Физика космических лучей / HEA11

5Characteristic acceleration timescale & rate for 2nd order Fermi

Acceleration as result of repeated collisions with "clouds":

Mean time between two collision = "scattering time": s

Average distance between two interactions (mean free path):

For energetic particle (v ' c) mean scattering time:

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6Evolution of Particle Spectra - recap

Remember kinetic equation derived in lecture 6:

n(E; t) dE number density of particle in interval dE at time t,

Q(E; t) injection term,

(E) := dEdt describing energy loss or gain (acceleration)

) for synchrotron losses (ignoring gains): dE=dt = c1E2

) for 2nd order Fermi (ignoring losses):

(E) = dEdt

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7Particle Spectra with Escape

Evolution of particle distribution assuming particles to be accelerated by 2nd order Fermi and to escape on energy-independent timescale esc:

Kinetic equation with escape term:

In steady-state @n(E; t)=@t = 0:

Solution for particle spectrum is a power-law:

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Caution: The above treatment is only approximately valid (i.e., in the limit of small tacc= esc) and hides broadening/momentum dispersion < ( p)2= t > that accompanies the energy increase in this stochastic process. The latter can be correctly obtained within a Fokker-Planck approach:

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Fokker-Planck Equation

Widely used for description of stochastic processes.

P (p;~ p~) := probability that particle with momentum p~ at time t changes momentum by p~ in time t.

f(~x; p;~ t):=phase space distribution function = probability to nd particle in phase space volume element d3xd3p.

Can write:

Z

f(p;~ t + t) = f(p~ p;~ t) P (p~ p;~ p~) d3( p)

Taylor expansion gives (to 2nd order in small p):

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Imposing normalization:

Z

P (p;~ p~) d3( p) = 1

and de ning "Fokker-Planck" coe cients:

Z

< pi > := P (p;~ p~) pi d3( p)

Z

< pi pj > := P (p;~ p~) pi pj d3( p)

) Fokker-Planck equation:

with 1st term (rhs) describing systematic energy gain/losses, and 2nd di u- sion part/dispersion/broadening.

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"Principle of detailed balance": if scattering process is reversible in the sense that:

P (p;~ p~) = P (p~ p;~ p~)

Integrating over d3( p) gives (treating p and p as independent variables):

Fokker-Planck eq. then reduces to a di usion equation in momentum space:

where

Dij := < pi pj >

2 t are the components of the di usion tensor.

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Example: Stochastic 2nd order Fermi for isotropic di usion in momentum space with f(p~) ! f(p) (i.e., f(p~)d~p = 4 p2f(p)dp; use spherical coordinates], and with constant escape:

with Dp(p) = < p p> =: D0p2 momentum space di usion coe cient.

2 t

Thus, to properly incorporate di usion in kinetic eq. before (Sec. 7), an additional term / @E@ E2 @E@n needs to be added.

Steady-state solution assuming power law ansatz f(p) = f0 p s:

(choose positive solution to ensure nite particle number).

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We have < E= t >= (4=3)(c= ) c2E for relativistic particles (E ' pc). Using "detailed balance"-relation for Fokker-Planck coe cients:

)D0 = 13 c c2

Power law index for f(p) = f0p s becomes:

Di erential particle distribution: n(p) / p2f(p) / p2 s / p with

with expansion for large D0 esc. This is close to our estimate from the kinetic equation.

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82nd order Fermi acceleration and its limits

E cient Acceleration of CR by 2nd order Fermi in the Galaxy?

Slow process: Acceleration time too long

) ISM clouds < Vc > 10 km/s, i.e. c 10 4, so < E=E >' c2 10 8 ) Distance or mean free path L 1 pc, so s L=c a few years.

) Acceleration timescale:

tCR=typical CR con nement time (leaky box) in the Galaxy.

Injection Problem:

) Coulomb losses independent of E, dominate at low energy

) Protons have to be injected with kinetic energy of a few 100 MeV ) need pre-acceleration/injection mechanism.

Non-universal index: power law index s sensitive to tacc and esc ) tacc depends on c

) esc depends on density of scattering clouds

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