Физика космических лучей / HEA11
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HIGH ENERGY ASTROPHYSICS - Lecture 11
PD Frank Rieger
ITA & MPIK Heidelberg
Wednesday
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Fermi Particle Acceleration
1Overview
Non-thermal particle acceleration - Fermi picture
2nd order Fermi mechanism
Acceleration rate & timescale
Particle Spectra
Fokker-Planck Equation
Scattering agent (plasma waves)
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2Non-thermal particle acceleration { Fermi picture
E. Fermi (1949):
Particle acceleration as consequence of elastic scattering o randomly moving magnetic clouds ("mirrors").
Principle:
Magnetic "clouds" de ect (elastically scatters) charged particles.
Clouds have random velocity in ISM.
Scattered/de ected particles gain energy in head-on collisions and lose energy in following collisions.
On average head-on collisions are more probable than following collisions.
) energy gain on average.
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Ef < Ei |
Ef > Ei |
Vc
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Magnetic cloud |
Ei |
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3Energy change of particles in elastic scattering events
Energy change E for particle with initial Ei and p~i interacting with massive
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cloud of speed Vc:
Elastic Scattering: In cloud frame K', particle energy is conserved, momentum
~ ~ ' E direction parallel to Vc reversed (noting pk = p~ Vc=Vc = p cos c2 v cos )
Lorentz-Transformation to cloud frame K' (cf. time and space trafo):
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Ei |
= c(Ei p~iVc) = c(Ei pi;kVc) |
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pi;0 k |
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Vc |
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Elastic scattering in frame K' implies: |
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Ef0 = Ei0 |
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pf;0 |
k = pi;0 |
k |
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Transforming back to lab. frame K:
Ef = |
c(Ef0 + pf;0 |
kVc) = c(Ei0 pi;0 kVc) |
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Vc2 |
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Vc |
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[Ei pi;kVc] pi;k |
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Ei |
Vc |
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= c |
1 + |
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Ei 2pi;kVc |
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c2 |
c2 |
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Energy change E:
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Ei 2pi;kVc |
Ei |
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E = Ef Ei = c |
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c2 |
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= ( c |
1)Ei + c |
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V 2 |
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c2 Ei 2pi;kVc |
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c |
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Vc2 |
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= 2 c |
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Ei pi;kVc |
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c2 |
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noting that ( c2 1) = c2 c2.
Fractional energy change for relativistic particle to rst order Vc=c 1 ( c 1):
E |
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Vc |
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cos Vc |
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V |
cos |
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i;k |
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Ei |
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V |
c |
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' 2 |
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= 2 |
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= 2 |
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= 2 |
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Ei |
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) Energy change E is proportional to initial energy (j~vij ' c).
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Gain or loss, depending on directions of ~vi and Vc. (loss for following cos 0, gain-for head-on cos 0).
) Energy change independent of charge and B.
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4Second order Fermi acceleration
Full picture: momentum directions are isotropized in cloud.
Ef , pf
out
Vc
in
Ei , pi
Magnetic cloud
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Average energy change for relativistic particles, E ' pc, taking interaction probability into account:
Initial particle energy as seen in cloud frame ( c = Vc=c):
Ein0 = c(Ein pinVc cos in) ' cEin(1 c cos in)
In cloud, direction is randomized, so in lab frame K:
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Eout = cEout0 (1 + c cos out0 ) |
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Elastic scattering in K0 |
implies E0 |
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in |
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Eout = c2Ein(1 c cos in)(1 + c cos out0 |
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Fractional change with E := Eout Ein: |
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E |
(1 |
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cos |
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out |
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Ein |
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1 c2 |
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1 c cos in + c cos out0 c2 cos in cos out0 (1 c2) |
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(cos 0 |
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1 c2 |
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cos 0 ) |
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cos |
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out |
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1 c2 |
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For ensemble of particles:
By assumption, escaping particles are isotropized in cloud frame K0, i.e.
<cos out0 >= 0
For cloud with velocity Vc, number of incident particles at in is:
dN / d / 2 sin ind in = 2 d cos in
During time t, number of particles reaching cloud is proportional to relative
velocity (cf. relativistic velocity addition for c 1): |
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dN |
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/ (v Vc cos in) t ' (c Vc cos in) t |
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d |
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for relativistic particles with v ' c |
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Angle average for incoming particles |
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< cos in > = |
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R 1 |
1 |
(cin |
Vc cos in) d cos in |
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(c Vc cos in) d cos in |
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11Rc cos in d cos in 11 Vc cos2 |
in d cos in |
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2Vc=3 |
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1 |
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R |
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11(c Vc cosR in) d cos in |
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2c |
3 |
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for more particles coming from front. |
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) Preference on average R |
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9
Thus averaged fractional energy change (see slide 8, eq. 1):
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(< cos 0 |
> < cos |
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E |
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c |
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1 c2 |
in |
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c2 c2 < cos in >< cos out0 > |
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1 c2 |
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1 |
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c2 |
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c2 |
4 c2 |
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3 |
(1 c2) |
(1 c2) |
3 (1 c2) |
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=43 c2 c2
For typically assumed low cloud speeds c 1:
E |
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c2 |
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E |
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Consequences:
) Stochastic process: Energy change on average.
) Average energy change is positive, particle gains energy < E >/ E. ) Average gain is of second order in Vc, i.e., < E >/ Vcc 2
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