# Shock Acceleration

#### Fermi Acceleration 1st order acceleration.

Let's imagine a shock moving through a plasma. In the reference system of the *unshocked* plasma the shock front approaches with speed ![$\vec{u\_1}$](https://render.githubusercontent.com/render/math?math=%5Cvec%7Bu_1%7D\&mode=inline) while the *shocked* plasma (left behind) moves at a slower velocity ![$\vec{V}$](https://render.githubusercontent.com/render/math?math=%5Cvec%7BV%7D\&mode=inline) where ![$|V| \< |u\_1|$](https://render.githubusercontent.com/render/math?math=%7CV%7C%20%26lt%3B%20%7Cu_1%7C\&mode=inline). If we now changed to the reference system where the shock-front is at rest the gas *unshocked* now appears to approach speed ![$-\vec{u\_1}$](https://render.githubusercontent.com/render/math?math=-%5Cvec%7Bu_1%7D\&mode=inline) while the *shocked* plasma recedes with speed ![${-\vec u\_2 = (\vec{V} - \vec u\_1)}$](https://render.githubusercontent.com/render/math?math=%7B-%5Cvec%20u_2%20%3D%20%28%5Cvec%7BV%7D%20-%20%5Cvec%20u_1%29%7D\&mode=inline). A test cosmic ray particle crossing from any side of the shock, will always face an encounter with plasma aproaching at speed ![$|V|$](https://render.githubusercontent.com/render/math?math=%7CV%7C\&mode=inline), hence ![$\beta$](https://render.githubusercontent.com/render/math?math=%5Cbeta\&mode=inline)now refers to this speed, the speed of the shocked (downstream) gas in the upstream reference system.

* The outcoming distribution of particles is not now 0, there is an asymmetry in the Fermi shock acceleration model, as particle in the upstream will re-enter the shock, only those going downstream can escape. Therefore the distribution follows the projection of an uniform flux on a plane:

![$$\frac{dn}{d\cos \theta\_2^\prime} = 2 \cos\theta\_2^\prime{\\;\\; \rm for\\;  0 \leq \cos\theta\_2^\prime \leq 1}$$](https://render.githubusercontent.com/render/math?math=%5Cfrac%7Bdn%7D%7Bd%5Ccos%20%5Ctheta_2%5E%5Cprime%7D%20%3D%202%20%5Ccos%5Ctheta_2%5E%5Cprime%7B%5C%3B%5C%3B%20%5Crm%20for%5C%3B%20%200%20%5Cleq%20%5Ccos%5Ctheta_2%5E%5Cprime%20%5Cleq%201%7D\&mode=display)

which gives:![$$\langle \cos\theta\_2^\prime\rangle = \frac{\int \cos\theta\_2^\prime \cdot \frac{dn}{d\cos\theta\_2^\prime} d\cos\theta\_2^\prime}{\int \frac{dn}{d\cos\theta\_2^\prime}d\cos\theta\_2^\prime} = \frac{2}{3}$$](https://render.githubusercontent.com/render/math?math=%5Clangle%20%5Ccos%5Ctheta_2%5E%5Cprime%5Crangle%20%3D%20%5Cfrac%7B%5Cint%20%5Ccos%5Ctheta_2%5E%5Cprime%20%5Ccdot%20%5Cfrac%7Bdn%7D%7Bd%5Ccos%5Ctheta_2%5E%5Cprime%7D%20d%5Ccos%5Ctheta_2%5E%5Cprime%7D%7B%5Cint%20%5Cfrac%7Bdn%7D%7Bd%5Ccos%5Ctheta_2%5E%5Cprime%7Dd%5Ccos%5Ctheta_2%5E%5Cprime%7D%20%3D%20%5Cfrac%7B2%7D%7B3%7D\&mode=display)

* The incoming angle distribution is also the projection of an uniform flux on a plen but this time with  ![$-1 \leq \cos\theta\_1 \leq 0$](https://render.githubusercontent.com/render/math?math=-1%20%5Cleq%20%5Ccos%5Ctheta_1%20%5Cleq%200\&mode=inline) and so ![$\langle \cos \theta\_1 \rangle = -2/3$](https://render.githubusercontent.com/render/math?math=%5Clangle%20%5Ccos%20%5Ctheta_1%20%5Crangle%20%3D%20-2%2F3\&mode=inline)

Particles entering the shockwave and exiting will have a gain of:![$$\xi = \frac{1 + \frac{4}{3}\beta + \frac{4}{9}\beta^2}{1 -\beta^2} - 1 \sim \frac{4}{3}\beta = \frac{4}{3}\frac{u\_1 - u\_2}{c}$$](https://render.githubusercontent.com/render/math?math=%5Cxi%20%3D%20%5Cfrac%7B1%20%2B%20%5Cfrac%7B4%7D%7B3%7D%5Cbeta%20%2B%20%5Cfrac%7B4%7D%7B9%7D%5Cbeta%5E2%7D%7B1%20-%5Cbeta%5E2%7D%20-%201%20%5Csim%20%5Cfrac%7B4%7D%7B3%7D%5Cbeta%20%3D%20%5Cfrac%7B4%7D%7B3%7D%5Cfrac%7Bu_1%20-%20u_2%7D%7Bc%7D\&mode=display)

#### Fermi Acceleration: Escape probability

The escape probability of loss rate of particles is given by the ratio of the incoming flux and the outgoing flux of particles.

* **Incoming rate.** Let's assume that the diffusion of particles is so effective that close to the shockwave the distribution of particles is isotropic. In this case the rate of encounters for a plane shock is the projection of an isotropic flux onto the plane shock. Let's assume ![$n$](https://render.githubusercontent.com/render/math?math=n\&mode=inline) to be the density of particles close to the shock, because it is isotropic it should follow:

![$${\rm d}n = \frac{n}{4\pi}{\rm d}\Omega$$](https://render.githubusercontent.com/render/math?math=%7B%5Crm%20d%7Dn%20%3D%20%5Cfrac%7Bn%7D%7B4%5Cpi%7D%7B%5Crm%20d%7D%5COmega\&mode=display)

assuming the particles moving at relativistic speed, the velocity across the shock is ![$c\cos\theta$](https://render.githubusercontent.com/render/math?math=c%5Ccos%5Ctheta\&mode=inline) therefore the rate of encounters of particles upstream with the shock is given by:![$$R\_{in} = \int {\rm d} n c \cos\theta = \int\_0^1 {\rm d} \cos \theta \int\_0^{2\pi} {\rm d} \phi\frac{cn(E)}{4\pi}\cos\theta = \frac{cn(E)}{4}$$](https://render.githubusercontent.com/render/math?math=R_%7Bin%7D%20%3D%20%5Cint%20%7B%5Crm%20d%7D%20n%20c%20%5Ccos%5Ctheta%20%3D%20%5Cint_0%5E1%20%7B%5Crm%20d%7D%20%5Ccos%20%5Ctheta%20%5Cint_0%5E%7B2%5Cpi%7D%20%7B%5Crm%20d%7D%20%5Cphi%5Cfrac%7Bcn%28E%29%7D%7B4%5Cpi%7D%5Ccos%5Ctheta%20%3D%20%5Cfrac%7Bcn%28E%29%7D%7B4%7D\&mode=display)

where ![$n(E)$](https://render.githubusercontent.com/render/math?math=n%28E%29\&mode=inline) is the CR number density.

* **Outgoing rate.** The outgoing rate is simply the number of particles escaping the system. In the shock rest frame, that's all particles not returning to the shockwave. In this reference system there is an outflow of cosmic-rays adverted downstream. Since particles are diffusing in all direction, the net outflow goes with the velocity of the downstream speed and is given simply by ![$R\_{out} = n(E) u\_2$](https://render.githubusercontent.com/render/math?math=R_%7Bout%7D%20%3D%20n%28E%29%20u_2\&mode=inline),

Therefore the escape probability is given by:![$$P\_{esc} = \frac{R\_{in} }{R\_{out} }= \frac{4 u\_2}{c}$$](https://render.githubusercontent.com/render/math?math=P_%7Besc%7D%20%3D%20%5Cfrac%7BR_%7Bin%7D%20%7D%7BR_%7Bout%7D%20%7D%3D%20%5Cfrac%7B4%20u_2%7D%7Bc%7D\&mode=display)

which for acceleration at shock gives:![$$\gamma = \frac{P\_{esc}}{\xi} = \frac{3}{u\_1/u\_2 -1}$$](https://render.githubusercontent.com/render/math?math=%5Cgamma%20%3D%20%5Cfrac%7BP_%7Besc%7D%7D%7B%5Cxi%7D%20%3D%20%5Cfrac%7B3%7D%7Bu_1%2Fu_2%20-1%7D\&mode=display)

So we get an estimate of the spectral index based on the relative velocities of the downstream and upstram gas in the sockwave.

#### Fermi acceleration: Kinematic relations at the shock

In order to derive the exact value of the spectral index we need to obtain a relation between ![$u\_1$](https://render.githubusercontent.com/render/math?math=u_1\&mode=inline) and ![$u\_2$](https://render.githubusercontent.com/render/math?math=u_2\&mode=inline) using the kinematics of a shock wave. This equations are the conservation of mass, the Euler equation for momentum conservation and conservation of energy:

* **Conservation of mass**:![$$\partial\_t \rho + \nabla \cdot (\rho\vec{u}) = 0$$](https://render.githubusercontent.com/render/math?math=%5Cpartial_t%20%5Crho%20%2B%20%5Cnabla%20%5Ccdot%20%28%5Crho%5Cvec%7Bu%7D%29%20%3D%200\&mode=display)
* **Conservation of momentum (Euler equation)**:

![$$\rho\frac{\partial \vec{u}}{\partial t} + \rho \vec{u} \cdot(\nabla\vec{u}) = \vec{F} - \nabla P$$](https://render.githubusercontent.com/render/math?math=%5Crho%5Cfrac%7B%5Cpartial%20%5Cvec%7Bu%7D%7D%7B%5Cpartial%20t%7D%20%2B%20%5Crho%20%5Cvec%7Bu%7D%20%5Ccdot%28%5Cnabla%5Cvec%7Bu%7D%29%20%3D%20%5Cvec%7BF%7D%20-%20%5Cnabla%20P\&mode=display)

where ![$\vec{F}$](https://render.githubusercontent.com/render/math?math=%5Cvec%7BF%7D\&mode=inline) is an external force, and ![$\nabla P$](https://render.githubusercontent.com/render/math?math=%5Cnabla%20P\&mode=inline) is a force due to a pressure gradient.

* **Conservation of energy**:

![$$\frac{\partial}{\partial t} \left(\frac{\rho u^2}{2} + \rho U + \rho\Phi\right) + \nabla\cdot\left\[\rho \vec{u}\left(\frac{u^2}{2} + U + \frac{P}{\rho} + \Phi\right)\right\] = 0 $$](https://render.githubusercontent.com/render/math?math=%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial%20t%7D%20%5Cleft%28%5Cfrac%7B%5Crho%20u%5E2%7D%7B2%7D%20%2B%20%5Crho%20U%20%2B%20%5Crho%5CPhi%5Cright%29%20%2B%20%5Cnabla%5Ccdot%5Cleft%5B%5Crho%20%5Cvec%7Bu%7D%5Cleft%28%5Cfrac%7Bu%5E2%7D%7B2%7D%20%2B%20U%20%2B%20%5Cfrac%7BP%7D%7B%5Crho%7D%20%2B%20%5CPhi%5Cright%29%5Cright%5D%20%3D%200\&mode=display)

where this equation accounts for the kinetic, internal, and potential energy ![$\Phi$](https://render.githubusercontent.com/render/math?math=%5CPhi\&mode=inline).

Let's assume a one-dimensional, steady shock in its rest frame (otherwise time derivates must be taking into account).

Then the first equation becomes simply:![$$\frac{d}{dx} (\rho u) = 0$$](https://render.githubusercontent.com/render/math?math=%5Cfrac%7Bd%7D%7Bdx%7D%20%28%5Crho%20u%29%20%3D%200\&mode=display)

and the Euler equation simplifies to:![$$\frac{d}{dx}(P + \rho u^2) = 0$$](https://render.githubusercontent.com/render/math?math=%5Cfrac%7Bd%7D%7Bdx%7D%28P%20%2B%20%5Crho%20u%5E2%29%20%3D%200\&mode=display)

In the energy equation we assume ![$\Phi = 0$](https://render.githubusercontent.com/render/math?math=%5CPhi%20%3D%200\&mode=inline):![$$\frac{d}{dx}\left(\frac{\rho u^3}{2} + (\rho U + P)u\right) = 0 $$$$\frac{d}{dx}\left\[ \rho u \left(\frac{u^2}{2} + U + \frac{P}{\rho}\right) \right\] = 0 $$](https://render.githubusercontent.com/render/math?math=%5Cfrac%7Bd%7D%7Bdx%7D%5Cleft%28%5Cfrac%7B%5Crho%20u%5E3%7D%7B2%7D%20%2B%20%28%5Crho%20U%20%2B%20P%29u%5Cright%29%20%3D%200%20%24%24%24%24%5Cfrac%7Bd%7D%7Bdx%7D%5Cleft%5B%20%5Crho%20u%20%5Cleft%28%5Cfrac%7Bu%5E2%7D%7B2%7D%20%2B%20U%20%2B%20%5Cfrac%7BP%7D%7B%5Crho%7D%5Cright%29%20%5Cright%5D%20%3D%200\&mode=display)

where U is the internal density per unit volume and we can write ![$ \rho U = \frac{P} {\Gamma -1}$](https://render.githubusercontent.com/render/math?math=%5Crho%20U%20%3D%20%5Cfrac%7BP%7D%20%7B%5CGamma%20-1%7D\&mode=inline), where ![$\Gamma = c\_p/c\_v$](https://render.githubusercontent.com/render/math?math=%5CGamma%20%3D%20c_p%2Fc_v\&mode=inline) is the [adiabatic index](http://en.wikipedia.org/wiki/Heat_capacity_ratio) or heat capacity ratio.

**Condition at the discontinuity of the shock wave**

Let's assume we are in the shock ref system. Applyting these equations at the discontinuity of the shock we have the three conditions at the discontinuity of the shock:![$$\begin{aligned}
\rho\_1 u\_1 \&= \rho\_2 u\_2\\\\
P\_1 + \rho\_1 u\_1^2 \&= P\_2 + \rho\_2 u\_2^2\\\\
\frac{\rho\_1 u\_1^2}{2} + \frac{\Gamma}{\Gamma - 1} P\_1 \&=\frac{\rho\_2 u\_2^2}{2} + \frac{\Gamma}{\Gamma - 1}  P\_2\\\\
\end{aligned}$$](https://render.githubusercontent.com/render/math?math=%5Cbegin%7Baligned%7D%0A%5Crho_1%20u_1%20%26amp%3B%3D%20%5Crho_2%20u_2%5C%5C%0AP_1%20%2B%20%5Crho_1%20u_1%5E2%20%26amp%3B%3D%20P_2%20%2B%20%5Crho_2%20u_2%5E2%5C%5C%0A%5Cfrac%7B%5Crho_1%20u_1%5E2%7D%7B2%7D%20%2B%20%5Cfrac%7B%5CGamma%7D%7B%5CGamma%20-%201%7D%20P_1%20%26amp%3B%3D%5Cfrac%7B%5Crho_2%20u_2%5E2%7D%7B2%7D%20%2B%20%5Cfrac%7B%5CGamma%7D%7B%5CGamma%20-%201%7D%20%20P_2%5C%5C%0A%5Cend%7Baligned%7D\&mode=display)

For a gas with ![$P = K \rho^\Gamma$](https://render.githubusercontent.com/render/math?math=P%20%3D%20K%20%5Crho%5E%5CGamma\&mode=inline) the speed of sound can be written as ![$c\_s = \sqrt{\Gamma P / \rho}$](https://render.githubusercontent.com/render/math?math=c_s%20%3D%20%5Csqrt%7B%5CGamma%20P%20%2F%20%5Crho%7D\&mode=inline) or ![$\rho c\_s^2 = \Gamma P$](https://render.githubusercontent.com/render/math?math=%5Crho%20c_s%5E2%20%3D%20%5CGamma%20P\&mode=inline). From the second condition we can write:![$$ P\_1 + \rho\_1 u\_1^2 =  \rho\_1 u\_1^2 \left( 1 + \frac{P\_1}{\rho\_1 u\_1^2}\right)  = \rho\_1 u\_1^2 \left( 1 + \frac{c\_s^2}{\Gamma u\_1^2}\right) = \rho\_1 u\_1^2 \left(1 + \frac{1}{\mathcal{M\_1}\Gamma}\right) $$](https://render.githubusercontent.com/render/math?math=P_1%20%2B%20%5Crho_1%20u_1%5E2%20%3D%20%20%5Crho_1%20u_1%5E2%20%5Cleft%28%201%20%2B%20%5Cfrac%7BP_1%7D%7B%5Crho_1%20u_1%5E2%7D%5Cright%29%20%20%3D%20%5Crho_1%20u_1%5E2%20%5Cleft%28%201%20%2B%20%5Cfrac%7Bc_s%5E2%7D%7B%5CGamma%20u_1%5E2%7D%5Cright%29%20%3D%20%5Crho_1%20u_1%5E2%20%5Cleft%281%20%2B%20%5Cfrac%7B1%7D%7B%5Cmathcal%7BM_1%7D%5CGamma%7D%5Cright%29\&mode=display)

For strong shocks ![$\mathcal{M\_1} \gg 1$](https://render.githubusercontent.com/render/math?math=%5Cmathcal%7BM_1%7D%20%5Cgg%201\&mode=inline) then the presure in the upstream is neglicable ![$P\_1 \sim 0$](https://render.githubusercontent.com/render/math?math=P_1%20%5Csim%200\&mode=inline)

We can isolate ![$\rho\_2$](https://render.githubusercontent.com/render/math?math=%5Crho_2\&mode=inline) and ![$P\_2$](https://render.githubusercontent.com/render/math?math=P_2\&mode=inline) as:![$$\rho\_2 = \frac{u\_1}{u\_2} \rho\_1$$$$P\_2 = P\_1 + \rho\_1 u\_1^2 - \rho\_1 \frac{u\_1}{u\_2}u\_2^2 = P\_1 + \rho\_1 u\_1 (u\_1 - u\_2) \sim \rho\_1 u\_1 (u\_1 - u\_2)$$](https://render.githubusercontent.com/render/math?math=%5Crho_2%20%3D%20%5Cfrac%7Bu_1%7D%7Bu_2%7D%20%5Crho_1%24%24%24%24P_2%20%3D%20P_1%20%2B%20%5Crho_1%20u_1%5E2%20-%20%5Crho_1%20%5Cfrac%7Bu_1%7D%7Bu_2%7Du_2%5E2%20%3D%20P_1%20%2B%20%5Crho_1%20u_1%20%28u_1%20-%20u_2%29%20%5Csim%20%5Crho_1%20u_1%20%28u_1%20-%20u_2%29\&mode=display)

Using these expression ot eliminate ![$\rho\_2$](https://render.githubusercontent.com/render/math?math=%5Crho_2\&mode=inline) and ![$P\_2$](https://render.githubusercontent.com/render/math?math=P_2\&mode=inline) from the third (enegy conservation) equation we have:![$$\frac{1}{2}u\_1^2 = \frac{1}{2}u\_2^2 + \frac{\Gamma}{\Gamma -1}\frac{P\_2}{\rho\_2} = \frac{1}{2}u\_2^2 + \frac{\Gamma}{\Gamma -1} u\_2 (u\_1 - u\_2)$$](https://render.githubusercontent.com/render/math?math=%5Cfrac%7B1%7D%7B2%7Du_1%5E2%20%3D%20%5Cfrac%7B1%7D%7B2%7Du_2%5E2%20%2B%20%5Cfrac%7B%5CGamma%7D%7B%5CGamma%20-1%7D%5Cfrac%7BP_2%7D%7B%5Crho_2%7D%20%3D%20%5Cfrac%7B1%7D%7B2%7Du_2%5E2%20%2B%20%5Cfrac%7B%5CGamma%7D%7B%5CGamma%20-1%7D%20u_2%20%28u_1%20-%20u_2%29\&mode=display)

grouping by powers of ![$u\_2$](https://render.githubusercontent.com/render/math?math=u_2\&mode=inline):![$$\left(\frac{\Gamma}{\Gamma -1} - \frac{1}{2}\right) u\_2^2 - \frac{\Gamma}{\Gamma -1} u\_2u\_1 + u\_1^2 = 0 $$](https://render.githubusercontent.com/render/math?math=%5Cleft%28%5Cfrac%7B%5CGamma%7D%7B%5CGamma%20-1%7D%20-%20%5Cfrac%7B1%7D%7B2%7D%5Cright%29%20u_2%5E2%20-%20%5Cfrac%7B%5CGamma%7D%7B%5CGamma%20-1%7D%20u_2u_1%20%2B%20u_1%5E2%20%3D%200\&mode=display)

multiplying by ![$2/u\_1^2$](https://render.githubusercontent.com/render/math?math=2%2Fu_1%5E2\&mode=inline):![$$\left(\frac{\Gamma + 1 }{\Gamma -1}\right) t^2 - \frac{2\Gamma}{\Gamma -1} t + 1 = 0$$](https://render.githubusercontent.com/render/math?math=%5Cleft%28%5Cfrac%7B%5CGamma%20%2B%201%20%7D%7B%5CGamma%20-1%7D%5Cright%29%20t%5E2%20-%20%5Cfrac%7B2%5CGamma%7D%7B%5CGamma%20-1%7D%20t%20%2B%201%20%3D%200\&mode=display)

where we defined ![$t \equiv u\_2/u\_1$](https://render.githubusercontent.com/render/math?math=t%20%5Cequiv%20u_2%2Fu_1\&mode=inline) this quadratic equation has the 2 solutions:![$$ t = 1 \rightarrow u\_1 = u\_2$$](https://render.githubusercontent.com/render/math?math=t%20%3D%201%20%5Crightarrow%20u_1%20%3D%20u_2\&mode=display)

ie, no shock at all, and a second solution given by:![$$ t = \frac{\Gamma - 1}{\Gamma + 1} \rightarrow \frac{u\_2}{u\_1} = \frac{\Gamma - 1}{\Gamma +1}$$](https://render.githubusercontent.com/render/math?math=t%20%3D%20%5Cfrac%7B%5CGamma%20-%201%7D%7B%5CGamma%20%2B%201%7D%20%5Crightarrow%20%5Cfrac%7Bu_2%7D%7Bu_1%7D%20%3D%20%5Cfrac%7B%5CGamma%20-%201%7D%7B%5CGamma%20%2B1%7D\&mode=display)

for a monatomic gas with 3 degrees of freedom the ratio of specific heats is ![$\Gamma = 1 + 1/f = \frac{5}{3}$](https://render.githubusercontent.com/render/math?math=%5CGamma%20%3D%201%20%2B%201%2Ff%20%3D%20%5Cfrac%7B5%7D%7B3%7D\&mode=inline), so![$$\frac{u\_2}{u\_1} = \frac{1}{4}$$](https://render.githubusercontent.com/render/math?math=%5Cfrac%7Bu_2%7D%7Bu_1%7D%20%3D%20%5Cfrac%7B1%7D%7B4%7D\&mode=display)

No matter how strong a shock wave is, a mono-atomic gas can only be compressed by a factor of 4. The spectral index is then:![$$\gamma = \frac{P\_{esc}}{\xi} = \frac{3}{u\_1/u\_2 -1} = 1$$](https://render.githubusercontent.com/render/math?math=%5Cgamma%20%3D%20%5Cfrac%7BP_%7Besc%7D%7D%7B%5Cxi%7D%20%3D%20%5Cfrac%7B3%7D%7Bu_1%2Fu_2%20-1%7D%20%3D%201\&mode=display)

If one keeps the factors of ![$1/\mathcal{M}^2$](https://render.githubusercontent.com/render/math?math=1%2F%5Cmathcal%7BM%7D%5E2\&mode=inline) (to prove if you are brave):![$$\gamma \sim 1 + \frac{4}{\mathcal{M}^2} \sim 1.1$$](https://render.githubusercontent.com/render/math?math=%5Cgamma%20%5Csim%201%20%2B%20%5Cfrac%7B4%7D%7B%5Cmathcal%7BM%7D%5E2%7D%20%5Csim%201.1\&mode=display)

Which matches remarkably to what we derived to be the differential spectral index at the accelerator:![$$ n(E) \propto E^{-(\gamma+1)} \sim E^{-2.1} $$](https://render.githubusercontent.com/render/math?math=n%28E%29%20%5Cpropto%20E%5E%7B-%28%5Cgamma%2B1%29%7D%20%5Csim%20E%5E%7B-2.1%7D\&mode=display)

#### Fermi acceleration: Maximum Energy

In an infinite planar shockwave, all particles upstream will encounter again the shochwave. However particles can advent downstream. In diffuse shock accelerations, particules will diffuse travelling a distance ![$l\_D$](https://render.githubusercontent.com/render/math?math=l_D\&mode=inline) upstream, until they are reached by the shock moving at spead ![$u\_1$](https://render.githubusercontent.com/render/math?math=u_1\&mode=inline) in the upstream reference system. Particles will cross when:![$$\begin{aligned}
l\_d \&\simeq \sqrt{D t\_d} \\\\
l\_d \&= u\_1 t\_d \\\\
t\_d \&\approx \frac{D}{u\_1^2}
 \end{aligned}$$](https://render.githubusercontent.com/render/math?math=%5Cbegin%7Baligned%7D%0Al_d%20%26amp%3B%5Csimeq%20%5Csqrt%7BD%20t_d%7D%20%5C%5C%0Al_d%20%26amp%3B%3D%20u_1%20t_d%20%5C%5C%0At_d%20%26amp%3B%5Capprox%20%5Cfrac%7BD%7D%7Bu_1%5E2%7D%0A%20%5Cend%7Baligned%7D\&mode=display)

Assuming a diffusion that depends on energy in the form of ![$D = D\_0 E^{\alpha}$](https://render.githubusercontent.com/render/math?math=D%20%3D%20D_0%20E%5E%7B%5Calpha%7D\&mode=inline) we can get that the maximum energy corresponds to:![$$E\_{max} \leq \left(\frac{u\_1^2 t\_d}{D\_0}\right)^{\frac{1}{\alpha}}$$](https://render.githubusercontent.com/render/math?math=E_%7Bmax%7D%20%5Cleq%20%5Cleft%28%5Cfrac%7Bu_1%5E2%20t_d%7D%7BD_0%7D%5Cright%29%5E%7B%5Cfrac%7B1%7D%7B%5Calpha%7D%7D\&mode=display)

where we can assume ![$t\_d$](https://render.githubusercontent.com/render/math?math=t_d\&mode=inline) to be the time during which the mechanism is working, ie the livetime of the shockwave ![$t\_d \sim t\_{age}$](https://render.githubusercontent.com/render/math?math=t_d%20%5Csim%20t_%7Bage%7D\&mode=inline). From the equation above we can conclude that the maximum energy:

* increases with time
* depends on: age, shock speed, magnetic field intensity and structure (through D)
* is not universal
* ![$D$](https://render.githubusercontent.com/render/math?math=D\&mode=inline) and therefore ![$l\_d$](https://render.githubusercontent.com/render/math?math=l_d\&mode=inline) increases with energy, and the each cycle energy increases, so the last cycle is the longest

We can rewrite the diffusion coefficient as: ![$$ D \sim \frac{l\_d^2}{t\_d} = l\_d v $$](https://render.githubusercontent.com/render/math?math=D%20%5Csim%20%5Cfrac%7Bl_d%5E2%7D%7Bt_d%7D%20%3D%20l_d%20v\&mode=display)where ![$v$](https://render.githubusercontent.com/render/math?math=v\&mode=inline) is the particle speed. A more detailed analysis gives ![$D = \frac{1}{3}l\_d v$](https://render.githubusercontent.com/render/math?math=D%20%3D%20%5Cfrac%7B1%7D%7B3%7Dl_d%20v\&mode=inline) where the factor 3 comes from the 3 dimensions. In other words, the diffussion coefficient can be understood as the product of the particle velocity ![$v \simeq c$](https://render.githubusercontent.com/render/math?math=v%20%5Csimeq%20c\&mode=inline) and its mean free path. At high energies, the mean free path between scatterings in the turbulent magnetic clous can be approximate as ![$l\_d = r\_L/r\_0$](https://render.githubusercontent.com/render/math?math=l_d%20%3D%20r_L%2Fr_0\&mode=inline), where ![$r\_0$](https://render.githubusercontent.com/render/math?math=r_0\&mode=inline) is the size of the magnetic cloud and ![$r\_L$](https://render.githubusercontent.com/render/math?math=r_L\&mode=inline) the Larmor radius of the particle. Assuming that ![$r\_L \gg r\_0$](https://render.githubusercontent.com/render/math?math=r_L%20%5Cgg%20r_0\&mode=inline) we can re-write:![$$ D = \frac{r\_L c}{3} \sim \frac{1}{3}\frac{E c}{Z eB} $$](https://render.githubusercontent.com/render/math?math=D%20%3D%20%5Cfrac%7Br_L%20c%7D%7B3%7D%20%5Csim%20%5Cfrac%7B1%7D%7B3%7D%5Cfrac%7BE%20c%7D%7BZ%20eB%7D\&mode=display)

Another way to see this, is to assume that mean diffusion path ![$l\_d$](https://render.githubusercontent.com/render/math?math=l_d\&mode=inline) cannot be smaller than the Larmor radius, since a magnetic field irregularities in a smaller scale than the Larmor radisu will be transparent. This is the regime of the Bohm diffusion and it is possible in higly turbulent magnetic fields, something that theoreticians think is possible when CR excite magnetic turbulence at shocks while being accelerated. This is called magnetic field amplification.

In that case, the diffusion coefficient depends linearly with energy (![$\alpha = 1$](https://render.githubusercontent.com/render/math?math=%5Calpha%20%3D%201\&mode=inline)) and the equation above can be rewritten as:![$$E\_{max} \leq 3 \frac{u\_1}{c} Z e B (u\_1 t\_{age})$$](https://render.githubusercontent.com/render/math?math=E_%7Bmax%7D%20%5Cleq%203%20%5Cfrac%7Bu_1%7D%7Bc%7D%20Z%20e%20B%20%28u_1%20t_%7Bage%7D%29\&mode=display)

where ![$t\_{age}$](https://render.githubusercontent.com/render/math?math=t_%7Bage%7D\&mode=inline) is the time in which the accelerator is working. Note that the product ![$(u\_1 t\_{age})$](https://render.githubusercontent.com/render/math?math=%28u_1%20t_%7Bage%7D%29\&mode=inline) is the radius of a expanding shell. Using some estimates on the time (![$t\_{age} \sim$](https://render.githubusercontent.com/render/math?math=t_%7Bage%7D%20%5Csim\&mode=inline) 1000 yrs as the typical SN shockwave) and ![$B\_{ISM} \sim 3 \mu$](https://render.githubusercontent.com/render/math?math=B_%7BISM%7D%20%5Csim%203%20%5Cmu\&mode=inline)G we can rewrite the maximum energy for SN shockwaves as:![$$E\_{max} \leq Z\times 3 \times 10^4 {\rm GeV}$$](https://render.githubusercontent.com/render/math?math=E_%7Bmax%7D%20%5Cleq%20Z%5Ctimes%203%20%5Ctimes%2010%5E4%20%7B%5Crm%20GeV%7D\&mode=display)

In order words, the shock-wave acceleration shown can accelerate CR **up to 100 Z TeV**, but not beyond this. Other acceleration mechanism are needed for the highest energy cosmic rays. We need very high magnetic fields (non-lineal acceleration mechanism). In these cases, even if this object cannot supply the all the galactic cosmic rays the energy per particle may be orders of magnitude higher than those possible in shock acceleration by blast waves.