11institutetext: 1 Laboratoire Univers et Théories, Observatoire de Paris,
Université PSL, Université Paris Cité, CNRS, F-92190 Meudon, France,
<EMAIL_ADDRESS>
2 Centre for Space Research, North-West University, Potchefstroom 2520, South
Africa<EMAIL_ADDRESS>
# On the evolution of the particle distribution and the cascade in a moving,
expanding emission region in blazar jets

Michael Zacharias1,2 Now at: Landessternwarte, Universität Heidelberg,
Königstuhl 12, D-69117 Heidelberg, Germany

(Received ? / accepted ? )

###### Abstract

Aims. There is a large variety in the models explaining blazar flares. Here,
we study the flare profile induced by a moving and expanding blob with special
emphasize on the $\gamma$-$\gamma$ pair production.

Methods. We first develop a simple semi-analytical model to study the
evolution of the particle distribution in the expanding blob and show the
influence of the pair production. In a second step, we produce a realistic
simulation using the OneHaLe code based upon parameters of PKS 1510-089.

Results. The semi-analytical model shows that the pair production
significantly influences the flare evolution, while the opening angle and the
expansion can prolong flares considerably. The simulation based on PKS
1510-089 indicate that flares of a moving expanding blob result in strongly
wavelength dependant light curves including delayed, secondary flares.

Conclusions. A moving, expanding blob can cause significant flaring events
with a large variety in light curve profiles. High-cadence multiwavelength
observations are necessary to derive the details causing the flare. Extended
observations beyond the initial burst may provide important information on the
opening angle and the particle content due to delayed secondary flares in some
energy bands.

###### Key Words.:

radiation mechanisms: non-thermal – galaxies: active – galaxies: jets – gamma-
rays: galaxies

##  1 Introduction

The emission of blazars — active galaxies with the jet pointing at Earth — is
typically explained with the so-called one-zone model, where a single zone is
responsible for most of the source’s radiative output. The spectral energy
distribution (SED) is characterized by two broad humps. The low-energy one is
explained by electron-synchrotron emission, while the high-energy hump can be
explained by inverse-Compton emission or hadronically induced processes, such
as proton synchrotron or synchrotron emission from the leptonic cascade. An
important role is played by photon fields external to the jet, such as the
accretion disk (AD), the broad-line region (BLR), and the dusty torus (DT), as
these fields may provide ample seed photons for particle-photon and photon-
photon interactions (see, e.g., Böttcher 2019; Cerruti 2020, for detailed
reviews). The one-zone model is well justified in flares, where the
variability time scale restricts the size of the emission region. However,
while the particle flow in this region is relativistic, the emission region
itself is typically assumed to remain stationary with respect to the black
hole (e.g., H.E.S.S. Collaboration et al. 2019), which allows one to ignore
various complications, like varying (external) photon fields, adiabatic
expansion and cooling, etc. On the other hand, the observation of stationary
features in radio VLBI observations (and other wavelengths where jets have
been resolved) suggests standing recollimation shocks (e.g., Weaver et al.
2022), where particles may be accelerated and radiate. In such a situation,
the emission region would indeed be stationary with respect to the black hole.

The same radio VLBI observations further reveal moving components launched
somewhere upstream from the radio core. The interaction of such moving
features with the standing features has been connected with multiwavelength
flaring events (e.g., Ahnen et al. 2017; H.E.S.S. Collaboration et al. 2021).
Indeed, numerical simulations by Fichet de Clairfontaine et al. (2021, 2022)
show that the interaction of moving and standing shocks can induce rapid
flaring events. It is, thus, important to study the flaring characteristics of
a moving, expanding emission region (or “blob”).

Recently, Boula & Mastichiadis (2022) and Tramacere et al. (2022) discussed
this model in detail and derived the expected time delays between the
$\gamma$-ray and the radio band expected from the progressive optical thinning
at radio frequencies of the expanding source (see also Saito et al. 2015). In
the present work, we specifically consider the effect of $\gamma$-$\gamma$
pair production on the evolution of the particle distribution and the photon
fluxes. Especially, bright external photon fields can have a major influence
as they provide a significant amount of absorption for the $\gamma$ rays of
the emission region (e.g., Zacharias 2021). However, as these photon fields
are only present in the direct vicinity of the black hole, the optical
thickness for $\gamma$ rays changes with time as the blob moves down the jet.
Additionally, the opening angle will strongly influence the escape time of
both photons and particles owing to the increase in radius of the emission
region. This could potentially prolong the pair production as the (internal)
photons have more time to interact before escape. This naturally competes with
the thinning of the photon density due to expansion. In any case, one may
obtain variability simply through the motion of an expanding blob through the
various radiation fields due to the varying injection of pairs.

Firstly, we derive a simple semi-analytical model that describes the time-
dependent evolution of the particle distribution (Sec. 2). In this section we
introduce the basic assumptions about the escape time scale and the
implication of the expansion. We then derive the particle distribution without
pair injection, followed by a very simple linear cascade model for a few
exemplary cases. In Sec. 3 we use the numerical code OneHaLe (Zacharias 2021;
Zacharias et al. 2022) to derive the light curves of a realistic simulation
based on the parameters of PKS 1510-089, which is known for its bright
external photon fields. We conclude in Sec. 4.

##  2 A simple electron evolution model

We derive simple semi-analytical models for the time-dependent evolution of
the electron distribution ignoring all energy dependencies. While this is a
major simplification, it allows us to study three distinct cases with an at-
most linear cascade. Using the time-dependent, one-zone radiation code
OneHaLe, we reproduce well two of the three cases. We discuss the failure of
the third case, as well as the influence of the choice of parameters on the
solutions.

###  2.1 Escape time

The blob travels with constant speed $\beta_{\Gamma}c$ corresponding to the
bulk Lorentz factor $\Gamma=(1-\beta_{\Gamma}^{2})^{-1/2}$ along the z-axis.
Within a conical jet, the opening angle is constant. VLBI observations suggest
an opening angle $\propto\alpha/\Gamma$ with $\alpha\sim 0.26$ (Pushkarev et
al. 2017), but we use $\alpha$ as a free parameter. The radius $R$ of the blob
thus evolves as a function of comoving time111Parameters in the observer’s
frame are marked by “obs”, in the black hole frame by a prime, while
parameters in the comoving frame and invariants are not marked. $t$ and jet
coordinate $z^{\prime}$:

$\displaystyle R(t)$
$\displaystyle=R_{0}+(z^{\prime}(t)-z_{0}^{\prime})\tan{(\alpha/\Gamma)}$
$\displaystyle=R_{0}+\Gamma\beta_{\Gamma}ct\tan{(\alpha/\Gamma)}$
$\displaystyle\approx R_{0}+\alpha ct,$ (1)

where in the last line we approximate for small angles and
$\beta_{\Gamma}\approx 1$. Boula & Mastichiadis (2022) expressed this equation
by an “expansion speed” $u_{exp}$, which relates to our equation as
$u_{exp}=\alpha c$. In their paper, they used $\alpha$ between 0.01 and 0.2.
It should be noted that one of the common estimates for jet expansion,
$\alpha=1$ (e.g., Ackermann et al. 2016), implies a radial expansion with the
speed of light, while $\alpha>1$ implies a superluminal expansion and thus a
causal disconnection of regions within the blob/jet.

On average, particles escape the emission region on an energy-independent time
scale

$\displaystyle t_{\rm esc}(t)=\eta_{\rm esc}R(t)/c$ (2)

with $\eta_{\rm esc}>1$. This resembles an advective motion of the plasma
below the speed of light mimicking the trapping of charged particles in the
magnetized blob. Calculating the average escape time scale for photons, one
obtains $\eta_{\rm esc,ph}=3/4$ (Böttcher & Chiang 2002).

In an expanding blob, the escape time increases. In turn, particles and
photons take longer to escape and only efficiently do so once the intrinsic
time $t$ since launch surpasses $\sim t_{\rm esc}(t)$. Namely,

$\displaystyle t$ $\displaystyle>\frac{\eta_{\rm esc}}{c}\left(R_{0}+\alpha
ct\right)$ $\displaystyle\Leftrightarrow t$ $\displaystyle>\frac{t_{\rm
esc}(0)}{1-\eta_{\rm esc}\alpha},$ (3)

with $t_{\rm esc}(0)=\eta_{\rm esc}R_{0}/c$. In case of $\eta_{\rm
esc}\alpha\rightarrow 1$, particles are effectively trapped in the blob
without a chance of a meaningful escape. In this case the escape of photons is
also significantly slowed down. Hence, a significant cascade could still
materialize at far distances from the black hole. It also suggests that a
cascade that has begun developing closer to the black hole (say, within the
external photon fields), can continue to grow even at far distances.

###  2.2 Particle density evolution without secondary injection

We are interested in the time-dependent evolution of the total particle
density, but not its detailed energy-dependent evolution, as this does not
change the overall density. Considering only time-dependent primary injection
and escape, but no secondary injections, the kinetic equation for the particle
distribution $n(t)$ becomes

$\displaystyle\frac{\,\partial n(t)\,}{\,\partial t\,}+\frac{n(t)}{t_{\rm
esc}(t)}=Q(t)$ (4)

with the analytical solution

$\displaystyle n(t)=\exp{\left(-\int^{t}\frac{\,\mathrm{d}t^{\prime}\,}{t_{\rm
esc}(t^{\prime})}\right)}\int\limits_{0}^{t}Q(t^{\prime\prime})\exp{\left(\int^{t^{\prime\prime}}\frac{\,\mathrm{d}t^{\prime}\,}{t_{\rm
esc}(t^{\prime})}\right)}\,\mathrm{d}t^{\prime\prime}\,.$ (5)

The particle injection rate $Q(t)$ is coupled to the particle injection
luminosity $L_{\rm inj}(t)$ in the form

$\displaystyle Q(t)=\frac{L_{\rm
inj}(t)}{V(t)E(\gamma)}=q_{0}\left(1+\frac{\eta_{\rm esc}\alpha}{t_{\rm
esc}(0)}t\right)^{-3-p},$ (6)

where $E(\gamma)$ is a function of the injection energy spectrum, which is of
no concern to us, while $V(t)$ is the volume of the spherical blob. With the
assumption $L_{\rm inj}(t)=L_{0}[R_{0}/R(t)]^{p}=L_{0}\left(1+\frac{\eta_{\rm
esc}\alpha}{t_{\rm esc}(0)}t\right)^{-p}$ and the definition
$q_{0}:=L_{0}/[V(0)E(\gamma)]$, the second equality of Eq. (6) is readily
achieved. The power-law index $p$ describes the decrease of the injection
luminosity as a function of $R(t)$. As the jet power is proportional to
$R(t)^{2}$, $p=2$ implies a constant jet particle injection power. While $p$
is a free parameter, we will mostly use $p=2$ below. For constant injection
and escape — that is a straight jet, $\alpha=0$ — the solution of Eq. (5) is
$n=q_{0}t_{\rm esc}(0)$ as expected.

With Eq. (2), the integrals in the exponentials of Eq. (5) are easily solved:

$\displaystyle n(t)$ $\displaystyle=q_{0}\left(1+\frac{\eta_{\rm
esc}\alpha}{t_{\rm esc}(0)}t\right)^{-\frac{1}{\eta_{\rm
esc}\alpha}}\int\limits_{0}^{t}\left(1+\frac{\eta_{\rm esc}\alpha}{t_{\rm
esc}(0)}t^{\prime\prime}\right)^{\frac{1}{\eta_{\rm
esc}\alpha}-(3+p)}\,\mathrm{d}t^{\prime\prime}\,$
$\displaystyle=\frac{q_{0}t_{\rm esc}(0)}{1-\eta_{\rm
esc}\alpha(2+p)}\left[\left(1+\frac{\eta_{\rm esc}\alpha}{t_{\rm
esc}(0)}t\right)^{-(2+p)}-\left(1+\frac{\eta_{\rm esc}\alpha}{t_{\rm
esc}(0)}t\right)^{-\frac{1}{\eta_{\rm esc}\alpha}}\right].$ (7)

This equation is positive for all times and $p$. For $p=(\eta_{\rm
esc}\alpha)^{-1}-2$ the density becomes

$\displaystyle n(t)=\frac{q_{0}t_{\rm esc}(0)}{\eta_{\rm
esc}\alpha}\left(1+\frac{\eta_{\rm esc}\alpha}{t_{\rm
esc}(0)}t\right)^{-\frac{1}{\eta_{\rm esc}\alpha}}\ln{\left(1+\frac{\eta_{\rm
esc}\alpha}{t_{\rm esc}(0)}t\right)}$ (8)

Equation (7) holds for all particle species as long as there is no secondary
injection or particle destruction. Below, we refer to Eq. (7) as the “standard
solution”.

Figure 1: Particle density as a function of time for various cases of
$\eta_{\rm esc}\alpha$ and $p$ as labeled. The dashed lines are the analytical
result, Eq. (7). The vertical red line is $t_{\rm esc}(0)$. The dash-double-
dotted line marks $t^{-3}$. Further parameters are $q_{0}=1\,$cm-3s-1,
$R_{0}=5\times 10^{15}\,$cm, and $\eta_{\rm esc}=3$. The solid lines are from
simulations using OneHaLe.

Figure 1 exemplifies Eq. (7) for two values of $p$, and various values of
$\eta_{\rm esc}\alpha$. The parameters $q_{0}$ and $R_{0}$ are chosen to
resemble typical one-zone parameters, while $\eta_{\rm esc}=3$ implies
$\alpha\in[0.0\bar{3},0.3]$ resembling the typical range of this parameter
(Pushkarev et al. 2017; Boula & Mastichiadis 2022). Interestingly, both $p$
and $\eta_{\rm esc}\alpha$ have a similar effect. A larger $p$ implies a
faster decrease in the injection luminosity and thus a faster decrease of the
density. More importantly though, larger opening angles imply a slower escape
of particles from the emission region, and thus a slower decline of the
density [see Eq. (3)]. The peak in each curve is attained at roughly $t\sim
t_{\rm esc}(0)$ (red vertical line), however with a slightly earlier peak time
for larger opening angles because of the quicker injection density decrease.
For $\alpha\rightarrow 0$, the lines would approach a constant as the blob
approaches the steady-state. The solid lines in Fig. 1 are derived from
simulations using OneHaLe, and confirm the analytical result.

The dash-double-dotted line in Fig. 1 indicates $n\sim t^{-3}$, which is the
expected evolution of the density if it were solely due to the increase in
volume of the blob. Most model lines are harder than this line implying a
continuous increase of particle number in the emission region, even for $p=2$.
Only for small opening angles (black and blue cases) for $p=2$, the model
lines are softer and the total number of particles decreases at large times.
This can be easily understood from Eq. (7), as the evolution at late times is
governed by the power-law index: $(2+p)$ or $1/\eta_{\rm esc}\alpha$,
whichever is smaller. If the injection switches off entirely, the density
drops $\propto t^{-1/\eta_{\rm esc}\alpha}$, which for large opening angles is
the same behavior as with continuous injection. This shows the aforementioned
trapping of particles in the (rapidly) expanding blob.

###  2.3 Linear cascade evolution

In this section we treat the additional injection of electrons through
$\gamma$-$\gamma$ pair production of a $\gamma$ ray colliding with a soft
photon. We do so by first deriving three simple (semi-)analytical scenarios.
These are compared to simulations in Sec. 2.3.4, where we discuss both the
success and failure of the approach.

We continue to ignore the energy dependency of the process, as we are only
interested in the rough time dependency of the total particle density. In
other words, we assume that pair production takes place. This clearly is a
very strong simplification, which may not hold in many cases. In this
scenario, the injection rate of pairs can be written as

$\displaystyle Q_{\gamma\gamma}(t)=\xi n_{\rm soft}(t)n_{\gamma}(t),$ (9)

with the soft photon density, $n_{\rm soft}(t)$, the $\gamma$-ray photon
density, $n_{\gamma}(t)$, and the correlation factor $\xi\sim\sigma_{T}c$
absorbing all constants and energy dependencies.

Both photon distributions can have various underlying production processes.
The soft photons are most likely electron-synchrotron photons, or thermal
photons from external sources, such as the AD, the BLR and the DT. The
$\gamma$ rays can be produced from electron-inverse-Compton emission, or
proton-dependent processes, such as proton-synchrotron or neutral-pion decay.
The internal photon densities thus depend on the underlying particle
distribution implying $n_{\rm phot}(t)\sim n_{e/p}(t)$, with the exception of
SSC radiation, which depends quadratically on the electron distribution. As we
only want to treat a linear cascade evolution — that is, $Q_{\gamma\gamma}$
shall depend at most linearly on the electron distribution — we will ignore
from now on the following combinations: electron-synchrotron and electron-
inverse-Compton, as well as external photon fields and electron-SSC.

In fact, the simplest case is hadronically induced $\gamma$ rays absorbed by
external photon fields, as this does not depend on the electron distribution
at all. Cases with a linear dependency on the electron distribution are
electron-synchrotron absorbing hadronically induced $\gamma$ rays, as well as
external photons absorbing electron-inverse-Compton radiation. This leaves us
with three distinct cases:

$\displaystyle Q_{\gamma\gamma}(t)$ $\displaystyle\sim\xi n_{\rm
ext}(t)n_{p}(t)$ (10) $\displaystyle Q_{\gamma\gamma}(t)$
$\displaystyle\sim\xi n_{e}(t)n_{p}(t)$ (11) $\displaystyle
Q_{\gamma\gamma}(t)$ $\displaystyle\sim\xi n_{\rm ext}(t)n_{e}(t).$ (12)

The external photon fields comprising AD and isotropic sources, such as the
BLR, are by themselves assumed to be time-independent. However, the motion of
the blob changes the distance to these external sources implying

$\displaystyle n_{\rm ext}(t)=\frac{n_{\rm AD}}{\left(1+\frac{\Gamma c}{R_{\rm
AD}}t\right)^{2}}+n_{i}\;H\left[\frac{z_{i}}{\Gamma c}-t\right].$ (13)

The first summand describes the AD field, which is roughly constant close to
the accretion disk, and falls off with distance-squared once the blob has
travelled a distance corresponding roughly to the AD radius, $R_{\rm AD}$
(Dermer & Schlickeiser 2002). The second summand represents isotropic photon
fields (in the black hole frame), such as the BLR and the DT, within a given
distance $z_{i}$ from the black hole.

####  2.3.1 Case 1: External and proton-induced photons

In this case, Eq. (10) is simply added to Eq. (4) and in Eq. (5). The solution
is thus separated into the primary injection following Eq. (7), and the
secondary injection leading to

$\displaystyle n_{e}^{\gamma\gamma}(t)$ $\displaystyle=\frac{\xi q_{0,p}t_{\rm
esc}(0)}{1-\eta_{\rm esc}\alpha(2+p)}\left(1+\frac{\eta_{\rm
esc}\alpha}{t_{\rm esc}(0)}t\right)^{-\frac{1}{\eta_{\rm esc}\alpha}}$
$\displaystyle\quad\times\int\limits_{0}^{t}\left[\left(1+\frac{\eta_{\rm
esc}\alpha}{t_{\rm esc}(0)}t^{\prime\prime}\right)^{\frac{1}{\eta_{\rm
esc}\alpha}-(2+p)}-1\right]$ $\displaystyle\qquad\times\left[\frac{n_{\rm
AD}}{\left(1+\frac{\Gamma c}{R_{\rm
AD}}t^{\prime\prime}\right)^{2}}+n_{i}\;H\left[\frac{z_{i}}{\Gamma
c}-t^{\prime\prime}\right]\right]\,\mathrm{d}t^{\prime\prime}\,.$ (14)

While the isotropic contribution can be easily integrated, the accretion disk
contribution leads to a hypergeometric integral without a simple analytical
solution.

Figure 2: Electron density as a function of time for various cases of
$\eta_{\rm esc}\alpha$ as labeled. Secondary electrons are produced from
proton-induced $\gamma$ rays absorbed by isotropic external photons. The thick
dash-dotted line is the analytical solution of Eq. (14) assuming only one
isotropic photon field with $n_{i}=1\times 10^{10}\,$cm-3, and $z_{i}=1\,$pc.
The dashed lines are the standard solution, Eq. (7). The vertical red line is
$t_{\rm esc}(0)$, while the horizontal dotted lines mark the maximum level of
the standard solution and when they are reached by the dash-dotted lines.
Further parameters are $p=2$, $q_{0,e}=q_{0,p}=1\,$cm-3s-1, $R_{0}=5\times
10^{15}\,$cm, $\eta_{\rm esc}=3$, and $\Gamma=10$. The solid lines mark
simulations with OneHaLe.

For isotropic external photons, that is $n_{\rm AD}=0$, the analytical result
for Eq. (14) becomes

$\displaystyle n_{e}^{\gamma\gamma}(t)=\frac{\xi q_{0,p}[t_{\rm
esc}(0)]^{2}n_{i}\left(1+\frac{\eta_{\rm esc}\alpha}{t_{\rm
esc}(0)}t\right)^{-\frac{1}{\eta_{\rm esc}\alpha}}}{[1-\eta_{\rm
esc}\alpha(2+p)]\;[1-\eta_{\rm esc}\alpha(1+p)]}$
$\displaystyle\times\begin{cases}\left[\left(\left(1+\frac{\eta_{\rm
esc}\alpha}{t_{\rm esc}(0)}t\right)^{\frac{1}{\eta_{\rm
esc}\alpha}-(1+p)}-1\right)-t\right]&\mbox{for}\ t\leq\frac{z_{i}}{\Gamma
c}\\\ \left[\left(\left(1+\frac{\eta_{\rm esc}\alpha}{t_{\rm
esc}(0)}\frac{z_{i}}{\Gamma c}\right)^{\frac{1}{\eta_{\rm
esc}\alpha}-(1+p)}-1\right)-\frac{z_{i}}{\Gamma c}\right]&\mbox{for}\
t>\frac{z_{i}}{\Gamma c}\end{cases}.$ (15)

The sum of Eqs. (7) and (15) is shown in Fig. 2. Compared to the standard
solution, Eq. (7), there are significantly more particles injected within the
boundaries of the external field, $z_{i}$. The boundary $z_{i}$ is noticeable
by the break in the dash-dotted and solid lines. The peak of the particle
density is attained later for larger opening angles compared to the standard
solution. Beyond $z_{i}$, the decay of the density again depends strongly on
the opening angle. The standard solution is only reached for small opening
angles.

Figure 3: Same as Fig. 2, except that the absorber is AD photons with $n_{\rm
AD}=1\times 10^{11}\,$cm-3, and $R_{\rm AD}=1\times 10^{16}\,$cm. Further
parameters are $p=2$, $q_{0,e}=q_{0,p}=1\,$cm-3s-1, $R_{0}=5\times
10^{15}\,$cm, $\eta_{\rm esc}=3$, and $\Gamma=10$.

The semi-analytical solutions of Eq. (14) for the AD, that is $n_{i}=0$, are
shown in Fig. 3. The behavior is comparable to the isotropic case except for
an earlier peak owing to $R_{\rm AD}\ll z_{i}$.

In both figures, 2 and 3, the horizontal dotted lines mark the maximum level
of the standard solution. They indicate the time delay for the semi-analytical
solution to drop below this density. Naturally, this happens later for the
isotropic field than for the AD field, and depends strongly on the opening
angle. In fact, the time delay can be orders of magnitude implying a
significantly more pronounced and longer high-state.

####  2.3.2 Case 2: Electron-synchrotron and proton-induced photons

Adding Eq. (11) to Eq. (4) we obtain with a slight rearrangement

$\displaystyle\frac{\,\partial n_{e}(t)\,}{\,\partial
t\,}+\left(\frac{1}{t_{\rm esc}(t)}-\xi n_{p}(t)\right)n_{e}(t)=Q_{e}(t).$
(16)

The linear cascade acts as a “catastrophic” injection with time scale
$t_{\gamma\gamma}(t)=[\xi n_{p}(t)]^{-1}$. We directly obtain the
generalisation of Eqs. (5) and (7) as

$\displaystyle n_{e}(t)$
$\displaystyle=\exp{\left(-\int^{t}\frac{\,\mathrm{d}t^{\prime}\,}{t_{\rm
esc}(t^{\prime})}+\xi\int^{t}n_{p}(t^{\prime})\,\mathrm{d}t^{\prime}\,\right)}$
$\displaystyle\quad\times\int\limits_{0}^{t}Q_{e}(t^{\prime\prime})\exp{\left(\int^{t^{\prime\prime}}\frac{\,\mathrm{d}t^{\prime}\,}{t_{\rm
esc}(t^{\prime})}-\xi\int^{t^{\prime\prime}}n_{p}(t^{\prime})\,\mathrm{d}t^{\prime}\,\right)}\,\mathrm{d}t^{\prime\prime}\,$
$\displaystyle=q_{0,e}\left(1+\frac{\eta_{\rm esc}\alpha}{t_{\rm
esc}(0)}t\right)^{-\frac{1}{\eta_{\rm
esc}\alpha}}\exp{\left(\xi\int^{t}n_{p}(t^{\prime})\,\mathrm{d}t^{\prime}\,\right)}$
$\displaystyle\quad\times\int\limits_{0}^{t}\left(1+\frac{\eta_{\rm
esc}\alpha}{t_{\rm esc}(0)}t^{\prime\prime}\right)^{\frac{1}{\eta_{\rm
esc}\alpha}-(3+p)}\exp{\left(-\xi\int^{t^{\prime\prime}}n_{p}(t^{\prime})\,\mathrm{d}t^{\prime}\,\right)}\,\mathrm{d}t^{\prime\prime}\,.$
(17)

As the protons follow Eq. (7), the integrals in the exponentials can be
performed:

$\displaystyle\int^{t}n_{p}(t^{\prime})\,\mathrm{d}t^{\prime}\,$
$\displaystyle=\frac{q_{0,p}t_{\rm esc}(0)^{2}}{[1-\eta_{\rm
esc}\alpha(2+p)]\,(1+p)\,(1-\eta_{\rm esc}\alpha)}$
$\displaystyle\quad\times\left[\left(1-\frac{1}{\eta_{\rm
esc}\alpha}\right)\,\left(1+\frac{\eta_{\rm esc}\alpha}{t_{\rm
esc}(0)}t\right)^{-(1+p)}\right.$
$\displaystyle\qquad+\left.\left(1+p\right)\,\left(1+\frac{\eta_{\rm
esc}\alpha}{t_{\rm esc}(0)}t\right)^{1-\frac{1}{\eta_{\rm
esc}\alpha}}\right].$ (18)

With this solution, the remaining integral in Eq. (17) can be solved
numerically. The result is shown in Fig. 4.

Figure 4: Same as Fig. 2, but for secondary electrons produced from proton-
induced $\gamma$ rays absorbed by electron-synchrotron photons, Eq. (17).
Further parameters are $p=2$, $q_{0,e}=1\,$cm-3s-1, $q_{0,p}=1\times
10^{3}\,$cm-3s-1, $R_{0}=5\times 10^{15}\,$cm, and $\eta_{\rm esc}=3$.

The cascade develops only after $t_{\rm esc}(0)$, while the opening angle
dictates the further evolution. Small opening angles result in a high number
density and a quick decay, while large opening angles seemingly keep the
cascade going for very long times without any significant decay. This is also
indicated by the dotted lines.

####  2.3.3 Case 3: External and electron-IC photons

This case is similar to the case 2, except that we have to replace $n_{p}(t)$
with $n_{\rm ext}(t)$, and the $\gamma$ rays stem from IC scattering of the
external photon fields. Then the electron density becomes

$\displaystyle n_{e}(t)$ $\displaystyle=q_{0,e}\left(1+\frac{\eta_{\rm
esc}\alpha}{t_{\rm esc}(0)}t\right)^{-\frac{1}{\eta_{\rm esc}\alpha}}$
$\displaystyle\quad\times\exp{\left(-\frac{\xi n_{\rm AD}R_{\rm AD}}{\Gamma
c\left(1+\frac{\Gamma c}{R_{\rm AD}}t\right)}+\xi
n_{i}t\;\tilde{H}\left[t,\frac{z_{i}}{\Gamma c}\right]\right)}$
$\displaystyle\quad\times\int\limits_{0}^{t}\left(1+\frac{\eta_{\rm
esc}\alpha}{t_{\rm esc}(0)}t^{\prime\prime}\right)^{\frac{1}{\eta_{\rm
esc}\alpha}-(3+p)}$ $\displaystyle\qquad\times\exp{\left(\frac{\xi n_{\rm
AD}R_{\rm AD}}{\Gamma c\left(1+\frac{\Gamma c}{R_{\rm
AD}}t^{\prime\prime}\right)}-\xi
n_{i}t^{\prime\prime}\;H\left[\frac{z_{i}}{\Gamma
c}-t^{\prime\prime}\right]\right)}\,\mathrm{d}t^{\prime\prime}\,,$ (19)

with the modified Heaviside function $\;\tilde{H}\left[x,a\right]=1$ for
$x\leq a$, and $\;\tilde{H}\left[x,a\right]=a/x$ for $x>a$.

Figure 5: Same as Fig. 2, but for secondary electrons produced from IC-induced
$\gamma$ rays absorbed by isotropic external photons, Eqs. (20) and (21), with
$n_{i}=4\times 10^{7}\,$cm-3, and $z_{i}=1\,$pc. Further parameters are $p=2$,
$q_{0,e}=1\,$cm-3s-1, $R_{0}=5\times 10^{15}\,$cm, $\eta_{\rm esc}=3$, and
$\Gamma=10$.

For isotropic external photon fields, that is $n_{\rm AD}=0$, Eq. (19) can be
analytically integrated:

$\displaystyle n_{e}\left(t\leq\frac{z_{i}}{\Gamma c}\right)$
$\displaystyle=\frac{q_{0,e}}{\xi n_{i}}\left(1+\frac{\eta_{\rm
esc}\alpha}{t_{\rm esc}(0)}t\right)^{-\frac{1}{\eta_{\rm
esc}\alpha}}\left(\frac{\eta_{\rm esc}\alpha}{\xi n_{i}t_{\rm
esc}(0)}\right)^{\frac{1}{\eta_{\rm esc}\alpha}-(3+p)}$
$\displaystyle\quad\times\exp{\left[\frac{\xi n_{i}t_{\rm esc}(0)}{\eta_{\rm
esc}\alpha}\left(1+\frac{\eta_{\rm esc}\alpha}{t_{\rm
esc}(0)}t\right)\right]}$
$\displaystyle\quad\times\left[\Gamma\left(\frac{1}{\eta_{\rm
esc}\alpha}-(2+p),\,\frac{\xi n_{i}t_{\rm esc}(0)}{\eta_{\rm
esc}\alpha}\right)\right.$
$\displaystyle\quad\left.-\Gamma\left(\frac{1}{\eta_{\rm
esc}\alpha}-(2+p),\,\frac{\xi n_{i}t_{\rm esc}(0)}{\eta_{\rm
esc}\alpha}\left(1+\frac{\eta_{\rm esc}\alpha}{t_{\rm
esc}(0)}t\right)\right)\right]$ (20) $\displaystyle
n_{e}\left(t>\frac{z_{i}}{\Gamma c}\right)$
$\displaystyle=q_{0,e}\left(1+\frac{\eta_{\rm esc}\alpha}{t_{\rm
esc}(0)}t\right)^{-\frac{1}{\eta_{\rm esc}\alpha}}\exp{\left(\xi
n_{i}\frac{z_{i}}{\Gamma c}\right)}$
$\displaystyle\quad\times\left\\{\exp{\left(\frac{\xi n_{i}t_{\rm
esc}(0)}{\eta_{\rm esc}\alpha}\right)}\frac{\left(\frac{\eta_{\rm
esc}\alpha}{\xi n_{i}t_{\rm esc}(0)}\right)^{\frac{1}{\eta_{\rm
esc}\alpha}-(3+p)}}{\xi n_{i}}\right.$
$\displaystyle\qquad\times\left[\Gamma\left(\frac{1}{\eta_{\rm
esc}\alpha}-(2+p),\,\frac{\xi n_{i}t_{\rm esc}(0)}{\eta_{\rm
esc}\alpha}\right)\right.$
$\displaystyle\qquad\left.-\Gamma\left(\frac{1}{\eta_{\rm
esc}\alpha}-(2+p),\,\frac{\xi n_{i}t_{\rm esc}(0)}{\eta_{\rm
esc}\alpha}\left(1+\frac{\eta_{\rm esc}\alpha}{t_{\rm
esc}(0)}\frac{z_{i}}{\Gamma c}\right)\right)\right]$
$\displaystyle\quad+\frac{t_{\rm esc}(0)}{1-\eta_{\rm
esc}\alpha(2+p)}\left[\left(1+\frac{\eta_{\rm esc}\alpha}{t_{\rm
esc}(0)}t\right)^{\frac{1}{\eta_{\rm esc}\alpha}-(2+p)}\right.$
$\displaystyle\qquad\left.\left.-\left(1+\frac{\eta_{\rm esc}\alpha}{t_{\rm
esc}(0)}\frac{z_{i}}{\Gamma c}\right)^{\frac{1}{\eta_{\rm
esc}\alpha}-(2+p)}\right]\right\\},$ (21)

with the incomplete Gamma-function $\Gamma(q,x)$. The result is shown in Fig.
5. Compared to Fig. 2, a strong pile-up is visible, which results from $\xi
n_{i}t>1$ close to $z_{i}$. This maybe due to an extreme choice of parameters,
but displays the importance of the immediate feedback of the cascade on the
$\gamma$ rays in this case compared to case 1. We note that the pile-up for
$\eta_{\rm esc}\alpha=0.1$ in Fig. 5 might be influenced by some approximation
inaccuracies for small values of $\eta_{\rm esc}\alpha$, and a smaller pile-up
is to be expected in this case.

Figure 6: Same as Fig. 2, but for secondary electrons produced from IC-induced
$\gamma$ rays absorbed by AD photons with $n_{\rm AD}=1\times 10^{10}\,$cm-3,
and $R_{\rm AD}=1\times 10^{16}\,$cm. Further parameters are $p=2$,
$q_{0,e}=1\,$cm-3s-1, $R_{0}=5\times 10^{15}\,$cm, $\eta_{\rm esc}=3$, and
$\Gamma=10$.

For AD photons, we need to numerically integrate Eq. (19) with the result
shown in Fig. 6. The result is fairly similar to case 1, which may again be
due to parameter choices, but also because the argument of the exponential is
a decreasing function with time. The latter reduces the amount of absorbing
photons before the $\gamma$ rays have fully developed resulting in a reduced
cascade compared to the isotropic case.

In both cases, the dotted lines are roughly as long as in case 1. In the
isotropic case, this is significantly influenced by the pile-up close to
$z_{i}$. With a slightly smaller size of the isotropic field, and a resulting
smaller pile-up in case 3, the duration of the high-state in case 3 would be
shorter than in case 1. No such strong influence is expected in the AD case.

####  2.3.4 Simulations

Table 1: OneHaLe input parameters to simulated cases 1 to 3. The first five parameters are the same in all simulations. The electron distribution parameters of case 1 correspond to the simulation used to reproduce the standard solution in Fig. 1.  Definition | Symbol | Case 1 | Case 2 | Case 3
---|---|---|---|---
|  |  | ISO | AD |  | ISO | AD
Initial distance to black hole | $z_{0}^{\prime}$ | [cm] | $1.0\times 10^{15}$
Initial blob radius | $R_{0}$ | [cm] | $5\times 10^{15}$
Escape time scaling | $\eta_{\rm esc}$ |  | $3$
Doppler factor | $\delta$ |  | $10$
Black hole mass | $M_{\rm BH}$ | [$M_{\odot}$] | $3\times 10^{8}$
Initial magnetic field | $B_{0}$ | [G] | $10$ | $10$ | $0.3$ | $0.1$ | $0.1$
e injection luminosity | $L_{e}$ | [erg/s] | $1.86\times 10^{43}$ | $1.86\times 10^{43}$ | $2.3\times 10^{43}$ | $1\times 10^{45}$ | $8\times 10^{44}$
Min. e Lorentz factor | $\gamma_{\rm e,min}$ |  | $2\times 10^{1}$ | $2\times 10^{1}$ | $2\times 10^{1}$ | $2\times 10^{1}$ | $2\times 10^{1}$
Max. e Lorentz factor | $\gamma_{\rm e,max}$ |  | $2\times 10^{4}$ | $2\times 10^{4}$ | $2\times 10^{5}$ | $2\times 10^{5}$ | $2\times 10^{5}$
e spectral index | $s_{e}$ |  | $2.5$ | $2.5$ | $2.5$ | $1.5$ | $1.5$
p injection luminosity | $L_{\rm p}$ | [erg/s] | $1.2\times 10^{50}$ | $6\times 10^{49}$ | $1\times 10^{50}$ | – | –
Min. p Lorentz factor | $\gamma_{\rm p,min}$ |  | $2$ | $2$ | $2\times 10^{3}$ | – | –
Max. p Lorentz factor | $\gamma_{\rm p,max}$ |  | $2\times 10^{10}$ | $2\times 10^{10}$ | $1\times 10^{9}$ | – | –
p spectral index | $s_{p}$ |  | $1.5$ | $1.5$ | $1.5$ | – | –
AD Eddington ratio | $\eta_{\rm Edd}$ |  | – | $3$ | – | – | $30$
BLR luminosity | $L_{\rm BLR}^{\prime}$ | [erg/s] | $5\times 10^{44}$ | – | – | $1.5\times 10^{46}$ | $3\times 10^{46}$
BLR Temperature | $T_{\rm BLR}^{\prime}$ | [K] | $2\times 10^{4}$ | – | – | $9\times 10^{4}$ | $1\times 10^{4}$
BLR radius | $R_{\rm BLR}^{\prime}$ | [cm] | $3\times 10^{18}$ | – | – | $3\times 10^{18}$ | $3\times 10^{18}$
DT luminosity | $L_{\rm DT}^{\prime}$ | [erg/s] | $8\times 10^{46}$ | – | – | – | $5\times 10^{46}$
DT Temperature | $T_{\rm DT}^{\prime}$ | [K] | $1\times 10^{3}$ | – | – | – | $3\times 10^{3}$
DT radius | $R_{\rm DT}^{\prime}$ | [cm] | $3\times 10^{18}$ | – | – | – | $3\times 10^{18}$

Figure 7: Small parameter study using OneHaLe. The black lines correspond in
style to those in Figs. 2 to 6, but for the case $\eta_{\rm esc}\alpha=0.5$.
The solid colored lines show simulations varying the input parameter as
labeled. In the bottom panels, the variation in the electron parameters
(magenta lines) preserves the initial injection rate of $\sim 1\,$cm-3s-1.

We use the OneHaLe code (Zacharias 2021; Zacharias et al. 2022, see the
appendix for a brief description) to model the time-dependent evolution of the
blob. In order to comply with some of the assumptions above, we have to
slightly tweak the code. The $\gamma$-$\gamma$ pair production injection rate
is given by (Aharonian et al. 1983; Cerruti et al. 2021)

$\displaystyle Q_{\rm\gamma\gamma}(\gamma)$
$\displaystyle=2\frac{3\sigma_{T}c}{32}\int\limits_{\gamma}^{\infty}\,\mathrm{d}\epsilon\,\frac{n_{\rm
ph}(\epsilon)}{\epsilon^{3}}\int\limits_{\frac{\epsilon}{4\gamma(\epsilon-\gamma)}}^{1}\,\mathrm{d}\tilde{\epsilon}\,\frac{n_{\rm
ph}(\tilde{\epsilon})}{\tilde{\epsilon}^{2}}$
$\displaystyle\quad\times\left[\frac{4\epsilon^{2}}{\gamma(\epsilon-\gamma)}\ln{\left(\frac{4\gamma\tilde{\epsilon}(\epsilon-\gamma)}{\epsilon}\right)}-8\epsilon\tilde{\epsilon}\right.$
$\displaystyle\quad+\left.\frac{2\epsilon^{2}(2\epsilon\tilde{\epsilon}-1)}{\gamma(\epsilon-\gamma)}-\left(1-\frac{1}{\epsilon\tilde{\epsilon}}\right)\left(\frac{\epsilon^{2}}{\gamma(\epsilon-\gamma)}\right)^{2}\right],$
(22)

with the photon distribution $n_{\rm ph}$, the photon energies $\epsilon$ and
$\tilde{\epsilon}$ normalized to the electron rest energy, and the electron
Lorentz factor $\gamma$. Eq. (22) requires $\epsilon\gg 1/\tilde{\epsilon}$.
This and $\tilde{\epsilon}<1$ ensure an absorber with low photon energies for
the $\gamma$ rays.

In order to suppress the severe cooling — due to the extreme parameter
settings, see Tab. 1 — we switch off electron IC cooling. While this is a
massive change, it is necessary to create sufficient synchrotron or IC
photons. Furthermore, the magnetic field is kept constant throughout these
simulations. Lastly, as we are only interested in the additional injection of
pairs from $\gamma$-$\gamma$ processes, the secondary injections from muon
decay and Bethe-Heitler pair production have also been disabled. The input
parameters for the simulations are given in Tab. 1, while the results are
shown as solid lines in Figs. 2 to 6. We emphasize that these parameters are
not chosen for realism but merely to reproduce as close as possible the semi-
analytical models.

In this regard, the external fields labeled “BLR” and “DT” should just be
considered as two separate external fields without any intended resemblance to
actual BLR and DT photon fields. In case 1, the external fields act as
absorbers. In case 3 they are needed for the IC process, while the absorption
is done by electron-synchrotron emission.

Cases 1 and 3 can be reproduced reasonably well. In case 1, the isotropic
photon field is made of two thermal fields with different temperature and
luminosity, while in the corresponding case in case 3 only one thermal field
is needed. In the latter case, the pile-up close to the edge of the external
field is present, but not as strong as in the analytical model222This is
probably due to the resolution of the time steps in the code. Given the
steepness of the analytical pile-up, the logarithmic time-stepping in the
simulation does not cover enough time steps to fully recreate the pile-up..
The accretion disk field is represented by a Shakura-Sunyaev disk Shakura &
Sunyaev (1973) with luminosity $L_{\rm AD}=\eta_{\rm Edd}L_{\rm Edd}$. In case
3, the AD field absorbs IC emission scattering both isotropic fields (but not
the AD field).

We have found no parameter set that could reproduce the semi-analytical result
in case 2. The simulations with OneHaLe with the parameters given in Tab. 1
show a pronounced maximum shortly after $t_{\rm esc}(0)$ followed by a
decrease depending on the opening angle. This seems reasonable, as the dashed
lines roughly correspond to the evolution of the $\gamma$ rays
(p-synchrotron). In turn, the number of “absorbees” (ie., $\gamma$ rays)
quickly reduces. The long evolution suggested by the semi-analytical model
would only be achievable if the absorber were to increase in such a way that
progressively a higher fraction of the $\gamma$ rays were absorbed. Then, at
late times, practically all $\gamma$ rays would be absorbed to keep the
cascade going. This is implausible and marks the limit of the simple
analytical model that ignores all the energy-dependencies of the cross-
section.

The influence of the parameters on the simulations is shown in Fig. 7, where 3
variations for each case have been derived. Clearly, reproducing the
analytical model required a specific set of parameters. This also shows the
limits of the approach. It’s nonetheless reassuring that 2 out of 3 cases have
been reproduced.

###  2.4 Interlude

The previous sections have shown the influence of the expansion of the moving
blob on the development of the pair cascade. The effects are two-fold.
Firstly, a significantly higher density can be achieved through the pair
production compared to no secondaries. Secondly, a larger opening angle can
drag out the peak of the density evolution. While the densities for the
standard solution peak around $t_{\rm esc}(0)$, the curves with pair injection
typically peak later than this time scale. The exception to this rule are
secondary injections involving AD photons, because of the time scale $R_{\rm
AD}/\Gamma/c<t_{\rm esc}(0)$. For small values of $z_{i}$, this could become
the case as well for the isotropic photon field. For internal photon fields
(case 2), however, the peak is always attained after $t_{\rm esc}(0)$, as the
internal fields only reach their maximum at this time, and the developing
cascade reinforces the interaction. Non-linear cascades, which we have not
treated here, might increase this effect, as both absorber and absorbee are
increased at later times.

The dotted lines in Figs. 2 to 6 mark the peak density of the standard
solution, and indicate at which time the models with pair injection drop below
this level. It typically happens at least an order of magnitude after $t_{\rm
esc}(0)$. This, of course, depends strongly on the amount of pairs produced,
the parameters, and the opening angle. This indicates the importance of pair
production and the geometry of the jet for the development and duration of a
flux high state.

Naturally, the inclusion of the energy dependency on the particle
distribution, as well as the pair production process can have a significant
influence. In the absence of efficient reacceleration of particles, the
particle cooling will lead to a drop in the pair production process, as less
energetic $\gamma$-ray photons will be produced. The adiabatic expansion of
the blob will also cause a reduction in the energy density of the particles
and the photon fields, similarly reducing the pair production. Hence, the time
evolution might be shorter than envisaged in the present semi-analytical
model.

##  3 Modeling based on PKS 1510-089

Table 2: OneHaLe input parameters for the PKS 1510-089 simulations. The steady-state simulation uses a stationary emission region. Parameters below the horizontal line are used in each simulation.  Symbol | steady-state | leptonic | hadronic
---|---|---|---
$z_{0}^{\prime}$ | [cm] | $7\times 10^{17}$ | $1\times 10^{15}$ | $1\times 10^{15}$
$R_{0}$ | [cm] | $7\times 10^{16}$ | $5\times 10^{14}$ | $5\times 10^{14}$
$\eta_{\rm esc}$ |  | $10$ | $3$ | $3$
$p$ |  | – | $2$ | $2$
$\delta$ |  | $22$ | $22$ | $22$
$B_{0}$ | [G] | $0.5$ | $20$ | $20$
$L_{e}$ | [erg/s] | $2.2\times 10^{43}$ | $1.5\times 10^{43}$ | $1.5\times 10^{43}$
$\gamma_{\rm e,min}$ |  | $900$ | $900$ | $900$
$\gamma_{\rm e,max}$ |  | $1\times 10^{5}$ | $1\times 10^{5}$ | $1\times 10^{5}$
$s_{e}$ |  | $3.4$ | $3.4$ | $3.4$
$L_{\rm p}$ | [erg/s] | – | – | $1.5\times 10^{43}$
$\gamma_{\rm p,min}$ |  | – | – | $2$
$\gamma_{\rm p,max}$ |  | – | – | $1\times 10^{9}$
$s_{p}$ |  | – | – | $2.2$
$M_{\rm BH}$ | [$M_{\odot}$] | $1.6\times 10^{8}$
$\eta_{\rm Edd}$ |  | $0.25$
$L_{\rm BLR}^{\prime}$ | [erg/s] | $5\times 10^{44}$
$T_{\rm BLR}^{\prime}$ | [K] | $1.1\times 10^{5}$
$R_{\rm BLR}^{\prime}$ | [cm] | $1.2\times 10^{17}$
$L_{\rm DT}^{\prime}$ | [erg/s] | $1\times 10^{45}$
$T_{\rm DT}^{\prime}$ | [K] | $1.8\times 10^{3}$
$R_{\rm DT}^{\prime}$ | [cm] | $1.94\times 10^{18}$
Figure 8: Model light curves based upon parameters describing PKS 1510-089 for
VHE $\gamma$ rays (top row), HE $\gamma$ rays (2nd row), X-rays (3rd row) and
the optical R-band (bottom row) for purely leptonic processes (left column)
and including hadronic processes (right column). Displayed models are either
with (thick dashed) or without (solid) $\gamma$-$\gamma$ pair production.
Colors are for various opening angles as labeled. The vertical lines mark
$t_{\rm esc}(0)$ (red), the crossing of the BLR edge (blue) and of the DT edge
(magenta); all times in the observer’s frame. For clarity, the launch of the
blob takes place at $t^{obs}=0.1\,$d.

The flat spectrum radio quasar (FSRQ) PKS 1510-089, located at a redshift of
$0.361$, is one of the most famous blazars. It was detected at very-high-
energy $\gamma$ rays with H.E.S.S. in 2009 (H. E. S. S. Collaboration et al.
2013), and has received significant attention since (e.g., Ahnen et al. 2017;
H.E.S.S. Collaboration et al. 2021). Modeling typically requires both the BLR
and DT photon fields (Barnacka et al. 2014), or multiple zones (Nalewajko et
al. 2012; Prince et al. 2019).

While no modeling of the source or a specific light curve is attempted here,
we use this source to study the impact of the pair cascade in a moving,
expanding blob to enhance flaring events in a realistic setting. The modeling
parameters are given in Tab. 2 and are based upon the works by Nalewajko et
al. (2012) and Barnacka et al. (2014). We derive multiple modelings: 1) a
steady-state model similar to the aforementioned works in order to provide a
baseline flux that may describe PKS 1510-089 on average; 2) a moving blob
using only leptonic processes similar to Saito et al. (2015); and 3) a moving
blob including hadronic interactions. For the moving blobs we again derive
models for various opening angles as before. We also derive specifically
models without $\gamma$-$\gamma$ pair production to show the difference. For
the moving-blob models 2) and 3) the launching position is chosen relatively
close to the black hole, where the jet has not expanded much (Zacharias et al.
2022). At this position the blob is still small in order to fit into the jet,
and its magnetic field is high compared to the steady-state one-zone model,
which is located further down in the jet.

One important addition to the simulations of the previous section is the
consideration of magnetic flux conservation. That is, $B(z)=B_{0}\,R_{0}/R(z)$
assuming a dominating toroidal guide field (Kaiser 2006). This also implies
that the larger the opening angle, the faster the magnetic field drops. This
may have a significant influence on the synchrotron emission compared to the
expectation from the simple model above.

###  3.1 Light curves

The multi-wavelength light curves are shown in Fig. 8. The wavebands are the
very-high-energy $\gamma$-ray (VHE) band ($E>30\,$GeV333With this threshold
definition, the EBL absorption plays a minor role and we ignore it.,
corresponding to the energy band of the forthcoming CTA observatory), the
high-energy $\gamma$-ray (HE) band ($100\,$MeV$<E<100\,$GeV, corresponding to
the energy band of the Fermi-LAT), the X-ray band ($2-10\,$keV, corresponding
to the range of Swift-XRT), and the optical R-band. It is immediately obvious
that the flare evolution depends strongly on the wavelength and the presence
of relativistic protons.

The models without pair production (solid lines) do not show major differences
between the leptonic and hadronic simulations. Except for the VHE band, which
is fully absorbed at early times, all light curves rise quickly to a maximum
at the initial escape time scale $t_{\rm esc}(0)$, followed by a subsequent
decay. The switch from IC/BLR cooling to IC/DT or synchrotron cooling during
the crossing of the BLR edge444The crossing of the BLR edge after only $\sim
0.1\,$d in the observer’s frame is a consequence of the relativistic motion of
the blob. The fore-shortening of the external radiation fields in the blob
frame, Eq. (1), and the boosting of the blob emission into the observer’s
frame result in $\Delta t^{\rm obs}=\Delta
z^{\prime}/(\delta\Gamma\beta_{\Gamma}c)$. results in a minor bump in the
X-rays and the R-band, which is absent in the HE band. The VHE light curve
shows a minor bump at this point, as the environment is now optically thin to
VHE photons. Except for the HE band, the decay of the light curve is faster
for larger opening angles, which is a consequence of the diminishing magnetic
field, and hence a faster decay of the synchrotron emission. One should note
here that the X-ray emission from the blob is also influenced by electron-
synchrotron emission, as the (initial) high magnetic field shifts the
synchrotron peak to higher energies compared to the steady-state model. As the
HE band is governed by inverse-Compton emission of external photon fields, the
magnetic field has basically no effect, and the decay of the light curve is
fully governed by the evolution of the electron density. In the hadronic model
and for large opening angles, the VHE light curve shows an additional bump
around the time of the DT-edge crossing. This radiation is electron-
synchrotron emission from secondary particles injected from muon decay.

Inclusion of the pair production has almost no consequence for the VHE band,
except for hadronic simulations of large opening angles, where the DT-edge-
crossing bump is much more pronounced and shows a plateau phase lasting for a
few hours. At that time, a peak also shows up in the X-ray domain for large
opening angles. It is induced by pair production, and is also visible –
although less pronounced – in the leptonic simulations. The peak flux in the
main flare is significantly higher compared to the case without pair
production. Similarly, the HE band exhibits higher peak fluxes compared to the
solid lines. In both bands, HE and X-rays, the peak fluxes are much more
pronounced in the hadronic simulations indicating the significant impact of
relativistic protons on the pair production. The R-band shows no change
between pair or no pair production implying that pairs are produced
predominantly at higher energies. While cooling will reduce the energy of the
secondary pairs, their density is insufficient to add to the synchrotron
emission of the primaries in the R-band.

Figure 9: Same as Fig. 8, but for the radio band at 15 GHz.

The most important effects in the light curves shown in Fig. 8 take place
within a few days in the observer’s frame. Figure 9 displays the first 100 d
in the evolution of the radio band at 15 GHz. The rise of the light curve
begins after a few days, when the flares in the other bands have already
ceased (see also Boula & Mastichiadis 2022; Tramacere et al. 2022). For small
opening angles, the radio flare diminishes within 50 d, while for large
opening angles the light curve keeps rising. While not shown, for $\eta_{\rm
esc}\alpha=0.9$ the peak is reached only after more than 1000 d. There is
barely a difference between the cases with and without pair production for the
same reason as in the R-band.

###  3.2 Luminosity evolution

Figure 10: Electron luminosity (top row), proton luminosity (2nd row),
radiation luminosity (3rd row), and total luminosity (bottom row) as a
function of time for pure electron (left column) and electron-proton
simulations (right column) for different opening angles (color code as
labeled) using the parameters of PKS 1510-089 as in Fig. 8. Simulations
without pair production are marked by solid lines, while dashed lines refer to
simulations with pair production. The vertical lines have the same meaning as
in Fig. 8. The black dot-dashed and black dash-double-dotted lines mark the
Eddington and AD luminosities, respectively. Note the logarithmic time axis.

An important measure for any blazar model is the luminosity of the emission
region. Power is contained in particles, magnetic field, and radiation. The
evolution of the luminosities as a function of time is shown in Fig. 10. By
design, the magnetic field luminosity is constant with a value of $L_{B}^{\rm
obs}=1.8\times 10^{44}\,$erg/s, which is why we do not display it in Fig. 10.
The total luminosity in the bottom row of that figure is the sum of the
electron, proton, magnetic field, and radiation luminosities. The individual
luminosities are calculated from the respective energy densities in the
comoving frame $u_{i}(t)$ according to $L_{i}^{\rm
obs}(t)=u_{i}(t)\Gamma^{2}\pi R(t)^{2}c$ deriving the bulk Lorentz factor
$\Gamma$ from the assumption $\Gamma=\delta$.

For small opening angles (black and blue lines), the respective luminosities
quickly rise to the maximum as the source is being filled with particles and
radiation, and then decrease. In turn, the total luminosity reaches a bottom
value which is given by the constant magnetic field luminosity. It also
implies that the initial particle and radiative powers are much larger than
the magnetic field power. For $\eta_{\rm esc}\alpha=0.5$, the particle powers
reach a constant level implying that the injection and escape of particles is
fully countered by the expansion. In the presence of protons (right column in
Fig. 10), the electrons slowly continue to rise in this case if pair
production is enabled (magenta dashed line). This implies continuous pair
creation even at very late times. For large opening angles (orange and yellow
lines), the particle luminosities keep rising, as the rapid expansion does not
allow for a meaningful escape of particles. The radiative luminosity in this
case continues to decrease initially, but reverses this trend for late time,
as the continuing accumulation of particles also increases the amount of
photons, which also remain longer and longer in the emission region.

Interestingly, the total luminosity is initially dominated by radiation, and
only after about 1 d in the observer’s frame does it change to particle or
magnetic field dominance. It implies that transfer of power into radiation is
initially very efficient, and that the particles are efficiently cooled. With
the decrease in the magnetic field strength, and the decrease of density of
internal and external photon fields due to expansion and motion, the radiation
production becomes less efficient over time and its luminosity drops more
rapidly than that of the particles.

In all cases, the comparison between particles and magnetic fields suggests
that the emission region is initially particle dominated. For large opening
angles, this is maintained throughout, while for small opening angles the
magnetic field dominates the particles at late times. This is contrary to most
models, where an initial magnetic field dominance is expected which changes to
particle dominance later (e.g., Zacharias et al. 2022, and references
therein). The reason is that we chose a relatively modest initial magnetic
field of only 20 G along with a linear decrease with radius. If the magnetic
field were initially mostly poloidal, one would expect a quadratic decrease
with radius (Kaiser 2006), and a much faster decrease. In this case, the
magnetic power would also drop as a function of time. While our initial value
is in line with the measurement in M 87 (Event Horizon Telescope Collaboration
et al. 2021), theoretical works suggest much larger values along with a rapid
decrease (Zdziarski et al. 2022). Testing such scenarios is left for future
work.

The black horizontal lines in Fig. 10 mark the Eddington luminosity and the AD
luminosity, respectively. As the accretion process fuels the jet, the jet
luminosity must be compared with these values. Apparently, all models remain
initially close to or below the limit of the Eddington luminosity. In case of
protons and pair production (right column, dashed lines), the Eddington
luminosity is initially surpassed by a factor 5. This is, however, due to the
large radiative power, which includes also the IC scattering of the external
fields. While the BLR and the DT are less luminous than the AD, they are
strongly beamed in the comoving frame. Thus, the short excess of the Eddington
luminosity can be expected, and is within reasonable bounds. The continued
increase of jet power for large opening angles has already been discussed, and
naturally surpasses the Eddington limit by far. Clearly, the model is not
realistic in these cases.

##  4 Conclusions

In this paper we have discussed the particle evolution within a moving,
expanding emission region within the jet of a blazar (Boula & Mastichiadis
2022; Tramacere et al. 2022). With a simple semi-analytical model, we have
shown that the opening angle has an important influence on the particle
distribution and the electron-positron cascade. With increasing opening angle
the high-state (including the cascade) may last much longer owing to the
increased escape time scale for both particles and photons.

With the help of the OneHaLe code (Zacharias 2021; Zacharias et al. 2022), we
have shown that the simple model can be reproduced for linear cascades induced
by external photon fields, while the linear cascade resulting from internal
processes was not reproduced successfully. Naturally, due to the neglect of
the energy dependencies of the particle evolution and the cascade, it required
quite specific, and extreme parameters in the simulation. Nonetheless, the
exercise has demonstrated the importance of the opening angle on the evolution
of the particle distribution.

The simple model may be improved by considering delta-function approximations
to the energy dependency of the particle distribution and the cascade. This
may provide a more reasonable view on the evolution providing a broader range
of application. These calculations are left to the interested reader.

In order to obtain realistic simulations, we have made use of parameters from
PKS 1510-089 (Nalewajko et al. 2012; Barnacka et al. 2014). The leptonic and
hadro-leptonic simulations were conducted for various opening angles. From the
resulting light curves in Fig. 8 we can derive the following conclusions. A
moving blob induces a rapid flare. The evolution depends critically on the
blob’s speed and on the initial size at launch. Observations in the X-ray and
VHE $\gamma$-ray are crucial to determine the opening angle and the particle
composition, as for large opening angles these bands may (or may not) show a
delayed flare when the blob moves out of the DT. These secondary flares are
significantly influenced by the pair production process and are more
pronounced if a hard and highly energetic proton distribution is present.
Interestingly, the optical band is also useful to obtain information on the
opening angle, as a small opening angle induces a longer decay phase due to
the slower decay of the magnetic field. In the HE $\gamma$-ray band, the flare
is bright and short with barely a difference between the various opening
angles. Importantly, the HE band peaks slightly before the other bands (cf.
the red vertical line in Fig. 8), whereas the X-ray and R-band peak at the
time corresponding to $t_{\rm esc}(0)$. In all bands, the (primary) flare is
asymmetric with a fast rise and a slow decay (Saito et al. 2015; Boula &
Mastichiadis 2022; Tramacere et al. 2022).

Truly simultaneous and high-cadence observations are paramount to obtain all
the potential information. The main flare lasts just for a few hours, and the
subtle differences between peak times and decay profiles might be hidden in
the statistical errors or a delay in observation. It should also be noted that
it is worthwhile to continue observing for a few days in the X-ray and VHE
$\gamma$-ray band to search for the secondary flare.

Radio observations may also provide indications for the opening angle. Large
values of $\eta_{\rm esc}\alpha$ result in years-long variations, while small
opening angles result in flares lasting a few tens of days. Single-dish radio
observations every few days for up to 100 d after the event at higher energies
should provide sufficient information on the opening angle. The downside could
be the higher chance of multiple events superposing each other.

The jet luminosities are mostly within reasonable bounds compared to the
Eddington luminosity. However, for large opening angles, the particle
luminosities keep increasing due to the lack of escape, which results in
unreasonable jet powers. Additionally, the magnetic field evolution is such
that its power remains constant. In turn, the initially partical-dominated
emission region becomes magnetic-field-dominated at later times and small
opening angles. This is contrary to standard expectations. A scenario, which
is more in line with the usual jet evolution, will be discussed elsewhere.

The above mentioned evolutionary details of the flare depend on the type of
object. The given results are based on parameters describing the FSRQ PKS
1510-089. In BL Lac objects with much weaker or even absent external fields,
the effects induced from the pair cascade are probably much less pronounced
(Zacharias et al. 2022). In this case, the solid lines in Fig. 8 might already
be an adequate description, and the secondary flares in the X-ray and VHE
$\gamma$-ray band are absent. Similarly, the details of the magnetic field
decay may have a significant influence on the evolution of the flux especially
in the R-band, and needs to be discerned from the expansion profile.

Lastly, using the standard one-zone description implies that at all times an
instantaneous particle spectrum was used. While shock acceleration can quickly
accelerate particles (Böttcher & Baring 2019), it would be useful to properly
include the particle acceleration in this model as it might have an important
influence on the light curve evolution at early, but also at late times. At
early times, the delayed injection of high-energetic particles would slow down
the flare evolution, and might also reduce the strength of the cascade
development. At late times, the weakened magnetic field might not be able to
accelerate particles to the highest energies, which might reduce or even
inhibit the development of the secondary flare. It would also substantially
reduce the particle luminosities. In order to adequately account for these
effects a two-zone model is needed (Weidinger & Spanier 2015; Chen et al.
2015; Dmytriiev et al. 2021), which is beyond the scope of this paper.

## Acknowledgement

I would like to thank Andreas Zech, Markus Böttcher, and Anton Dmytriiev for
fruit- and beerful discussions on the content of the manuscript. I extend my
gratitude to the anonymous referee for valuable comments that helped to
improve the manuscript. MZ acknowledges postdoctoral financial support from
LUTH, Observatoire de Paris. Simulations for this work have been performed on
the TAU-cluster of the Centre for Space Research at North-West University,
Potchesftroom, South Africa. The OneHaLe code is available upon request to the
author.

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##  Appendix A The OneHaLe code

The OneHaLe code is a time-dependent, one-zone, hadro-leptonic radiation code
that solves the Fokker-Planck equation of the particle distribution for
protons, charged pions, muons and electrons (including positrons). In each
time step, the radiation transport equation is solved allowing for the direct
feedback of the particle and photon interactions. The Fokker-Planck equation
for the particle distribution of species $i$ is

$\displaystyle\frac{\,\partial n_{i}(\chi,t)\,}{\,\partial t\,}$
$\displaystyle=\frac{\,\partial\,}{\,\partial\chi\,}\left[\frac{\chi^{2}}{(a+2)t_{\rm
acc}}\frac{\,\partial n_{i}(\chi,t)\,}{\,\partial\chi\,}\right]$
$\displaystyle-\frac{\,\partial\,}{\,\partial\chi\,}\left(\dot{\chi}_{i}n_{i}(\chi,t)\right)+Q_{i}(\chi,t)-\frac{n_{i}(\chi,t)}{t_{\rm
esc}}-\frac{n_{i}(\chi,t)}{\gamma t^{\ast}_{i,{\rm decay}}}.$ (23)

The distributions are given as a function of normalized momentum
$\chi=\gamma\beta$, with the particle Lorentz factor $\gamma$ and its
corresponding speed $\beta$ normalized to the speed of light. The first term
on the right-hand-side is momentum diffusion representing Fermi-II
acceleration using hard-sphere scattering with the ratio $a$ of shock to
Alfvèn speed. The second term marks continuous momentum gains and losses.
Gains are achieved through Fermi-I acceleration, while losses depend on the
particle species and include synchrotron, adiabatic, Bethe-Heitler, pion
production, and inverse-Compton processes. The third term marks the injection
term, while the forth term marks the catastrophic escape of particles. The
last term is the decay term for unstable particles.

The acceleration time scale is given as a multiple of the escape time scale:
$t_{\rm acc}=\eta_{\rm acc}t_{\rm esc}$. It merely marks the reacceleration of
particles in the acceleration zone, and does not provide “first-principle”
acceleration, which typically requires a much smaller zone (e.g. Weidinger &
Spanier 2015; Chen et al. 2015; Dmytriiev et al. 2021). The initial
acceleration is mimicked through the primary injection term $Q_{i}$, which for
protons and electrons takes the form of a power-law between minimum and
maximum Lorentz factors, $\gamma_{\rm i,min}$ and $\gamma_{\rm i,max}$,
respectively, with spectral index $s_{i}$. The injection of pions and muons is
directly calculated from the respective interactions (using the template
approach of Hümmer et al. 2010) and decays. Secondary electrons are injected
from muon decay, Bethe-Heitler pair production, and $\gamma$-$\gamma$ pair
production.

Further details are given in Zacharias (2021) and Zacharias et al. (2022).
These references provide the gory details about the particle and photon
integrals solved in the code.

###  A.1 Acceleration and cooling time scales

In this section we briefly discuss the influence of the expanding blob
scenario on the acceleration and cooling time scales for illustrative
purposes. These are not necessarily considered in the discussion and
simulations of the main part.

For the (re-)acceleration of particles, OneHaLe includes acceleration terms
based on hard-sphere scattering. This is parameterized by an energy-
independent time scale $t_{\rm acc}(t)=\eta_{\rm acc}t_{\rm esc}(t)$, where
$\eta_{\rm acc}$ is a free parameter. Hence,

$\displaystyle t_{\rm acc}(t)=\eta_{\rm acc}t_{\rm
esc}(0)\left(1+\frac{\eta_{\rm esc}\alpha}{t_{\rm esc}(0)}t\right).$ (24)

The cooling of particles also becomes time-dependent, as several important
variables are a function of radius. For instance, with the conservation of
magnetic flux, the magnetic field $B(t)$ can be written as

$\displaystyle B(t)=\frac{B_{0}}{\left(1+\frac{\eta_{\rm esc}\alpha}{t_{\rm
esc}(0)}t\right)^{b}},$ (25)

with the initial magnetic field $B_{0}$, and the power $b\in[1,2]$ of the
radial dependence ($b=1$ for a toroidal guide field, $b=2$ for poloidal guide
field, Kaiser 2006). The synchrotron cooling time scale then becomes:

$\displaystyle t_{\rm syn}(\gamma,t)=\frac{6\pi
m_{e}c^{2}}{c\sigma_{T}B_{0}^{2}}\left(\frac{m}{m_{e}}\right)^{3}\gamma^{-1}\left(1+\frac{\eta_{\rm
esc}\alpha}{t_{\rm esc}(0)}t\right)^{2b},$ (26)

with the particle Lorentz factor $\gamma$, the rest mass of the electron
$m_{e}$, the rest mass of the particle $m$ ($m_{p}$ for protons), and the
Thomson cross section $\sigma_{T}$.

Following the prescription of Schlickeiser (2009), the SSC cooling time scale
in the Thomson regime can be written as

$\displaystyle t_{\rm
ssc}(\gamma,t)=\frac{m_{e}c^{2}}{3c_{1}c_{2}^{2}\sigma_{T}P_{0}R_{0}B_{0}^{2}}\gamma^{-1}F(t)^{-1}\left(1+\frac{\eta_{\rm
esc}\alpha}{t_{\rm esc}(0)}t\right)^{2b-1},$ (27)

with $c_{1}=0.684$, $c_{2}=1.856\times 10^{-20}\,$erg1/2cm3/2, and
$P_{0}=2\times 10^{24}\,$erg-1s-1. The function

$\displaystyle
F(t)=\int\limits_{0}^{\infty}\gamma^{2}n_{e}(\gamma,t)\,\mathrm{d}\gamma\,$
(28)

depends on the instantaneous electron distribution function $n_{e}(\gamma,t)$,
which may have a complicated time-dependency resulting in a non-linear
evolution of the particle distribution (Zacharias & Schlickeiser 2012). If we
assume that the energy part of the particle energy distribution is time-
independent, the time-dependency derived above for the particle distribution,
Eq. (7), can be included here (cf. Fig. 11).

The expansion of the blob results in adiabatic cooling with time scale

$\displaystyle t_{\rm
adi}(\gamma,t)=\frac{\Gamma}{3(1-\gamma^{-2})}\frac{t_{\rm esc}(0)}{\eta_{\rm
esc}\alpha}\left(1+\frac{\eta_{\rm esc}\alpha}{t_{\rm esc}(0)}t\right),$ (29)

where we followed the prescription of Zdziarski et al. (2014).

As we assume $\Gamma=\mbox{const.}$, the cooling due to external isotropic
photon fields such as the BLR is constant in time, as long as the blob is
within the distance of the external field:

$\displaystyle t_{\rm
ext}(\gamma,t)=\frac{m_{e}c^{2}}{c\sigma_{T}\Gamma^{2}u_{\rm
ext}^{\prime}}\gamma^{-1}\;H\left[\frac{z_{\rm ext}^{\prime}}{\Gamma
c}-t\right],$ (30)

with the energy density of the external photon field in the black hole frame
$u_{\rm ext}^{\prime}$, and the radius of the external photon field $z_{\rm
ext}^{\prime}$. We approximated again $\beta_{\Gamma}\approx 1$.

Figure 11: Time scales in arbitrary units as labeled versus time in the
comoving frame. The lines are indicative of the time-dependency. The vertical
red line marks $t_{\rm esc}(0)$. Free parameters are $\eta_{\rm
esc}\alpha=0.5$, $\eta_{\rm esc}=3$, $R_{0}=5\times 10^{15}\,$cm, $b=1$,
$p=2$, $\Gamma=10$ and $z_{\rm ext}^{\prime}=3\times 10^{18}\,$cm.

Figure 11 shows the evolution with time of the cooling and escape time scale.
The adiabatic and acceleration (not shown) time scales behave exactly like the
escape time scale, while the external-Compton cooling simply stops once the
blob leaves the respective region. The synchrotron cooling time scale (with
$b=1$) increases rapidly for $t>t_{\rm esc}(0)$. The SSC cooling time scale
(including the time-dependency of Eq. (7)) initially decreases due to the
increase in particle density. This reverses at $t\sim t_{\rm esc}(0)$, as the
maximum density has been passed and particles overall leave the emission
region.