UHECRs and Multiple Shock Acceleration in Active Galactic ... - TAUP
UHECRs and Multiple Shock Acceleration in Active Galactic ... - TAUP
UHECRs and Multiple Shock Acceleration in Active Galactic ... - TAUP
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Diffusive shock acceleration<br />
• Test particle - diffusion - n acceleration shock cycles<br />
E<br />
n ( x + 1) ⋅ E0<br />
• Energy ga<strong>in</strong>: fraction of <strong>in</strong>itial energy<br />
• Average energy ga<strong>in</strong> per collision:<br />
n<br />
=<br />
ΔE<br />
= E − E = x⋅<br />
E<br />
< ΔE<br />
/ E >≅ ( 2V<br />
/ c)<br />
• Lead<strong>in</strong>g to a power-law energy behaviour<br />
∞<br />
i=<br />
n<br />
n(<br />
E)<br />
−σ<br />
( 1−<br />
P ) = ... ∝<br />
N(<br />
> E)<br />
= ∑ E<br />
σ = (r+2)/(r-1), r = V1/V2 = (γ +1) / ( γ -1)<br />
esc<br />
for mono-atomic gas:<br />
γ =5/3 r = 4 E -2<br />
Note: for non-relativistic shocks σ is ‘constant’ ,<br />
<strong>in</strong>dependent of scatter<strong>in</strong>g media or shock <strong>in</strong>cl<strong>in</strong>ation<br />
0<br />
0<br />
Probability of scatter<strong>in</strong>g x av.no. scatter<strong>in</strong>gs x ΔE<br />
Upstream Downstream<br />
(e.g. Krymskii ‘77, Bell ’78, Drury ’83 )
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