---
abstract: 'The peak of the de-absorbed energy distribution of the TeV emitting blazars, all of the BL Lacertae (BL) class, can reach values up to $\sim 10$ TeV. In the context of synchrotron-self Compton (SSC) models of relativistic uniformly moving blobs of plasma, such high energy peak emission can be reproduced only by assuming Doppler factors of $\delta \sim 50$. However, such high values strongly disagree with the unification of FR I radio galaxies and BLs. Additionally, the recent detections of slow, possibly sub-luminal velocities in the sub-pc scale jets of the TeV BLs MKN 421 and MKN 501 suggest that the jets in these sources decelerate very early to mildly relativistic velocities ($\Gamma\sim$ a few). In this work we examine the possibility that the relativistic flows in the TeV BLs are longitudinally decelerating. In this case, modest Lorentz factors ($\Gamma \sim 15$), decelerating down to values compatible with the recent radio interferometric observations, can reproduce the $\sim $ few TeV peak energy of these sources. Furthermore, such decelerating flows are shown to reproduce the observed broadband BL - FR I luminosity ratios.'
address: 'Laboratory for High Energy Astrophysics, NASA Goddard Space Flight Center, Code 661, Greenbelt, MD 20771, US'
author:
- 'Markos Georganopoulos & Demosthenes Kazanas'
title: Decelerating Flows in TeV Blazars
---

galaxies: active — quasars: general — radiation mechanisms: nonthermal — X-rays: galaxies

Introduction
============

The blazar TeV emission reaches the Earth partially absorbed by the Diffuse InfraRed Background (DIRB) radiation which pair-produces with the TeV photons [@stecker92]. The de-absorbed spectra depend on the source redshifts and the still elusive energy distribution of the DIRB. However, for all expected DIRB forms, both the intrinsic peak energy $E_p$ and peak luminosity $L_p$ of the TeV spectral component are higher than those observed. Even for the nearby ($z=0.031$) MKN 421, $E_p$ can increase by a factor of $\sim 10$ after de-absorption to $\sim $ a few TeV [@dejager02]. The de-absorbed spectrum of H1426+428 at z=0.129 is even more extreme, characterized by $E_p \sim 10$ TeV [@aharonian02].

The BL synchrotron spectra exhibit a break at energies $\epsilon'_{b} 
\sim 10^{-4}-10^{-6}$ (primes denote energies on the flow rest frame, all energies normalized to $m_ec^2$) such that the greatest fraction of the synchrotron luminosity is at energies $\epsilon' > \epsilon'_{b}$. For this reason and because of the K-N decrease in the Compton scattering cross section, electrons with $\gamma \gsim 1/\epsilon'_{b}$ will inverse Compton scatter a decreasing fraction of the available photons; as a result, the maximum IC luminosity will occur at energies $\epsilon'_p \simeq 1/
\epsilon'_{b}$, or $\epsilon'_{p}\epsilon'_{b}\simeq 1$. The values of $\epsilon'_p, \, \epsilon'_{b}$ observed at the lab frame are $\epsilon_p 
= \delta \epsilon'_p$ and $\epsilon_{b} = \delta \epsilon'_{b}$, yielding the following relation between $\delta$ and the observed energies $\epsilon_p, \, \epsilon_{b}$ $$\delta \gsim (\epsilon_{b}\; \epsilon_{p})^{1/2} =   40\;(\nu_{b,16} 
\;E_{p,\,10\,\rm TeV})^{1/2},  \label{d_constr}$$ where $\nu_{b,16}$ is the observed synchrotron break frequency in units of $10^{16}$ Hz, $E_{p,\,10\,\rm TeV}$ is the energy of the de-absorbed TeV peak in units of 10 TeV and $\delta=1/\Gamma(1-\beta\cos\theta)$, with $\beta$ the dimensionless flow speed and $\theta$ its angle to the observer’s line of sight. These values of $\delta (\simeq 
\Gamma)$ are in strong conflict [@chiaberge00] with the unification of BLs and FR I radio galaxies [@urry95], which require Lorentz factors $\Gamma \sim 3-7$. Additional constrains for the inner jet flow of the TeV blazars come from the small, possibly sub-luminal apparent velocities observed interferometrically in the sub-pc scale jet of MKN 421 and 501 [@marscher99; @piner99; @edwards02], suggestive of a decelerating flow in the inner jet of TeV blazars [@marscher99].

Decelerating relativistic flows: application in TeV blazars
===========================================================

Motivated by the above analysis and the unification of the multiwavelength spectra of the hotspots of FR II radio galaxies and quasars in terms of decelerating relativistic flows [@georganopoulos03], we propose that the same considerations are applicable in resolving the conflict of the high $\delta$’s with TeV blazar unification. So, we assume that a power law electron distribution is injected at the base of a relativistic flow that decelerates while at the same time cooling radiatively. The highest frequencies of its synchrotron component originate preferentially at its fast base where the electrons are more energetic and its Lorentz factor largest. As both the flow velocity and electron energy drop with radius, the locally emitted synchrotron spectrum shifts to lower energies while its beaming pattern becomes wider. The observed synchrotron spectrum is the convolution of the comoving emission from each radius weighted by the beaming amplification at each radius. At small angles the observed spectrum is harder than that observed at larger angles. This is the result of the progressively smaller $\Gamma$ of the flow with distance in combination with the concomitant lower electron energies due to cooling.

The inverse Compton emission of such a flow behaves in a more involved way: Electrons will upscatter the locally produced synchrotron seed photons, giving rise to a local SSC emission with $\delta-$dependence similar to to that of synchrotron. However, the electrons of a given radius scatter not only the locally produced synchrotron photons, but also those produced downstream in the flow. The energy density of the latter, will appear Doppler boosted in the fast (upstream) part of the flow by $\sim 
\Gamma_{rel}^2$ [@dermer95], where $\Gamma_{rel}$ is the relative Lorentz factor between the fast and slow part of the flow. With their maximum energy being lower (because of cooling) and their energy density amplified they can now contribute to the IC emission at energies higher than expected on the basis of uniform velocity models (see section 1). Also the $\delta-$dependence of this [*upstream Compton (UC)*]{} emission will be different from that of SSC and more akin to that of external Compton (EC) [@dermer95].

![image](f1_georganopoulos.ps){width="2.75in"} ![image](f2_georganopoulos.ps){width="2.75in"}

In the left panel of fig. 1 we plot the SED for a flow decelerating from $\Gamma=15$ to $\Gamma=4$ for two angles $\theta=3^{\circ}$ and $\theta=
6^{\circ}$ over a distance $Z = 2\times 10^{16}$ cm. The radius of the cylindrical flow is set to $R = Z = 2\times 10^{16}$ cm, while the electron distribution at the base of the flow is $n(\gamma)\propto \gamma^{-2}$, $\gamma \leq 3\times 
10^7$ and the magnetic field $B=0.1$ G, half its equipartition value. At $\theta=3^{\circ}$ this model achieves a peak energy for the high energy component at $\sim 10$ TeV, using a modest initial Lorentz factor of $\Gamma=15$. Note the strong dependence of synchrotron peak energy on $\theta$ discussed above. Finally, note that, in contrast to uniform velocity homogeneous SSC models, the Compton component is more sensitive to orientation than synchrotron, an indication that UC scattering dominates the $\sim $ TeV observed luminosity.

We now turn our attention to the problem of the unification of BLs with FR I sources. A comparison of a sample of FR I nuclei to BLs of similar extended radio power shows that FR I’s are overluminous by a factor of $10-10^4$ compared to their expected luminosity [@chiaberge00; @trussoni03], under the assumption that BLs are characterized by Lorentz factors $\Gamma\sim15$, and that FR I’s are seen under an average angle of $60^{\circ}$. In particular the average BL to FR I nucleus luminosity ratio at radio, optical and X-ray bands was found to be: $\log (L_{BL}/L_{FR\;I})_R \approx 2.4$, $\log (L_{BL}
/L_{FR\;I})_{opt} \approx 3.9$, $\log (L_{BL}/L_{FR\;I})_{X} \approx 3.5$. In the right panel of fig. 1 we plot as vertical bars the luminosity separation of BLs and FR Is according to [@chiaberge00; @trussoni03]. We also plot the SED of a decelerating flow with physical parameters similar to the one described above but with $\gamma_{max}= 2\times 10^{5}$ to produce an SED synchrotron peak similar to those of the intermediate BLs that correspond in extended radio power to the FR Is of [@chiaberge00; @trussoni03]. As can be seen, the range of the model SEDs between $\theta=60^{\circ}$ (FR I) and $\theta=1/\Gamma$ (BL) reproduce well the observed range in luminosities, while a uniform velocity model of $\Gamma = 15$ (as demanded by fits of the $\gamma-$ray spectra) would produce at $\theta=60^{\circ}$ an SED many orders of magnitude smaller than shown in the figure.

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