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Introduction

Studying cosmic-ray acceleration implies trying to answer three questions: What is the source of power of cosmic rays? What is the actual mechanism? and can it reproduce the spectral index observed?

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Energy Density of Galactic Cosmic Rays

In cosmic ray physics we called spectrum to the flux per stero radian, so the relationship between them is:$$\Phi (E) = \int_\Omega {\rm d} \Omega I(E) = 4\pi I(E)$$

For all-hemispheres. So we can relate the number density of CR with the spectrum by:$$n(E) = \frac{4\pi}{v}I(E)$$

since the flux is just the number density times the velocity.

And so kinetic energy density of CR, $\rho_{CR}$ is therefore the integral of the energy density spectrum, $E\cdot n(E)$:$$\rho_{CR} = \int E n(E) {\rm d} E =4\pi \int \frac{E}{v}I(E) {\rm d} E$$

assuming for the Galactic CR (and that $v = c$ for relativistic particles):$$ I(E) \approx 1.8 \times 10^4 \left(\frac{E}{1\; \rm{GeV}}\right)^{-2.7} \frac{\rm{nucleons}}{\rm{m}^2\; \rm{s\;sr\;GeV}}$$

we can calculate the energy density for cosmic-rays from above the solar modulations up the knee, which is given by:$$ \rho_{CR} = \frac{4\pi}{c} \frac{1.8}{1 - 1.7} \left[\left(\frac{E_{max}}{1\;\rm{GeV}}\right)^{1 -1.7} - \left(\frac{E_{min}}{1\;\rm{GeV}}\right)^{1 -1.7}\right] \approx 1\; {\rm ev\; cm}^{-3}$$In [7]:

import scipy.constants as cte
from astropy.constants import pc

cspeed = cte.value("speed of light in vacuum") * 1e2 # in cm/s

emin = 1. #GeV
emax = 1e5 # 100 TeV
rho = 4 * np.pi /cspeed * 1.8 / (1 - 1.7) * (np.power(emax,1-1.7) - np.power(emin,1-1.7)) * 1e9 # in ev cm-3

Latex(r"The energy density is $\rho_{CR} \approx %.2f$ ev/cm$^{3}$" %rho)

Out[7]:The energy density is $\rho_{CR} \approx 1.08$ ev/cm$^{3}$

This energy density is comparable with the energy density of the CMB $\rho_{CMB} \approx 0.25$ eV/cm$^{3}$

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Power required for galactic cosmic-rays

If we assume this value to be the constant value over the galaxy, the power required (called luminosity in astrophysics) to supply all the galactic CR and balance the escape processes is:$$\mathcal{L}_{CR} = \frac{V_D \rho_{CR}}{\tau_{esc} }\sim 4\times 10^{40} {\rm erg\;s}^{-1}$$

where $V_D$ is the volume of the galactic disk$$V_D = \pi R^2 d \sim \pi (15 {\rm kpc})^2(300 {\rm pc}) \sim 6 \times 10^{66} {\rm cm}^3.$$In [8]:

Out[8]:Galactic Volume is 6.2e+66 cm$^{-3}$In [9]:

Out[9]:Power $\mathcal{L}_{CR} \sim$ 1e+40 erg s$^{-1}$

It was emphasized long ago (Ginzburg & Syrovatskii 1964) that supernovae might account for this power. For example a type II supernova gives an average power output of:$$\mathcal{L}_{SN} \sim 3 \times 10^{42} {\rm erg\;s}^{-1}$$

Therefore if SN transmit a few percent of the energy into CR it is enough to account for the total energy in the cosmic ray beam. That was called the SNR paradigm

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Power required for > 100 TeV

The derivation above was considered using the CR flux with an integral spectral index of $\gamma = \alpha -1 = 1.7$ which describes well the CR up to the knee. This is the bulk of cosmic ray density. The power required for the high energy part will be significantly less due to the steeply falling primary cosmic ray spectrum. For example assuming an integral index of $\gamma = 1.6$ for $E < 1000$ TeV and $\gamma = 2$ for higher energy we get:$$\begin{aligned} \sim 2 \times 10^{39} {\;\rm\;erg/s\; for\; } E &> 100 {\rm\; TeV}\\ \sim 2 \times 10^{38} {\;\rm\;erg/s\; for\; } E &> 1 {\rm\; PeV}\\ \sim 2 \times 10^{37} {\;\rm\;erg/s\; for\; } E &> 10 {\rm\; PeV} \end{aligned}$$

which are considerably less than the total power required for all the cosmic-rays. This power might be available from a few very energetic sources.

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