High-Energy Astrophysics Lecture 4: Particle acceleration and ...

High-Energy Astrophysics
Lecture 4: Particle acceleration and
energy loss
Robert Laing
Principal energy loss processes
Adiabatic losses (particles expand and do p dV
work). Loss rate ∝E:
-(dE/dt) ad = (∇.v)E/3 (ultrarelativistic limit)
Electron energy distribution n(E)dE = n 0 E -k dE.
How does it evolve with time?
Synchrotron and inverse Compton losses per
electron for isotropic pitch angle distribution:
-(dE/dt) sync = (4/3)σTcγ2umag -(dE/dt) iC = (4/3)σ T cγ 2 u rad
Same mathematics (both loss rates ∝E 2 ).
Adiabatic losses (relativistic)
Ultrarelativistic gas has a different equation of
state:
dU = -pdV
U = 3nkTV
p = U/3
dU = nVdE = -(1/3)nEdV
dE/dt = -(1/3)(nE/N)dV/dt
For a (non-relativistic) velocity field v,
dV/dt = (∇.v)V, so
dE/dt = -(1/3) (∇.v)E
Particle acceleration and energy loss
How does the energy spectrum of synchrotronemitting
electrons evolve? Synchrotron, inverse
Compton and adiabatic losses.
Where must particle acceleration take place?
Fermi acceleration at strong shocks
Other acceleration mechanisms
Relation to cosmic rays
Adiabatic losses (non-relativistic)
Non-relativistic gas:
dU = -pdV
U = (3/2)nkTV
p = nkT
dU = nVdE = -(2/3)nEdV
dE/dt = -(2/3)(nE/N)dV/dt
For a velocity field v,
dV/dt = (∇.v)V, so
dE/dt = -(2/3) (∇.v)E
Adiabatic losses in a spherical expansion
Ultrarelativistic particles (e.g. in a supernova
remnant)
Uniform expansion v = v0 (r/r0 )
∇.v = 3(v0 / r0 )
dE/dr = -E / r0 E = E0 (r0 / r)
Compare the non-relativistic case:
E = E0 (r0 / r) 2
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Adiabatic losses: consequences
Adiabatic losses cause rapid dimming of
synchrotron emission as sources expand.
Therefore, bright sources (supernova remnants,
radio galaxies) cannot just be expanding plasma
clouds: energy must be transported in a loss-free
way to the region where it is dissipated.
Adiabatic losses in a jet - consequences
The surface-brightness (∝jr) is predicted to fall
rapidly with distance. This is not observed.
Either jets decelerate, or particles must be
accelerated, or both.
For a decelerating, relativistic jet, adiabatic losses
only:
j ∝(γβ) -(5α/3 + 2) r -(7α/6 + 3) (Bperp )
j ∝(γβ) -(2α/3 + 1) r -(10α + 9)/3 (Bpar )
Synchrotron and inverse Compton
lifetimes
Characteristic lifetime τ = E/(dE/dt)
Synchrotron
τ = E/ (4/3)σTcγ2umag = 3.3 x 107 (B/nT) -3/2 (ν/GHz) -1/2 years
Some examples:
Cygnus A hot-spots, 5GHz: τ ≈ 104 years
M87 knots, X-ray: τ ≈ 10 years
Maximum lifetime for inverse Compton losses
(locally):
τ = E/ (4/3)σTcγ2urad = 2.3 x 1012 / γ years
Adiabatic losses in a jet
Conical jet with constant (non-relativistic) bulk
velocity v:
E-1 dE/dt = -(2/3)v/r
E = E0 (r0 / r) 2/3
Energy spectrum evolves:
K0 E -(2α+1)
0 dE0 r 2
0 = K E-(2α+1) dE r2 K = K0 (r / r0 ) -(4α/3 + 2)
Magnetic field B par ∝r -2 or B perp ∝r -1
Emissivity j ∝ KB 1+α ∝r -(7α/6 + 3) = r -3.6 (α = 0.5; B perp )
∝r -(10α+9)/3 = r -4.7 (α = 0.5; B par )
Surface-brightness variation along a jet
Synchrotron radiation losses
Observations
Adiabatic evolution
Brightness decrease
along the jet is far
too rapid
Synchrotron and inverse-Compton
losses cause spectral steepening at
high frequencies.
Adiabatic losses cause the spectrum to
shift in the log S - log ν plane
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M87 jet UV-IR
0.3μm
0.8μm
1.6μm
Optical synchrotron emission from a hotspot
in a powerful radio galaxy
Synchrotron lifetimes: consequences
If τ 10 20 eV
They must function both in the Galaxy and in
extragalactic sources.
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Cosmic-ray energy spectrum Fermi acceleration at strong shocks
The best-understood acceleration mechanism.
Shocks are formed in:
The inner regions of jets (superluminal
components)
Hot-spots (where jets terminate)
Elsewhere in powerful jets
Supernova remnants
But probably not in low-power, decelerating jets
(which appear to be transonic)
3C279: images Supersonic jet in a radio galaxy
Supersonic jet in a quasar
A shell supernova remnant bounded by a
strong shock
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Transonic jet in a weak radio galaxy:
X-ray/radio overlay
Fermi acceleration: basics
Fundamental idea is that a particle gains energy
when it crosses a shock, and that it can be
scattered so that it does so many times.
Consider non-relativistic shocks and thermal
plasma (ratio of specific heats = 5/3).
Assume shocks perpendicular to fluid flow.
Material passing through a shock becomes hotter
and denser (I.e. KE is converted into internal
energy).
Shock propagates at speed V into stationary fluid
(e.g. ISM around a supernova remnant).
Strong shocks
Assume fluid behaves like a perfect gas in the limit
where the post-shock fluid is much hotter than the
pre-shock fluid (a strong shock). Then the ratio of
the pre- and post-shock velocities is:
v2 /v1 = (γH - 1)/ (γH + 1) = 1/4 for a non-relativistic
fluid (γH = 5/3). v1 = V in our case.
Transonic jet: radio-X-ray jet spectrum
Shock geometry
In pre- and post-shock frames
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Energy gain on shock crossing
Particles in post-shock fluid with velocities v >> V.
A fraction of these can cross back into the preshock
fluid.
Energy of one of these particles in the pre-shock
frame is
E’ = γ(E +px V) = E +px Vrel if non-relativistic
where γ is the Lorentz factor, px is the momentum
component normal to the shock and Vrel = 3V/4 is
the relative velocity.
Change of particle energy beween rest frames is
ΔE = p Vrel cos θ where θ is the angle w.r.t. shock
normal. ΔE > 0 for particle to cross shock.
Average energy gain
Assume isotropic. Speed ∝cosθ, so probability
distribution of angles for particles crossing the
shock (0 < θ < π/2) is:
and average energy gain is
Energy spectrum
Number of crossings
Number with energy > E n
Number with energy E n
Energy gain on shock crossing - 2
If the particles are already mildly relativistic, E = pc
and ΔE/E = V rel cos θ / c.
Key assumption: there is an elastic scattering
process which isotropises the particle
distribution in the pre-shock fluid with no loss of
energy.
Therefore, a fraction of particles can be rescattered
back into the post-shock fluid. Repetition
of this process causes a fraction of particles to
undergo many shock crossings and therefore to
attain high energies.
Multiple shock crossings
Energy gain for a double crossing is ΔE/E = V/c;
mean particle energy increased by factor β = 1 +
V/c.
Suppose probability of scattering to achieve
another double crossing is p. Then probability of n
crossings is p n , after which energy is increased by a
factor β n .
Start with N 0 particles with energy E 0 . A fraction p n
is accelerated up to at least energy E n = β n E 0 , so
N(> E n ) = N 0 p n .
The scattering probability
Downstream of the shock, fluid moves away from
the shock at speed V/4. Rate of particle loss is
NV/4 (N is the total number density). Particle speed
across the shock = vcosθ and probabilty distribution
of angles is p(θ) = (1/2)sinθ (isotropy) so the rate at
which particles enter the downstream region is
and fraction leaving shock region is
Hence p = 1 - V/c and ln p ≈ -V/c
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Final energy distribution
N(E) ∝ E (-V/c)(c/V)-1 = E -2
This corresponds to the flattest spectrum observed
in optically-thin regions (frequency spectral index α
= 0.5).
Synchrotron and inverse Compton losses tend to
steepen the spectrum.
Relativistic shocks (gamma-ray bursts, parsecscale
extragalactic jets; pulsar-driven SNR)
produce N(E) ∝ E -2.1
Limits on the maximum energy
Acceleration timescale < shock lifetime.
Acceleration timescale < synchrotron lifetime τ =
E/(dE/dt)
Scattering mechanism fails at high energies (e.g. if
gyro-radius r G = γmv/eB is bigger than the size of
the acceleration region).
Other acceleration processes
Direct acceleration by coherent electromagnetic
fields. Relevant near rotating compact objects
(certainly pulsars, perhaps black holes).
Stochastic acceleration by MHD turbulence.
Relevant in transonic and subsonic flows?
The scattering mechanism
Scattering off Alfven waves (plasma sound waves
excited by particles streaming faster than the Alfven
speed - the effective sound speed in a magnetised
plasma).
Tangled magnetic field on either side of the shock
(recall that a tangled, but anisotropic, magnetic field
can still produce high polarization).
Complications
Oblique shocks - essentially the same as
perpendicular ones.
Back-reaction of high-energy particles on the
shock - not yet solved.
Injection - particles have to be at least mildly
relativistic before this mechanism becomes
efficient.
Time-dependence - are the injection and escape
rates sufficiently constant?
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