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Friday, May 22, 2020

Cosmic ray energy and pressure

The pressure Pc and energy density Uc associated with the isotropic cosmic ray moment distribution are given by

Pc(x,t)=04π3vf0dp
Uc(x,t)=04πp2ϵf0dp,
where 
ϵ=(γ1)mc2v=pγmγ=p2m2c2+1.

a. The lower limit (0) should be the injection momentum pinj.
b. The upper limit () may cause divergence.
c. Imposing cutoffs at high and low momentum is necessary.
d. Generally, the definition for pressure tensor is m(vi¯vi)(vj¯vj)f0(v,x)d3v. For isotropic distribution function ¯vi is zero and ¯vivj=0. We recover the above definition.
e.g. 
vxvyf0d3v=vsin(θ)cos(ϕ)vsin(θ)sin(ϕ)2πv2sin(θ)dθdϕ=0.

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