A&A 433, 765-776 (2005)
DOI: 10.1051/0004-6361:20042161
A. Mastichiadis^{1} - R. J. Protheroe^{2} - J. G. Kirk^{3}
1 - Department of Physics, University of Athens, Panepistimiopolis,
15783, Zografos, Greece
2 -
Department of Physics and Mathematical Physics, University of Adelaide,
North Terrace, Adelaide, SA 5005, Australia
3 -
Max-Planck-Institut für Kernphysik, Postfach 103980,
69029 Heidelberg, Germany
Received 12 October 2004 / Accepted 15 December 2004
Abstract
We present calculations of the spectral and temporal radiative
signatures expected from ultrarelativistic protons in compact sources.
The coupling between
the protons
and the leptonic component is assumed to occur via Bethe-Heitler
pair production. This process is treated by modeling the results of
Monte-Carlo simulations and incorporating them in a time-dependent
kinetic equation, that we subsequently solve numerically.
Thus, the present work is, in many
respects, an extension of the leptonic "one-zone'' models to include
hadrons. Several examples of
astrophysical importance are presented, such as the signature
resulting from the cooling of relativistic protons
on an external black-body field and that of their cooling in the presence of
radiation from
injected electrons. We also investigate and refine the threshold conditions
for the
"Pair Production/Synchrotron'' feedback loop
which operates when relativistic protons
cool efficiently on the synchrotron radiation of the internally produced
Bethe-Heitler pairs. We demonstrate that an additional component of injected
electrons lowers the threshold for this instability.
Key words: radiation mechanisms: non-thermal - radiative transfer - galaxies: active
The spectral energy distribution (SED) of powerful AGN such as flat-spectrum radio quasars and blazars has a double humped appearance with the low energy part extending from the radio to UV (or in extreme cases to X-rays), and a high energy part extending from X-rays to -rays. In AGN with relativistic jets closely aligned to the line of sight the emission is dominated by non-thermal radiation, with the low energy hump being mainly synchrotron radiation. If the alignment is not so close, a thermal component of UV radiation from an accretion disk may dominate. The non-thermal components can be strongly variable, probably originating in the jet.
These observations indicate that the jets of blazars act as efficient particle accelerators. Furthermore the gamma-ray observations in the GeV (Hartman et al. 1999) and TeV regime (Horan & Weekes 2004) imply that the accelerated particles can reach very high energies. Models involving electron radiation can adequately explain both this high energy emission and the coordinated multiwavelength campaigns (Tavecchio et al. 2001; Mastichiadis & Kirk 1997; Krawczynski et al. 2002). The usual assumption is then that the high energy part of the SED is due to inverse Compton scattering of the low energy part of the SED (synchrotron self-Compton model) possibly supplemented by inverse Compton scattering of external photons (external Compton models) for example from the disk, either directly or scattered by clouds. Despite these successes, the question of the role of a possible relativistic hadronic component remains an open one.
In principle, sites of electron acceleration may accelerate protons as well. Consequently, models in which the high energy part, and some fraction of the low energy part, of the SED is due to acceleration and interaction of protons in the jet have also been proposed. Some of these models invoke interactions with ambient matter (Beall & Bednarek 1999; Pohl & Schlickeiser 2000; Schuster et al. 2002) but they require high mass densities in the jet to be viable. Here we concentrate on hadronic models in which the protons interact with low energy photons via Bethe-Heitler pair production.
As with leptonic models, the target photons may be produced inside the emission region in the jet or may originate from outside of the jet, e.g., from an accretion disk (Atoyan & Dermer 2001; Bednarek & Protheroe 1999; Protheroe 1997; Neronov & Semikoz 2002). For internally produced target photons, synchrotron emission by a co-accelerated population of electrons is assumed (Mannheim 1993,1995; Mannheim & Biermann 1992). The high energy hump of the SED then results from electromagnetic cascading of gamma-rays from decay and electrons from Bethe-Heitler pair production and decay in the radiation and magnetic field of the blob. Neutrinos and cosmic rays would also be emitted as a result of the neutrinos from the and decays, and neutrons produced in interactions if the threshold for pion photoproduction is exceeded (Protheroe & Szabo 1992; Kirk & Mastichiadis 1989; Eichler & Wiita 1978; Sikora et al. 1989; Begelman et al. 1990; Waxman & Bahcall 1999; Atoyan & Dermer 2003; Sikora et al. 1987; Protheroe 2004; Mannheim et al. 2001; Giovanoni & Kazanas 1990; Szabo & Protheroe 1994). For protons to be accelerated to energies sufficient to exceed the Bethe-Heitler and photo-pion-production thresholds, relatively high magnetic fields are required. In proton synchrotron blazar models the magnetic field is sufficiently high such that the high energy part of the SED has a major contribution also due to synchrotron radiation by protons (Aharonian 2000; Mücke & Protheroe 2001; Reimer et al. 2004; Mücke & Protheroe 2000). All of the work described above assumes the emission has reached a steady state, and that the target photon fields are steady. In reality, the strong variability displayed by these sources mandates a time-dependent calculation, that, ideally, should be done self-consistently, with internally produced radiation fields contributing alongside external ones to the target radiation field.
Time-dependent codes that solve the kinetic equations describing electrons and photons and their interactions have been developed and successfully applied to AGN (Mastichiadis & Kirk 1997; Krawczynski et al. 2002). However, codes of this type that also account for hadronic interactions have been neglected. One reason for this is that whereas the modeling of leptonic processes is relatively straightforward (e.g., Lightman & Zdziarski 1987; Coppi & Blandford 1990), photo-hadronic and hadron-hadron interactions are much more complex. To date, all attempts have used approximations of uncertain accuracy (e.g., Stern & Svensson 1991) but the use of Monte-Carlo event generators which model in detail electromagnetic (Protheroe & Johnson 1996; Szabo & Protheroe 1994) and hadronic interactions (Mücke et al. 2000) opens up the possibility of extracting accurate descriptions of the fundamental interactions suitable for incorporation into a kinetic code.
Motivated by these developments we investigate the consequences of the presence of relativistic hadrons in compact sources by incorporating new results from Monte-Carlo simulations into a time-dependent code which follows the evolution of relativistic hadrons, electrons and photons by solving the appropriate kinetic equations. In the present paper we investigate as a first step, the case in which the only channel of coupling between hadrons and leptons is the Bethe-Heitler pair-production process, leaving the investigation of photo-meson production for a future paper. Although this is not a complete description of hadronic models it nevertheless enables one to draw useful and interesting new results.
The present paper is structured as follows: in Sect. 2 we present the numerical code that solves simultaneously in a self-consistent manner the coupled, time-dependent kinetic equations for each species, i.e. protons, electrons and photons. In Sect. 3 we present the Monte-Carlo results for the Bethe-Heitler process and show how these can be incorporated in the kinetic equations. In Sect. 4 we present some results for the case in which relativistic protons interact with an external black-body radiation field. In Sect. 5, we present a numerical analysis of the "Pair-Production/Synchrotron'' instability. The case of simultaneous injection of relativistic protons and electrons is examined in Sect. 6 and the main conclusions are summarized in Sect. 7.
The kinetic equations describing a homogeneous source
region containing protons, electrons and photons were
formulated and solved numerically by
Mastichiadis & Kirk (1995, henceforth MK95).
We follow the same method, using an improved description
of the microscopic processes. The equations to be solved can
be written in the generic form
(1) |
The unknown functions n_{i} are the differential
number densities of the three species, normalised as follows:
(2) | |||
(3) | |||
(4) |
The physical processes to be included in the kinetic equations are:
(11) |
For an isotropic target comprising monoenergetic photons of energy the effective cross-section for interaction of a proton of energy is given by
(12) |
(15) |
(16) |
Figure 1: The angle-averaged Bethe-Heitler pair-production cross section (solid curve), the angle-averaged Bethe-Heitler pair-production inelasticity (dashed curve), and (dotted curve) are plotted as a function of for . | |
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Examination of the integrand in Eq. (14) shows that the
square of the
total CM frame energy is distributed as
(17) |
The distribution in energy
of electrons (of
either charge),
Figure 2: The distribution of electron energies is shown for x=10^{-6} and (dashed histogram), 10^{7} (solid histogram) and 10^{9} (dotted histogram). | |
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The pair-production spectra were calculated using the results of the Monte Carlo code described in Sect. 2.1. Protons of a specific energy were allowed to interact with isotropic monoenergetic target photons and the energies of the products were tabulated. The photon target energies used were x_{0}=10^{-6}, 10^{-4}, 10^{-2} and 1 (in units of electron rest mass). Proton energies ranged from (so the threshold requirement could be met) up to in logarithmic steps of 0.1.
The pair-creation rate is then given
(20) |
(21) |
We note that MK95 used the approximation corresponding to . For a proton distribution which decreases as increases, this overestimates the production rate of electrons of a given energy.
Since the proton loses a small amount of energy
(typically given by
in each pair-producing collision
we can treat the losses as a continuous
process and write
(23) |
(26) |
(27) |
(28) |
In this case, the photons produced by electrons created in the BH process are not important as targets and the resulting photon spectrum is quite simple. It is a single power-law of (number) spectral index -3/2, extending up to an energy , that is approximately the inverse of the temperature of the black-body field, i.e. , where . The explanation of this spectrum is straight-forward: protons pair-produce on the black-body field and, since they are produced with high energies, the pairs cool on the black-body photons initially by Compton scattering in the Klein-Nishina regime. This produces -rays that are above the threshold for pair production on the black-body photons and, therefore, are re-absorbed. Lower energy pairs, which cool by Compton scattering in the Thomson regime, produce photons which are below the threshold for photon-photon pair production. This naturally produces an electron distribution function and thus a photon spectrum . The value of , therefore, is set by the condition , where is the optical depth for photon-photon pair production on the background black-body field.
Since for the present case there is
neither escape of electrons from the system
nor any sink of energy, (e.g. synchrotron self-absorption)
other than photon escape,
the radiated luminosity
(or, equivalently,
)
equals the luminosity injected
in pairs once a state steady is achieved, i.e.
,
where
As an illustrative case we take
.
Here also
K, while
,
.
The dotted line in
Fig. 3 depicts the electron injection function
which shows a broad maximum. The same
figure shows the photon
spectrum which is obtained from the cooling of
these electrons and the ambient black-body field.
The dashed line shows the
spectrum in the case where the
pair production has been artificially
switched off.
The spectrum is flat and peaks at high energies.
We note that both the electron production
function and the unabsorbed photon spectrum extend to
about two orders of magnitude above
,
an effect that is due to the kinematics of the Bethe-Heitler
pair production - see Fig. 3.
At lower energies
it produces the characteristic -3/2 power-law.
The full line shows the
final photon spectrum which includes
absorption. It has still the same power-law,
however all the details of the electron injection
spectrum which were evident in the unabsorbed spectrum
have disappeared due to the intense attenuation.
Nevertheless the overall luminosity is conserved (as it
should be) and as the electromagnetic cascade
redistributes the power to lower energies,
the flux is increased there.
Figure 4 shows the effects of losses on the
steady-state proton distribution for these parameters (
)
and two
higher values of
,
compared to the loss-free case.
Figure 3: Steady-state photon spectrum resulting from a power-law proton injection and subsequent proton-photon pair production on a black-body photon field in the case where photon-photon pair production (i) has been ignored (dashed line) and (ii) has been taken into account (full line). The dotted line curve depicts the electron/positron distribution function at production. For this particular run the protons were assumed to be injected with a power-law of slope , between the limits and . Also the values and have been assumed. The black-body photon field parameters were K ( ) and . Synchrotron losses were neglected. | |
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Figure 4: Proton steady-state spectra for various external black-body photon compactnesses. All cases are taken for the same injection proton parameters ( ) and for the same temperature of the external black-body field ( ) The plotted proton spectra are shown when (no-loss case, full line), (dotted line), (short-dashed line) and (long-dashed line). | |
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The efficiency of the BH process, i.e., the ratio of the proton
power turned into pairs to the total power injected as protons, is
shown in Fig. 5 as a function of the maximum Lorentz factor of
injection
.
The inelasticity of this process is small
(
), so that high efficiency can only be achieved if
a proton interacts many times before escape. This is indeed the case
for high black-body compactnesses - the efficiency exceeds
for
and
.
Figure 5: Proton efficiency, i.e. fraction of power lost to pairs to total power injected in protons as a function of the upper cutoff of the proton distribution . The rest of the proton injection parameters are kept constant ( , , ), while the external black-body field has temperature . and compactnesses ( bottom to top) . | |
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In the more realistic case, where
synchrotron losses cannot be neglected, the
spectrum of the electrons becomes more complicated,
as now the electrons cool by a combination of
synchrotron radiation and inverse Compton scattering.
Figure 6 shows the obtained spectra
in the cases where the magnetic compactness
,
defined according
to
= | (30) |
Figure 6: Steady state photon spectra for the same parameters used in Fig. 3 but including synchrotron radiation. The magnetic field used was B=100 G and the magnetic compactnesses were (full line), (dashed line) and . The black-body compactness was . | |
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Kirk & Mastichiadis (1992, henceforth KM92)
have shown that ultra-relativistic protons
can, under certain conditions, become unstable
to various types of radiative instabilities. They showed
explicitly the necessary conditions for one of them to
occur, namely the Pair-Production/Synchrotron instability
(henceforth PPS).
To understand the basic idea, assume that protons are
confined in a region of characteristic radius R
where a magnetic field of strength B is also present.
Assume, moreover, that the protons are relativistic and
have Lorentz factors such that if they photo-pair produce,
the synchrotron photons radiated from the created pairs
are
sufficiently energetic for the protons
to produce more pairs on them.
Making the simplifying
assumptions that (i) the created pairs have the same
Lorentz factors as the protons and (ii) the synchrotron
photons are all emitted at the critical frequency,
KM92 showed that in order for protons to be able to
initiate this loop they should have
Lorentz factors above a critical value given by
(31) |
MK95
presented a numerical simulation of the PPS instability
in the case where protons are accelerated from low momenta
by a Fermi-type acceleration scheme. Once the protons
(assumed to have a density exceeding the critical number density)
reached energies above
,
the
conditions for the instability loop were complete and the internally
produced photons increased, saturating the acceleration, and
driving the system to equilibrium.
However, the code used by Mastichiadis & Kirk (1995) is limited by
the
simplifying assumptions mentioned in the previous paragraph.
Here we re-examine
this problem with the
improved version of the code that uses, as
described
in Sect. 3, the Bethe-Heitler pair-production spectra as given by
Monte-Carlo code and the full synchrotron emissivity. The objective
is to find an accurate estimate of the critical
proton number density above which the PPS instability occurs,
i.e., a numerical version of the analytical (but approximate)
Fig. 1 of KM92.
Figure 7: Plot of the internally produced photon compactness as a function of time for values of around the critical value . In each curve the value of is increased 2% from its previous value. For these runs . | |
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To make our results directly comparable with those of the aforementioned figure, we have set the parameters to the values prescribed there. For this we took a source size R=10^{15} cm, a magnetic field B=10^{3} Gauss and a proton distribution function of the form with from the lowest allowed proton energy to a maximum energy . We note that in this case the proton distribution is held constant throughout each run, i.e. protons do not evolve. For various values of , we run the code for different values of the only remaining free parameter ( ). According to KM92, the time evolution of the photon and electron distribution functions is of the form with s>0 when the protons are in the unstable regime. Thus in order to verify numerically the existence of the PPS instability we seek a value above which the internally produced electron/positron pairs and photons start increasing with time.
The onset of instability for can be seen in Fig. 7 which depicts the photon compactness as a function of time t (expressed, as always, in units of ) for various values of around . When the protons are in the stable regime there is some pair production between the protons and the synchrotron photons produced from the initial electron distribution but the system, for times larger than the synchrotron cooling time, settles to a steady-state. Thus for we get s=0. However, as can be seen from the figure, as increases (from bottom to top) the photons start to grow exponentially, with s increasing with increasing . It is worth mentioning that each curve corresponds to a value of that is larger by only 2% than its previous value; therefore, this figure depicts the rapid onset of the instability.
Figure 8 shows the behaviour of s as a function
of the proton normalisation
for three
values of
.
It is evident
that once the instability sets in, s is a very sharp function
of
.
Therefore, we find that increasing
approximately by a factor
of 2 above its critical value, s becomes
greater than one, i.e. the density of photons starts growing
on a timescale shorter than the crossing time of the source.
This, as we shall see in the next section, has catastrophic
consequences for the high energy protons as the
spontaneously growing
photons make them lose their energies.
Figure 8: Behaviour of the instability growth index s as a function of for values of ( right to left). | |
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Figure 9 shows the marginal stability curve
as is obtained from the present code (in practice we have
calculated the values of
which correspond to s=.05.
Due to the steep dependence of s on
this value
can be considered as very close to
the marginal stability one). Note
that this curve is in very good agreement
with
the
curve estimated by
KM92 (plotted here as a dotted line)
for values of close to the threshold, but exceeds it by a factor of about 2 for
at high energy.
This difference can be attributed to the overestimation
of the electron production rate in the
BH process as discussed in Sect. 2. For maximum values of the proton Lorentz
factor close to
,
all the BH interactions occur
close to threshold, so that the assumption used by MK is
accurate.
(The small difference in the shape of the KM curve and our
present result in this energy range can be attributed
partly to the kinematics
of Bethe-Heitler pair production and partly to the fact that
the full expression for the synchrotron emissivity was
used.)
However, once the upper cut-off of the proton distribution
substantially exceeds
,
the effect of pairs
injected with
becomes important. These
are not taken into account in the approximation used by MK92,
but are treated accurately in the simulation-based method used here.
It is evident that the basic concept of the feedback loop
remains unaltered by our more accurate treatment. The
quantitative implications, however, are discussed below.
Figure 9: Plot of the numerically obtained marginal stability as a function of (full line). The dashed line shows the s=1 locus, while the dotted line depicts the marginal stability results of KM92. Note that the KM92 approximation is reasonably good at low . | |
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In order to see the effects of the PPS instability
when proton losses are taken into account
we assume once again (see Sect. 4) that
protons are injected in a region of radius R with
a power-law (Eq. (24)). However, to make the picture
less complicated, we assume that there is no external
photon field present. Thus one expects that the proton
spectrum will reach an equilibrium state given by
the no-loss solution (Eq. (25)). This is indeed true as
long as this steady-state solution is below the critical
density for the PPS instability, i.e. when
,
with
the normalisation of the protons and
given from Fig. 9.
In this case also there
is a very small number of internally produced photons.
However, as soon as the condition
holds,
the criteria for the PPS instability are satisfied.
As a result,
the photons increase exponentially, protons lose
energy due to pair production and the system
behaviour depends on the choice of the parameters
and
.
Figure 10: Plot of the evolution of the system in the case where protons are injected with , while the combination of the proton injection rate ( ) and is such that the steady-state proton distribution in the no-loss case corresponds to an unstable proton distribution. The solid line shows the photon compactness when all processes are taken into account, while the long dashed and dot-dashed lines show the photon compactness when certain processes are omitted (for details see text). The dotted and short dashed lines show the evolution of the first and last proton occupation number bin in the case where all the relevant processes are taken into account. | |
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As a first example, we show in Fig. 10 the case where with a proton injection rate of . The proton compactness parameter, according to Eq. (9), is . The above combination of and , according to Fig. 8, corresponds to a feedback loop that causes the photon density to increase with s=0.7. The dotted and short-dashed lines show, respectively, the evolution of the lowest and highest differential density bins of the proton energy. In agreement with the analytical solution of Eq. (5) in the no-loss case, the number of particles in these bins increases very quickly (in about one ) to a steady state, which, however, corresponds to an unstable proton configuration. The long-dashed line shows the photon compactness when only synchrotron radiation is taken into account. This increases as with s=0.7 until it reaches a steady state. The dot-dashed line shows the effects on the photon compactness caused from the addition of inverse Compton scattering and photon-photon pair production. As these processes are quadratic with respect to the photon and electron/positron number densities they do not affect the slope of the compactness at early stages, i.e. as long as . However, at the later stages of evolution these processes become important and, because they are quadratic, they cause the photon compactness to increase even faster and reach saturation sooner. Finally the solid line shows the evolution of the photon compactness when, in addition to the above processes, photon trapping due to the high density of created pairs is taken into account. This leads to higher photon compactnesses as relatively more energy is extracted from the high energy protons. This curve must be considered the "correct'' one as it contains all the relevant processes. On the other hand the photon increase causes the high energy protons (short-dashed line) to lose energy and settle in a new steady state. These losses do not affect naturally the low energy protons (dotted line) which maintain their original steady state.
At the other extreme, one can envisage a case where protons
are
injected slowly, but have a very long
escape time.
In this case we take
and
which corresponds
to the same no-loss steady state as before.
The result is shown in Fig. 11, where
quasi-periodic behaviour typical of a relaxation
oscillator is apparent.
The protons accumulate in the source
and, once
their density
rises
above the critical density,
the photon
density grows rapidly
(on the timescale of a few times
)
and deplete the high energy
part of the proton spectrum. Once the protons have
lost their energy, there is nothing left to sustain the
loop and the photons escape from the system. This cycle is
repeated
as the protons accumulate again
in the source.
The behaviour of the lowest and highest proton energy bins
is also shown (dotted and dashed lines respectively).
It is clear that photons and high energy protons are
anticorrelated, in the sense that when one population is
high, the other is low. These cycles are similar to those
found by Stern & Svensson (1991) using Monte-Carlo techniques.
Figure 11: Plot of the evolution of the system in the case where a low proton injection rate ( ) combines with a slow protonescape time ( ) in such a way that the steady-state proton distribution in the no-loss case to correspond to an unstableproton distribution. The rest of the parameters are as in the previous figure. Solid line shows , while dotted and short dashed lines show the evolution of the proton spectrum at and respectively. | |
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As a last case we examine the situation where electrons and protons are
injected simultaneously, i.e., both proton and electron kinetic
Eqs. (5) and (6) have an external
injection term. We assume that both of these terms are in power-law form,
so that,
in addition to Eq. (24) that
describes proton injection, we prescribe electron injection
using a similar expression:
Depending on the particular choice of parameters, high energy electron injection can result in a synchrotron and/or an inverse Compton component (see, for example, Mastichiadis & Kirk 1997). When protons are injected as well, these will interact with the aforementioned photons causing a secondary injection of Bethe-Heitler pairs. This leads to a non-linear situation. To see this, one should compare the compactness of the externally injected electrons (Eq. (10)) with the corresponding compactness of the internally produced pairs via the Bethe-Heitler process (Eq. (29)). As the latter is, in general, a function of both and , when the system operates in the non-linear regime, in the sense that the cooling of the protons occurs mainly on the internally produced photons.
To investigate the effect described above we proceed as follows: we keep the electron injection parameters constant and change only the normalisation of the proton injection rate in Eq. (24) - or, equivalently, we change the proton compactness parameter (Eq. (9)). When the resulting photon spectrum is simply that produced from the cooling of the externally injected electrons. However as we increase , the quantity increases as well and, at some stage, it becomes comparable to . Above this point the system enters the nonlinear regime, as the cooling of the protons occurs primarily on its own radiation. Finally, above some critical proton compactness, a loop analogous to the one described in Sect. 5 operates, and the protons convert a substantial fraction of their energy content to electron/positron pairs and radiation. This is depicted in Fig. 12 which shows the evolution of the photons when is kept constant, while varies. The first curve (dotted line) assumes that no protons are injected. Due to the particular choice of the parameters the primary electrons cool fast and the system quickly reaches a steady state. In the next two cases, the injected proton compactness is (long-dashed line) and (short-dashed line) respectively. Again a steady state is quickly reached. However the photon compactness increases as increases because of the BH pairs created and subsequently cooled. In both of these cases the feedback loop does not operate in the sense that the proton losses remain low - or, equivalently, . However, for the last two cases which are for (full line) and (dot-dash line) the effects of the feedback are evident. In the case of the solid curve, and for the first 300 or so light crossing times, there is a gradual increase of pairs and photons in the system until their numbers are built to a level that allows a catastrophic release of the energy stored in protons. After that, a steady state is reached, but at a level much higher than that of the previous cases: . It is worth mentioning that this loop operates at a proton compactness which is below the critical threshold obtained in Sect. 5. Therefore, the presence of external electrons helps to initiate the catastrophic proton energy losses at lower proton densities. Finally, the uppermost curve corresponds to a proton injection which is above the critical threshold for the PPS instability and the photons grow very quickly as discussed in Sect. 5 - see also Fig. 10.
The photon spectra corresponding to the steady states obtained
in each of these runs are shown in Fig. 13. The bottom curve
(dotted line) corresponds to
the injection of
electrons only
whereas
the four others are for electron and proton injection,
corresponding
to the curves of Fig. 12.
The extra component
due to BH pair production is apparent on the two lower spectra that include
protons (long and short-dashed lines).
The two uppermost curves
correspond to the steady-state spectra when the loop
was able to extract most of
the proton energy. Due to the resulting high photon
compactness the spectra
are strongly absorbed at energies above 1 MeV due to photon-photon pair production.
Figure 12: Plot of the evolution of the photons in the case where protons and electrons are injected simultaneously with the same slope and the same high Lorentz factor cutoff, i.e. in a region of radius R=10^{15} cm immersed in a magnetic field of strength B=10^{3} G. The injected electron compactness is and the evolution of the photons is shown for proton compactnesses ( bottom to top) , 200, 400, 800 and 1600. Both species have an escape time equal to . All except the highest curve correspond to stable proton distributions in the no-loss case. Nevertheless the solid curve shows a loop that eventually leads to catastrophic proton losses. The uppermost curve (dot-dashed line) corresponds to a proton distribution that is unstable even in the no-loss case. Therefore the photons increase quickly and drive the system to an equilibrium. | |
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Figure 13: Plot of the photon spectra corresponding to the steady states obtained for the runs shown in Fig. 12. The bottom curve (dotted) shows the spectrum obtained from the cooling of the injected electrons when no protons are injected. As the injected proton compactness increases the photon spectrum is modified due to the presence of the radiation from the created pairs (long and short-dashed curves). Finally when the loop operates the photon spectrum is strongly absorbed above 1 MeV by photon-photon pair creation (solid and dot-dashed curves). | |
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The above findings
are summarised in Fig. 14 which shows the photon
compactness of the system versus the proton compactness for various
injected electron compactnesses. For low values of
(i.e. less than 30), the photons of the low-frequency part of the SED
produced come almost exclusively from
the presence of the electrons, i.e. the protons cannot significantly
influence the low-frequency behaviour of
the system. We note here that for synchrotron proton blazar model
fits to the SED of BL Lac Objects observed at gamma-ray energies
(e.g. Mücke et al. 2003) the proton compactness is
-10^{-2} implying that Bethe-Heitler pair production, and
any associated instability, is unimportant in these models.
For intermediate values of
(i.e. between 30 and 300) the internally produced Bethe-Heitler pairs make their presence
visible in that their cooling increasingly dominates the photon
spectrum. Finally at even higher compactnesses the protons become
unstable and cool efficiently on their
own radiation.
Figure 14: Plot of the photon compactness versus the proton compactness for various injected electron compactnesses. The curves correspond to , 10 and 100 ( from bottom to top). The other parameters are the same as those used in Figs. 12 and 13. The dotted line represents the critical proton compactness in the no-loss case as this was described in Sect. 5. | |
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In the present paper we have examined some consequences arising from the presence of ultrarelativistic hadrons in compact sources. This was done with the help of a numerical code that was constructed to follow the evolution of the system through the solution of three coupled, time-dependent kinetic equations for protons, electrons and photons respectively. All the relevant basic processes involving electrons and photons in astrophysical pair-plasmas were included. The coupling between the hadronic and the leptonic component was assumed to occur via Bethe-Heitler pair-production. For this process, detailed electron/positron pair-production spectra were obtained with the help of a Monte-Carlo code. These were then incorporated into the kinetic equations which were subsequently solved numerically, revealing effects mainly due to synchrotron and inverse Compton losses.
The choice of the kinetic equation approach allowed us to study various aspects of the behaviour of such a system. Thus we showed that the presence of an external black-body radiation field can extract energy efficiently from the relativistic protons only when the photon compactness is high.
Of special interest are the non-linear cases, i.e., cases in which the protons cool not on a prescribed external photon field, but on the radiation of the internally produced Bethe-Heitler pairs. In the present paper we have verified the existence of the "Pair-Production/Synchrotron'' loop previously studied analytically. We showed that this is a very efficient way of channelling proton energy into electron/positron pairs and radiation. and that the coexistence of relativistic electrons in the system does not stabilise the system but, on the contrary, lowers the critical density threshold, i.e. it facilitates the efficient transfer of energy from the hadronic component to the leptonic/photonic one. This, and the fact that the threshold and critical density conditions can be greatly relaxed if the protons are in relativistic bulk motion, makes this loop a promising candidate for some AGNs (Kazanas & Mastichiadis 1999) and GRBs (Kazanas et al. 2002).
Of the various processes omitted in the present treatment, the most important is that of photo pion-production. This, however, does not affect the results of the present paper because, in the examples shown, the initial conditions were chosen so as to avoid the onset of this process. Other processes involving protons, such as proton synchrotron radiation and proton-proton interactions, etc., are negligible for the parameters of the particular examples given in this paper.
Acknowledgements
A.M. would like to thank R.J.P. for hospitality during his stay in the University of Adelaide. This research was funded in part by a Grant from the Special Funds for Research of the University of Athens. A.M and J.K. acknowledge the EC funding under contract HPRCN-CT-2002-00321 (ENIGMA network). The research of RJP is supported by an Australian Research Council Discovery Grant.