# Escape-limited Model of Cosmic-ray Acceleration Revisited

###### Key Words.:

Acceleration of particles – ISM: cosmic rays – ISM: supernova remnants – Galaxies: jets###### Abstract

Context: The spectrum of cosmic rays (CRs) is affected by their escape from an acceleration site. This may have been observed not only in the gamma-ray spectrum of young supernova remnants (SNRs) such as RX J1713.73946, but also in the spectrum of CRs showering on the Earth.

Aims:The escape-limited model of cosmic-ray acceleration is studied in general. We discuss the spectrum of CRs running away from the acceleration site. The model may also constrain the spectral index at the acceleration site and the ansatz with respect to the unknown injection process into the particle acceleration.

Methods: Analytical derivations. We apply our model to CR acceleration in SNRs and in active galactic nuclei (AGN), which are plausible candidates of Galactic and extragalactic CRs, respectively. In particular, for young SNRs, we take account of the shock evolution with cooling of escaping CRs in the Sedov phase.

Results:The spectrum of escaping CRs generally depends on the physical quantities at the acceleration site, such as the spectral index, the evolution of the maximum energy of CRs and the evolution of the normalization factor of the spectrum. It is found that the spectrum of run-away particles can be both softer and harder than that of the acceleration site.

Conclusions: The model could explain spectral indices of both Galactic and extragalactic CRs produced by SNRs and AGNs, respectively, suggesting the unified picture of CR acceleration.

## 1 Introduction

The origin of cosmic rays (CRs) has been one of the long-standing problems. The number spectrum of nuclear CRs observed at the Earth, , shows a break at the “knee” energy ( eV), below which the spectral index is about (Cronin, 1999). Because of the energy-dependent propagation of CRs, the spectral shape at the source is different from that observed at the Earth. Taking into account the propagation effect, the source spectral index has been well constrained as in various models (e.g., Strong & Moskalenko, 1998; Putze et al., 2009). This value of has been also inferred in order to explain the Galactic diffuse gamma-ray emission (e.g., Strong et al., 2000). This fact may give us valuable insights on the acceleration mechanism of CRs.

Mechanisms of CR acceleration have also been studied for a long time and the most plausible process is a diffusive shock acceleration (DSA) (Krymsky, 1977; Axford et al., 1977; Bell, 1978; Blandford & Ostriker, 1978). Very high-energy gamma-ray observations have revealed that the existence of high-energy particles at the shock of young supernova remnants (SNRs), which supports the DSA mechanism as well as the paradigm that the Galactic CRs are produced by young SNRs (e.g., Enomoto et al., 2002; Aharonian et al., 2004, 2005; Katagiri et al., 2005). Recent progress of the theory of DSA has revealed that the back-reactions of accelerated CRs are important if a large number of nuclear particles are accelerated (Drury & Völk, 1981; Malkov & Drury, 2001). There are several observational facts which are consistent with the predictions of such nonlinear model (Vink & Laming, 2003; Bamba et al., 2003, 2005a, 2005b; Warren et al., 2005; Uchiyama et al., 2007; Helder et al., 2009). The model predicts, however, the harder spectrum of accelerated particles at the shock than (corresponding to ) where is the momentum of CRs, in particular, near the knee energy (Malkov, 1997; Berezhko & Ellison, 1999; Kang et al., 2001). This fact apparently contradicts with the source spectral index of inferred from the CR spectrum at the Earth. Even in the test-particle limit of DSA, such a soft source spectrum requires a shock with the small Mach number (), which is unexpected for young SNRs.

There are several models of DSA, depending on the boundary conditions imposed. Different models predict different spectra of CRs dispersed from the shock region. So far, the age-limited acceleration has been frequently considered as a representative case (§ 2). In this model, all the particles are stored around the shock while accelerated. When the confinement becomes inefficient, all the particles run away from the region at a time. Then, the source spectrum of CRs which has just escaped from the acceleration region is expected to be the same as that at the shock front. Therefore, this model predicts that the source spectrum is the same as that of accelerated particles, which is typically harder than the observed one. In this paper, we consider an alternative model, the escape-limited acceleration, to explain the observed CR spectrum at the Earth (§ 3). This model is preferable to the age-limited acceleration when we consider observational results for young SNR RX J1713.73946, of which TeV -ray emission is more precisely measured than any others (§ 4.1).

The nature of CRs with energies much higher than the knee energy is also still uncertain. While CRs below the second knee ( eV) may be Galactic origin, the highest energy CRs above eV are believed to be extragalactic. Possible candidates are active galactic nuclei (AGNs) (e.g., Biermann & Strittmatter, 1987; Takahara, 1990; Rachen & Biermann, 1993; Pe’er et al., 2009), gamma-ray bursts (Waxman, 1995; Vietri, 1995; Murase et al., 2006), magnetars (Arons, 2003; Murase et al., 2009) and clusters of galaxies (Kang et al., 1997; Inoue et al., 2007). The intermediate energy range from eV to eV is more uncertain. Both the Galactic and extragalactic origins are possible and it may just a transition between the two. As the extragalactic origin, AGNs (Berezinsky et al., 2006), clusters of galaxies (Murase et al., 2008) and hypernovae (Wang et al., 2007) have been proposed so far.

Among these possibilities, AGN is one of the most plausible candidates for accelerators of high-energy CRs, because it can explain the UHECR spectrum above eV assuming the proton composition. In such a proton dip model, the source spectrum of UHECRs is with –2.7, depending on models of the source evolution (Berezinsky et al., 2006). The required source spectral index of –2.7 can be explained by several possibilities. First, it can be attributed to the acceleration mechanism itself. One can consider non-Fermi acceleration mechanisms (Berezinsky et al., 2006) or the two-step diffusive shock acceleration in two different shocks (Aloisio et al., 2007). Second, the index can be attributed to a superposition of many AGNs with different maximum energies, and one can suppose that AGNs with different luminosities may have different maximum energies (Kachelriess & Semikoz, 2006). Recently, Berezhko (2008) proposed another possibility under the cocoon shock model. In this cocoon shock scenario, different maximum energies can be interpreted as maximum energies of escaping particles at different ages of AGN jets. Although it is very uncertain whether efficient CR acceleration occurs there, this scenario would also be one of the possibilities to be investigated in detail.

The organization of the paper is as follows. After the brief introduction of the age-limited model of the CR acceleration (§ 2), we study the escape-limited model in § 3. For a simple understanding, the general argument in a stationary, test-particle approximation is given in § 3.1. Then, we derive the formulae of the maximum energy of accelerated particles in § 3.2, and of the spectrum of escaping particles in § 3.3. We consider the applications to young SNR and AGN in § 4 and § 5, respectively. Section 6 is devoted to a discussion.

## 2 Maximum Attainable Momentum in the Age-limited Acceleration

For comparison with the escape-limited acceleration, we briefly summarize the case of the age-limited acceleration. In this case, the maximum momentum of accelerated particles, , is determined by , where and are the age of the shock and the acceleration time scale, respectively. When we consider DSA, is given by (Drury, 1983)

(1) |

where and are the diffusion coefficient as a function of the momentum of accelerated particles and the velocity of the background fluid, respectively. Subscripts 1 and 2 represent upstream and downstream regions, respectively. For simplicity, we assume the Bohm-type diffusion, i.e.,

(2) |

where , and are the magnetic field strength, the gyro-factor and the proton mass, respectively. Taking into account that the fluid velocities are related to the shock velocity, , as , where is the compression ratio of the shock, we derive (e.g., Aharonian & Atoyan, 1999)

(3) |

## 3 Escape-limited Acceleration

In the frame work of DSA, accelerated particles are scattered by the turbulent magnetic field, and go back and forth across the shock front. Upstream turbulence may be excited by the accelerated particles themselves (Bell, 1978), and the magnetic field strength of such turbulence is theoretically expected to be strong (e.g., Lucek & Bell, 2000). There are observational evidences suggesting that CRs are responsible for substantial amplification of the ambient magnetic field in the precursors of shock fronts in SNRs, and that such magnetic turbulence well confines the particles around the shock front (Vink & Laming, 2003; Bamba et al., 2003, 2005a, 2005b; Yamazaki et al., 2004; Parizot et al., 2006; Uchiyama et al., 2007), leading to the efficient CR acceleration.

The spectrum of accelerated particles is affected by the spatial and spectral structures of the magnetic turbulence through the process in which the particles escape from the shock toward far upstream region. There are mainly two scenarios of the escape model considered so far; one causes the effect on the boundary in the momentum space, and the other causes the effect on the spatial boundary. The former comes from significant decay of the wave amplitude below the wave number of the spectrum of the turbulence (Reynolds, 1998; Drury et al., 2009). Particles with the Lorentz factor above satisfying the resonance condition, , where is the cyclotron frequency, are not confined around the shock front and escape into far upstream region. In this context, the escape flux was calculated previously (e.g., Ptuskin & Zirakashvili, 2005; Drury et al., 2009). The latter effect has been recently discussed by several authors (Ptuskin & Zirakashvili, 2005; Reville et al., 2009; Caprioli et al., 2009). The turbulence generation may be connected with the flux of accelerated particles themselves. Hence, in the region far from the shock front, the flux of high-energy particles is small and wave excitation is less significant. If the accelerated particles reach the region, they are dispersed into the far upstream region. Let be the distance from the shock beyond which the amplitude of the upstream turbulence becomes negligible. Characteristic spatial length of particles penetrating into the upstream region is given by . As long as , the particles are confined without the significant escape loss, and they are accelerated to higher energies. On the other hand, when their momentum increases up to sufficiently high energies satisfying , their acceleration ceases and they escape into the far upstream. Therefore, the maximum momentum of accelerated particles in this scenario is given by the condition . In the following of the paper, we consider the escape-limited model, where the maximum energy is essentially determined by .

### 3.1 A simple case of stationary, test-particle approximation

In order to take an essential feature of the escape-limited acceleration, we calculate the escape flux and the maximum attainable energy of accelerated particles in the simplest case (see also Caprioli et al., 2009). Let us consider the stationary transport equation

(4) |

with the boundary condition, , where () is the upstream escape boundary. The fluid velocity is given by

(5) |

where and are constants. The solution to the transport equation in the test-particle approximation is derived as (Caprioli et al., 2009)

(6) |

where is given by

(7) |

and . The escape flux at is

(8) | |||||

Let us introduce a new variable and a new function . Then, we expand around its maximum value at . To do this, we calculate the first and the second derivatives as

(9) |

(10) |

In the following, we consider the case of Bohm diffusion, . Then, one can find

(11) |

(where ) so that we obtain

(12) | |||||

where . Note that the quantity given by Eq.(11) also plays the role of maximum momentum of the accelerated particles at the shock () because one can see for while for (Caprioli et al., 2009).

Finally, going back to the function , we obtain

(13) |

One can clearly see, from Eq. (13), that particles with momentum around escape the shock region most efficiently.

### 3.2 The maximum energy of accelerated particles

We have seen in § 3.1 that in the stationary, test-particle case, the quantity given by Eq. (11) plays the role of maximum momentum of the accelerated particles at the shock. Taking this fact into our mind, we assume that in the more general escape-limited case, the maximum momentum, , is determined by

(14) |

Given that

(15) |

which is the same as in the age-limited case [Eq. (2) in § 2], we obtain

(16) |

Since , we find from Eqs. (3) and (16)

(17) |

Hence, as long as which is expected in the early phase of the shock, we don’t need the escape-limited acceleration. However, if the shock evolves so that , we have to consider the effect of the particle escape, and we cannot simply apply the well-known result of age-limited acceleration, Eq. (3).

### 3.3 The spectrum of CRs dispersed from an accelerator

In this subsection, we derive the time-integrated spectrum of CR particles which is dispersed from an accelerator. The derivation is essentially identical to that of Ptuskin & Zirakashvili (2005). However, our argument is simpler and more general, so the final form of the spectrum (Eqs. (27) and (28)) is more general. Note that our formalism is applicable not only to DSA but also to arbitrary acceleration processes.

The proton production rate, , at a certain epoch labeled by a parameter , is defined as the number of protons with momentum between and which is produced in the interval between and . Here is the parameter which describes the dynamical evolution of the accelerator — it can be either simply the age, or the position of the shock front. It is expected that contains the term of exponential cutoff at the momentum which depends on [see, for example, Eq. (LABEL:eq:N)]. The number of protons with momentum between and that is escaping from the accelerator at the epoch between and is denoted by , and we assume

(18) |

where

(19) |

and . This is because we expect that particles with momentum around is most efficiently escaping from the source. Indeed, as shown in § 3.1, Eqs. (18) and (19) are good approximation in the test-particle, stationary case. Here, we simply expect these assumptions are also correct in general.

The time-integrated spectrum of protons which have escaped at the source, , is obtained by

(20) |

In order to derive a simple analytical form, we approximate Eq. (19) as

(21) |

If we use a general mathematical formula for -functions for an arbitrary function :

(22) |

where , then Eq. (21) is rewritten as

(23) |

where is the inverse function of , that is, and . Using Eqs. (18) and (23), we calculate as

(24) |

This is the most general analytical formula of the spectrum of protons dispersed from an accelerator.

In the following of the paper, the form of is assumed to have

which is a power-law with the index and the exponential cut-off at . Substituting Eq. (LABEL:eq:N) into Eq. (24), we obtain

(26) |

In particular, if and are written by the power-law forms, such as and , then, , so that Eq. (26) becomes

(27) |

where

(28) |

This is the simplest form of the spectrum of CRs which are dispersed from the acceleration region.

Generally speaking, in order to obtain the energy spectrum of accelerated particles, time-dependent kinetic equation should be solved. Instead, we have assumed that at an arbitrary epoch, the spectral form is given by Eq. (LABEL:eq:N). This assumption is justified if the spectrum at the given epoch is dominated by those which are being accelerated at that time, in other words, if the particle spectrum does not so much depend on the past acceleration history. For example, in the case of the spherical expansion, accelerated particles suffer adiabatic expansion after they are transported downstream of the shock and lose their energy (e.g., Yamazaki et al., 2006), so that the contribution of the previously accelerated particles is negligible. Strictly speaking, even if we consider the energy loss via adiabatic expansion, the energy spectrum of accelerated particles does depend on the past acceleration history in some cases. When we use the shock radius as , the final form of Eqs. (27) and (28) is correct as long as (see Appendix), which is satisfied in the cases considered in § 4 and § 5. Otherwise, the form of Eq. (LABEL:eq:N) is no longer a good approximation, and the final form of is different from Eq. (28) (see Appendix).

## 4 Application to Young Supernova Remnants

### 4.1 Inconsistency of age-limited acceleration with observed results of RX J1713.73946

RX J1713.73946 is a representative SNR from which bright TeV -rays have been detected. The H.E.S.S. experiment measured the TeV spectrum and claimed that its shape was better explained by the hadronic model (Aharonian et al., 2006, 2007). Furthermore, evidences of amplified magnetic field ( mG) are derived from the width of synchrotron X-ray filaments (Parizot et al. 2006; see also Vink & Laming, 2003; Bamba et al., 2003, 2005a, 2005b) and from time variation of synchrotron X-ray hot spots (Uchiyama et al., 2007). These facts also support the hadronic origin of TeV -rays, because the leptonic, one-zone emission model (e.g., Aharonian & Atoyan, 1999) cannot explain the TeV-to-X-ray flux ratio. Hence it is natural to assume that the TeV -rays are produced by the hadronic process although there are several arguments against this interpretation (Butt, 2008; Katz & Waxman, 2008; Plaga, 2008).

In the age-limited case, Eq. (3) reads

(29) |

where we adopt , and is the magnetic field strength in units of mG.

H.E.S.S. observation has revealed that the cutoff energy of TeV -ray spectrum is low (Aharonian et al., 2006, 2007), so that in the one-zone hadronic scenario the maximum energy of protons, is estimated as 30–100 TeV (Villante & Vissani, 2007). If TeV and mG, then Eq. (29) tells us , implying far from the “Bohm limit” () which is inferred from the X-ray observation (Parizot et al., 2006; Yamazaki et al., 2004) or expected theoretically (Lucek & Bell, 2000; Bell, 2004; Reville et al., 2007; Giacalone & Jokipii, 2007; Inoue et al., 2009; Ohira et al., 2009b, c). This statement is recast if we involve recent results of X-ray observations. The precise X-ray spectrum of RX J1713.73946 is revealed, which gives cm s (Tanaka et al., 2008). Then, Eq. (29) can be rewritten as (Yamazaki et al., 2009)

(30) |

Hence, in order to obtain TeV, we need G in the context of the hadronic scenario of TeV -rays, which contradicts current estimates of the magnetic field.

A possible solution is to consider the escape-limited acceleration. One can find from Eq. (17) that if we take pc, the maximum energy becomes

(31) | |||||

which is consistent with the observed gamma-ray spectrum. In the following, we consider the model of escape-limited acceleration under simple assumptions, estimating the evolution of the number density and the maximum momentum of accelerated particles so as to discuss the spectral index, , of .

### 4.2 Evolution of

Time evolution of the maximum momentum of accelerated particles, , has been so far discussed in many contexts (e.g., Ptuskin & Zirakashvili, 2003). One way to estimate is to use Eq. (16). In this approach, a key parameter is the magnetic field, which is likely amplified around the shock front (Vink & Laming, 2003; Bamba et al., 2003, 2005a, 2005b; Yamazaki et al., 2004; Uchiyama et al., 2007) and may depend on various physical quantities such as the shock velocity, the ambient density, and so on. At present, the evolution of the magnetic field is not well understood despite many works (e.g., Niemiec et al., 2008; Riquelme & Spitkovsky, 2009; Ohira et al., 2009a; Luo & Melrose, 2009). In addition, the evolution of another parameter is also unknown. These facts prevents us from predicting rigorously.

Here we adopt a different phenomenological approach based on the assumption that young SNRs are responsible for observed CRs below the knee (Gabici et al., 2009). The maximum energy is expected to increase up to the knee energy ( eV) until the end of the free expansion phase, , and decreases from that epoch. As seen in § 4.1, is limited by the escape at , that is . Then, to reproduce the observed CR spectrum from GeV to the knee, we may assume a functional form of

(32) |

where , so that at . For later convenience, we change variables from to the SNR radius in order to take , where is a variable appeared in § 3.3. We further assume the dynamics of as

(33) |

where is the shock radius at . Then, we obtain

(34) |

so that we have (see the last paragraph of § 3.3). If we adopt ad hoc, in addition to that is expected in the Sedov phase, then we find as a phenomenologically required value in the escape-limited model. Note that, even if the SNR dynamics is modified by the CR escape, is almost the same as one in the adiabatic case and its effect on is expected to be so small that our conclusion is not affected qualitatively.

### 4.3 Dynamics of SNR shock waves

In this subsection, we consider the dynamics of the SNR shock in order to estimate the evolution of the normalization factor of the spectrum . A simple treatment of the dynamics of the SNR shock from the free expansion to the adiabatic expansion (Sedov) phase has been given by several authors (Ostriker & McKee, 1988; Drury et al., 1989; Bisnovatyi-Kogan & Silich, 1995). Here we extend their method taking account of the cooling by CR escape. The total mass of the SNR shock shell is calculated as

(35) |

where and are the ejecta mass and the density of ambient gas, respectively. The equation of motion of the thin shell is given by (Ostriker & McKee, 1988; Bisnovatyi-Kogan & Silich, 1995)

(36) |

where the gas velocity is related to the shock velocity and the adiabatic index as . Quantities and are the pressures of the post-shock gas and the ambient gas, respectively. For strong shocks, one can neglect . The explosion energy consists of the internal energy , the kinetic energy, and the energy which is carried by the escaping CRs. Then Eq. (36) is rewritten as

(37) |

Since , the left-hand side of Eq. (37) becomes

(38) |

where . Hence we obtain

(39) | |||||

Once and are given, we can integrate Eqs. (35) and (39) to derive the dynamics of the SNR, . We mainly consider the evolution of in the Sedov phase. Hence, in the following, we simply assume that is constant with because even in the case of the core-collapse supernova, the wind region is about a few pc and has been already passed by the SNR shock until the beginning of the Sedov phase. The energy carried by escaping particles is written as

(40) |

where is the Heaviside step function, i.e., for while for . Substituting Eqs. (27), (28), and (34) into Eq. (40), we obtain

(41) | |||||

where is the ratio of the total energy of escaping CRs to the explosion energy. Note that is assumed to be constant. Integrating Eq. (39) with the aid of Eqs. (35) and (41), the dynamics of the SNR shock is determined. To make more realistic discussion, we should perform fluid simulations with CR back reaction, which is beyond the scope of this paper (e.g., Drury et al., 1989; Völk et al., 2008).

### 4.4 Evolution of

In this subsection, we discuss the evolution of the normalization factor, , of the spectrum of accelerated particles. At present, the injection process for CR acceleration at the shock is not well understood. Hence, we consider two representative scenarios of the injection process to model the amount of the accelerated particles.

At first, we consider the same injection model as that of Ptuskin & Zirakashvili (2005). The model requires that the CR pressure at the shock is proportional to the fluid ram pressure, that is, . The CR pressure at the shock is given by

(42) |

where we neglect the contribution of non-relativistic particles. Then one can find that is related to as

(43) |

where we have used the fact that the distribution function of CRs at the shock front is essentially for . Note that is approximately proportional to when , because in this case only weakly depends on . Hereafter, the cases of and are called PS and PH, respectively. Since , the normalization factor of the CR spectrum inside the SNR shock is calculated as

(44) |

where the mechanical energy of the ejecta is (up to the numerical coefficient)

(45) |

If the explosion is adiabatic, is constant with during the Sedov phase because .
However, if the modification of SNR dynamics via CR cooling is taken into account (§ 4.3), then is no longer proportional to , and it is obtained by solving Eq. (39).
In Figs. 1 and 2, we show as a function of time.
Figure 1 is for the case PS in which we adopt , , , and , while Fig. 2 is for the case PH with parameters , , and .
One can find that after , is around , so that is approximately given by
with constant
^{1}^{1}1
We search for approximate solutions to
Eqs. (39), (41) and (48)
which determine (and ) for given parameters
, , and .
The procedure is as follows.
For a trial value of which is assumed to be constant,
we solve Eqs. (39) and (41) to obtain so as to calculate and as functions of .
It is found that is almost time-independent after , hence we can derive its average value in the epoch .
Then, we update the value of using Eq. (48) with
the average value of .
We repeat this procedure until the iteration converges.
.
For the case PS, the effect of CR cooling appears at the late time because low-energy CRs have a large fraction of the total energy of the escaping CRs.
On the other hand, for the case PH, the effect emerges at an earlier epoch.

Next, we consider the thermal leakage (TL) model (Malkov & Völk, 1995). This model requires the continuity of the distribution function to the downstream Maxwelian at the injection momentum , namely

(46) |

Using this fact, we derive

(47) |

where is the spectral index in the non-relativistic regime, that is, for . We further assume the constant and . Then we obtain , so that . Although the value of is uncertain, one can expect . In the test particle approximation, one derives . If the nonlinear model of shock acceleration is considered, the spectrum is softer than in the non-relativistic regime (Berezhko & Ellison, 1999). In particular, if , then , so that the value of becomes similar to those of PS case.

In summary, the evolution of the normalization factor of the spectrum of accelerated particles is given by , with

(48) |

### 4.5 The spectrum of escaping particles

We obtain from Eqs. (27), (28), and (48), the index of the momentum spectrum of escaping particles as

(49) |

In the following of this section, we discuss which injection model is suitable to make the galactic CR spectrum observed at Earth. We adopt as a typical value (see § 4.2) and to make the Galactic CR spectrum. In this paper, we assume –2.4, where we believe that the deviation from is significant. This would be tested not only from Galactic CR observations but also observations of extragalactic galaxies. Note that, even if we adopt , our results are qualitatively unchanged.

In the case PS, is smaller than because , so that the model predicts the harder spectrum of escaping particles than that of the source. However, since , the difference is small. In order to reproduce the observed Galactic CR spectrum , the source spectrum should be . Hence, the PS model requires at the source. This condition is satisfied if we consider the diffusive shock acceleration at moderate Mach number (Fujita et al., 2009). It is also possible to derive if we consider the effects of neutral particles (Ohira et al., 2009b, c).

In the case PH, because the value of is small, always near 2, which is the value predicted by the diffusive shock acceleration theory in the strong shock, test-particle limit. In particular, if — indeed, even in the test-particle limit where the cooling via CR escape can be neglected, the value of is not exactly zero unless (see Fig. 2) — then, one can find . This is what Berezhko & Krymsky (1998) and Ptuskin & Zirakashvili (2005) showed. Note that from Fig. 2, is negative for a long time, so that . Therefore, this model PH cannot reproduce the observed Galactic CR spectrum at the Earth.

In the case TL, neglecting , we obtain

(50) |

Hence if , then . For the test particle acceleration, we have , so that the source spectrum should be in order to obtain . For the nonlinear acceleration, one typically expects and , so that . Therefore, the nonlinear model seems to explain the inferred source spectrum marginally. Possible rigorous determination of the value of may give a constraint on the theory of nonlinear acceleration.

## 5 Application to AGN Cocoon Shocks

In this section, taking account of a constraint derived from the spectrum at the Earth, we study the origin of CRs with energies larger than eV and the acceleration mechanism of them at AGNs. There are many works which discuss UHECR production in AGNs (e.g., Biermann & Strittmatter, 1987; Takahara, 1990; Rachen & Biermann, 1993; Berezinsky et al., 2006; Berezhko, 2008; Pe’er et al., 2009). Many of them focus on UHECR acceleration in radio galaxies including Fanaroff-Riley (FR) I and II galaxies, which typically have powerful jets. In the context of DSA, one can basically suppose three acceleration zones; internal shocks in jets, hot spots, and cocoon shocks. The former two are most widely discussed scenarios but the detailed study of DSA at such mildly relativistic shocks has not yet been achieved. In this section, we concentrate on the cocoon shock scenario proposed by Berezhko (2008), where the non-relativistic DSA theory can be applied.

In this scenario, extragalactic CRs with energies larger than the second knee ( eV) may be accelerated at the outer cocoon shock running into the intergalactic medium (IGM). As powerful jets penetrating into a uniform ambient medium with a density , the heads of the cocoon advances into the IGM with a velocity . At the same time, the cocoon expands sideways with a velocity . Since the typical cocoon shock is non-relativistic, we apply the escape-limited model considered in previous sections. Although we hereafter focus on this scenario, note that it is very uncertain whether the efficient acceleration occurs there since observed non-thermal emission is much weaker than that from hot spots and lobes.

In the following, we investigate whether the CR spectrum above the second knee can be explained by the AGN cocoon shock scenario with the same parameters for young SNRs explaining the CR spectrum below the knee, which have been discussed in § 4. Similar to the previous calculations in § 4, we hereafter calculate the values of and in order to derive the spectral index, . Here we adopt , where is a variable appeared in § 3.3.

First, let us consider evolution of the maximum momentum, , in a phenomenological way. In the young SNR case (§ 4.2), we have phenomenologically expected . Then, by using the Sedov-Taylor solution ( and ), we can easily obtain

(51) |

where is a phenomenologically introduced parameter since one can expect in general. Note that , however, does not depend on but on as long as is generated by plasma instabilities such as CR streaming instabilities. In the cocoon shock scenario, and are replaced with and , respectively.

In order to obtain the value of , the dynamics of the AGN cocoon is necessary. A simple consideration of the cocoon dynamics for the constant density IGM tells us that is almost time-independent and evolves as (Begelman & Cioffi, 1989), so that the cocoon radius evolves as and the jet radius evolves as . Then, we obtain and for and , respectively.

Next, let us consider the time dependence of the normalization factor of the spectrum of accelerated CRs, .
The volume of the acceleration region swept by the cocoon shock is , where is the total area of the shock surface.
If we assume the elliptical shape of the cocoon^{2}^{2}2Berezhko (2008) assumed a kind of spherical cocoon, i.e., , and used which is similar to the relation obtained in the case of model PS.
However, when the cocoon becomes spherical, and evolve according to the adiabatic solution as in the Sedov-Taylor solution for SNRs (Begelman & Cioffi, 1989; Fujita et al., 2007)., then ,
so that .
In the cases PS and PH, is related to .
The dependence of is written as
, where we neglect for simplicity the evolution of the acceleration efficiency which was considered in Berezhko (2008).
Then, in the case PS, leads to

(52) |

while in the case PH,