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Secondary to Primary Cosmic Rays

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Primary-to-Secondary ratios

Since we know the partial cross-section of spallation processes we can use the secondary-to-primary abundance ratios to infer the gas column density traversed by the average cosmic ray.

Let us perform a simply estimate of the Boron-to-Carbon ratio. Boron is chiefly produced by Carbon and Oxygen with approximately conserved kinetic energy per nucleon (see Superposition principle), so we can relate the Boron source production rate, $Q_B(E)$ to the differential density of Carbon by this equation:$$Q_B(E) \simeq n_{H} \beta c \sigma_{\rightarrow B} N_C$$

where, $n_H$ denotes the average interstellar gas number density and $N_C$ is the Carbon density and $\beta c$ is the Carbon velocity and $\sigma_{\rightarrow B}$ is the spallation cross-section of Carbon into Boron.

The Boron density is related to the production rate by the lifetime of Boron in the Galaxy, $\tau$, before it escapes or losses itself energy by spallation:$$Q_B = \dot{N}_B = \frac{N_B}{\tau}$$

where we used $\dot{N}_B = \frac{N_B}{\tau}$ assuming a constant per unit time lifetime (see next Leaky Box model). So we can write:$$\frac{N_B}{N_C} \simeq n_{H} \beta c \sigma_{\rightarrow B}\tau$$

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Boron-to-Carbon ratio

The plot below represents the 2014 measurements from PAMELA and AMS satellites of the Boron-to-Carbon ratio. The decrease in energy of the Boron-to-Carbon ratio suggests that high energy CR spend less time than the low energy ones in the Galaxy before escaping.

Above about 10 GeV/nucleon the experimental data can be fitted to a test function, therefore the Boron-to-Carbon ratio can be expressed as:$$\frac{N_B}{N_C} = n_{H}\beta c \sigma_{f,B} \tau =0.4 \left(\frac{E}{\rm{GeV}}\right)^{-0.3}$$

For energies above 10 GeV/nucleon we can approximate $\beta \sim 1$, which leads, using the values of the cross-section, to a life time gas density of:$$ n_H \tau \simeq 10^{14}\left(\frac{E}{\rm{GeV}}\right)^{-0.3} \; \rm{s}\;\rm{cm}^{-3} $$

Boron Lifetime

But what is this Boron lifetime? The lifetime $\tau$ for Boron includes the catastrophic loss time due to the partial fragmentation of Boron, $\tau_{f,B}$ and the escape probability from the Galactic confinement volume, $T_{esc}$. The fragmentation cross section is $\sigma_{f,B} \approx 250$ mbarn so we find that:In [4]:

Latex("The boron lifetime is approx: %.2e s cm$^{-3}$" %(1/0.250/1e-24/2.998e+10))

Out[4]:The boron lifetime is approx: 1.33e+14 s cm$^{-3}$$$n_H \tau_{f,B} = \frac{n_H}{n_H \beta c \sigma_{f,B}} \simeq 1.33 \times 10^{14}\; \rm{s}\;\rm{cm}^{-3} $$

which is a good match with the loss time bound at $\sim$ 1 GeV but is larger at higher energies and does not depend on energy. For example at 1 TeV it is an order of magnitude larger:$$ n_H\tau(1\; \rm{TeV}) \simeq 10^{14} 1000^{-0.3} \sim 1.3 \times 10^{13} \rm{s\;cm}^{-3}$$

Borom escape

It could be that Borom escape the leaky box, but that time will be $\tau_{esc} = \frac{H}{c}$ which will be roughly:$$\tau_{esc} = \frac{300\;{\rm pc}}{c} \simeq 3\times 10^{10}\; {\rm s} $$

which is too small compared to the effective lifetime found. This seems to indicate that CR do not travel in straight lines. Let's assume that the overall process is a convination of both the borom fragmentation and another process with a lifetime $T$. By summing the inverse of these processes (being exponential processes):$$\tau^{-1} = n_H \beta c \sigma_{f,B} + T^{-1}$$

and solving for $T$ we have that:$$n_H T = \frac{n_H}{\frac{1}{\tau} - \frac{1}{\tau_{f,B}}} \simeq \frac{10^{14} \; \rm{s}\;\rm{cm}^{-3}}{\left(\frac{E}{\rm{GeV}}\right)^{-0.3} -0.7} \simeq 10^{14}\left(\frac{E}{\rm{GeV}}\right)^{-0.55} \rm{s}\;\rm{cm}^{-3}$$

There no other physical loss process for Boron, so $T$ really must be the escape of the galactic confinement (leaky box). But if the box has a size $H$, $T_{esc}$ will be H/c which is the time required by CR generated in the Galactic plane to escape the box of height $H$! However we know that $T \gg H/c$. So there must be something else confining the CR in the galaxy... what could it be?

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