# Secondary to Primary Cosmic Rays

#### 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*, ![$Q\_B(E)$](https://render.githubusercontent.com/render/math?math=Q_B%28E%29\&mode=inline) to the differential density of Carbon by this equation:![$$Q\_B(E) \simeq n\_{H} \beta c \sigma\_{\rightarrow B} N\_C$$](https://render.githubusercontent.com/render/math?math=Q_B%28E%29%20%5Csimeq%20n_%7BH%7D%20%5Cbeta%20c%20%5Csigma_%7B%5Crightarrow%20B%7D%20N_C\&mode=display)

where, ![$n\_H$](https://render.githubusercontent.com/render/math?math=n_H\&mode=inline) denotes the average interstellar gas number density and ![$N\_C$](https://render.githubusercontent.com/render/math?math=N_C\&mode=inline) is the Carbon density and ![$\beta c$](https://render.githubusercontent.com/render/math?math=%5Cbeta%20c\&mode=inline) is the Carbon velocity and ![$\sigma\_{\rightarrow B}$](https://render.githubusercontent.com/render/math?math=%5Csigma_%7B%5Crightarrow%20B%7D\&mode=inline) 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, ![$\tau$](https://render.githubusercontent.com/render/math?math=%5Ctau\&mode=inline), before it escapes or losses itself energy by spallation:![$$Q\_B = \dot{N}\_B = \frac{N\_B}{\tau}$$](https://render.githubusercontent.com/render/math?math=Q_B%20%3D%20%5Cdot%7BN%7D_B%20%3D%20%5Cfrac%7BN_B%7D%7B%5Ctau%7D\&mode=display)

where we used ![$\dot{N}\_B = \frac{N\_B}{\tau}$](https://render.githubusercontent.com/render/math?math=%5Cdot%7BN%7D_B%20%3D%20%5Cfrac%7BN_B%7D%7B%5Ctau%7D\&mode=inline) assuming a constant per unit time lifetime (see next Leaky Box model). So we can write:![$$\frac{N\_B}{N\_C} \simeq n\_{H} \beta c \sigma\_{\rightarrow B}\tau$$](https://render.githubusercontent.com/render/math?math=%5Cfrac%7BN_B%7D%7BN_C%7D%20%5Csimeq%20n_%7BH%7D%20%5Cbeta%20c%20%5Csigma_%7B%5Crightarrow%20B%7D%5Ctau\&mode=display)

#### 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.![](https://github.com/zemrude/PHYS-F-467/raw/12f2d10ab5e7b0128f70a9bcf8407ece300332c6/images/boron-carbon.png)

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:![$$\frac{N\_B}{N\_C} = n\_{H}\beta c \sigma\_{f,B} \tau  =0.4 \left(\frac{E}{\rm{GeV}}\right)^{-0.3}$$](https://render.githubusercontent.com/render/math?math=%5Cfrac%7BN_B%7D%7BN_C%7D%20%3D%20n_%7BH%7D%5Cbeta%20c%20%5Csigma_%7Bf%2CB%7D%20%5Ctau%20%20%3D0.4%20%5Cleft%28%5Cfrac%7BE%7D%7B%5Crm%7BGeV%7D%7D%5Cright%29%5E%7B-0.3%7D\&mode=display)

For energies above 10 GeV/nucleon we can approximate ![$\beta \sim 1$](https://render.githubusercontent.com/render/math?math=%5Cbeta%20%5Csim%201\&mode=inline), which leads, using the values of the cross-section, to a life time gas density of:![$$ n\_H \tau \simeq 10^{14}\left(\frac{E}{\rm{GeV}}\right)^{-0.3} \\; \rm{s}\\;\rm{cm}^{-3} $$](https://render.githubusercontent.com/render/math?math=n_H%20%5Ctau%20%5Csimeq%2010%5E%7B14%7D%5Cleft%28%5Cfrac%7BE%7D%7B%5Crm%7BGeV%7D%7D%5Cright%29%5E%7B-0.3%7D%20%5C%3B%20%5Crm%7Bs%7D%5C%3B%5Crm%7Bcm%7D%5E%7B-3%7D\&mode=display)

**Boron Lifetime**

But what is this Boron lifetime? The lifetime ![$\tau$](https://render.githubusercontent.com/render/math?math=%5Ctau\&mode=inline) for Boron includes the **catastrophic loss** time due to the partial fragmentation of Boron, ![$\tau\_{f,B}$](https://render.githubusercontent.com/render/math?math=%5Ctau_%7Bf%2CB%7D\&mode=inline) and the **escape probability** from the Galactic confinement volume, ![$T\_{esc}$](https://render.githubusercontent.com/render/math?math=T_%7Besc%7D\&mode=inline). The fragmentation cross section is ![$\sigma\_{f,B} \approx 250$](https://render.githubusercontent.com/render/math?math=%5Csigma_%7Bf%2CB%7D%20%5Capprox%20250\&mode=inline) 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![$^{-3}$](https://render.githubusercontent.com/render/math?math=%5E%7B-3%7D\&mode=inline)![$$n\_H \tau\_{f,B} = \frac{n\_H}{n\_H \beta c \sigma\_{f,B}} \simeq 1.33 \times 10^{14}\\; \rm{s}\\;\rm{cm}^{-3} $$](https://render.githubusercontent.com/render/math?math=n_H%20%5Ctau_%7Bf%2CB%7D%20%3D%20%5Cfrac%7Bn_H%7D%7Bn_H%20%5Cbeta%20c%20%5Csigma_%7Bf%2CB%7D%7D%20%5Csimeq%201.33%20%5Ctimes%2010%5E%7B14%7D%5C%3B%20%5Crm%7Bs%7D%5C%3B%5Crm%7Bcm%7D%5E%7B-3%7D\&mode=display)

which is a good match with the loss time bound at ![$\sim$](https://render.githubusercontent.com/render/math?math=%5Csim\&mode=inline) 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:![$$ n\_H\tau(1\\; \rm{TeV}) \simeq 10^{14} 1000^{-0.3} \sim 1.3 \times 10^{13} \rm{s\\;cm}^{-3}$$](https://render.githubusercontent.com/render/math?math=n_H%5Ctau%281%5C%3B%20%5Crm%7BTeV%7D%29%20%5Csimeq%2010%5E%7B14%7D%201000%5E%7B-0.3%7D%20%5Csim%201.3%20%5Ctimes%2010%5E%7B13%7D%20%5Crm%7Bs%5C%3Bcm%7D%5E%7B-3%7D\&mode=display)

**Borom escape**

It could be that Borom escape the leaky box, but that time will be ![$\tau\_{esc} = \frac{H}{c}$](https://render.githubusercontent.com/render/math?math=%5Ctau_%7Besc%7D%20%3D%20%5Cfrac%7BH%7D%7Bc%7D\&mode=inline) which will be roughly:![$$\tau\_{esc} = \frac{300\\;{\rm pc}}{c} \simeq 3\times 10^{10}\\; {\rm s} $$](https://render.githubusercontent.com/render/math?math=%5Ctau_%7Besc%7D%20%3D%20%5Cfrac%7B300%5C%3B%7B%5Crm%20pc%7D%7D%7Bc%7D%20%5Csimeq%203%5Ctimes%2010%5E%7B10%7D%5C%3B%20%7B%5Crm%20s%7D\&mode=display)

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 ![$T$](https://render.githubusercontent.com/render/math?math=T\&mode=inline). By summing the inverse of these processes (being exponential processes):![$$\tau^{-1} =  n\_H \beta c \sigma\_{f,B} + T^{-1}$$](https://render.githubusercontent.com/render/math?math=%5Ctau%5E%7B-1%7D%20%3D%20%20n_H%20%5Cbeta%20c%20%5Csigma_%7Bf%2CB%7D%20%2B%20T%5E%7B-1%7D\&mode=display)

and solving for ![$T$](https://render.githubusercontent.com/render/math?math=T\&mode=inline) we have that:![$$n\_H T = \frac{n\_H}{\frac{1}{\tau} - \frac{1}{\tau\_{f,B}}} \simeq \frac{10^{14} \\; \rm{s}\\;\rm{cm}^{-3}}{\left(\frac{E}{\rm{GeV}}\right)^{-0.3} -0.7} \simeq 10^{14}\left(\frac{E}{\rm{GeV}}\right)^{-0.55} \rm{s}\\;\rm{cm}^{-3}$$](https://render.githubusercontent.com/render/math?math=n_H%20T%20%3D%20%5Cfrac%7Bn_H%7D%7B%5Cfrac%7B1%7D%7B%5Ctau%7D%20-%20%5Cfrac%7B1%7D%7B%5Ctau_%7Bf%2CB%7D%7D%7D%20%5Csimeq%20%5Cfrac%7B10%5E%7B14%7D%20%5C%3B%20%5Crm%7Bs%7D%5C%3B%5Crm%7Bcm%7D%5E%7B-3%7D%7D%7B%5Cleft%28%5Cfrac%7BE%7D%7B%5Crm%7BGeV%7D%7D%5Cright%29%5E%7B-0.3%7D%20-0.7%7D%20%5Csimeq%2010%5E%7B14%7D%5Cleft%28%5Cfrac%7BE%7D%7B%5Crm%7BGeV%7D%7D%5Cright%29%5E%7B-0.55%7D%20%5Crm%7Bs%7D%5C%3B%5Crm%7Bcm%7D%5E%7B-3%7D\&mode=display)

There no other physical loss process for Boron, so ![$T$](https://render.githubusercontent.com/render/math?math=T\&mode=inline) really must be the escape of the galactic confinement (leaky box). But if the box has a size ![$H$](https://render.githubusercontent.com/render/math?math=H\&mode=inline), ![$T\_{esc}$](https://render.githubusercontent.com/render/math?math=T_%7Besc%7D\&mode=inline) will be H/c which is the time required by CR generated in the Galactic plane to escape the box of height ![$H$](https://render.githubusercontent.com/render/math?math=H\&mode=inline)! However we know that ![$T \gg H/c$](https://render.githubusercontent.com/render/math?math=T%20%5Cgg%20H%2Fc\&mode=inline). So there must be something else confining the CR in the galaxy... what could it be?