Off-Axis Emission from Relativistic Plasma Flows

Abstract

We show that there is no universal law describing how the spectra and luminosity of synchrotron and inverse Compton radiation from relativistic jets change with increasing observation angle. Instead, the physics of particle acceleration leaves pronounced imprints in the observed spectra and allows for a freedom in numerous modifications of them. The impact of these effects is the largest for high-energy radiation and depends on the details of particle acceleration mechanism(s), what can be used to discriminate between different models. Generally, the beam patterns of relativistic jets in GeV-TeV spectral domain are much wider than the inverse Lorentz factor. The off-axis emission in this energy range appear to be brighter, have much harder spectra and a much higher cut-off frequency compared to the values derived from Doppler boosting considerations alone. The implications include the possibility to explain high-latitude unidentified EGRET sources as off-axis but otherwise typical relativistic-jet sources, such as blazars, and the prediction of GeV-TeV afterglow from transient jet sources, such as Gamma-Ray Bursts. We also discuss the phenomenon of beam-pattern broadening in application to neutrino emission.

Full text

arXiv:astro-ph/0605456v1 18 May 2006
Off-axis emission from relativistic plasma flows
E.V. Derishev
1
, F.A. Aharonian
2
, Vl.V. Kocharovsky
1
1
Institute of Applied Physics, 46 Ulyanov st., 603950 Nizhny Novgorod, Russia
2
Max-Planck-Institut f¨ur Kernphysik, Saupfercheckweg 1, D-69117 Heidelberg, Germany
ABSTRACT
We show that there is no universal law describing how the spectra and lumi-
nosity of synchrotron and inverse Compton radiation from relativistic jets cha nge
with increasing observation angle. Instead, the physics of particle acceleration
leaves pronounced imprints in the observed spectra and allows for a freedom in
numerous modifications of them. The impact of these effects is the larg est f or
high-energy radiation and depends on the details o f particle acceleration mecha-
nism(s), what can be used to discriminate between different models. Generally,
the beam patterns of relativistic jets in GeV-TeV spectral domain are much wider
than the inverse Lorentz factor. The off-a xis emission in this energy range appear
to be brighter, have much harder spectra and a much higher cut-off frequency
compared to t he values derived from Doppler boosting considerations alone.
The implications include the possibility to explain high-latitude unident ified
EGRET sources as off-axis but otherwise typical relativistic-jet sources, such
as blazars, and the prediction of GeV-TeV afterglow from transient jet sources,
such as Gamma-Ray Bursts. We also discuss the phenomenon of b eam-pa tt ern
broadening in application to neutrino emission.
Subject headings: gamma rays: bursts — gamma rays: theory — ISM: jets a nd
outflows — neutrinos — radiation mechanisms: non-thermal — shock waves
1. Intr oduction
Fast-moving plasma outflows are core elements in models of bright and rapidly variable
astrophysical sources, such as Active Galactic Nuclei (AGNs), Gamma-Ray Bursts (GRBs),
and microquasars (see, for example, Urry & Padovani 1995 for a r eview on AGNs, Zhang &
M´esz´aros 2004; Piran 2005 for reviews on GR Bs). Relativistic flows offer a simple solution to
the gamma-ray transparency problem fo r compact objects: thanks to the Lorentz boosting,
variability timescales – as seen in the lab frame – decrease, the energies of individual photons

– 2 –
and bolometric brightness increase, whereas the pair-production opacity can be made smaller
than unity for a source of any size.
Relativistic flows can be formed by hot plasma left behind a relativistic shock or exist
in the form of jets. In either case the question about properties of emission fro m these flows
naturally divides into two. Firstly, one has to know the distribution of radiating particles in
the plasma comoving f r ame – a problem far f r om complete solution as it depends on details
of the acceleration mechanism. Secondly, the resulting photon field needs to be recalculated
for the lab frame, i.e., Lorentz-transformed from the comoving frame. This is regarded as a
routine and obvious procedure and usually r eceives little attention.
So far, all calculations were made under silent assumption that the radiation is isotropic
in the comoving frame. In many situations the aforementioned isotropy is a good guess,
as is discussed in more detail in the following section. In this paper, however, we would
like to emphasize that this assumption is model-dependent, and t hat highly relativistic jets
and shock waves violate it in many situations. Abandoning the isotropy assumption has
a profound effect on the predicted spectra and timing of the astrophysical sources with
relativistic plasma flows (hereafter we will call them jets for short, that does not mean we
exclude shocks from consideration).
θ
θ
V
jet
Fig. 1.— D efinition o f a ng les in the jet comoving frame (θ
′
) and in the lab frame (θ).
For convenience, let us remind the Lorentz transformations for the quantities to be
used in this paper: emission and observation angles (θ
′
and θ, respectively), frequencies,

– 3 –
and intensities, i.e., energy flux per unit solid angle and per unit frequency. (Prime stands
for the jet-frame in the case we need to introduce both the lab-frame and the jet-frame
quantities.) The reference frames are defined as shown in Fig. 1; note that the polar axes in
the comoving and laboratory frames are directed opposite to each other. Then, we have the
following relations
for angles: cos θ =
β − cos θ
′
1 − β cos θ
′
or tan
θ
2
tan
θ
′
2
=
1
Γ(1 + β)
, (1)
for frequencies: ν = Γ (1 − β cos θ
′
) ν
′
≡ δν
′
, (2)
for intensities: I(ν, θ) = δ
n
I
′
(ν
′
, θ
′
) . (3)
Here β is the jet velocity in units of the speed of light c, Γ the jet Lorentz factor, and
δ = Γ(1 − β cos θ
′
) =
1
Γ(1 − β cos θ)
=
sin θ
′
sin θ
(4)
the Doppler factor. In the last equation, n = 2 for a continuous jet and n = 3 fo r a
relativistically moving blob (Lind & Blandford 1985). The flux density per unit frequency,
F
ν
, changes with the observation angle as intensity.
Throughout t he paper we will often use the asymptotic form of Eqs. (1), (2) and (3) for
the case, where both the observation angle and the emission angle are small (θ, θ
′
≪ 1 ). In
terms of observation angle, this condition is equivalent to Γ
−1
≪ θ ≪ 1. In this case
δ ≃
Γθ
′ 2
2
≃
2
Γθ
2
, θθ
′
≃
2
Γ
. (5)
In this pa per we consider only uniform jets, i.e., those with Lorentz fa ctor independent
on direction and distance from the orig in. However, we use this simplification only when
considering particular implications in order to present conclusions in a clear fo rm.
The paper is organized as follows. First, we discuss the conditions causing strong off-axis
emission from relativistic jets (Sect. 2), then we analyze motion and radiation of particles
responsible for this emission (Sect. 3 and 4). After discussing some general results in Sect. 5,
we move on to consider implications for particular astrophysical objects (Sect. 6).
2. Conditions for anisotropic emission in the jet frame
Almost in every model of compact objects with relativistic flows the observed radiation
comes from particles, which are themselves relativistic in the comoving frame (such a flow is
sometimes called a hot jet). Photons produced by a relativistic part icle continue to stream

– 4 –
in the direction of the particle’s motion, hence the photon field in the jet frame has the
same degree of anisotropy as the particle distribution does. There always exists a continuous
anisotropic supply of relativistic part icles – those, entering the flow from surrounding medium
through the shock front or the shear layer at the jet’s boundary. If their scattering length
ℓ
s
is shorter than or comparable to their radiation length ℓ
r
(both are measured in the jet
frame), then the resulting emission can be treated as isotropic. Otherwise, the radiation
is essentially anisotropic in the jet frame; in the extreme case it is directed opposite to
the velocity of the flow and confined within a narrow cone with the opening angle ∼ Γ
−1
.
Let’s define the critical energy E
′
cr
so that ℓ
s
(E
′
cr
) = ℓ
r
(E
′
cr
). Since ℓ
s
/ℓ
r
is a monotonically
growing f unction of energy
1
, the super-critical particles (those with energies E > E
cr
) do not
get isotropized and produce anisotropic emission.
One can notice that the very nature of diffusive acceleration implies that ℓ
s
. ℓ
r
. Indeed,
the diffusive acceleration proceeds t hro ugh multiple passages of an accelerated particle from
the jet to surrounding medium and back, and in each round the particle’s energy increases by
a factor ∼ 2, even in the case of relativistic shock (e.g., Achterberg et al. 2001). The particle
should be able to preserve at least a half of its energy over the scattering length to keep
accelerating, so that the condition ℓ
s
. ℓ
r
is automatically satisfied, providing seemingly
firm ground to the isotropy assumption.
However, energetic particles emerge in the jet in several other ways as well. They can be
accelerated by preceding shocks and survive till t he shocked gas slows down to sub-relativistic
speeds, or be the secondary electrons from inelastic interactions of higher-energy protons, or
be injected in the surrounding medium by the jet itself, as in the case of e
−
e
+
-pair loading
ahead of GRB shocks (e.g., Madau & Thompson 2000). Also, super-critical particles can be
produced via non-diffusive converter acceleration mechanism (Derishev et al. 2003; Stern
2003), in which the energy gain per shock crossing is not limited to the fa ctor ∼ 2. The
sup er-critical particles do not get isotropized by definition, and hence they cont ribute to
the anisotropy of photon field in the jet frame. Although both electrons and protons can
be super-critical, the case of electrons is of much larger importance since the acceleration
of protons is almost always limited by their diffusive escape of by the accelerator’s lifetime
rather than by radiative losses.
It is safe to claim that the emission from relativistic jets is partly contributed by super-
critical particles, though how large is this part is the question to be considered separately
for various sources. Anyway, smallness o f the anisotropic part in the particle distribution
1
The scattering length of a particle in the magnetic field grows linearly proportional to the particle’s
energy or faster, whereas the radiation length either decreases or grows no fa ster than E/ ln(E).

– 5 –
does not mean its contribution to the observed emission is negligible in all spectral domains
and at all viewing angles.
3. Motion of super-critical particles and their emission
Suppose that super-critical particles are injected in the jet with highly anisotropic angu-
lar distribution, having the opening angle φ
0
(in the comoving frame) and elongated counter
to the jet’s velocity. Over their radiation length, the particles get deflected by an angle φ,
which is a decreasing function of energy. The characteristic width of the beam pattern in
the jet frame (θ
b
) equals to that of the par ticle distribution:
θ
′
b
(ε) =



1 f or ε < 1
φ(ε) for 1 < ε < ε
cr 2
φ
0
for ε > ε
cr 2
.
(6)
To simplify the notatio ns here and below, we introduce the following dimensionless variables:
ε is the particle’s energy in units of the critical energy E
′
cr
(so that ε
cr
≡ 1), x the distance
in units ℓ
s
(E
′
cr
), which is the scattering length at the critical energy. The second critical
energy ε
cr 2
is defined as the energy of particles whose r.m.s. deflection angle equals to the
initial width of the particle distribution: φ(ε
cr 2
) = φ
0
.
Equation (6) has simple physical meaning. The sub-critical particles (ε < 1) have
enough time to get fully isotropized, whereas for super-critical ones t he width of the distri-
bution function is equal to their r.m.s. deflection angle φ, unless φ < φ
0
. Above the second
critical energy ε
cr 2
, the particles loose energy before being significantly deflected, and the
distribution function preserves its intrinsic width φ
0
. The latter can be as small as φ
0
= Γ
−1
(if the distribution is isotropic in the lab frame): for example, the converter acceleration
mechanism essentially implies isotropisation of accelerated particles each time when they
get into the surrounding medium (Derishev et al. 2003).
Typically, the plasma in relativistic flows is collisionless, i.e., the only scattering mech-
anism is deflection of charged particles by the magnetic field. For a broad class of the
magnetic-field g eometries t he dependence of the r.m.s. deflection angle o n the distance
travelled by the particle allows the following representation (for small deflection angles and
approximately constant energy):
φ =
x
p
ε
. (7)
This comprises two widely used limiting cases: small-angle scattering in the purely chaotic
magnetic field (p = 1/2) and regular deflection in the quasi-uniform magnetic field (p =

– 6 –
1), which corresponds to Bohm-like diffusion. Values in the range 1/2 < p < 1 cover
intermediate situations, including scattering in the turbulent mag netic field with power-law
power spectrum.
The number of po ssible energy loss mechanisms, on the other hand, is large. Each of
them implies different dependence of the energy loss rate o n the particle energy and (in
general) on the propagation angle. Moreover, in different energy domains prevalent and
negligible mechanisms can swap their roles. To keep the paper reasonably concise, we focus
our analysis on three representative cases, marked below by the dominant energy loss channel.
Synchrotron or inverse-Compton (including self-Compton) radiation in the
Thomson regime. In this case the energy loss rate is proportional to the square of the
particle’s energy:
dε
dx
= −ε
2
⇒ x
r
=
1
ε
, (8)
where x
r
is the normalized radiation length. Substituting Eq. (8) into Eq. (7) we find that
the beam-pattern width is
θ
′
b
= ε
−(p+1)
(9)
for particles in the energy range 1 < ε < ε
cr 2
.
The typical frequency of emitted photons is ν
′
= ε
2
ν
′
cr
, and the second critical energy is
ε
cr 2
= φ
−
1
p+1
0
.
Inverse-Compton radiation in the Klein-Nishina regime. If t he spectrum of
radiation being comptonized is a power-law, F
′
ν
∝ ν
′ q
, where −1 < q < 1, then
dε
dx
= −ε
1−q
⇒ x
r
= ε
q
. (10)
For positive q the radiation length increases with increasing energy, but anyway slower than
the scattering length. Thus, substitution of Eq. (10) into Eq. (7 ) gives the beam-pattern
width
θ
′
b
= ε
pq−1
, (11)
which exhibits regular (monotonically decreasing) dependence on the particle energy.
The typical frequency of emitted photons is ν
′
= εν
′
cr
, and the second critical energy is
ε
cr 2
= φ
−
1
1−pq
0
.
Comptonization of exter nal (isotropic in the lab frame) radiation in the
Thomson regime. In this case the radiating particles loose their energy through interaction
with photons, whose distribution is highly anisotropic in the jet frame. Because of this, the

– 7 –
energy loss rate depends on the particle’s propagation angle, but – for an ultrarelativistic
particle – it does not depend on t he particular choice of reference frame. Making use of this
fact, we calculate t he energy lo ss rate in the lab frame, where it is simply
˙
E ∝ −E
2
, and
find from Eq. ( 5) that E ∝ φ
2
ε. We get
dε
dx
= −φ
4
ε
2
⇒ x
r
=
1
φ
4
ε
, (12)
where the proportionality coefficient is uniquely defined by the condition x
r
(1) = 1. Solving
Eqs. (7) and (12) for the beam-pattern width gives
θ
′
b
= ε
−
p+1
4p+1
. (13)
If the external radiation has a logar ithmically narrow spectrum, then the typical f r e-
quency of emitted photons is ν
′
= θ
′ 2
b
ε
2
ν
′
cr
= ε
6p
4p+1
ν
′
cr
. The second critical energy is
ε
cr 2
= φ
−
4p+1
p+1
0
.
The three radiation mechanisms describ ed above give rise to nine qualitatively different
situations, depending on what is the main energy lo ss channel (this determines the beam-
pattern width) and what type of emission is observed (this determines the spectral shape).
Clearly, the physics of the off-axis emission is still r icher due to a possibility of interplay
between various radiation mechanisms.
4. Off-axis spectra and luminosity
In what follows, we take a power-law injection of super-critical particles, |d
˙
N/dE| ∝ E
−s
for E > E
cr
, where
˙
N(E) ∝ E
1−s
is the injection rate integrated over energies larger than E,
and s > 1 to keep the number of injected particles finite. In the jet frame, this transforms
into the power-law injection with the same index at energies ε > 1 and presumably narrow
angular distribution (Γ
−1
≤ φ
0
≪ 1). Generalization for the case of arbitrary injection is
straightforward, though cumbersome; we skip it for the sake of brevity.
Because of their large scattering length, the super-critical particles have practically
no chance to leave the j et. Instead, they loose energy and form a cooling distribution
dN/dε =
˙
N(ε)/ ˙ε
l
, where ˙ε
l
is the total energy lo ss rate, mainly associated with the dominant
radiation mechanism. We also introduce ˙ε
e
– the energy loss rate via the mechanism t hat
produces the observed emission. The two can be the same, in which case ˙ε
l
≃ ˙ε
e
, otherwise
|˙ε
l
| > |˙ε
e
|.

– 8 –
For a cooling distribution, the angle-averaged spectrum in the comoving frame can be
derived in the standard way:
hF
′
ν
(ν
′
)i ∝ ˙ε
e
dN
dε

dν
′
dε

−1
∝
˙ε
e
˙ε
l
ν
′
2−s−x
x
. (14)
Here we assumed that emission from each particle is monochromatic with frequency ν
′
∝ ε
x
(a caution is necessary with this assumption, as discussed below) and the ratio ˙ε
e
(ε)/ ˙ε
l
(ε) is
regarded as a function of the frequency, corresponding to emission from particles of energy
ε.
The radiation of super-critical particles is concentrated within a narrow cone with open-
ing angle θ
′
b
(ε): outside of this cone the radiation is virtually absent, whereas the apparent
flux density within the cone is larger than the angle-averaged one by the factor 4 θ
′ −2
b
. Thus,
F
′
ν
(ν
′
) =
4
θ
′ 2
b
hF
′
ν
(ν
′
)i ∝

˙ε
e
θ
′ 2
b
˙ε
l

ν
′
2−s−x
x
(15)
for ν
′
cr
< ν
′
< ν
′
max
, where ν
′
max
is a function of emission angle θ
′
b
.
Emission observed at an arbitrary angle to the jet axis has two components: the radi-
ation from sub-critical particles, whose spectrum cuts off at ν
cr
, and is continued to higher
frequencies by the radiation from super-critical particles. The true cut-o ff in the off-axis
spectrum at ν
max
is due to the fact t hat an observer, looking at a given angle θ to the jet
axis, cannot see radiation fro m energetic particles, whose r .m.s. deflection angle is smaller
than 2/(Γθ).
All the changes in the off-axis spectrum as compared to the ordinary head-on emission
are entirely due to the factor θ
′ −2
b
, which is a rising function of particles’ energy (and hence
– of frequency). Therefore, the off-axis spectrum is always harder, a nd in most cases – much
harder, than the head-on spectrum. There is a subtle point in the assumption that each
particle emits mono chromatic r adiation. It wo r ks well unless the spectrum given by Eq.
(15) is harder than the low-frequency asymptotic in the spectrum of an individual particle.
All the hard spectra actually are determined by the low-frequency emission of the most
energetic particles and have the corresponding spectral index. Such a spectrum covers the
frequency range ν
′
cr
< ν
′
< ν
′
max
, extending also below ν
′
cr
up to the point, where it intersects
with the (softer) spectrum of sub-critical radiation.
To find the observed luminosity one has to take the appropriate energy loss rates from
Eqs. (8), (10), and (12), substitute the beam-pa tt ern width θ
′
b
in Eq. (15) with the corre-
sponding function of energy and then – energy with fr equency, and finally apply the Lorentz
transformations given by Eqs. (2) and (3). So, an observer in the lab frame, whose line of

– 9 –
sight makes a small angle 1/Γ ≪ θ ≪ 1 with the jet axis, sees the following spectrum:
F
ν
(ν, θ) ∝ ν
α
for ν
cr
< ν < ν
max
, (16)
where both the critical frequency, ν
cr
= δν
′
cr
, and the cut-off frequency, ν
max
= δν
′
max
, are
functions of the observation angle. The values of spectral index α , as well as the ratio
ν
max
/ν
cr
, can be found in Tab. 1, where the summary on the resulting spectra for nine
different cases is presented.
Another important aspect of the off-axis emission is the way its appearance changes
with the observation angle. It can be characterized by dep endence of the cut-off frequency
on the viewing angle,
ν
max
(θ) = δ(θ)ν
′
max
(θ
′
) =
ν
max
ν
cr
δ(θ)
δ(0)
ν
cr
(0) ∝ θ
a
, (17)
and by the jet luminosity taken at the cut-off, L
peak
(θ) = ν
max
F
ν
(ν
max
, θ),
L
peak
(θ) =

ν
max
ν
cr

α+1
δ
n+1
(θ)ν
′
cr
(θ
′
)F
′
ν
(ν
′
cr
, θ
′
) =

ν
max
ν
cr

α+1

δ(θ)
δ(0)

n+1
L
peak
(0) ∝ θ
b
.
(18)
The values of indices a and b are presented in Tab. 1.
So fa r , we considered only narrow jets, i.e., those having opening angle smaller than
or of the order of Γ
−1
. Fo r a wide relativistic flow, one needs to integrate over observation
angles, which are different for different portions of the flow.
5. Discussion
The off-axis emission is intrinsically high-energy phenomenon. In the case of Bohm
diffusion, for instance, the critical energy for electrons, whose acceleration is limited by the
synchrotron losses, is
E
′
cr
=
3
2
(m
e
c
2
)
2
√
e
3
B
′
, (19)
and the associated cut-off frequency o f their synchro t ron emission is at
ν
′
cr
≃
0.5
π
eB
′
m
e
c

E
′
cr
m
e
c
2

2
, hν
′
cr
≃
9
4
m
e
c
2
α
f
≃ 310 m
e
c
2
, (20)
where α
f
is the fine structure constant. In the observer’s frame the cut-off is blueshifted
to GeV range. However, a diffusion faster that the Bohm one results is a smaller cut-off

– 10 –
frequency. For example, in the case of random small-angle scattering
hν
′
cr
≃

ℓ
c
r
g0

2/3

α
f
B
′
B
cr

1/3
m
e
c
2
α
f
, (21)
where r
g0
= m
e
c
2
/eB
′
is the “cold” gyroradius, ℓ
c
the correlation length of the magnetic
field, and B
cr
≃ 4 .4 × 10
13
G. The factor ℓ
c
/r
g0
can be as small as unity (if ℓ
c
< r
g0
,
then electrons radiate in the undulator regime and their cut-off frequency increases with
decreasing magnetic-field scale), and inverse Compton losses further decrease the value of
ν
′
cr
. In the case of GRBs, where B
′
∼ 10
5
− 10
6
G, we find from Eq. (21) that the cut-off
can be located at just few MeV, so that the radiation above the peak in G RB spectra can
be interpreted as off-a xis synchrotron emission.
An important factor to be kept in mind when considering the off-axis emission is two-
photon pair production. Absorption of high-energy photons in this process rapidly makes
a source opaque with increase of the viewing angle, effectively limiting its maximum value.
There is one particular situation, where interference from the two-photon absorption is always
important: it is inverse Compton off-axis emission in the Klein-Nishina regime in the case,
where it is the dominant radiatio n mechanism. Indeed, the off-axis emission implies fast
cooling of radiating electrons, i.e., the probability that they interact with target photo ns
is close to unity. The same is true for the comptonized high-energy photons, since t he
cross-sections for electron-photon and photon-photon interactions are of the same order of
magnitude in the Klein-Nishina limit.
As easy to see from Tab. 1, a spectral index of the off-axis emission, as a rule, exceeds
−1. In fact, this is always the case as long as the injection spectrum is hard (s < 2). As
the injection gets softer, there appear exceptions. The first to break this rule (what happens
at any s > 2) is IC emission in the Klein-Nishina regime fo r the case, where it dominates
energy losses, the spectrum of comptonized radiation is F
ν
∝ ν, a nd the magnetic field is
quasi-uniform (p = 1). The above preconditions, taken together, make this situation rather
unlikely. The more common synchrotron emission, on the other hand, is quite resistant in
the hardening trend: only very soft injection with s > 4 can make its spectral index smaller
than −1. So, in the vast majority of situations, the luminosity at cut-off, L
peak
, is roughly
the same as the bolometric luminosity of the jet.
Since it has many applications, it is int eresting to discuss the synchrotron emission in
more detail. In the case where it is the dominant radiation mechanism, the spectral index
between ν
cr
and ν
max
increases by 2 (for the Bohm-like diffusion) or by 1.5 (for random
small-angle scattering) relative to what would be the spectral index of ordinary head-on
emission. For an injected particle distribution with indices s < 10/3 or s < 7/3 (the Bohm-
like diffusion and the small-angle scattering, respectively), the resulting spectrum formally

– 11 –
appears to be harder than the low-frequency asymptotic for the synchrotron emission of an
individual particle. In practice, this means that the spectrum is determined by the low-
frequency emission of t he most energetic part icles. The observed cut-off frequency depends
on the viewing angle as ν
max
∝ θ
−1
for the Bohm-like diffusion and ν
max
∝ θ
−2/3
for the
small-angle scattering, that is, much weaker than dictated by the Lorentz transformations
alone (ν
max
∝ θ
−2
).
Prevalence of the external Compton losses reverses the above dependence and even
cancels out the effect of jet dimming with increasing viewing angle. Indeed, one finds from
Tab. 1 that the cut-off frequency increases with viewing angle as ν
max
∝ θ
3
or ν
max
∝ θ
2
for
the Bohm-like diffusion and the small-angle scattering, respectively. Under a widely used
assumption that the particle injection function has spectral index s = 2, the peak luminosity
L
peak
of a continuous jet appears to be independent on the viewing angle for any p . Moreover,
a hard injection with the index s < 2 makes an off-axis jet to appear brighter than when it
is viewed head-on.
An intermediate situation takes place in the case where self-Compton ra diatio n in the
Klein-Nishina r egime dominates the energy losses. Here the cut-off frequency may increase or
decrease with the viewing angle, depending on whether the spectral index q of the radiation
being comptonized is positive or negative.
The off-axis radiation is not necessarily electromagnetic in its nature; for instance, it can
be neutrino emission. The only practical source of neutrinos in relativistic jets is the decay
of charged pions, which are produced in photo-pionic reactions or in inelastic collisions of
nucleons. To be precise, we note that decaying charged pions give muons plus only one half of
the total number of muon neutrinos and anti-neutrinos. Another half and all of the electron
neutrinos and anti-neutrinos come from subsequent decays of secondary muons. In this
way, neutrinos are born alongside with energetic photons, electrons, and positrons, which
altogether carry about a half of the energy of decaying pions. This arg ument apparently
leads to the conclusion that the neutrino luminosity of a relativistic jet is at most as large
as its electromagnetic luminosity.
Once again, the common wisdom does no t work with the off-axis emission. A situation
is p ossible, where the jet is opaque for the high-energy photons, which therefore get repro-
cessed through electromagnetic cascade, producing isotropic in the jet-comoving frame soft
electromagnetic radiation. The latter is strongly beamed in the laboratory frame due to
jet’s motion. Neutrinos, on the other hand, preserve t heir initial anisotropy in the comoving
frame and can be efficiently emitted at larger angles to the jet axis. When observed at large
viewing angles, such a jet looks as an over-efficient neutrino source.

– 12 –
Since the o ff -axis neutrino emission can originate only from anisotropic angular distri-
bution of the parent pions (muons), it requires substantially anisotropic – in jet frame –
source o f energetic nucleons. It is possible if acceleration of protons is radiative-loss limited
or if there is a neutron component in the jet (Derishev, Kocharovsky, & Kocharovsky 199 9),
which moves with the Lorentz factor different from that of the bulk matter. In either case
the decay length of pions (muons) must be less than their scattering length not to let them
isotropize. In terms of diffusion coefficient D(E), which depends only on the particle’s energy
in the ultra-relativistic limit, this condition means
D(E) > D
i
(E) =
1
3
t
i
m
i
E , (22)
where the index i stands either for charged pions (π) or muons (µ), t
π
≃ 2.6 × 10
−8
s and
t
µ
≃ 2.2 × 10
−6
s are their lifetimes, m
π
and m
µ
their masses. If D
π
< D < D
µ
, then only
a half of muon neutrinos contribute to the off-axis emission, whereas the beam-pattern for
electron neutrinos and the rest of muon ones is similar to that of o r dinary emission.
In the case of Bohm diffusion, Eq. ( 22) translates simply into an upper limit f or the
magnetic field strength:
B < B
i
=
m
i
c
e t
i
. (23)
Here B
π
≃ 600 G and B
µ
≃ 5 G, so that the above condition is true for any potential
neutrino sources except arguably for the GRB internal shocks, where the diffusion should be
orders of magnitude faster than the Bohm diffusion to fulfill Eq. (23).
6. Implications
Many astrophysical sources with relativistic jets change their appearance in presence of
the off-axis emission. The difference is negligible at low frequencies, but becomes dramatic
for high-energy photons (typically X- and gamma-rays). Unfortunately, it is practically im-
possible t o make definitive and unequivocal predictions from the first principles since the
properties of the off -axis emission strongly depend on details of both radiation and accel-
eration mechanisms, with uncertainties in geometry further increasing the range of possible
solutions. The problem, however, has a silver lining from the observatio nal perspective:
the very same diversity of unique observatio nal signatures provides a means to determine
physical parameters in a source.
In accordance with the above note, this section is not to present a comprehensive analysis
of the properties of off-axis emission for various sources, but ra ther to give an idea of what

– 13 –
one expects in typical situations, that is do ne below using primarily GRBs as a representative
example.
GRBs are a complex phenomenon (see, e.g., [] and [] for a review), which can be de-
composed into qualitatively different prompt-emission and afterglow phases. During t he
prompt phase, which lasts from a fraction of a second to few hundred seconds, G RBs usually
have highly irregular lightcurves and relatively hard emission. The afterglow is characterized
by gradually decaying smooth lightcurve with occasional rises and regular softening of t he
emission.
In the following discussio n we assume for definiteness that peaks in observed GRB
spectra correspo nd to transition f r om sub-critical to super-critical radiation regimes, so that
the radiation above the peak is mainly due to off-axis emission. Such an interpretation
implies that both sub- and super-critical particles form a single distribution. This is possible
if t he radiating electrons are secondary particles from inelastic interactions of high-energy
protons, or produced via a non-diffusive (for example, converter) acceler ation mechanism.
The prompt emission of GRBs is thought to be the synchrotron radiation or ig inating
from a successio n of internal shocks, i.e., those developing within the fireball at a distance
of the order of D ∼ 10
12
cm from the central engine. R adiation from a large number of such
shocks contributes to observed flux at any moment of time; in effect, they can be treated as
a continuous jet. One can imagine two situations: a jet, whose opening angle θ
0
is smaller
than the viewing angle θ, and the opposite case of small viewing angle, θ < θ
0
.
The t iming properties of the prompt emission are similar in both cases. Since the
Doppler factor for off-axis jets is smaller, the variability timescale must be longer. However,
the total duration of a burst t
GRB
is not affected: it is determined by the lifetime of central
engine, at least as far a s geometrical delay for light propagation is smaller than the lifetime,
i.e., θ
2
D/2c . t
GRB
. Even for short bursts, the latter condition corresponds to relatively
large viewing angles θ . 0.1.
For the case of small viewing angle, observed spectra are affected in two ways. The
off-axis emission from edge portions of t he jet (those propagating at angles much larger
than 1/Γ to the line of sight) can contribute to: (1) t he bolometric luminosity and (2) a
high-energy tail above the cut-off frequency. For the synchrotron-self-Compton emission, no
matter whether the synchrotron or inverse Compton losses are dominant, the peak luminosity
L
peak
(θ) normally
2
decreases with increase o f the viewing angle faster than θ
−2
, making the
2
The opposite requires either hard injection with s < 2 or the low-frequency as ymptotic in the spectrum
of comptonized radiation harder than F
ν
∝ ν
1/3
, that is a so urce of emission other than the synchrotron.

– 14 –
first effect negligible. On the contrary, if the jet looses energy mostly to external Compton
radiation, then the edge portions of the jet dominate the overall bolometric luminosity as far
as s < 14/5, that is, f or any reasonable injection. Although prevalence of external Compton
losses in the jet’s radiative balance or an injection with the index s < 2 are not favored by
current GRB theories, we conclude that the edge portions of the jet cannot safely be ignored
even when calculating the bolometric luminosity.
If the cut-off frequency ν
max
increases with increasing viewing angle, then the off-axis
emission f r om edges of a wide jet significantly changes the observed (composite) spectrum,
causing a high-energy tail to appear instead of an exponential cut-off. Parts of the jet viewed
at different angles contribute to this tail with luminosities ∝ θL
peak
(θ) ∝ θ
b+1
, concentrated
mostly around f r equency ν(θ) ≃ ν
max
(θ) ∝ θ
a
. The envelope of individual contributions
gives the power-law tail:
νF
ν
∝ ν

θL
peak
d θ
d ν

∝ ν
2+b
a
, (24)
where θ ∝ ν
1/a
and a > 0. As follows from Tab. 1, the condition a > 0 can be satisfied in a
consistent synchrotron-self-Compton model, for example if comptonization proceeds in the
Klein-Nishina regime and the spectral index of comptonized r adiation is positive (i.e., q > 0),
so that emergence of the power-law tail should be considered a common phenomenon.
In the case of large viewing angle, θ ≫ θ
0
, every portion of a j et moves at approximately
the same angle to the line of sight, so that the situation is in almost every respect equivalent
to the case of narrow jet, which was considered in Sect. 4. The only correction to be made
is to take into account that the jet subtends an angle, which is much larger than 1/Γ. For
an idealized (uniform with sharp edges) jet the corrected dep endence of luminosity on the
viewing angle is
L
peak
(θ) = (Γθ)
b
(Γθ
0
)
2
L
peak
(0) , (25)
where b < −2 and θ, θ
0
≫ Γ
−1
. As the viewing angle exceeds the opening ang le of the jet,
the bolometric luminosity drops by a factor ∼ (Γθ
0
)
−2−b
.
Taking into consideration the off-axis emission, it is interesting to discuss a possibility
that the X- ray flashes (XRFs) are normal GRBs, whose jets are not pointing to the observer.
The peak energy in XRF spectra is smaller than in GRB spectra, implying that a < 0 and,
consequently (see Tab. 1 ) , that b < −3. Therefore, in the case of idealized jet, the XRFs
and GRBs are members of separate source populations, whose average brightness differs by a
factor Γθ
0
≪ 1 or smaller. Apart from this difference in brightness, the XRF spectra in their
low-energy (below the peak) part should possess the intrinsic feature of off -axis emission –
a paucity of soft photons.
Finally, let us discuss of a tempting possibility to explain unusually weak GRBs as off-

– 15 –
axis jet. Unlike XRFs, weak GRBs have luminosities many orders of magnitude smaller than
their normal counterparts, but radiate in the same spectral range. The off-axis synchrotron
emission can account for the properties of weak GRBs if their main radiative mechanism is
self-Compton in the Klein-Nishina regime, and their low-frequency spectral index q is close
to zero. Indeed, in this case the cut-off frequency is nearly independent on the viewing angle,
whereas the observed luminosity can drop as fast as θ
−3
.
Unlike the prompt GRB emission, the afterglow comes fr om a single blast wave, which
forms when the GRB ejecta plunge into surrounding interstellar gas and which in most cases
can be approximated by a thin spherical shell. Despite the simple geometry, dynamics of
this blast wave is complicated by various factors, such as inhomogeneous external medium,
formation of multiple sub-shocks, late energy injection, etc., which are beyond the scope
of this paper. However, a principal part of the pro blem – namely, obtaining the Green
function for a radiating spherical shell – can be formulated in model-independent terms.
Physically, the Green function G(R, t, ν) is the spectral flux density, measured by a distant
observer as a function of time, provided the radiation comes from an instantaneous release
of unit energy in a spherical shell of radius R expanding with velocity v(R). By definition,
Z
G(R, t, ν) dν dt = 1 and G ≡ 0 for any t < 0. The spectrum o f any thin blast wave can
be represented as
F
ν
(t, ν) =
Z
∞
0
λ(R) G(R, t − t
e
, ν) dR , (26)
where λ(R) is the energy lost for radiation per unit distance, and t
e
(R) =
Z
R
0
dR
v(R)
−
R
c
the time, as measured by the distant observer, it takes for the blast wave to expand to the
radius R. If necessary, Eq. (26) includes another integration to take into account the radial
structure of the blast wave.
To find t he Green function, we note that the area of spherical segment that comes into
the observer’s view during the time interval dt is equal to 2πRcdt. This segment moves at an
angle θ(t) = arccos(1 −ct/R) to the line of sight and its contribution t o the detected fluence
is proportional to θ
k(α+1)
δ
n+1
dt ∝ δ
−b/2
dt, where the index n (see Eq. (3) for definition) is
equal to 2. Indeed, physically an element of the blast wave is a blob, whose luminosity scales
with n = 3, and whose apparent lifetime is proportional to δ
−1
, so that its fluence scales
with n = 2. So, we obtain
G(R, t, ν) =

2
b + 2
R
βc

(1 + β)
1+b/2
− (1 − β)
1+b/2


−1
Θ(t) Θ

2R
c
− t

1 − β + β
ct
R

b/2
f(R, ν, θ) ,
(27)
where Θ(t) is the step function, f(R, ν, θ) the spectral energy distribution, normalized to

– 16 –
unity, and the factor in square brackets ensures that the Green function as a whole is nor-
malized to unity.
Since we are interested in the ultra-relativistic case, where β → 1, it is convenient to
use the approximate expression for the Green function,
G(R, t, ν) ≃
h
−(b + 2)Γ
2
c
R
i
Θ(t)

1 + 2Γ
2
ct
R

b/2
f(R, ν, θ) , (28)
which is also valid fo r a blast wave with finite angular extent as long as the opening angle is
much larger than Γ
−1
.
An important thing to learn from Eq. (2 8) is that a radiating shell fades away rather
gradually after the shock has passed it, producing what may be called a geometrical, or re-
tarded, afterglow. Due to the geometrical delay, the retarded emission fro m early afterglow
coexists in time with ordinary emission from la te afterglow, and its bolometric luminosity
asymptotically decreases as t
b/2
. In absence of the off-axis emission, b = −6 and the lu-
minosity of geometrical afterglow rapidly decays to a level, indiscernible against the much
brighter ordinary afterglow. For the off-axis emission, the index b is typically in the r ange
−4 < b < −2, that corresponds to decay rate of the geometrical afterglow between t
−2
and t
−1
. For comparison: the bolometric luminosity of ordinary afterglow behaves as t
−3/2
and t
−12/7
for adiabatic and fully radiative shocks, respectively, propagating into uniform
medium, or as t
−1
and t
−4/3
if the shocks propagate into a wind with density profile ρ ∝ R
−2
.
The properties of off-axis emission, which are discussed above in application to GRBs,
show up also in AGNs (and microquasars, as far as they can be considered a scaled-down
version of AGNs). Thus, we limit o ur a nalysis of AGNs to only one specific point – the
observational bias against detection of jets pointing away from the line of sight .
With present-day telescopes AGN surveys are sensitivity-limited, i.e., we detect only
those, whose apparent brightness is above certain threshold. Let us suppose that the bolo -
metric luminosity changes with the viewing angle as θ
−b
and the sources are uniformly
distributed in space. Then the number of detectable sources decreases with viewing angle as
N(θ) ∝ θ
4−3b
2
. (29)
In absence of the off-axis emission the bolometric luminosity decreases with the viewing angle
as θ
−6
, so that N(θ) ∝ θ
−7
. Among hundreds of known blazars one can hardly expect to find
even a single source with θ > 2/Γ, in accordance with the existing observational data. On
the other hand, the radiogalaxies are observed with randomly directed jets, that comes at no
surprise since the radio-emission is not beamed. The off-axis emission is much less beamed

– 17 –
than the ordinary radiation from AGN inner jets and – in this resp ect – resembles radio-
emission, though it occupies the opposite end of electromagnetic spectrum. The number of
detectable off-a xis sources increases with increasing index b and they dominate the entire
source population for any b > −4/3.
It turns out that a situation, typical for blazars (the r adiative losses are mostly due to
inverse Compton scattering of external photons), provides also an extreme example of the
off-axis synchrotron emission, with apparent luminosity almost independent on the viewing
angle. It means that the majority of blazars, which are not detected at present because
of large inclination o f their jets to the line of sight, will show up when observed in t he
right spectral range. For the synchrotro n off- axis emission from MeV blazars the most
favorable (fr om the observational point of view) spectral domain is around 100 MeV, within
the operational range of GLAST and AGILE. The inverse Compton component of the off-
axis emission can be detected with modern ground-based Cherenkov telescopes, which are
sensitive to photons down to 30-100 GeV. Some of these off-axis blazarz may have already
been detected by EGRET as unidentified high-latitude sources.
7. Conclusions
A number of processes lead to generation of super-critical particles in relativistic flows.
Having the scattering length of the order of the radiation length o r exceeding it, these
particles do not isotropize upon ent ering the r elativistic flow and radiate their energy while
preserving a certain degree of anisotropy. This anisotropy counteracts the beaming, which
results from t he Lorentz boost, so that the emission produced in such a way has a wider
beam pattern in the laboratory frame than that of sub-critical particles and can be called
the off-axis emission. The properties of the off-axis emission under various conditions are
summarized in Table 1. Among many implications considered in this paper, the following
are of major importance from the observational point of view.
The j et sources, which are observed off-axis owing to the effect of beam pattern broad-
ening should exhibit very hard spectra. Indeed, for the super-critical particles the r .m.s.
deflection angle (and hence the width o f the beam pattern) is a function of their energy.
An observer situated at a large angle to the jet axis effectively sees the particle distribution
devoid of its low-energy part , whose emission can only be seen at smaller viewing angles.
Therefore, the off-axis emission is the hardest possible – in most cases it is essentially as
hard a s the spectrum of an individual particle. An off-axis j et (for example, AGN, GRB, or
microquasar) is likely to be a source of gamma-ray radiation above several MeV and up to
TeV range without any bright X- r ay or optical counterpart. Apparently, the off-axis AGNs

– 18 –
can account a t least for some of unidentified extragalactic EGRET sources.
The space-borne gamma-ray telescopes with wide field of view, such as GLAST and AG-
ILE, have t he greatest chance to detect synchrotron radiation from off-axis j ets. The inverse
Compton component of the off- axis emission can be detected by ground-based Cherenkov
telescopes. However, they have very limited surveying capabilities, so that the best observing
strategy in search o f the off-axis emission would be to look at known AGNs whose jet are
pointing away fr om the line of sight.
In transient sources (for example, GRBs) broader beam pattern means larger geometrical
delay, which is proportional to the square of angle between the jet axis and the line of sight.
The luminosity of in the retarded off-axis emission decays rather slowly in time, allowing
observations in GeV-TeV range when the prompt emission is over. Moreover, if the temporal
index of geometrical afterglow is lar ger than -3/2 (that is right in the middle of the typical
range), then the integral signal-to-noise ratio continually grows with observing time as long
as the off-axis emission is present. This opens an interesting possibility for observatio n of
GRBs with ground-based Cherenkov telescopes, which normally have to slow response to
catch the prompt radiation. A serendipitous discovery of orphan GRB afterglows in the
TeV range is also possible, because the beam pattern is broa der f or high-energy photons.
It should be noted that there is observational evidence for delayed GRB emission, a t least
in the case of GRB940217 (Hurley et al., 1994), which can be interpreted as geometrical
afterglow due to the off-axis emission.
The off-axis emission is intrinsically high-energy phenomenon and in some cases it may
experience two-photon absorption within the source, esp ecially at large viewing a ngles. In
opaque sources the electromagnetic radiation from super-critical particles is reprocessed
through the electromagnetic cascade, looses its identity and becomes collimated, but the
neutrino signal from them still comes out. Such jets, when viewed o ff-axis, appear over-
efficient neutrino sources, where the ratio of neutrino luminosity to the electromagnetic one
can be almost ar bitra rily large. The off-axis neutrinos can be detected by the next generation
of cubic-kilometer scale high-energy neutrino detectors, and may provide unique information
on the details and relative importance of various particle acceleration processes.
8. Acknowledgments
E.V. Derishev acknowledges the support from the President of the Russia n Federation
Program for Support of Young Scientists (grant no. MK-2752.2005.2) . This work was
also supported by the RFBR grants no. 05-02-1752 5 and 04-02-16987, the President of

– 19 –
the Russia n Federation Program for Support of Leading Scientific Schools (grant no. NSh-
4588.2006.2), and the program ”Origin and Evolution of Stars and Galaxies” of the Presidium
of the Russian Academy of Science.
REFERENCES
Achterberg, A., Gallant, Y.A., Kirk, J.G., Guthmann, A.W. 2001, MNRAS, 328, 393
Derishev, E.V., Kocharovsky, V.V., & Kocharovsky, Vl. V. 1999, ApJ, 521, 640
Derishev, E.V., Aharonian, F.A., Kocharovsky, V.V., Kocharovsky, Vl.V. 200 3, Phys. Rev.
D, 68, 043003
Hurley, K. et al. 1994, Nature, 3 72, 652
Lind, K.R., & Blandford, R.D. 1985, ApJ, 295, 358
Madau, P., & Thompson, C. 2000, ApJ, 534 , 23 9
Piran, T. 2005, Rev. Mod. Phys., 76, 114 3
Stern, B.E. 2003, MNRAS, 352, L35
Urry, C.M., & Padovani, P. 19 95, PASP, 107, 803
Zhang, B., & M´esz´aros, P. 2004, IJMPA, 19, 2385
This preprint was prepared with the AAS L
A
T
E
X macros v5.2.

– 20 –
Dominant energy loss mechanism
Observed
emission
Synchrotron or IC IC in the KN Regime External Compton
Synchrotron
or IC
α = p + 1 −
s
2
k =
2
p + 1
a = −
2p
p + 1
b =
2
p + 1
−
s
p + 1
− 2n
α =
3
2
−

p −
1
2

q −
s
2
k =
2
1 − pq
a =
2pq
1 − pq
b =
3 + q − s
1 − pq
− 2n
α =
3(p + 1)
4p + 1
−
s
2
k = 2 +
6p
p + 1
a =
6p
p + 1
b =
6(2p + 1)
p + 1
−
4p + 1
p + 1
s − 2n
IC in the
KN Regime
α = 2(p + 1) − q − s
k =
1
p + 1
a = −
2p + 1
p + 1
b =
1 − q − s
p + 1
− 2n
α = 3 − 2pq − s
k =
1
1 − pq
a =
2pq − 1
1 − pq
b =
2 − s
1 − pq
− 2n
α =
6(p + 1)
4p + 1
− q − s
k = 1 +
3p
p + 1
a =
3p
p + 1
− 1
b =
3p
p + 1
−
4p + 1
p + 1
(q + s) + 5 − 2n
External
Compton
α =
4
3
(p + 1) −
4p + 1
6p
s
k =
6p
(p + 1)(4p + 1)
a = −
2p
p + 1
−
2
4p + 1
b =
4p − 2
(p + 1)(4p + 1)
−
s
p + 1
− 2n
α =
4p + 1
6p
(3 + q − 2pq − s) −
p + 1
3p
k =
6p
(1 − pq)(4p + 1)
a =
6p
(1 − pq)(4p + 1)
− 2
b =
4p − 2
(1 − pq)(4p + 1)
+
1 + q − s
1 − pq
− 2n
α =
2(p + 1)
3p
−
4p + 1
6p
s
k =
6p
p + 1
a =
4p − 2
p + 1
b =
4p + 1
p + 1
(2 − s) − 2n
Ta ble 1: The summary on indices, which describe the spectrum of off-axis emission, F
ν
∝ ν
α
, and its extent in frequency,
ν
max
/ν
cr
= (Γθ/2)
k
. The table also presents the angular dependence of the cut-off frequency and luminosity at the
peak, ν
max
∝ θ
a
and L
peak
∝ θ
b
, respectively. The viewing angle is in the rang e Γ
−1
≪ θ ≪ 1. For the details on
evaluation of these indices see text.