Radiative losses and cut-offs of energetic particles at relativistic
  shocks
Mon. Not. R. Astron. Soc. 000, 1–10 (????) Printed 4 November 2018 (MN LATEX style file v2.2)
Radiative losses and cut-offs of energetic particles at
relativistic shocks
Paul Dempsey⋆ and Peter Duffy⋆
UCD School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland.
Accepted 2007 March 28. Received 2007 March 21; in original form 2007 January 24
ABSTRACT
We investigate the acceleration and simultaneous radiative losses of electrons in the
vicinity of relativistic shocks. Particles undergo pitch angle diffusion, gaining energy as
they cross the shock by the Fermi mechanism and also emitting synchrotron radiation
in the ambient magnetic field. A semi-analytic approach is developed which allows us
to consider the behaviour of the shape of the spectral cut-off and the variation of that
cut-off with the particle pitch angle. The implications for the synchrotron emission of
relativistic jets, such as those in gamma ray burst sources and blazars, are discussed.
Key words: relativistic shock acceleration, radiative losses.
1 INTRODUCTION
The role of radiative losses in determining the spectra from
non-thermal sources has been well understood in the non-
relativistic shock limit since the work of Webb et al. (1984)
and Heavens & Meisenheimer (1987). Their results were in
broad agreement with the natural expectation that there
would be a cut-off in the spectrum, at the shock, and at a
momentum where acceleration and loss timescales are equal,
with the shape of this cut-off depending critically on the
momentum dependence of the particle scattering. Subse-
quently, as the particles are advected downstream, and are
no longer efficiently accelerated by the shock, the spectra
steepens at momenta where the particles have had sufficient
time to cool. At a strong, nonrelativistic shock the differen-
tial number density of particles at energies where radiative
cooling is unimportant is a power law with N(E) ∝ E−0.5
with a corresponding intensity of Iν ∝ ν−0.5 for the emitted
synchrotron radiation. At higher momenta, where cooling
becomes important, the spectrum steepens so that the ra-
diation, beyond a break frequency νb, is Iν ∝ ν−1 up to a
critical frequency, νc corresponding to cut-off of the particle
spectrum. The position of νb depends on position away from
the shock; decreasing downstream as the particles have more
time to cool. The observed emission is therefore dependent
on the spatial resolution with which the source is observed
as discussed in Heavens & Meisenheimer (1987). The results
in the existing literature refer only to non-relativistic flows
and are of great use in analysing the spectra from super-
novae and the jets of some active galaxies (AGN). How-
ever, a number of objects of astrophysical importance, such
as AGN jets, microquasars and gamma-ray bursts, contain
⋆ E-mail: paul.dempsey@ucd.ie ; peter.duffy@ucd.ie
flows which have bulk relativistic motion and the purpose
of this paper is to examine the breaks, cut-offs and emission
for such sources.
While the first order Fermi process at relativistic shocks
contains the same basic physics as in the nonrelativistic case,
i.e. scattering leading to multiple shock crossings competing
with a finite chance of escape downstream, the anisotropy
of the particle distribution complicates the analysis consid-
erably (Kirk & Schneider (1987), Heavens & Drury (1988)
and Kirk et al. (2000)). The inclusion of self-consistent syn-
chrotron losses will, as in the nonrelativistic limit, modify
the spectrum at high momenta but we would also expect
pitch angle effects to become apparent in the position of
the cut-off and the emission itself. In order to motivate our
treatment of this problem we first discuss the nonrelativistic
shock limit in section 2, including the emission from a spa-
tially integrated source. Section 3 then presents the analysis
of synchrotron losses at relativistic shocks with particular
emphasis on the shape of the momentum cut-off. We con-
clude with a discussion in section 4.
2 NONRELATIVISTIC SHOCKS
The effect of synchrotron losses on the energetic particle dis-
tribution in the presence of nonrelativistic shocks is demon-
strated rigorously in Webb et al. (1984). However a simpler
approach is described in Heavens & Meisenheimer (1987)
provided synchrotron losses are not considered important
at the injection energies. We will follow this approach here,
although we shall introduce a slightly different definition of
the cut-off momentum.
In the presense of a magnetic field charged particles
emit synchrotron radiation with an energy loss rate given
c© ???? RAS
http://arxiv.org/abs/0704.0168v1
2 Paul Dempsey and Peter Duffy
= −asB2p2 = −λp2 (1)
where as is a positive constant. The radiative loss timescale
is therefore tloss = 1/(λp). In the steady state, and in the
presence of a nonrelativistic flow u, energetic particles obey a
transport equation describing advection, diffusion, adiabatic
compression and radiative losses,
= 0. (2)
In the presence of a nonrelativistic shock front where
the upstream flow speed is u− and that downstream is u+
the acceleration timescale is
tacc =
u− − u+
. (3)
At momenta for which tacc ≪ tloss the phase space den-
sity will be a simple power law with f ∝ p−s where
s = 3u−/(u− − u+).
2.1 Momentum Cut-off
The spectrum will steepen at momentum p∗ where
tacc(p
∗) = tloss(p
∗). In the case of momentum independent
diffusion this gives
u− − u+
. (4)
In the case of a relativistic shock this result no longer
strictly holds since the acceleration timescale defined above
is only valid for nonrelativistic flows. Nevertheless, we will
use this definition of p∗ throughout the paper for the sake
of comparison.
However, we require a general definition of the cut-off
momentum that can be applied in the relativistic limit. An
obvious alternative is to define the momentum at which the
local spectral index, ∂ ln f/∂ ln p, becomes s + 1 but, as we
shall see, it is necessary to perform a Laplace transform of
the transport equation to proceed with this problem and
it is more straightforward to define the cut-off in terms of
spectral steepening of the Laplace transformed spectrum.
In order to motivate such a definition we solve the nonrel-
ativistic shock acceleration problem in the presence of syn-
chrotron losses by first making the substitutions W ≡ p4f
and y ≡ 1/p so that the transport equation, either upstream
or downstream of the shock where adiabatic losses are zero,
becomes
(λW ) = 0. (5)
Taking the Laplace transform with respect to y
Ŵ (k, z) =
W (y, z) exp(−yk)dy (6)
and using the fact that losses prevent any particles achieving
infinite energy, i.e. W (0, z) = 0, the transformed transport
equation is
+ λkŴ = 0. (7)
in the case of a momentum independent diffusion coefficient.
Since the distribution function must be bounded infinitely
far upstream and downstream, the solution becomes
Ŵ± = A±(k) exp
1 + ω±k
where we have introduced
4λ±κ±
. (9)
The isotropic and anisotropic parts of the particle dis-
tribution function must match up at the shock giving,
f−(p, 0) = f+(p, 0) (10)
. (11)
Multiplying the isotropic boundary condition by p4, making
the substitutions as above and taking the Laplace transform
with respect to 1/p gives
Ŵ−(k, z = 0) = Ŵ+(k, z = 0) (12)
which in turn gives
A−(k) = A+(k) ≡ A(k). (13)
The flux continuity condition (11) becomes
A(k) = A0k
1 + ω−k − (s− 3)
1 + ω+k
1 + ω−k
1 + ω+k
In the absence of synchrotron losses, ω± = 0, we have
Ŵ±(k) ∝ k3−s which, upon inversion, gives f(p) ∝ p−s as
expected. We can therefore define a function, Q̂, by Ŵ =
k3−sQ̂. Recalling that k is the Laplace transformed variable
of inverse momentum we define the cut-off momentum, pcut,
to occur at the point where
∂ ln Q̂
∂ ln k
k=pcut
= −1. (15)
As an illustrative example, consider a power law distribution
with a sharp maximum momentum, f(p) ∝ p−sH(pmax − p)
with H the Heaviside function. In this case we have
W = y
y − 1
⇒ Ŵ = e−
pmax Γ(s− 3)k3−s
where Γ is the Gamma function. With Q̂ ∝ exp(−k/pmax)
we then have pcut = pmax as required physically in this sim-
ple case.
Returning to the solution of the shock problem we have
from equation 14
= exp
1 + ω−k − (s− 3)
1 + ω+k
1 + ω−k
1 + ω+k
. (17)
Defining
Ω = ω+/ω− (18)
(s2 + 2 s2Ω+ Ω2s2 − 6Ω2s+ 2Ω s+ 9Ω2 − 8Ω )
× (s− 3) (19)
c© ???? RAS, MNRAS 000, 1–10
Radiative losses and cut-offs of energetic particles at relativistic shocks 3
 1e−04  0.01  1  100  10000
Figure 1. The cut-off momentum, pcut as a function of Ω for
fixed equilibrium momentum, p∗ = 1, and spectral index, s = 4.
Salzer Sum
 0.001
 0.01
 0.01  0.1  1  10
Figure 2. Laplace inversion for s = 4 and Ω = 1. Using just M =
6 in the Salzer summation the inversion has already converged.
The first approximation M = 1 is exactly the Laplace function
Ŵk. We can see how fast the Salzer summation Post-Widder
inversion converges as M = 2 is a very reasonably approximation
to the actual solution.
gives
pcut =
s4(2 + 2Ω)− s3(5 + 11Ω) + s2(5 + 8Ω− 2χ)
(s2(1− Ω) + 6Ωs − 9Ω)2
s(33Ω + χ)− 36Ω
(s2(1− Ω) + 6Ωs − 9Ω)2
. (20)
This is always greater than p∗,
s+ (s− 3)Ω
as can be seen from figure 1. The minimum value for pcut
occurs for Ω = 1, and is given by pcut =
2(s−1)
The Laplace inversion (see Appendix for details) for s =
4 and Ω = 1 is shown in figure 2. Using just M = 6 in the
Salzer summation the inversion has already converged. The
first approximation M = 1 is exactly the Laplace function
Ŵk. We can see how fast the Salzer summation Post-Widder
inversion converges as M = 2 is a good approximation to the
actual solution.
Figure 3 shows how the particle distribution varies with
Ω; if Ω 6= 1 the cut-off is broader. While p∗ is independent
of Ω, pcut increases as the distribution broadens.
The data in figure 3 can be fitted by an exponential tail
Ω=3.2Ε7
Ω=1 Ω=0 Ω=3.2Ε7
log p
flog p4
 0.001
 0.01
 0.1  1  10
Figure 3. Particle Distributions of Various Ω
Ω p∗ pcut β
0 1 1.25 2.25
1 1 1 2.25
9 1 1.4 2
16 1 1.53 1.8
25 1 1.63 1.75
∞ 1 2 1.5
Table 1. Parameters for fitting particle spectra
to the distribution of the form
− (p/pcut)β
where β ∼ 2. Table 1 shows how β varies with Ω for a
shock of natural spectral index s = 4, with β attaining its
maximum value of 2.25, i.e. the cut-off is sharpest, when Ω =
1. When Ω ≫ 1 particles can diffuse in the upstream without
losing any energy, allowing a greater spread in momentum
above pcut.
2.2 The Integrated Distribution Function and
Synchrotron Spectra
When the source cannot be fully resolved observationally,
we must include the contribution from all particles within
some distance z′ of the shock in calculating the spatially
integrated emission. In the case of steady emission from a
jet pointing towards us, or a completely unresolved source,
z′ is essentially the source size in the optically thin limit. For
log  p
log p4 f
z’=0z’=1z’=10z’=50
 0.01
 0.01  0.1  1  10
Figure 4. The spatial variation of the particle distribution for
Ω = 1. z′ =
c© ???? RAS, MNRAS 000, 1–10
4 Paul Dempsey and Peter Duffy
log p
log p4 f
 0.001
 0.01
 0.01  0.1  1  10
Figure 5. The spatially integrated particle distribution for Ω = 1
from z′ = 0 to z′ = 50. Note that as well as the particle cut-off
there is a spectral break earlier were they spectrum softens from
p−4 to p−5. This is analogous to synchrotron ageing.
simplicity we assume that the magnetic field downstream of
the shock is constant although the model can be generalised
for more complex cases.
The integrated Laplace distribution function is
Ŵ dz
k3−s Q̂
1 + ω+k
1− exp
1 + ω+k
. (23)
When z′ is very small the result is k3−s with a cut-off at
high k as expected. As z′ tends to infinity at low k we have
k2−s so the spectrum is steepened , with the same high k
cut-off. For finite values of z′ the spectrum starts as k3−s
before turning into k2−s and finally cutting off. We shall see
later that this result also holds in real momentum space.
Figure 4 shows how the cut-off tends to lower momenta
as we go further downstream. However what is most often
observed a result of the integrated distribution is shown in
figure 5. While the cut-off momentum is the same as at
the shock, the distribution changes from an initial p−4 to
a p−5 spectrum at some critical momentum, pb, which de-
pends on z′. Here we will consider only synchrotron emission
from an ordered magnetic field (parallel to the flow). Let
4πνm3ec
where ν is the frequency, q is the charge on
the electron, me is the electron mass and B is the magnetic
field strength. Then given a spatially integrated particle dis-
tribution f ∝ p−sg(p, µ) the total power emitted per unit
frequency is (Rybicki & Lightman 1986)
Ptot(ω) ∝
1− µ2
1− µ2
!(s−5)/2
1− µ2
AF (x) dx (24)
where F is the first synchrotron function
F (x) ≡ x
(y) dy. (25)
In the case of non-relativistic diffusive shock accelera-
Emission at ShockνΙ(ν)
 0.001
 0.01
 0.001  0.01  0.1  1  10  100
Figure 6. A telescope with very high resolution may be able
to observe synchrotron radiation at the shock. The only spectral
feature here is the cut-off at νcut. Before νcut the spectrum has
shape I(ν) ∼ ν−1/2. For this result Ω = 1.
bνΙ(ν)
z’=50
 0.001
 0.01
 0.001  0.01  0.1  1  10  100
Figure 7. Spatially integrated emission for Ω = 1. The black
dashed curve shows the synchrotron spectrum of an unresolved
object containing a strong shock. The only feature is the spectral
cut-off at νcut. Before νcut the spectrum has shape I(ν) ∼ ν
The solid curve illustrates the synchrotron spectrum of a partially
resolved source with emission from the shock (z′ = 0) to some
downstream distance (z′ = 50).
tion f downstream of a shock is assumed to be isotropic in
which case g is independent of µ.
Figure 6 shows the emission at the shock. The only fea-
ture here is the cut-off hump, which is of course related to
the particle cut-off, before which Iν ∼ ν−(s−3)/2. Figure 7
shows a more realistic plot, that of emission from an ex-
tended region. If the region is infinite in size then the cut-off
remains the sole feature but the spectrum before the cut-off
is different Iν ∼ ν−(s−2)/2. If the region has finite size then
a second feature, the spectral break νb, appears. Before the
break the spectrum goes as Iν ∼ ν−(s−3)/2 while after it it
is Iν ∼ ν−(s−2)/2. Again this is related to the momentum
break pb we see in the particle distribution in figure 5.
3 RELATIVISTIC SHOCK ACCELERATION
WITH LOSSES
In the case of a relativistic shock, the particle transport
equation describing advection, pitch angle diffusion and
losses becomes
Γ (u+ µ)
+ λg(µ)
∂(p4f)
where µ is the cosine of the pitch angle of the particle
and the flow velocity is constant upstream and downstream
of the shock. λ =
UB and g(µ) = 1 − µ2 for syn-
chrotron losses in an ordered magnetic field, λ = 4σTUB/3
and g(µ) = 1 for synchrotron losses in a tangled magnetic
field, or λ = 4σTUrad/3 and g(µ) = 1 for inverse Compton
c© ???? RAS, MNRAS 000, 1–10
Radiative losses and cut-offs of energetic particles at relativistic shocks 5
losses. Equation (26) holds separately upstream and down-
stream with the conditions that the distribution is isotropic
infinity far downstream, there are no particles infinitely far
upstream and the distribution is continuous at the shock. Al-
though we will derive equations for general momentum inde-
pendent pitch-angle diffusion and an arbitrary magnetic field
alignment, the figures and results produced throughout the
rest of this paper are for isotropic diffusion Dµµ = D(1−µ2)
in an ordered (longitudinal) magnetic field with λ/D = 0.1.
Guided by the treatment of the nonrelativistic case we
set W = p4f and y = 1/p so that
Γ(u+ µ)
D(µ)(1− µ2)∂W
− λg(µ)∂W
. (27)
Taking the Laplace Transform with respect to y and assum-
ing W (0, µ, z) = 0
Γ(u+ µ)
D(µ)(1− µ2)∂Ŵ
− λg(µ)kŴ . (28)
With the spatial and pitch angle variables separable we look
for solutions of the form
Ŵ (k, µ, z) =
ai(k)Xi(k, z)Qi(k, µ) (29)
putting this back into the reduced transport equation we get
Γ(u+ µ)
Qi = (DQi)Xi (30)
where we have defined the differential operator D via
DΦ = ∂
D(µ)(1− µ2)∂Φ
− λg(µ)kΦ. (31)
Separating X and Q we get the usual
= Λi(k) =
Qi(u+ µ)
DQi (32)
⇒ Xi(k, z) = exp
Λi(k)z
and we have an equation for Q(k, µ)
DQi − Λi(k)Qi(u+ µ) = 0. (34)
Expanding out the differential operator we get
D(µ)(1− µ2)∂Qi
− (Λi(u+ µ) + kλg(µ))Qi = 0
which has regular singularities at µ = ±1 and so it should
be possible to find solutions for Qi on [−1, 1] for all k ∈ C.
3.1 Determining the Eigenfunctions
We know that along the real axis, k = x ∈ R, each Qi
satisfies
D(µ)(1− µ2)∂Qi
− (Λi(u+ µ) + xλg(µ))Qi = 0.
We define an inner product by:
〈ζ, ξ〉 =
(u+ µ)ζ
ξdµ. (37)
u  =.3; u  =.076
u  =.5; u  =.129
u  =.7; u  =.189+
−Λ  (  )k
 1e−04
 0.001
 0.01
 0.001  0.01  0.1  1  10
Figure 8. The zeroth order downstream eigenvalue for shock
speeds u− = .3, .5, .7. Along the x-axis we have plotted the log-
arithm of k+ while along the y-axis we have the logarithm of
−Λ0(k+). When k+ = 0 we have Λ0(0) = 0.
It can be shown that the Qi(x) are orthogonal and either
real or purely imaginary, and the Λi(x) are real and distinct.
We can normalise the eigenfunctions such that
〈Qi, Qj〉 = δi,j (38)
or considering them as real
〈Qi, Qj〉 = δi,j(1/2 − i) / |1/2− i| ≡ ηi,j . (39)
Then we have (see Appendix for details)
j 6=i
Λj − Λi
g(µ)QiQjdµ
Qjηj,j (40)
g(µ)QiQidµ
ηi,i. (41)
We solve equation 36 at x = 0 using the Prüfer transforma-
tion as in Kirk et al. (2000). We then use equations 40 and
41 to find Qi(x, µ) and Λi(x) for x > 0 using Runge-Kutta
methods.
Figures 8, 9, 10 and 11 show the zeroth downstream
eigenvalues and eigenfunctions for shocks speeds of .3, .5
and .7. This eigenfunction is the dominant component in the
downstream distribution function at the shock of such mildly
relativistic shocks, where we are close to isotropy. Further
downstream, where the contribution of higher eigenfunctions
are more strongly damped, so the anisotropy for some z > 0
is essentially that of the zeroth eigenfunction. While Λ0(k+)
is initially zero, note from figure 8 that it decreases linearly
until a certain point which, as we will see later, is close to
the cut-off momentum. This will play a major role in the
integrated distribution function and emission. Figures 9,
10 and 11 show how anisotropy arises in the zeroth order
eigenfunction which is isotropic for k = 0. Given that k is
related to the momentum these figures show that, since this
is the dominant eigenfunction, the anisotropy will increase
with increasing energy.
c© ???? RAS, MNRAS 000, 1–10
6 Paul Dempsey and Peter Duffy
Figure 9. The zeroth order downstream eigenfunction for shock
speed u− = .3. Along the x-axis we have plotted the logarithm
of k+ while along the y-axis we have µ+. Up the z-axis we have
plotted Q0(k+, µ+), which also defines the grayscale. Note the
anisotropy increases with k+.
Figure 10. The zeroth order downstream eigenfunction for shock
speed u− = .5. Along the x-axis we have plotted the logarithm
of k+ while along the y-axis we have µ+. Up the z-axis we have
plotted Q0(k+, µ+), which also defines the grayscale. Note the
anisotropy increases with k+.
3.2 Shock matching conditions
Starting from
Ŵ (k, µ, z) =
ai(k)Xi(k, z)Qi(k, µ) (42)
ai(k) exp
Λi(k)z
Qi(k, µ)) (43)
we note that upstream (z < 0) we have ai = 0 for all i
such that Λi 6 0 and that downstream (z > 0) we have
ai = 0 for all i such that Λi > 0. The distribution function
Figure 11. The zeroth order downstream eigenfunction for shock
speed u− = .7. Along the x-axis we have plotted the logarithm
of k+ while along the y-axis we have µ+. Up the z-axis we have
plotted Q0(k+, µ+), which also defines the greyscale. Note the
anisotropy increases with k+.
is continuous at the shock,
f−(y−, µ−, 0) = f+(y+, µ+, 0) (44)
with (y−, µ−) related to (y+, µ+) by a Lorentz transforma-
tion of velocity urel = (u− − u+)/(1− u−u+),
y− = Γrely+(1 + urelµ−). (45)
In terms of W the matching condition becomes
rel(1 + urelµ−)
W−(y−, µ−, 0) = W+(y+, µ+, 0) (46)
and we now need to express this in terms of Ŵ , the Laplace
transform with respect to y. Taking k−y− = k+y+, multi-
plying the matching condition for W by exp(−k+y+) and
integrating over y+ gives
rel(1 + urelµ−)
Ŵ−(k−, µ−, 0) = Ŵ+(k+, µ+, 0). (47)
Guided by the discussion for the nonrelativistic case, we
use the expansion
i (k±, µ±) (48)
so that the matching condition for the Laplace transformed
spectrum at the shock reduces to
rel(1 + urelµ−)
i (k−)Q
i (k−, µ−) =
i (k+)Q
i (k+, µ+). (49)
In order to solve for the particle spectrum, we multiply by
(u++µ+)Q
j (k+, µ+) j > 0 and integrate over µ+. Then for
a fixed k− we have
i (k−)
(1 + urelµ−)
i (k−, µ−)×
(u+ + µ+)Q
j (k+, µ+)dµ+ = 0. (50)
Defining a matrix S with elements
i,j =
(1 + urelµ−)
i (k−, µ−)(u+ + µ+)Q
j (k+, µ+)dµ+
we need to find the spectral index s, such that detS = 0.
The Laplace inversion is then carried out numerically (see
Appendix for details). As motivated by the nonrelativistic
case, we define the cut-off to be the point at which
d(lnR)
d(ln k)
= −1 (52)
where R =
biQi. Figures 12, 13 and 14 plot
d(lnR)/d(ln k) at the shock against k as measured down-
stream. The results are summarised in table 2.
Figures 12, 13 and 14 show how the cut-off momentum
becomes increasing anisotropic as the shock speed increases.
The distribution can be fitted approximately by
f ≈ p−s exp
Γrelpcut(µ+)
where β is typically 2. However it is difficult justify the use
of the factor
Γrel in general as our results are only for
mildly relativistic shocks. This fit justifies our definition of
pcut instead of using the equilibrium momentum p
∗. Figure
c© ???? RAS, MNRAS 000, 1–10
Radiative losses and cut-offs of energetic particles at relativistic shocks 7
d(ln k)
d(ln R)
 0.001  0.01  0.1  1  10  100
Figure 12. Plotted along the x-axis we have the logarithm of
momentum k+ while along the y-axis we have d(lnR+)/d(ln k+)
for u− = .3 and R+ =
i (k+)Q
i (k+, µ+). Note the cut-off
depends on µ+
d(ln k)
d(ln R)
 0.001  0.01  0.1  1  10  100
Figure 13. Plotted along the x-axis we have the logarithm of
momentum k+ while along the y-axis we have d(lnR+)/d(ln k+)
for u− = .5 and R+ =
i (k+)Q
i (k+, µ+). Note the cut-off
depends on µ+
17 illustrates this approximation for a .7c shock. β seems
to be pitch angle dependent varying between 1.75 and 2.2,
but typically 2. In fact for the .3c and .5c shock cases β
showed much less variation about 2. For the shock speeds
we have chosen, with the Juttner-Synge equation of state,
the spectral indices in the absence of losses are close to 4.
Figures 15, 16 and 17 illustrated a feature that was
not present in the non-relativistic case. The pitch angle de-
pendence of the cut-off momentum leads to a difference in
the isotropy levels between particles above and below some
d(ln k)
d(ln R)
 0.001  0.01  0.1  1  10  100
Figure 14. Plotted along the x-axis we have the logarithm of
momentum k+ while along the y-axis we have d(lnR+)/d(ln k+)
for u− = .7 and R+ =
i (k+)Q
i (k+, µ+). Note the cut-off
depends on µ+
u− .3 .5 .7
u+ .076 .129 .189
Γrel 1.027 1.089 1.23
p∗ .404 1.143 2.26
Non-Rel pcut .621 1.79 3.719
pcut(µ+ = −1.0) .541 1.42 2.566
pcut(µ+ = −0.5) .595 1.71 3.612
pcut(µ+ = 0) .64 1.929 4.375
pcut(µ+ = 0.5) .682 2.138 5.105
pcut(µ+ = 1) .741 2.533 6.773
Table 2. Summary of Cut-Off Momenta
aniso
p pcut−off
 0.01
 0.001  0.01  0.1  1  10
Figure 15. The downstream function evaluated at the shock for a
shock speed of .3. Along the x-axis we have plotted the logarithm
of momentum p+ while along the y-axis we have the logarithm of
W = p4f .
critical momentum paniso. Indeed there is a clear pattern of
greater levels of anisotropy at high energies as the shock
speed increases, despite the fact that the results presented
here are only for mildly relativistic shocks.
3.3 The Spatially Integrated Distribution
While the method we follow in this paper finds the upstream
particle distribution directly, it is easy to find the down-
aniso
cut−off
 0.01
 0.001  0.01  0.1  1  10
Figure 16. The downstream function evaluated at the shock for a
shock speed of .5. Along the x-axis we have plotted the logarithm
of momentum p+ while along the y-axis we have the logarithm of
W = p4f .
c© ???? RAS, MNRAS 000, 1–10
8 Paul Dempsey and Peter Duffy
cut−off
aniso
W µ=−1
 0.01
 0.001  0.01  0.1  1  10  100
.185 exp(−(p/(2.566 sqrt(1.23)))^1.75)
.185 exp(−(p/(4.375 sqrt(1.23)))^1.80)
.185 exp(−(p/(6.773 sqrt(1.23)))^2.20)
Figure 17. The downstream function evaluated at the shock for a
shock speed of .7. Along the x-axis we have plotted the logarithm
of momentum p+ while along the y-axis we have the logarithm
of W = p4f . The lines are data while the points are the best fit
described in the text.
break
aniso
cut−off
 0.01
 1000
 0.001  0.01  0.1  1  10
Figure 18. The downstream function integrated for a shock
speed of .3 between z′ = 0 and z′ = 100 where z′ = Dz/Γ+.
Along the x-axis we have plotted the logarithm of momentum p+
while along the y-axis we have the logarithm of W = p4f .
stream distribution by using the matching condition, as dis-
cussed in the previous section. The downstream distribution
is, in many respects, more important physically as it will
be responsible form most of the spatial integrated emission.
As it can be difficult to spatially resolve observational data
from non-thermal emitters, we must consider the emission
from an extended region of space. Our eigenfunction expan-
sion allows us to do this quite easily. The spatially averaged
distribution from a downstream region [z0, z1] in terms of
Laplace variables is
R[z0,z1](k+, µ+) =
ai(k)
Λ+i (k+)z1
− exp
Λ+i (k+)z0
i (µ+, k+)
Λ+i (k+)
. (54)
In the case of a source which is completely spatially unre-
solved this reduces to
R[0,∞](k+, µ+) = −Γ+
ai(k)
Q+i (µ+, k+)
Λ+i (k+)
. (55)
Of course the optical depth of the emitting region will also
have an effect on the spectrum of unresolved sources by re-
ducing z1.
cut−off
aniso
break
 0.01
 0.001  0.01  0.1  1  10
Figure 19. The downstream function integrated for a shock
speed of .5 between z′ = 0 and z′ = 100 where z′ = Dz/Γ+.
Along the x-axis we have plotted the logarithm of momentum p+
while along the y-axis we have the logarithm of W = p4f .
cut−off
aniso
break
 0.01
 0.001  0.01  0.1  1  10
Figure 20. The downstream function integrated for a shock
speed of .7 between z′ = 0 and z′ = 100 where z′ = Dz/Γ+.
Along the x-axis we have plotted the logarithm of momentum p+
while along the y-axis we have the logarithm of W = p4f .
Using the same numerical Laplace inversion as in the
non-relativistic case we have calculated the distribution
functions and synchrotron emission. Figures 18, 19 and 20
show the spatially integrated distribution functions for a fi-
nite emission region. Now there are there features: a momen-
tum break, pb, due to spatial effect; an anisotropic break,
µ=−.5
νΙ(ν)
 1e−06
 1e−05
 1e−04
 0.001
 0.01
 1e−07  1e−06  1e−05  1e−04  0.001  0.01  0.1  1  10  100  1000
Figure 21. Synchrotron emission from the particle distribution
shown in 18 measured in the downstream medium. In plotting
our µ = ±1 we used µ = ±.9999 as there is no emission from an
ordered field along µ = ±1.
c© ???? RAS, MNRAS 000, 1–10
Radiative losses and cut-offs of energetic particles at relativistic shocks 9
µ=−.5
νΙ(ν)
 1e−06
 1e−05
 1e−04
 0.001
 0.01
 1e−06  1e−04  0.01  1  100  10000
Figure 22. Synchrotron emission from the particle distribution
shown in 19 measured in the downstream medium. In plotting
our µ = ±1 we used µ = ±.9999 as there is no emission from an
ordered field along µ = ±1.
µ=−.5
νΙ(ν)
 1e−06
 1e−05
 1e−04
 0.001
 0.01
 1e−06  1e−04  0.01  1  100  10000
Figure 23. Synchrotron emission from the particle distribution
shown in 20 measured in the downstream medium. In plotting
our µ = ±1 we used µ = ±.9999 as there is no emission from an
ordered field along µ = ±1.
paniso, due to relativistic effects; and a cut-off, pcut, due
to energy losses. Given that the magnetic field is constant
throughout this region it is trivial to produce the associ-
ated synchrotron emission plots of figures 21, 22 and 23. It
should be noted that in the emission plots Iν is measured in
the downstream frame, but since Iν/ν
3 is a Lorentz invariant
the transformation is trivial. The synchrotron emission also
includes the same three features we observed in the particle
distribution; namely a break frequency beyond which the ef-
fect of synchrotron cooling becomes important, a frequency
at which pitch-angle or anisotropic effects play a role and an
upper cut-off beyond which there is virtually no emission.
4 DISCUSSION
Particle acceleration and self-consistent synchrotron radia-
tion have been considered previously by Kirk et al. (1998)
using a zonal model. They were successful in explaining the
radio to X-ray spectrum of Mkn 501. However such zonal
models typically depend on isotropic particle distributions.
We have shown, however, that for particles near the high
energy cut-off this is not true even for mildly relativistic
flows. The computational resources available restricted our
results to be below .7c. However even for the mildly rela-
tivistic shock velocities we see a clear pattern of high energy
anisotropy emerging resulting in synchrotron emission which
is also anisotropic. This could be extremely important in the
modelling of the inverse Compton hump in γ-rays observed
in TeV Blazars (Aharonian et al. 2006). As a second implica-
tion of the particle anisotropy, in the presence of losses, the
idealised situation, of a two sided strongly polarised iden-
tical jet system can be considered. Each jet contains only
forward external shocks, and the jet which is directed to-
wards the observer is inclined at an angle θ = cos−1(−µ)
to the line of sight (magnetic field direction same as that
of shock). Then we will observe the emission from particles
in the jet directed towards us which have pitch angle µ and
from particles in the jet directed away from us which have
pitch angle −µ. While at low energies the only difference
between the observed emission of the two jets will be as a
result of the effects of beaming, at energies near the syn-
chrotron cut-off the details of the acceleration mechanism
will amplify this difference, depending on viewing angle.
Although the work in this paper is limited to an ide-
alised form of diffusion, and mildly relativistic shocks, it
illustrates previous unexamined features which could be im-
portant in the modelling of relativistic, γ-ray sources such
as microquasars, blazars and GRBs. We have parameterised
the exponential shape of the distribution cut-off and identi-
fied new pitch angle dependent features between break and
cut-off frequencies. Further work is needed to examine both
momentum dependent scattering and high Lorentz factor
flows.
ACKNOWLEDGMENTS
Paul Dempsey would like to thank the Irish Research Coun-
cil for Science, Engineering and Technology for their finan-
cial support. He would also like to thank Cosmogrid for ac-
cess to their computational facilities. We are grateful for
discussions with Felix Aharonian. Peter Duffy would like to
thank the Dublin Institute for Advanced Studies for their
hospitality during the completion of this work. We would
like to thank the referee for comments that improved the
quality of this paper.
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Heavens A.F., Drury L.O’C., 1988, MNRAS, 235, 997
Heavens A.F., Meisenheimer K., 1987, MNRAS, 225, 335
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2000, ApJ, 542, 235
Kirk, J. G., Rieger, F. M., & Mastichiadis, A., 1998, A&A,
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Kirk J.G., Schneider P., 1987, ApJ, 323, L87
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This paper has been typeset from a TEX/ LATEX file prepared
by the author.
APPENDIX A: INVERSE LAPLACE
TRANSFORMS
While Heavens & Meisenheimer (1987) invert the Laplace
transform analytically for particular cases here we use nu-
merical methods as we will need to when dealing with rela-
tivistic flows.
Formally the inverse Laplace transform is the Bromwich
integral, which is a complex integral given by:
f(t) = L−1 [F (s)] = 1
Z γ+i∞
F (s) ds (A1)
where γ is to the right of every singularity of F (s). If the sin-
gularity of F (s) all ly in the left half of the complex plane
γ can be set to 0 and this reduces to the inverse Fourier
transform, which is easy to do. However for complicated or
numerical Laplace functions the Bromwich integral is ex-
tremely difficult to solve. The four main numerical inversion
techniques are Fourier Series Expansion, Talbot’s method,
Weeks method and methods based on the Post-Widder for-
mula. However some of these methods converge rather slowly
and a lot of work has gone into creating acceleration meth-
ods. Numerical Laplace inversion is a area of active research
and the choice of inversion technique is as much an art as
a science at the moment. In this paper the Post-Widder
based method was chosen and only these methods shall by
described below. Let F (s) be the Laplace transform of f(t)
then Widder (1932) showed that fn(t) → f(t) where
fn(t) =
(−1)n
((n+ 1)/t). (A2)
The advantage of this method in our case in that we see
that the Laplace transform of the solution times the Laplace
coordinate is the zeroth order approximation to the actual
solution.
W0(y) =
Ŵ (1/y) ⇒ W0(p) = pŴ (p).
When dealing with numerical results however it is easier to
use the Gaver-Stehfest algorithm (Abate & Valkó 2004). It
is an algorithm based on the Post-Widder method with the
Gaver approximants, {fn(t) : n > 0}, defined as
fn(t) ≡
(n+ 1) ln(2)
2(n+ 1)
f̂ ((n+ 1 + k) ln(2)/t) . (A3)
However the convergence for both these methods is slow.
A test of methods for accelerating this convergence can be
found in Valkó & Abate (2004) and two are found to be
quite good: the non-linear Wynn’s Rho Algorithm and the
linear Salzer summation. Again a choice has to be made
and here we present only Salzer summation: f(t,M) → f(t)
where
f(t,M) =
Wkfk−1(t) (A4)
Wk = (−1)k+M
. (A5)
The Post-Widder method based on differentiation was im-
plemented in Maple with the Salzer acceleration. It was used
to produce the results in the non-relativistic limit as we
have an analytic form of the Laplace function to work with.
The Salzer accelerated Gaver-Stehfest algorithm was imple-
mented in C/C++ code for use with the numerical output
from the relativistic approach discussed above.
APPENDIX B: DERIVING THE
EIGENSYSTEM DIFFERENTIAL EQUATIONS
The solutions, Qi, to equation 36
D(µ)(1− µ2)∂Qi
− xλg(µ)Qi = Λi(u+ µ)Qi (B1)
for real x, are orthogonal, with weight u+ µ and have real,
distinct eigenvalues Λi (Boas 1983). Taking the derivative of
this equation with respect to x gives
D(µ)(1− µ2) ∂
− xλg(µ)∂Qi
= Λi(u+ µ)
(u+ µ) + λg(µ)
Qi. (B2)
Since solution to Sturm Liouville equations form an orthog-
onal basis, we can write
qmQm (B3)
which gives
D(µ)(1− µ2) ∂
qmxλg(µ)Qm
qmΛm(u+ µ)Qm
qmΛi(u+ µ)Qm +
(u+ µ) + λg(µ)
Multiplying by Q∗j and integrating over µ gives
qm(Λm − Λi)〈Qj, Qm〉
〈Qj , Qi〉+ λ
g(µ)QiQ
j dµ. (B5)
Taking j = i implies equation 41 and j 6= i implies equation
c© ???? RAS, MNRAS 000, 1–10
	Introduction
	Nonrelativistic Shocks
	Momentum Cut-off
	The Integrated Distribution Function and Synchrotron Spectra
	Relativistic Shock Acceleration with Losses
	Determining the Eigenfunctions
	Shock matching conditions
	The Spatially Integrated Distribution
	Discussion
	Inverse Laplace Transforms
	Deriving the eigensystem differential equations
