Astrophysical Jets from accreting Black Holes

II.2 Diffusive shock acceleration
● In reality p is a random variable, but for v >> U and initial
momentum identically
n
pn~ i=1 p0[1 4
3
p 0
U 1−U 2
v i
]⇒ ln pn ~
p0 4
3 U n
1−U 2 i=1
● The probability of crossing the shock front n times is
n
Pn~ i=11−
4U 2
n
⇒ ln P
v
n~−4U 2 i=1 i
⇒ P n= p −3U 2 /U 1−U 2 n
=
p0 p −3/r −1
n
p0 1
v i
1
=−3
vi U 2
ln
U1−U 2
pn
p0