Физика космических лучей / HEA12
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13Energy gain per cycle
Average fractional energy change per crossing (cf. lect. 11, now c ! v1 c v2 ):
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(< cos 0 |
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2< cos |
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> < cos 0 > |
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in |
1 c2 |
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Have di erent angular average now (compared to cloud - 2nd order Fermi):
{Rate of particle crossing dN=d / c cos with cos < 0 (otherwise 0)
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0 |
cos in(c cos in) d cos in |
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) Angle average : < cos in >= |
R 1 |
01(c cos in) d cos in |
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and similarly for returning cycle part (/ cos 2;in for cosR2;in > 0 etc). |
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< cos 0 |
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{ Momentum direction is isotropised in each frame |
>= 0. |
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2 v1 v2 |
(non- |
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Average fractional energy change per crossing < E=E >' 3 |
c |
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relativistic case), so per cycle (crossing & re-crossing) |
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E |
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4 |
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v1 v2 |
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Energy change is positive, E / E, gain is rst order in c
) 1st order Fermi acceleration
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14Accelerated particle spectrum
Average energy gain per cycle:
<E >= 4 v1 v2 E := E
3 c
Energy after n cycles given injection with E0:
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= (1 + )nE |
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Number of cycles needed to reach En: |
= ln(1 + ) |
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n = ln |
E0 |
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En |
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Number of particles left after n cycle given nite escape probability:
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= N0 eln(1 Pesc) |
ln(E=E0) |
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N( E) = N0 (1 Pesc) |
ln(1+ ) |
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= N0 |
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for (v1 v2) c, i.e., 1, we can linearize (ln[1 + x] x): |
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ln(1 Pesc) |
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Pesc |
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4v1 |
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ln(1 + ) |
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(1 1=r) |
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ln(1 Pesc)
ln(1+ )
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Di erential spectrum is |
obtained by di erentiation, since |
N( E) = |
R N(E)dE: |
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N(E) / |
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r + 2 |
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with = |
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E0 |
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r 1 |
r 1 |
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Di usive shock acceleration produces power-law spectrum.
Spectrum only depends on compression ration r and not on v = v1 v2.
For a strong shock, r = 4 ) = 2.
Quasi-universal power-law!
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15Acceleration Timescale & Rate (non-relativistic shocks)
Energy gain per cycle:
< E >= E with = |
4 |
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v1 v2 |
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c |
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But duration of cycle? "Mean residence time in down/upstream"?
{depends on timescale for isotropization (di usion)
{average distance traveled by a particle?
cf. isotopic scattering with random walk: mean squared net displacement:
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> +2 < ~r1~r2 > +2 < ~r1~r3 > +:::::: 6= 0 |
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R = ~r1 + ~r2 |
+ ::::~rN ; < R >= 0, but < R |
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~ 2 |
>' N |
2 |
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with ld = p |
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< R |
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= root mean squared net displacement. |
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< R2 > |
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{ Spatial di usion coe cient (Einstein) [units: length2/time]: |
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1 |
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< R2 > |
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D := |
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lim |
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t |
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{Di usion length (=typical distance traveled by particle in available time ti):
p
) ld;i ' 2Diti
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In meantime, in upstream the shock has passed distance: l1 = v1t1 ) t1 = l1 ,
v1
in downstream plasma has been advected by l2 = v2t2 ) t2 = l2 .
v2
Hence, equating ld;i = li, di usion length scale upstream and downstream:
q
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ld;1 = 2D1t1 = |
2D1l1=v1 = |
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) ld;1 = |
2D1 |
and similarly; ld;2 = |
2D2 |
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v1 |
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Respective mean residence times for energetic particle: |
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ldi |
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noting tci = |
<vjj> |
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with < vjj >= R0 |
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v cos d cos = 2 v ' 2 c |
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t |
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2ld;2 |
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Cycle time: |
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tcycle = tc1 + tc2 = |
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c |
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= c |
v11 |
+ v22 |
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2ld; |
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2ld; |
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D |
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25
Characteristic acceleration time scale:
E E
tacc := < E=dt > = < E > tcycle
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3 c |
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E |
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4 |
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D1 |
D2 |
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4 v1 v2 E |
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c v |
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tacc = v1 |
v2 |
v11 |
+ v22 |
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26
16Estimate for acceleration time scale
Assumption: D1 ' D2.
Bohm limit for spatial di usion coe cient:
DB := |
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and ' rgyro = |
E |
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3 |
ZeB |
with E the particle energy.
) Minimum acceleration timescale for strong shock r = 4, v2 = v1=4:
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v1 v41 |
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v12 |
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3DB |
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3DB |
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20DB |
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tacc |
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20 |
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) tacc ' |
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ZeBc |
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17Example - SNR
For typical young supernova remnant shock, v1 104 km/s and B 20 G (ISM, without m.f. ampli cation).
Acceleration timescale to achieve particle energies E=100 TeV [SI] =160 erg [cgs] (required for VHE emission):
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tacc ' |
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100 yr |
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3 ZeB |
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Shocks can be e cient acceleration mechanisms!
Maximum particle energy Emax limited by
{energy losses (e.g., radiative)
{time: age of the system (CR knee - problem)
{particle escape (geometry)
Maximum energy depends on physical conditions of system.
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