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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.

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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.