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Particle Acceleration in
Compact Objects
Demosthenes Kazanas
NASA
Goddard Space Flight Center
There is plenty of evidence for the presence
of particle acceleration in compact
objects:
i.
High (and low) energy emission from
pulsars.
ii.
High (and low) energy emission from
plerionic SN remnants.
iii. Emission from 109 – 1027 Hz in Active
Galactic Nuclei.
Outline
Direct particle acceleration by electric fields (in
EM gaps).
Bulk acceleration of particles in MHD flows.
Stochastic Acceleration (shocks, turbulence).
Dynamic Effects of accelerated particles
(effects on accretion disks, outflows).
The Seven Highest-Confidence Gammaray Pulsars
Broad-band spectra



Power peaked in g-rays
No pulsed emission
above 20 GeV
Increase in hardness
with age

High-energy turnover
Increase in hardness
with age

Thermal component
appears in older
pulsars
 Must
distinguish between acceleration of
individual particles and acceleration in
bulk.
 These two are generally distinct
processes, however, there are cases in
which they are intimately related.
 The most obvious evidence of the
presence of acceleration of particles is
that of pulsars.
Rotating Magnetic Field
Formation of an outflow
The rotation of the highly conducting
neutron star crust generates enormous
potential differences over the surface
 E || B above the surface => very large
outward-acting unbalanced electric stresses
 A charged magnetosphere is
spontaneously built up in order to shortout the parallel component of the electric
field (Goldreich-Julian 1969): 

GJ
The density of charge carriers can be
easily estimated:
nGJ
BLC
  BLC
E



e4
RLC e4
ec 4
This density co-rotates with the pulsar out to
the light cylinder. Beyond that the magnetic
stresses cannot confine the plasma and must
open-up, i.e. the dipole is not a valid
magnetospheric solution
The flux of the open field lines at the
light cylinder defines the polar cap of
the
pulsar, as the region of the last open
field line
B* R
2
pc
 BLC R
2
LC
1/ 2
 R* 

R pc  R* 
 RLC 
The pulsar slow down can then be
worked out in a simple way. It does not
require the pulsar to be misaligned; the
same slow down works out for a purely
aligned magnetosphere.
I  R n c  B R
2
pc GJ
2
2
*
v
 R* 
V  R pc B  R pc 

c
 c 
W  I V  R*6 4 B 2
3/ 2
 R* 
B
 R* B
 c 
2
Surface Fields and Currents
The presence of a sufficiently dense plasma
cancels all parallel E. Discrepancy between
the actual charge density from that of GJ
leads to gaps (Polar Cap; Outer Gap models).
Particles can be accelerated at gaps and lead to
the creation of photons. The resulting spectra
depend on the ensuing interactions (A. Harding)
The EM potentials available are of order of
1018 (P/ms) B15 eV. As such they
could produce galactic cosmic rays up to the
energy of ankle (Arons 02).
The problem of pulsar magnetospheric
emission, as is the case with all problems
that involve magnetic fields (which cannot
be shorted out) is a global one. One has to
solve for the currents and the resulting
magnetic fields over all space before we can
decide the dynamics and radiation emission
from a pulsar.
The magnetosphere is determined by the
balance between the current and electrostatic forces in the magnetosphere. These
are given by the “Pulsar Equation”
   1    

2
(1  x ) 2 
 2   2x
  RLC
AA
x x z 
x
 x
2
2
2
The parameters involved are
Poloidal electric
current:
 Magnetic flux:
 Force-free:
 Space charge
density:

A  A( )  RB

1
c
J  B  e E  0
    2 Bz  AA
e  

2
4

c
1

x


Contopoulos, Kazanas & Fendt 1999; Gruzinov 2005
The solution is smooth, contains a return
Current, it contains a zero charge line and
it provides the wind asymptotic structure.
Emission is expected at places where MHD
is violated (polar cap, zero charge line,
return current boundary, but not the
Light Cylinder).
Pulsar Winds/The s-Problem
For the geometry of the magnetic lines beyond
the Light Cylinder (split monopole) for which
Bp= 1/R2, B= 1/ R ,  = 1/R2.
Therefore their ratio, s ~106 near the LC should
be independent of the radius R .
However, the spectra of the Crab nebula need a
value s ~3 10-3 to fit the observed spectrum and
for Vela one needs s ~1.
The asymptotic (split) monopole geometry
of CKF allows a crack at this problem:
The energy conservation equation along a field
line has the form:

R v 
  g *
g  1 
RLC c 

While the flux freezing condition reads
v
v p B
R


c
RLC
c Bp
Under force-free conditions
the energy equation reads
R
B  
Bp
RLC
  R
g  1  
  RLC



2
 vp
1 
c


  g *

Leading eventually to:
 2
R 
g  g *  2 
RLC 

2
(Contopoulos & DK 2002)
1/ 2
R

RLC
Under conditions of a monopole geometry the Lorentz
factor of the flow increases linearly with distance. This
happens as long as the effects of inertia are negligible.
Beyond this point the field geometry should deviate
From monopolar and possibly part of it collimate and
part form an equatorial wind. The wind terminates at a
shock which is responsible for the nebular emission.
(The extent of monopole geometry is debatable. It may
extend only up to the fast magnetosonic point; then the
maximum g will be only ~s1/3 ).
Plerion Components
rN
TORUS
rs
Vexp
SHOCK
JET
Vlahakis & Konigl 2001

Linearly increase in
Lorentz factor is a
property of general
MHD flows of
geometries different
from monopolar
(Vlahakis&Konigl
2001)
The MHD outflow acceleration and the s-Problem are related
issues. They demand the simultaneous solution of the
conservation equations along with the transverse force balanc.e
First axisymmetric wind
solutions by Blandford
& Payne; extended to
Relativistic case by Li,
Chieuh, Begelman (92)
and Contopoulos (94).
Solutions known only for
self-similar geometry.
Flow acceleration
depends on assumptions
used. LCB find logarithmic
acceleration with height.
Contopoulos (94) finds final
velocity similar to that at
the accretion disk at the
base of the flow (Vlahakis
& Konigl 04 for a more
recent study).
The relativistic outflows produce shocks, which
accelerate particles and lead to radiation emission. Blazar
emission is thought to be derived this way.
The apparently thin
photon spectra indicate
emission from large
distances and suggests
association with jet
flows (Mastichiadis &
Kirk 1997).
Particle acceleration (in shocks, converging
flows, turbulence) is the result of an
interplay between particle energy gains in
scattering and particle transport. The
exponentially small probability of
undergoing N interactions with the plasma
before escape, coupled with exponentially
increasing energy with the number of
scatterings lead to power law distributions.
The geometry of particle transport across a plane shock. The
upstream velocity is u1 and the downstream u2=u. The particle
velocity is v. The shaded region shows the fraction of particles that
make it upstream and have a chance to accelerate.
Generic description of the acceleration process.
Application to plane parallel shocks (r is the compression ratio, P(p) is the integral spectrum).
Probability of N returns:
P(N ) / e¡N®
Energy of particle after N returns:
p(N )=p0 / eN¯
µ ¶
µ ¶
µ ¶¡®=¯
1
p
®
p
p
N = Ln
; Ln[P(p)] = ¡ Ln
; P(p) =
¯
p0
¯
p0
p0
For a plane parellel shock (Jones & Ellison 1991)
®
3u2
=
;
¯ u1 ¡ u2
µ ¶¡3u2=(u1¡u2) µ ¶¡3=(r¡1)
p
p
P(p) =
=
p0
p0
Effects of acceleration on dynamics
The presence of relativistic particles
can affect the dynamics of the flow:
Relativistic particles reduce the fluid
adiabatic index and increase the shock
compression ratio r. This hardens the
spectra; most kinetic energy is
converted to relativistic particles that
dominate the pressure.
Particle (relativistic) escape from the
system also increases the compression
ratio of the shock with similar effect.
(Ellison et al 2000)
In the vicinity of a compact object, the strong gravitational
field could separate the relativistic and the non-relativistic
populations, provided that cooling does not ; this can cause
outflows similar to those inferred in compact objects (DK &
Ellison 86); Subramanian et al (99), provided that the
accelerated particles do not lose energy on time scales shorter
than free-fall.
Separation can also take place through the production of
neutral particles (neutrons) that can increase the power of
relativistic outflows (Contopoulos & DK 94).
Plasma production outside an Acc. Disk from n -> p e. For a large
black hole, most neutron produced protons are relativistic while for a
small one most are non-relativistic. The critical value is M~108 M_o
The Radio Jets of GRS 1915+105
The Radio Jets of GRS 1915+105
The Radio Jets of GRS 1915+105
Acceleration in Accretion Disks can
result from particle-wave interactions
(e.g. Dermer, Miller, Li ’96). Acceler.
Time scales are quite short and should
Produce accelerated populations.
Accretion Disks could accelerate
particles by their shearing motion
(Subramanian et al. ’99). This leads
to 2nd order acceleration.
Slope and Maximum Energies


The slope of accelerated
population depends on the
interplay between energy
gain per interaction and
escape probability (e.g. the
Comptonization parameter t
kT/mc2). For shocks this is
3/(r-1) (integral slope).
The acceleration rate is happening on the gyro-period at
the given field ~
E(eV)/B(G)


Maximum energy is given by
the balance between acceleration and losses or escape
from the system. For
electrons this energy is ~TeV
(blazars), while for protons it
gets close to 1020 eV.
Eventually, the max.energy is
roughly ~R (v/c) B, where R
is the size of the system, v the
velocity and B the magnetic
field.
Conclusions - Questions






Particle Acceleration is a ubiquitous process in compact objects (spectra,
superluminal motions).
Particles can get accelerated in EM gaps (deviations from MHD
conditions). Energy/particle ~ Potential drop across gap.
MHD acceleration in rotating magnetospheres. Conversion of magnetic
to kinetic energy of high efficiency (depends on current distribution).
Lorentz factors of ~10 – 106 possible.
Particle acceleration possible in turbulent, shocked plasmas. Conversion
of KE to relativistic particles with high efficiency. Max. energy depends
on particulars of system.
Why don’t we see prominent non-thermal emission in the spectra of
accreting binary sources? Why are most AGN radio quiet?
Does acceleration take place in the Acc. Disks of AGN, GBHC? If yes,
do the accelerated particles play any role in the dynamics of these disks?
Are observational tests to distinguish between these possibilities?
The geometry of particle transport across a shock. The upstream
velocity is u1 and the downstream u2=u. The particle velocity is v.
The shaded region shows the fraction of particles that make it
upstream and have a chance to accelerate.
tion (BP81, Eq. 18) for the photon occupation number n(r; º
1
c
1
@n
vb ¢ rn + r ¢ ( rn) + (r ¢ vb ) º
= ¡~j(r; º);
3
·
3
@º
; º) (PB81) is given in terms of n(r; º)
1
1
@n
F (r; º) = ¡
rn ¡ vb º ;
3· (r)
3
@º