Cosmic ray energy and pressure

The pressure $P_c$ and energy density $U_c$ associated with the isotropic cosmic ray moment distribution are given by

\[ P_c(x,t) = \int_0^{\infty} \frac{4\pi}{3} vf_0 dp \]

\[ U_c(x,t) = \int_0^{\infty} 4\pi p^2 \epsilon f_0 dp ,\]

where 

\[ \epsilon = (\gamma-1)mc^2 \qquad v=\frac{p}{\gamma m} \qquad \gamma = \sqrt{\frac{p^2}{m^2c^2}+1}. \]

a. The lower limit ($0$) should be the injection momentum $p_{inj}$.

b. The upper limit ($\infty$) may cause divergence.

c. Imposing cutoffs at high and low momentum is necessary.

d. Generally, the definition for pressure tensor is $\int m(v_i-\bar{v_i})(v_j-\bar{v_j})f_0(v,x)d^3v$. For isotropic distribution function $\bar{v_i}$ is zero and $\bar{v_i v_j}=0$. We recover the above definition.

e.g. 

\[ \int v_xv_y f_0 d^3v = \int v\sin(\theta)\cos(\phi) v\sin(\theta)\sin(\phi) 2\pi v^2 \sin(\theta)d\theta d\phi =0 .\]