Cosmic-Ray Spectrum

Cosmic rays are high energy charge particles mostly coming from outside our solar system.

Cosmic-Ray Spectrum

Cosmic Rays (CRs) are charge particles, mostly protons, accelerated at sites within the galaxy and constantly hitting the Earth's atmosphere. Below energies of 1 GeV, however, these particles hitting the atmosphere are mostly of solar origin. We can plot the amount of cosmic-ray particles, ie flux, as a function of their energy. In particle astrophysics this is what is the cosmic-ray spectrum. The figure below shows the cosmic-ray flux versus energy:

Cosmic ray spectrum as a function of their kinetic energy. Data have been assembled by T. K. Gaisser.

As can be seen when plotted in $\log-\log$ scale the spectrum looks like a straight line. This is what is called a power-law, and it practically means that the cosmic-ray flux follows a this expression:

$$I(E) \propto E^{-\alpha}$$

where the parameter $\alpha$ is called the spectral index. However, it is not a featureless power-law, we can see some changes in their slope. In particular there are two evident changes in the spectral index. The knee at energy $Z \times 1 {\rm\; PeV}$, where Z is the charge of the cosmic ray, and the ankle at about $5 \times 10^{19} {\rm \; eV}$.

The changes in slope can be put in more evidence by multiplying by powers of the energy. The plot below shows the same spectrum but now the flux is multiplied by $E^{2.5}$.

The cosmic ray spectrum by Ralf Engel (see ttps://indico.cern.ch/event/660851/)

The plot above has two scales. The scale below shows the energy in the laboratory reference system, and the scale give the equivalent center-of-mass energy (see image below).

Center-of-mass collision vs laboratory collision

As it is made clear in the plot, even if cosmic rays are more energetic than whatever can be produced in man-made accelerators, the energy available for generating new particles is considerable less important when comparing the energy at the center-of-mass reference system.

Cosmic Ray Energy

The example above is a good example on how important it is to correctly define which energy are we using to plot a cosmic-ray spectrum. Generally there are four different ways to describe the spectra of the cosmic ray radiation:

• By particles per unit rigidity. Propagation and deflection on magnetic fields depends on the rigidity.

• By particles per energy-per-nucleon. Fragmentation of nuclei propagating through the interstellar gas depends on energy per nucleon, since that quantity is approximately conserved when a nucleus breaks up on interaction with the gas.

• By nucleons per energy-per-nucleon. Production of secondary cosmic rays in the atmosphere depends on the intensity of nucleons per energy-per-nucleon, approximately independently of whether the incident nucleons are free protons or bound in nuclei.

• By particles per energy-per-nucleus. Air shower experiments that use the atmosphere as a calorimeter generally measure a quantity that is related to total energy per particle.

For $E > 100 {\rm \; TeV}$ the difference between the kinetic energy and the total energy is negligible and fluxes are often presented as particle per energy-per-nucleus.

For $E > 100 {\rm \; TeV}$ the difference is important and it is common to present nucleons per kinetic energy-per-nucleon. This is the usual way of presenting the spectrum for nuclei with different masses: the conversion in energy per nucleus is not trivial.

Observation of Cosmic Rays

As can be seen, the energy window and scale of fluxes of cosmic rays is enormous. This means that it is impossible to have single detection technique that can cover the wide energy range and different experiments are used. Up to energies of $10^{14}\rm{eV}$, the CRs spectrum can be directly detected above the atmosphere on stratospheric balloons or in outer space by experiments on board satellites. At higher energies, however, the flux of CRs particles is so low that it will require a large collection area impossible to send to space. At these energies CRs measurements are accessible from ground detection infrastructures. These experiments don't detect CRs directly, they observed the showers of secondary particles created by interaction of CRs in the atmosphere (see Cosmic Ray Showers).

Primary Cosmic-Ray Spectrum

The word primary might sound misleading, and it won't be totally clear until we talk about the composition of cosmic rays. Let us say for now, that we know that some of the nuclei of cosmic-rays have been accelerated by astrophysical sources, and we call them primaries, while other nuclei are generated during the propagation of these primary cosmic rays. This definition is far from universal and sometimes physicists, specially those dedicated to CRs air-showers, refer to primary CRs to those arriving at the atmosphere as opposed to the secondary CRs generated by their interactions. The energy spectrum these primary nucleons from GeV to ~ 100 TeV is given by the following expression:

$$I(E) \approx 1.8 \times 10^4 \left(\frac{E}{{\rm 1 \; GeV}}\right)^{-2.7} \frac{{\rm nucleons}}{{\rm m^2\;s\;sr\;GeV}}$$

Where $\alpha \equiv 1 + \gamma = 2.7$ is the differential spectral index and $\gamma$ is called the integral spectral index. We will review the cosmic-ray composition in more detail in the follow section Cosmic-Ray Composition, so let's just say for the moment that cosmic rays are mostly protons, in particular about 70% are protons and 25% He, and the rest are heavier nuclei. There are also electrons with about 2%. One important thing to notice is that the primary spectrum of cosmic rays for each component, seems to follow the same power-law. This is clearly visible in the figure below, where the ratios among the different components remain constant as function with energy.

An interesting parametrization for the region below the knee is presented in Gaisser, 2019 and shown below, including the composition and the recently observed by PAMELA and AMS hardening of the spectrum in the region at around 240 GeN/nucleon.

Source: Particle Data Group

Aside from this hardening it is noticeable that the spectrum of cosmic ray below the knee is quite regular and slopes are parallel for all components. Nonetheless a differet slope is observed for H and He. The spectum resembles a straight line in a $\log - \log$plot. At higher energies there are two relevant changes of slope : the knee and the ankle.

The Knee

At energies of about $5 \times 10^{15} {\rm\; eV}$there is a steepening in the spectrum from where the spectral index $\gamma \sim 1.7$ goes to $\gamma \sim 2$. This feature is known as the knee of the cosmic-ray spectrum. The reason for this hardening of the spectrum is not fully known. Already in 1959, Bernard Peters concluded that the origin of the knee could be due to a consequence of the breakdown of an acceleration mechanism, or an increased rate of escape of cosmic rays from our Galaxy at high energies.

Cosmic ray spectrum indicating the features resulting in a change of the spectral index

A third explanation could be a change in CR interactions at $\sqrt{s} \sim {\rm TeV}$ which means that the feature is not in the original CR spectrum, but a consequence of the interactions of CR with the atmosphere, as at these energies, CR are only detected indirectly. The first two explanations will produce a rigidity dependent knee, ie the position of the knee for different nuclei depends on $Z$, while the third explanation will depend on $A$. Experimentally the rigidity dependence is favored suggesting that the knee is the cut-off of protons, while the rest of nuclei follow subsequently at energies $Z\times 3\times 10^{15}\rm{eV}$. This phenomenological model can explain another feature known as the second knee, or Iron knee happening at energies about $4\times 10^{17} {\rm eV}$, where the spectral index increases again.

It should be mentioned that cosmic-ray composition at these energies is very tricky business. As we already said experimentally at these energies we cannot observe cosmic-rays in a direct way. In order to estimate the CRs composition experiments need to start looking at the CRs interactions with the atmosphere (see Cosmic Ray Showers). This imposes limitations on the precision of the CRs composition. In particular different models of hadronic interactions have to be assumed and the results are model dependent.