# Shear Acceleration in Expanding Flows

###### Abstract

Shear flows are naturally expected to occur in astrophysical environments and potential sites of continuous non-thermal Fermi-type particle acceleration. Here we investigate the efficiency of expanding relativistic outflows to facilitate the acceleration of energetic charged particles to higher energies. To this end, the gradual shear acceleration coefficient is derived based on an analytical treatment. The results are applied to the context of the relativistic jets of active galactic nuclei. The inferred acceleration timescale is investigated for a variety of conical flow profiles (i.e., power law, Gaussian, Fermi-Dirac) and compared to the relevant radiative and non-radiative loss timescales. The results exemplify that relativistic shear flows are capable of boosting cosmic-rays to extreme energies. Efficient electron acceleration, on the other hand, requires weak magnetic fields and may thus be accompanied by a delayed onset of particle energization and affect the overall jet appearance (e.g., core, ridge line and limb-brightening).

^{†}

^{†}slugcomment: ApJ, to appear

## 1 Introduction

The non-thermal radiation seen from astrophysical objects bears witness to the presence of energetic charged particles that have experienced efficient acceleration within these sources. In the galactic domain, new high-resolution observations of supernova remnants have brought fresh momentum to the theory of diffusive shock acceleration (e.g., Bell13, , for review), while short-term variability seen in the context of active galactic nuclei (AGN) has motivated deeper studies of one-shot (gap- or reconnection-type) particle acceleration scenarios (see e.g., rieger12, , for review of the case of M87). Complementary, new observational results in the radio and VHE domain and progress in our understanding of turbulence modeling have given new impetus to turbulent shear acceleration and emission scenarios (e.g., aloy08, ; Sahayanathan09, ; liang13, ; grismayer13, ; ohira13, ; laing13, ). The present study focuses on the potential of accelerating energetic charged particles in expanding relativistic outflows in a regime appropriate for Active Galactic Nuclei (AGN). It follows an earlier analysis where the implications for the high-speed (bulk flow Lorentz factors ) gamma-ray-burst (GRB) regime has been investigated rieger05 (), provides a general derivation of the relations presented there, and extends it to the AGN context.

In contrast to GRBs, AGN are in fact seen to exhibit relativistic outflows extending up to hundreds of kilo-parsecs. In particular, for the ”blazar” sub-class of AGN, radio VLBI/VLBA observations of their inner (pc-scale) jets frequently reveal significant apparent superluminal motion () of individual jet components propagating away from the core. When the fastest measured radio jet components are put together, the velocity distribution shows a peak , with a tail extending up to Lister09 (); Piner12 (), suggesting that (radio) flow speeds in AGN jets may reach bulk flow Lorentz factors up to . On the other hand, measured speeds of the (parsec-scale) radio components in the VHE-detected, high-frequency-peaked BL Lac objects (HBLs) appear to be consistently well below those found in the above-noted, radio-selected samples Piner13 (). If representative, the apparent lack of significant superluminal speeds in the parsec-scale radio jets of TeV-HBLs in fact would seem to suggest that the (radio) bulk flow Lorentz factor in these objects is only modest (), in contrast to common constraints on their (sub-pc-scale) bulk Lorentz (Doppler) factors based on radiative modeling of their nuclear high-energy emission. In principle, such a difference could be accounted for by some change in directionality (e.g., jet bending, intrinsic helical motion) or by the presence of some velocity gradient in the flow, such as a (longitudinally) decelerating flow George03 (); Levinson07 (), or a (transversally) structured jet with a fast-moving spine and a slower moving sheath Ghisellini05 (). The former might be caused by radiative Compton drag, while the latter scenario of a fast inner flow (spine) encompassed by a slower outer flow (sheath) is generically expected in MHD models for the formation of relativistic jets with an ergospheric-driven jet surrounded by a slow moving disk wind (see e.g. McKinney06, ; Porth10, ; Hawley15, ). The implied flow velocity gradients could possibly facilitate the acceleration of energetic charged particles to higher energies once, e.g., particles are efficiently scattered across the flow rieger06 (), and this is what is studied here.

## 2 Shear acceleration in spherical coordinates

In the comoving frame , the acceleration coefficient in a gradual shear flow can be cast into the form (e.g., web89, , eq. 3.27)

(1) |

where denotes the comoving particle momentum, with for the energetic particles considered here, is the mean scattering time, and is the shear coefficient. In the strong scattering limit for quasi-isotropic diffusion in a turbulent environment (i.e. , with the relativistic gyro-frequency measured in the comoving frame) we have (see web89, , eq. 3.34)

(2) |

where , with , is the (covariant) fluid shear
tensor given by ^{1}^{1}1Note that this fixes a typographical sign error in eq.[A3] in (rie04, ).

(3) |

In this, denotes the (covariant) metric tensor and the covariant derivative. For spherical coordinates , with the azimuthal and the polar angle, one has

(4) |

The only non-vanishing connection coefficients (Christoffel symbols of the second kind) are then given by

(5) | |||||

(6) |

Restricting ourselves to a time-independent, relativistic radial bulk flow velocity profile of the form

(7) |

where is the bulk Lorentz factor, the fluid four divergence becomes

(8) |

while the only non-vanishing components of the fluid four acceleration are

(9) | |||||

(10) |

For the non-vanishing components of the shear tensor one then finds

(11) | |||||

(12) | |||||

(13) | |||||

(14) | |||||

(15) | |||||

(16) | |||||

(17) |

Noting that , the relativistic shear coefficient becomes

(18) |

which for non-relativistic flow speeds (i.e., ) and independent of (and ), i.e. , reduces to

(19) |

It can be shown that this expression corresponds to the (non-relativistic) viscous transfer coefficient derived by Earl et al. (1988) (their eq. 7) when the latter is expressed in spherical coordinates and the corresponding velocity profile is applied.

For an energy-dependent scattering timescale of the form , the shear flow acceleration coefficient, eq. (1), is given by

(20) |

so that the characteristic acceleration timescale for gradual shear becomes

(21) |

where in the presence of a background magnetic field the particle mean free path formally has to be smaller than the gyro-radius to satisfy the strong scattering. Equation (21) exemplifies the characteristic inverse dependence, , on the particle mean free path. This is related to the fact that in a shear flow the average energy gain per scattering increases with increasing particle mean free path rieger06 ().

Consider the simplified case where the radial flow velocity is only a function of polar angle , so that in four-vector notation the flow speed is given by

(22) |

where is the bulk Lorentz factor of the flow. The associated (comoving) timescale for the shear flow acceleration of particles then becomes

(23) |

where is the radial coordinate measured in the cosmological rest frame, is the particle mean free path, and is the particle momentum in the
comoving (jet) frame. As the jet flow is diverging and streamlines are separating, the acceleration
timescale increases with the square of the radial coordinate .

## 3 Flow velocity profiles and related energy losses

By means of application, let us consider three different bulk flow velocity profiles parameterized in terms of (cf. also zha02, ; kum03, ; zha04, , for instantiation in the case of GRBs), i.e., a power-law model, where is power-law function of outside a core of opening angle , i.e.,

(24) |

with , a Gaussian profile with

(25) |

and a Fermi-Dirac-type (top-hat) profile

(26) |

with , and where denotes the Lorentz factor at the jet axis ( for AGN), and typically rad. Particle energization in these flow profiles then competes with conventional energy-loss processes.

### 3.1 Adiabatic Losses

For the corresponding adiabatic energy changes one finds (using eq. 8)

(27) |

where the last relation holds for independent of . This gives the ratio of viscous gain versus adiabatic losses to

(28) |

Hence, one expects viscous shear energization in the diffusion regime () for the present application (radially expanding flows, no azimuthal component) to be important only in the relativistic regime. Figures 1 and 2 show examples of the flow profile and energization ratio assuming . For this case, at a given , only particles with (power-law profile), (Gaussian profile), and (Fermi-Dirac) can get efficiently accelerated. Efficient acceleration thus needs energetic seed particles and is usually difficult to achieve for electrons unless the magnetic field is weak.

Depending on the shape of the velocity profile, particles are more easily accelerated (i.e., require
less injection energy) at different angular scales, i.e., not necessarily in the innermost () region. This becomes particularly apparent for the chosen Gaussian and the Fermi-Dirac
type profile, where the shear gain to adiabatic loss ratio (Fig. 2) peaks at . This could in principle introduce more complex emission features (see below) and
support, for example, some ridge-line structure or a limb-versus centrally-brightened morphology
(see e.g. Nagai14, ; Bocc16, , for recent exemplary findings in the context of 3C 84 and Cygnus A,
respectively) in cases where potential differences in Doppler boosting are
effectively compensated by more efficient particle acceleration.

For illustration, Fig. 4 shows two possible (optically-thin) intensity maps (,
with the line of sight element) for a jet possessing a Fermi-Dirac bulk flow profile and inclined
at viewing angle in the case where particle injection across the jet is kept constant or
varied with polar angle approximately following Fig. 2. For simplicity, it is assumed that the
number density of non-thermal electrons (with differential energy distribution described by a power
law of index ) is proportional to the plasma density
(where is the distance along the jet axis) and that the comoving emissivity is synchrotron type,
, with , (perpendicular
scaling, with ) and and the Doppler factor.
In such a case, efficient shear acceleration leads to the appearance of a more prominent off-axis (ridge-line)
structure.

If the above-mentioned conditions (Eq. 28) are satisfied, acceleration always proceeds (up to a factor of the order of unity) on timescales shorter (faster) than the relevant dynamical timescale . Hence, as a particle moves out along , it will (in the absence of significant radiative losses) continue to get further accelerated to higher energies until it eventually leaves the flow by cross-field escape.

### 3.2 Synchrotron Losses

In the presence of magnetic fields, energetic charged particles will undergo synchrotron losses given by

(29) |

with the particle’s rest mass, its charge, and its Lorentz factor. We assume the background magnetic field to scale with radius with (typically ). The ratio of shear gain to synchrotron losses then becomes

(30) |

where is the comoving gyro-radius of the particle. Using characteristic (conical jet-type) scaling numbers in the AGN context (and ), this gives

(31) | |||||

Hence, if scales with the gyro-radius, energetic protons () are expected to experience efficient acceleration right from the start, almost independently of the magnetic field scaling. On the other hand, for the chosen magnetic field dependence (only on ), electrons () will only be efficiently accelerated if the magnetic field becomes sufficiently weak, e.g., for on scales . This is illustrated in Fig. 3 where the synchrotron ratio factor is plotted for the considered flow profiles. In reality, however, one may expect the magnetic field to also reveal some -dependence, probably decreasing with , thereby facilitating the acceleration of particles further away from the axis.

### 3.3 Inverse Compton Losses

For electrons, inverse Compton scattering (Thomson regime) could lead to additional cooling effects,

(32) |

where is the Thomson cross-section and the energy density of the target photons as seen in the comoving frame. A non-reducible photon field is provided by the CMB ( erg cm ). With , the ratio of shear energization to Compton losses becomes

(33) | |||||

using the same scaling for the magnetic field as above. The Compton ratio factor is illustrated in Fig. 3. Accordingly, once the magnetic field becomes sufficiently weak so that electron acceleration can overcome synchrotron losses (), shear energization is expected to continue out to larger distances until Compton losses set in ().

## 4 Discussion and Conclusion

Shear flows are ubiquitous in astrophysical environments and potential sites of non-thermal
particle acceleration. Relativistic outflows are in fact known to be conducive to efficient Fermi
type shear particle acceleration (e.g., Ostrowski2000, ; rie04, ). In the present paper, we
have explored the potential of expanding relativistic outflows to boost energetic particles to
higher energies. To this end, a set of simplified (azimuthally symmetric) conically expanding
flow profiles (power-law, Gaussian and Fermi-Dirac-type) has been examined where the
outflow bulk Lorentz factor is solely a function of polar angle. When applied to the AGN context,
the results show that for the acceleration mechanism to overcome radiative and non-radiative
losses and to work efficiently, the injection of pre-accelerated seed particles is required. This
could in general be achieved by first-order shock or stochastic second-order Fermi processes.
In this sense, shear acceleration would resemble a two-stage process for further particle
energisation beyond the common limit. Depending on the shape of the flow profile, particles
are more easily accelerated (i.e., require less injection energy) at different angular scales, i.e.,
not necessarily in the innermost core region close to the axis. This could in principle introduce
different jet emission features (e.g., core versus off-axis ridge-line structures or limb-brightening)
and allow for a variety in jet appearance. Once operative, gradual shear acceleration proceeds
on a timescale inversely proportional to the particle mean free path, .
For a gyro-dependent particle mean free path, , this gives the same
scaling as synchrotron losses, so that once started,
synchrotron losses will not be able to further constrain particle acceleration, while for electrons
Compton losses in an expanding flow might perhaps do so on larger scale.

From a methodological point of view, numerical studies focusing on the diffusive transport and
acceleration of particles in turbulent fields (e.g. osullivan09, ) and the excitation of a
large-scale shear dynamo (e.g. yousef08, ) become of particular interest to adequately
quantify the potential of shear particle acceleration on the relevant scales.

We note that shear acceleration in AGN jets seems in principle capable of accounting for continued
acceleration and related extended emission. The inverse dependence on the particle mean free
path makes shear acceleration a preferred mechanism for the acceleration of hadrons and provides
some further weight to the relevance of AGN jets for our understanding of the origin of the recently
detected high-energy (PeV) neutrinos (e.g. Becker14, ; Tavecchio15, ) and the production of
extreme cosmic-rays (e.g. Lemoine13, ).

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