeed provides a high enough CR flux to match the measured data. The spectral behavior has here a minor contribution.

Especially Fig.  proves very good agreement with experimental data. This benchmark test is a first supporting argument for our simulation method. However, the discrepancies to PAMELA data of our averaged flux are evident, and thus, possible reasons for it shall be discussed in this section.

At first, statistics may cause an issue in our analysis. We can find an average of the whole set of simulations, < dF/dT >, but the variance is not to be determined. This measure is important to decide whether or not statistic parameters cause the failure at lower energies. Of course, the variance of CR observables may be presented here, but its statistical interpretation is limited. The variance would only represent the spread of CR observables, which is induced by randomly selected positions of the 21 SNRs, hence, decreasing behavior is expected when more SNRs are considered.

By putting more emphasis on time during the simulation, i.e. production rate or lifetime, the then calculated variance would be physically more meaningful. The GALPROP manual gives hints on how this could be implemented and will be investigated soon.

Furthermore, a close-by source that happens to coincide with one of the Earth's closest grid points would sophisticate the entire calculation as well, particularly when rather few simulations are performed.

Another aspect regarding numerical stability arises when simulating SNR as CR sources as δ-functions. Despite the fact that sharp decreases in spatial source distributions are an uneasy way of treatment in numerical simulations, all our simulations seem to remain stable when variations of numerical time steps are considered.

Secondly, physical issues shall be taken into account.

In the current configuration the Kolmogorov diffusion scalar coefficient is used with value of αdiff = 0.33. This coefficient is worth investigating as it directly affects the steepening of the spectrum. There might be a value that reproduces low energy data better than the actual one, see e.g. .

This behavior could be explained by missing SNRs with a flatter spectrum that have not yet been observed. With a higher sensitivity experiments like HAWK and CTA it may be possible to retrieve a more complete sample of SNRs. Other experiments like IceCube and KM3NET could resolve the parts of the gamma ray signal that hadronizes, see e.g. , and .

A further aspect would be to consider the scalar diffusion as a broken power law; the kink would occur at ∼ 100 GeV, as already pointed out by .

With unreached accuracy the diffusion process is presented in a tensor structure. Naturally, this opens a complex field of studies but first steps have been already taken and been presented in publications using GALPROP's successor DRAGON for CR propagation and the PICARD code, see e.g.  and .

All the above mentioned aspects might lead to a better description of data by our analysis. First indications of a verification of our method already exist.

In this proceeding we have introduced a procedure to predict the CR flux measured at Earth up to the knee. The tool for propagating CR particles from their source to Earth, i.e. GALPROP, allows to simulate protons as well as heavier nuclei. As possible sources for explaining the CR spectrum we investigated SNRs that feature a hadronic part in their spectrum . In terms of simulation, we distributed the SNRs randomly on grid points, which is how the GALPROP code is structured, according to a given SNR distribution function.

The obtained results with this method show that our proton flux < dF/dT > differs in slope in the low energy regime, for E < 106 MeV, with data. However, for higher energies as well as for < B/C > ratio we find a good agreement with data.

Our objective of this proceeding was to introduce our approach along with its techniques. As there is still a lot of work in progress, we are not able to present a parameter optimization nor do we want to draw far-reaching conclusions. Especially the normalization of the spectrum requires further effort as this will give definite answers wether SNRs with a hadronic part in their gamma ray spectrum are capable of explaining the observed CR flux.

Our future plans cover adding temporal features to the simulation in order to enhance the quality of our statistics, including more suitable SNRs when observed by experiments with higher sensitivities, and also implementing an inhomogeneous grid through the Galaxy.

Monte-Carlo simulations can be performed, as well, that will propagate the CR through the Galactic medium. One approach would be the also publicly available code CRPropa . First estimations have been made to analyze Galactic CR of kinetic energies E ∈ (10, 100) TeV with these Monte-Carlo techniques and calculation times are considered reasonable. Lower energies might require to use a diffuse approximation instead, since solving the equations of motion it that case turns out be extraordinary time consuming.