Short-time Variability of Blazars via Non-linear, Time-dependent ...

Short-time Variability of Blazars via Non-linear, Time-dependent Synchrotron-Self Compton Radiative Losses Christian R¨oken∗ Universit¨ at Regensburg, Fakult¨ at f¨ ur Mathematik, 93040 Regensburg, Germany

Katharina Proksch Georg-August-Universit¨ at G¨ ottingen, Institut f¨ ur Mathematische Stochastik, 37077 G¨ ottingen, Germany

Sebastian Sch¨oneberg and Florian Schuppan Ruhr-Universit¨ at Bochum, Institut f¨ ur Theoretische Physik, Lehrstuhl IV: Weltraum- und Astrophysik, 44780 Bochum, Germany (Dated: September 2016) A leptonic one-zone model accounting for the radiation emission of blazars is presented. This model describes multiple successive injections of mono-energetic, ultra-relativistic, interacting electron populations, which are subjected to synchrotron and synchrotron-self Compton radiative losses. The electron number density is computed analytically by solving a time-dependent, relativistic transport equation. Moreover, the synchrotron and synchrotron-self Compton intensities as well as the corresponding total fluences are explicitly calculated. The lightcurves and fluences are plotted for realistic parameter values, showing that the model can simultaneously explain both the specific short-time variability in the flaring of blazars and the characteristic broad-band fluence behavior.

arXiv:1609.00941v1 [astro-ph.HE] 4 Sep 2016

ABSTRACT.

Contents

I. Introduction II. The Relativistic Transport Equation A. Formal Solution of the Relativistic Transport Equation  B. Solution of the Relativistic Transport Equation for t ∈ R≥0 | tj ≤ t < tj+1 with j ∈ {1, ..., m} 1. NID Asymptotics 2. FID Asymptotics C. Solution of the Relativistic Transport Equation for {t ∈ R≥0 }

2 3 3 5 6 7 9

III. Synchrotron and SSC Intensities A. Synchrotron Intensity B. SSC Intensity

11 11 12

IV. Short-time Variability and Broad-band Fluence of Blazars A. Numerical Implementation B. Lightcurves and Fluences

13 13 14

V. Summary and Conclusions

15

A. Laplace Method

16

B. NID-FID Transition Time

17

C. Constants of Integration – Initial and Transition Conditions and Updating

18

D. Components of G(t | 0 ≤ t < ∞) and Initial Values Gi

19

E. The CS-Function

20

∗

e-mail: [email protected]

2 F. Lorentz Transformation

21

G. Synchrotron and SSC Fluences 1. Synchrotron Fluence 2. SSC Fluence

22 22 24

Acknowledgments

24

References

25

I.

INTRODUCTION

Blazars are among the most energetic phenomena in nature, representing the most extreme type in the class of active galactic nuclei [36]. Their main jet outflow in form of magnetized plasmoids, which are assumed to arise in the Blandford-Znajek and Blandford-Payne processes [6, 7], constitutes the major radiation zones. These plasmoids pick up – and interact with – particles of interstellar and intergalactic clouds along their trajectories [26], giving rise to the emission of a series of strong flares. A blazar spectrum consists of two broad non-thermal radiation components in different energy domains. The low-energy spectral component, ranging from radio to optical or X-ray energies, is usually attributed to synchrotron radiation of relativistic electrons interacting with ambient magnetic fields. The origin of the highenergy spectral component, covering the X-ray to γ-ray regime, is still under debate. It can be modeled by inverse Compton radiation coming from low-energy photon fields that interact with the relativistic electrons [10]. This process can be described either by a synchrotron-self Compton (SSC) model, where the electrons scatter their self-generated synchrotron photons, or by so-called external Compton models, where the seed photons are generated in the accretion disk [16], the broad-line region [34], or the dust torus [8] of the central black hole. Instead of such a leptonic scenario, the high-energy component can also be modeled by hadronic scenarios via proton-synchrotron radiation or the emission of secondary products of neutral pions formed in interactions of protons with ambient matter, which then decay into two γ-rays [11, 13, 37]. Mixed models including leptonic as well as hadronic components are also considered in the literature (see, e.g., [12, 38]). The main feature of the non-thermal blazar emission is a distinct variability at all frequencies with different variability time scales ranging from years down to a few minutes. The shortest variability time scales are usually observed for the highest energies of the spectral components, as in PKS 2155-304 [2, 3] and Mrk 501 [4] in the TeV range, or Mrk 421 [15] in the X-ray domain. So far, mostly multi-zone models explaining the multifrequency short-time variability of blazars have been proposed (see, e.g., [5, 18–20, 25, 35]). These require an internal structure of the source that is caused by collisions of moving and stationary shock waves, which lead to the formation of various radiation zones in the emission region. Moreover, they employ synchrotron and inverse Compton radiative losses as well as finite light-travel times. In the framework of a one-zone model, the variability phenomenon with time scales reaching from years down to months was studied in [24] using purely numerical methods. Here, we propose a fully analytic leptonic one-zone model with multiple interacting electron populations and non-linear, time-dependent SSC radiative losses (for a previous work, see [29]). This model explains the specific short-time variability down to the minute time scale and the typical shape of the broad-band fluence. More precisely, we assume that the blazar radiation emission originates in spherically shaped and fully ionized plasmoids, which feature intrinsic randomly-oriented, but constant, large-scale magnetic fields and propagate relativistically along the general direction of the jet axis. These plasmoids pass through – and interact with – clouds of the interstellar and intergalactic media, successively and instantaneously picking up multiple monoenergetic, spatially isotropically distributed, ultra-relativistic electron populations, which are subjected to linear, time-independent synchrotron radiative losses via interactions with the ambient magnetic fields and to nonlinear, time-dependent SSC radiative losses in the Thomson limit. We point out that because the SSC cooling is a collective effect, that is, the cooling of a single electron depends on the entire ensemble within the source, the injection of further particle populations into an already cooling system alters the overall cooling behavior [29, 39]. More importantly, it results in intrinsic radiation loss times that meet the requirements for the observed short-time variability in the flaring of blazars down to the order of minutes [23, 24, 28, 31]. The article is organized as follows. In Section II, an approximate analytical solution of the time-dependent, relativistic transport equation of the volume-averaged differential electron number density is derived. Based on this solution, the optically thin synchrotron and SSC intensities as well as the corresponding total fluences are computed in Section III and Appendix G. It is shown in Section IV that the model can simultaneously account for both the specific short-time variability of blazars down to the minute time scale and their characteristic broad-band fluence behavior. Besides, the non-linear coupling between injections and the occurrence of the

3 shortest variability time scales for the highest energies are demonstrated. Appendices A-F contain supplementary material, which is required for the computations of the electron number density and the synchrotron and SSC intensities. II.

THE RELATIVISTIC TRANSPORT EQUATION

The relativistic transport equation describing the temporal evolution and energy dependence of the volumeaveraged differential electron number density n = n(γ, t) for m ultra-relativistic, mono-energetic, instantaneously injected, spatially isotropically distributed electron populations in the rest frame of a non-thermal radiation source with dominant magnetic field self-generation and radiative losses L = L(γ, t) reads [22] m
X ∂n ∂ − Ln = qi δ(γ − γi ) δ(t − ti ) , ∂t ∂γ i=1

(1)

where δ(·) is the Dirac distribution, the qi denote the injection strengths, γi := Ee,i /(me c2 )  1 the normalized initial electron energies, and ti the injection times for all i : 1 ≤ i ≤ m. In this work, radiative losses in form of both a linear, time-independent synchrotron cooling process (with a constant magnetic field) and a non-linear, time-dependent SSC cooling process in the Thomson limit are considered   Z ∞ 2 2 L = Lsyn. + LSSC = γ D0 + A0 γ n dγ . (2) 0

The quantities D0 and A0 are positive cooling constants, which depend on the magnetic field strength B = kBk = const. In the present context, the term linear means that the synchrotron losses do not depend on the electron number density n(γ, t), thus, resulting in a linear term in the partial differential equation (PDE) (1), whereas the SSC loss term depends on an energy integral including n(γ, t), yielding a nonlinearity in (1). Note that, except for TeV blazars, where Klein-Nishina effects drastically reduce the SSC cooling strength above a certain energy threshold, the dominant contribution of the SSC energy loss rate originates in the Thomson regime [31]. Therefore, restricting this study to GeV blazars, it is sufficient for the evolution of the electron number density to employ SSC radiative losses in the Thomson limit. As a consequence, the initial electron energies are bounded from above by γi < 1.94 × 104 b−1/3 , which in turn bounds the maximum SSC photon energy in each injection interval by s,i,max. = 4 0 γi4 < 1.3 × 104 b−1/3 , where 0 = 2.3 × 10−14 b and b is the magnetic field strength B = b Gauss normalized to one Gauss. We also point out that the only accessible energy in synchrotron and SSC cooling processes is the kinetic energy of the electrons. Thus, γ denotes the kinetic component of the normalized total energy Etot. /(me c2 ), i.e., γ=

Etot. − me c2 = γtot. − 1 . me c2

Hence, γtot. ∈ [1, ∞) implies γ ∈ [0, ∞). For this reason, the lower integration bound in (2) is zero. Furthermore, the Dirac distributions δ(γ − γi ) and δ(t − ti ) in Eq.(1) describe a sequence of idealized, discrete energy and time points (γi , ti )i∈{1,...,m} . They are to be understood in the distributional sense, that is, the Dirac distribution is a linear functional on the space of smooth functions ϕ on R with compact support  C0∞ (R) := ϕ | ϕ ∈ C ∞ (R) , supp ϕ compact . More precisely, for k ∈ R, one can rigorously define the Dirac distribution as the mapping δ : C0∞ (R) → R , ϕ 7→ ϕ(0) with the integral of the Dirac distribution against a test function given by Z ∞ ϕ(k) δ(k) dk = ϕ(0) for all ϕ ∈ C0∞ (R) . −∞

A.

Formal Solution of the Relativistic Transport Equation

In terms of the function R(γ, t) := γ 2 n(γ, t) and the variable x := 1/γ, we can rewrite Eq.(1) in the form m

∂R ∂R X +J = qi δ(x − xi ) δ(t − ti ) , ∂t ∂x i=1

(3)

4 where Z

∞

J = J(t) := D0 + A0 0

R(x, t) dx . x2

(4)

Defining the strictly increasing, continuous function G = G(t) via 0
k0

with k, k0 ∈ R, which has a jump discontinuity at k = k0 . A detailed derivation of this solution can be found in Appendix A. In order to determine the function G, we substitute solution (7) into (4) and, by using (5), obtain the ordinary differential equation (ODE) m X qi H(G − Gi ) dG = D0 + A0
2 . dt i=1 G − Gi + xi

(8)

This equation can be regarded as a compact notation for the set of m piecewise-defined ODEs  dG q1  = D0 + A0 for G1 ≤ G < G2 
2   dt  G − G1 + x1      !     dG q q 1 2   = D0 + A0 for G2 ≤ G < G3 
2 +
2   dt G − G1 + x1 G − G2 + x2    ..    .            dG = D + A  0 0  dt

.. .

.. .

q1 G − G1 + x1


2 +

.. .

q2 G − G2 + x2


2 + . . . +

!

qm G − Gm + xm


2

for

Gm ≤ G < ∞ . (9)

In Section II B, an approximate analytical solution for the general case of j injections, i.e., for the interval Gj ≤ G < Gj+1 , where j ∈ {1, ..., m} and G1 = 0, Gm+1 = ∞, is presented in detail. Having a separate analytical solution for each ODE of (9), these are glued together successively requiring continuity at the transition points (cf. Section II C). With (7), the electron number density results in   m X
1 1 n(γ, t) = γ −2 R(γ, t) = γ −2 qi H G(t) − Gi δ − − G(t) + Gi γ γi i=1 (10) =

m X i=1

qi H (t − ti ) δ γ −

γi
1 + γi G(t) − Gi

! .

5 B.

 Solution of the Relativistic Transport Equation for t ∈ R≥0 | tj ≤ t < tj+1 with j ∈ {1, ..., m}

For Gj ≤ G < Gj+1 , Eq.(8) gives j X dG qi = D0 + A0
2 . dt i=1 G − Gi + xi

(11)

Next, we introduce the time-dependent sets S1 and S2 that contain injections i : 1 ≤ i ≤ j with G − Gi  xi and G − Gi  xi , respectively. In order to rewrite Eq.(11) in terms of these sets, we extrapolate the domains of definition to G − Gi < xi and G − Gi ≥ xi . However, for the computations below, we still use the former much less than and much greater than relations when we carry out approximations. Then, Eq.(11) can be formally represented by X dG = D 0 + A0 dt

k∈S1

qk G − Gk + xk


2 + A0

ql

X

G − Gl + xl

l∈S2


2 .

(12)

With the generalized geometric series ∞ X 1 = (n + 1) z n , (1 − z)2 n=0

|z| < 1 ,

we can express the ODE (12) as −2 −2  X qk  G − Gk A0 X Gl − xl dG = D0 + A0 1+ + 2 ql 1 − dt x2k xk G G k∈S1

= D0 + A0

l∈S2

  n p ∞ ∞ X X Gl − xl (n + 1) qk G − Gk A0 X X (p + 1) q − + . l x2k xk G2 G p=0 n=0

k∈S1

l∈S2

This equation is suitably approximated by considering only the leading- and next-to-leading-order terms    X qk  dG G − Gk A0 X Gl − xl ≈ D0 + A0 1 − 2 q + 1 + 2 l dt x2k xk G2 G k∈S1

l∈S2

(13) = M1 (j, S1 ) + M2 (j, S1 ) G +

N1 (j, S2 ) N2 (j, S2 ) + , G2 G3

where the constants  X qk  X qk 2 Gk M1 (j, S1 ) := D0 + A0 1+ , M2 (j, S1 ) := −2 A0 , 2 xk xk x3k k∈S1

N1 (j, S2 ) := A0

X l∈S2

ql , and N2 (j, S2 ) := 2 A0

k∈S1

X

ql (Gl − xl )

l∈S2

depend on the current numbers of elements of S1 and S2 . Note that the explicit dependences are suppressed in the subsequent calculations for simplicity if possible and given if necessary. An approximate analytical solution of (13) can be obtained by first computing the solutions for the near-injection domain (NID) Gj . G and the far-injection domain (FID) Gj  G < Gj+1 , and then extrapolating these up to the transition point GT = GT (j) := Gj + xj , which defines the transition between S1 and S2 for the jth injection and corresponds to the transition time tT = tT (j) specified in Appendix B, where they are glued together continuously. We point out that the FID only plays a role in the construction of G(t | tj ≤ t < tj+1 ) if tT (j) < tj+1 . In the following, the general strategy is to use the leading- and next-to-leading-order contributions in (13) only if one set S1 or S2 has to be taken into account, whereas only the leading-order terms are considered if both sets have to be employed.

6 1.

NID Asymptotics

In the NID Gj . G, at least the jth injection is an element of S1 . Further, one may – but does not necessarily need to – have injections in S2 . Therefore, we distinguish the two cases S2 = ∅ and S2 6= ∅. First, if S2 = ∅, (13) yields dG = M1 (j) + M2 (j) G dt

for


Gj ≤ G < min Gj+1 , GT (j) .

(14)

This ODE can be easily solved using separation of variables. We obtain Z dG 1 = ln (M1 + M2 G) = t + c1 , c1 = const. ∈ R , M1 + M2 G M2 which is equivalent to G(t) =


M1 − c2 exp −|M2 | t |M2 |


with c2 := exp −|M2 | c1 /|M2 | ∈ R>0 . The constant c2 is fixed by applying the initial condition G(t = tj ) = Gj , giving  
M1 − Gj exp |M2 | tj c2 = |M2 | and, hence, G(t) =

M1 (j) − |M2 (j)|




M1 (j) − Gj exp −|M2 (j)| (t − tj ) |M2 (j)|

for


tj ≤ t < min tj+1 , tT (j) .

(15)

In the case S2 6= ∅, by considering the leading-order terms only, (13) results in
N1 dG = P1 + 2 for Gj ≤ G < min Gj+1 , GT (j) , (16) dt G P where P1 = P1 (j, S1 ) := D0 + A0 k∈S1 qk /x2k . Again employing separation of variables and extending the integrand with the multiplicative identity P1 /P1 and the neutral element N1 − N1 , we find r r !   Z Z G2 dG 1 dG 1 N1 P1 = G − N1 = G− arctan G = t + c3 , (17) N1 + P1 G2 P1 N1 + P1 G2 P1 P1 N1 where c3 = const. ∈ R. Since arg(arctan) ≥ 0 for all G : Gj ≤ G < min(Gj+1 , GT ), one way of deriving an approximate analytical solution of this transcendental equation is to determine G asymptotically for small and large arguments of the arctan-function, extrapolate these solutions up to an intermediate transition point such that the entire domain of definition is covered, and apply a branch-gluing at the transition point requiring p continuity. For small arguments P1 /N1 G  1, we use the third-order approximation arctan (x)|x1 ≈ x−x3 /3, as the linear term in Eq.(17) and the linear contribution in the approximation of the arctan-function cancel each other. This leads to the asymptotic solution G(t) ' (3 N1 t + c)1/3 ,

c = const. ∈ R .

(18)

p For large arguments P1 /N1 G  1, the arctan-function can be well approximated by π/2. We directly obtain an asymptotic solution of the form G(t) ' P1 t + c0 , By means of the condition

p

c0 = const. ∈ R .

P1 /N1 G = 1, we derive the NID transition value r N1 (n) GT = . P1

(19)

(20)

7 This value specifies the upper and lower bounds of the domains of validity of (18) and (19), respectively. The corresponding transition time can be directly computed by substituting (20) into (17), where the constant c3 is fixed via the initial condition G(t = tj ) = Gj r r ! 1 N1 P1 c3 = Gj − arctan Gj − tj . P1 P1 N1 This yields (n) tT

1 = tj + P1

r

!  r  N1 P1 π 1 − + arctan Gj − Gj . P1 4 N1 (n)

(21) (n)

We remark that in general, one does not have the ordered sequence Gj < GT < GT because GT can in principle p (n) be larger than GT . Then, for P1 /N1 Gj < 1 and tj < tT < tT , the solution of Eq.(17) is approximately given by (
1/3
(n) 3 N1 (j) t + c4 (j) for tj ≤ t < min tj+1 , tT (j) (22) G(t) =

(n) P1 (j) t + c5 (j) for min tj+1 , tT (j) ≤ t < min tj+1 , tT (j) , c4 , c5 = const. ∈ R , for

p

(n)

P1 /N1 Gj < 1 and tT ≤ tT

by


1/3 G(t) = 3 N1 (j) t + c6 (j) and for

p


for tj ≤ t < min tj+1 , tT (j) ,

c6 = const. ∈ R ,

(23)

P1 /N1 Gj ≥ 1 by G(t) = P1 (j) t + c7 (j)


for tj ≤ t < min tj+1 , tT (j) ,

c7 = const. ∈ R .

(24)

The integration constants c4 , ..., c7 are determined in Appendix C. In order to compute the
constants M1 , M2 , N1 , and P1 , we have to constantly check during the evolution of G ∈ Gj , min(Gj+1 , GT (j)) whether an injection belongs to S1 or S2 . To this end, we specify two different kinds of updates. The first kind occurs when a new injection enters the system, while the second kind is due to NID-FID transitions. We start with the initial classification for the update at the time of the jth injection Gj − Gi < xi Gj − Gi ≥ xi

for the ith injection being in S1 for the ith injection being in S2 ,

where i ∈ {1, ..., j}. Next, since all transition values GT (i) are known, they can be arranged as an ordered j-tuple representing an increasing time sequence. Assuming that a certain number of GT (i)-values is contained in the time interval under consideration, whenever G reaches one of them, both sets S1 and S2 have to be updated, i.e., the corresponding injection is removed from S1 , while at the same time it is added to S2 . 2.

FID Asymptotics

In the FID Gj  G < Gj+1 , at least the jth injection is an element of S2 . Moreover, S1 contains only injections of lowly-energetic electrons. As the latter yield just a small contribution to the radiation emission, they are neglected. Thus, by taking into account the leading- and next-to-leading-order terms with respect to S2 , (13) becomes dG N1 (j) N2 (j) = D0 + + dt G2 G3

for GT (j) ≤ G < Gj+1 .

(25)

An approximate analytical solution of this ODE can be obtained in a similar way as the NID solution by first applying separation of variables, which leads to an integral equation of the form Z G3 dG = t + d1 , d1 = const. ∈ R , (26) D0 G3 + N1 G + N2

8 and subsequently deriving asymptotic solutions that are extrapolated up to an intermediate transition point, where they are glued together continuously. In more detail, since N1 G  N2 , we employ a first-order geometric series approximation in N2 /(N1 G) in the integrand of (26) Z Z Z G 1 N1 G + N2 G 1 dG G3 dG = − dG = − 3 3 D0 G + N1 G + N2 D0 D0 D 0 G + N1 G + N2 D0 D0 D0 G3 1+ N1 G + N2 =

≈

G 1 − D0 D0

G 1 − D0 D0

Z

dG G 1 −  −1 = 2 D0 D0 N2 D0 G 1+ 1+ N1 N1 G

Z

dG D0 1+ N1



N2 G− 2 N1

2

D0 N22 − 4 N13

≈

Z

dG ∞  2 X

D0 G 1+ N1

G 1 − D0 D0

n=0

Z D0 1+ N1

N2 − N1 G

n

dG  2 . N2 G− 2 N1 (27)

Note that in the first step, we included the multiplicative identity D0 /D0 and the neutral element (N1 G + N2 ) − (N1 G + N2 ) in the numerator of the integrand, giving the splitting into two terms. Then, by means of simple algebraic manipulations, we expressed the integrand in a form suitable for the substitution of the geometric series. For reasons of computational simplicity, we dropped the small second-order contribution −D0 N22 /(4 N13 ) in the last step. The final integral in (27) is solved by an arctan-function. Thus, introducing the parameter
1/2 3/2 β = β(j) := D0 N2 (j)/ 2 N1 (j) , Eq.(26) results in    1/2 2 N1 G N1 G − 3/2 arctan β −1 = t + d1 . (28) D0 N2 D0 Since arg(arctan) > 0 for all G : GT ≤ G < Gj+1 , we find an approximate analytical solution of this transcendental equation by first computing the asymptotics of G for small and large arguments of the arctan-function, secondly extrapolating these up to an intermediate transition point, and then constructing a continuous function on the entire domain via branch-gluing. Again using the third-order approximation of the arctan-function for small arguments β (2 N1 G/N2 − 1)  1, the associated asymptotic solution becomes G(t) ' (3 N1 t + d)1/3 +

N2 , 2 N1

d = const. ∈ R .

Also, because arctan (x)|x1 ≈ π/2, the asymptotic solution for large arguments β (2 N1 G/N2 − 1)  1 reads G(t) ' D0 t + d0 , Connecting them continuously at the transition point r (f )

GT =

d0 = const. ∈ R .

N1 N2 + , D0 2 N1

(29)

which is derived from the transition condition β (2 N1 G/N2 − 1) = 1 and corresponds to the transition time " r   #! N2 π 2 N1 (Gj + xj ) 1 N1 (f ) −Gj − xj + + 1 − + arctan β −1 , (30) tT = tT + D0 2 N1 D0 4 N2
(f ) we obtain for β 2 N1 Gj /N2 − 1 < 1 and tT < tT < tj+1 the piecewise-defined function  (f )  3 N (j) t + d (j)
1/3 + N2 (j) for tT (j) ≤ t < tT (j) 1 2 2 N1 (j) G(t) =  (f ) D0 t + d3 (j) for tT (j) ≤ t < tj+1 , d2 , d3 = const. ∈ R ,
(f ) for β 2 N1 Gj /N2 − 1 < 1 and tT < tj+1 < tT G(t) = 3 N1 (j) t + d4


1/3

+

N2 (j) 2 N1 (j)

for tT (j) ≤ t < tj+1 ,

d4 = const. ∈ R ,

(31)

(32)

9
and otherwise for β 2 N1 Gj /N2 − 1 ≥ 1 G(t) = D0 t + d5 (j)

for tT (j) ≤ t < tj+1 ,

d5 = const. ∈ R .

(33)

We remark that similar to the calculation of the transition time (21) of the NID, the transition time (30) was computed fixing the integration constant r   ! 1 2 N1 (Gj + xj ) N1 Gj + xj − d1 = −tT + arctan β −1 D0 D0 N2 by imposing the boundary condition G(t = tT ) = GT on (28) and substituting the transition value (29). The determination of the integration constants d2 , ..., d5 is given in Appendix C. Further, the constants N1 and N2 are specified by identifying which injections are initially (that is now at t = tT ) in S2 , verifying the condition GT (j) − Gi ≥ xi for all i ∈ {1, ..., j}. We do not consider the remaining injections, as S1 is of no relevance for the calculation of these constants. Nonetheless, an injection being in S1 switches over to S2 if GT (i) ∈ [GT (j), Gj+1 ). But since the electrons of these injections are ab initio lowly-energetic, the associated updates can be safely disregarded. C.

Solution of the Relativistic Transport Equation for {t ∈ R≥0 }

The complete solution of Eq.(8) is derived as follows. Beginning
with the single-injection domain (SID), the solution branch G(t | 0 ≤ t < t2 ) is given for 0 ≤ t < min t2 , tT (1) by Sol.(15) and for tT (1) ≤ t < t2 by p (f ) • Sol.(31) for D0 /(A0 q1 ) x1 < 1 and tT (1) < tT (1) < t2 , p (f ) D0 /(A0 q1 ) x1 < 1 and tT (1) < t2 < tT (1) , • Sol.(32) for p • Sol.(33) for D0 /(A0 q1 ) x1 ≥ 1 with j = 1. We point out that in this domain, Eq.(8) can also be solved by directly applying separation of variables, resulting in [33] s ! Z Z G G2 G 1 dG (A0 q1 )1/2 D0 = arctan dG = − − G = t + e1 , (34) 3/2 D0 G 2 + A0 q1 D0 D0 D0 A0 q 1 D0 G 2 D0 1+ A0 q 1 where e1 = const. ∈ R and G := G + x1 . Using the asymptotic-branch-gluing method with the transition time " "s #!# r π A0 q1 D0 1 (s) −x1 + 1 − + arctan x1 tT = D0 D0 4 A0 q 1 leads for

for

p

p

(s)

D0 /(A0 q1 ) x1 < 1 and 0 < tT < t2 to the approximate solution ( (s) (3 A0 q1 t + e2 )1/3 − x1 for 0 ≤ t < tT G(t) = (s) D0 t + e3 for tT ≤ t < t2 , e2 , e3 = const. ∈ R ,

(35)

(s)

D0 /(A0 q1 ) x1 < 1 and t2 ≤ tT to G(t) = (3 A0 q1 t + e4 )1/3 − x1

for 0 ≤ t < t2 ,

e4 = const. ∈ R ,

(36)

p and otherwise for D0 /(A0 q1 ) x1 ≥ 1 to G(t) = D0 t + e5

for 0 ≤ t < t2 ,

e5 = const. ∈ R .

The integration constants e1 , e2 , e4 , and e5 are fixed by the initial condition G(t = t1 = 0) = G1 = 0 s ! x1 (A0 q1 )1/2 D0 e1 = − arctan x1 3/2 D0 A0 q 1 D 0

e2 = e4 = x31 e5 = 0 ,

(37)

10 (s)

(s)


whereas the integration constant e3 is determined by the transition condition G t = tT | 0 ≤ t < tT
(s) (s) tT | tT ≤ t < t2 (s)

e3 = 3 A0 q1 tT + x31


1/3

=G t=

(s)

− x1 − D0 tT .


In the double-injection domain (DID), the solution branch G(t | t2 ≤ t < t3 ) is given for t2 ≤ t < min t3 , tT (2) by • Sol.(15)

for S2 = ∅ ,

• Sol.(22)

p (n) P1 (2)/N1 (2) G2 < 1 , and t2 < tT (2) < tT (2) , p (n) for S2 6= ∅ , P1 (2)/N1 (2) G2 < 1 , and t2 < tT (2) ≤ tT (2) , p for S2 6= ∅ and P1 (2)/N1 (2) G2 ≥ 1 ,

• Sol.(23) • Sol.(24)

for S2 6= ∅ ,

and for tT (2) ≤ t < t3 by (f )

• Sol.(31)

for β(2) (2 N1 (2) G2 /N2 (2) − 1) < 1 and tT (2) < tT (2) < t3 ,

• Sol.(32)

for β(2) (2 N1 (2) G2 /N2 (2) − 1) < 1 and tT (2) < t3 < tT (2) ,

(f )

• Sol.(33)

for β(2) (2 N1 (2) G2 /N2 (2) − 1) ≥ 1 , p (n) always setting j = 2. We remark that in the NID case P1 (2)/N1 (2) G2 < 1 with t3 < tT (2) < tT (2), only the first branch of (22) has to be taken into account. The initial value G2 is fixed by requiring continuity of the SID and DID solution branches at the time of the second injection, yielding p 
(s) (s) for D0 /(A0 q1 ) x1 < 1 and 0 < tT < t2  G t = t2 | tT ≤ t < t2 ; Sol.(35)     p
(s) G2 = G t = t2 | 0 ≤ t < t2 ; Sol.(36) for D0 /(A0 q1 ) x1 < 1 and 0 < t2 ≤ tT      p
 G t = t2 | 0 ≤ t < t2 ; Sol.(37) for D0 /(A0 q1 ) x1 ≥ 1 . In addition to the initial updating of constants at t = t2 , we have to perform NID-FID updates at t = tT (1) and t = tT (2) if tT (1), tT (2) ∈ [t2 , t3 ). Assuming that both transition times are contained in this interval with the order t2 < tT (1) < tT (2) < t3 , we have to update the initial sets S1 and S2 twice. More precisely, at the time of the second injection t = t2 , both the first and the second injection are contained in S1 while S2 is empty. During the progression toward the upper bound t3 , the first injection switches from S1 to S2 at t = tT (1), whereas the second injection continues to be in S1 . At t = tT (2), also the second injection switches over to S2 , leaving S1 empty. This amounts to the following updating sequence:   2 2 X X

2 Gj qj qj • t2 ≤ t < tT (1) : M1 2, S2 = ∅ = D0 + A0 1 + , M 2, S = ∅ = −2 A 2 2 0 2 3 , x x x j j j j=1 j=1 2 X


qj N1 2, S2 = ∅ = N2 2, S2 = ∅ = 0 , and P1 2, S2 = ∅ = D0 + A0 x2j j=1

• tT (1) ≤ t < tT (2) :

 

A0 q 2 2 G2 2 A0 q2 M1 2, S2 = {1} = D0 + 1 + , M2 2, S2 = {1} = − , 2 x2 x2 x32

N1 2, S2 = {1} = A0 q1 , N2 2, S2 = {1} = −2 A0 q1 x1 ,
A0 q2 and P1 2, S2 = {1} = D0 + x22

• tT (2) ≤ t < t3 :

2 X


M1 2, S2 = {1, 2} = D0 , M2 2, S2 = {1, 2} = 0 , N1 2, S2 = {1, 2} = A0 qk , k=1

11 2 X

N2 2, S2 = {1, 2} = 2 A0 qk (Gk − xk ) , and P1 2, S2 = {1, 2} = D0 . k=1

Repeating this procedure for all injection numbers j : 3 ≤ j ≤ m results in the formal representation of G on the complete time interval [0, ∞) G(t | 0 ≤ t < ∞) = H(t) H(t2 − t) GSID (t | 0 ≤ t < t2 )

+

m X


h
H(t − ti ) H(ti+1 − t) H tT (i) − t χ(S2 = ∅) GNID t | ti ≤ t < tT (i) ; S2 = ∅

i=2

(38)  
i + 1 − χ(S2 = ∅) GNID t | ti ≤ t < tT (i) ; S2 6= ∅

+

m X



H t − tT (i) H(ti+1 − t) GFID t | tT (i) ≤ t < ti+1 ,

i=2

where ( χ(S2 = ∅) :=

1 0

for S2 = ∅ for S2 = 6 ∅

is the characteristic
function. The SID contribution
GSID (t | 0 ≤ t < t2 ), both NID contributions GNID
t | ti ≤ t < tT (i) ; S2 = ∅ , GNID t | ti ≤ t < tT (i) ; S2 6= ∅ , the FID contribution GFID t | tT (i) ≤ t < ti+1 , and the initial constants Gi are given explicitly in Appendix D. Note that the updating of constants is suppressed for readability. It could, however, be written down explicitly similar to the updating of the integration constants presented in Appendix C.

III.

SYNCHROTRON AND SSC INTENSITIES

In this section, the optically thin synchrotron intensity and the optically thin SSC intensity in the Thomson limit are calculated. The optically thick component of the synchrotron intensity is not considered because it was shown in [30] that, for all frequencies and times, it provides only a small contribution to the SSC intensity, which usually dominates the blazar radiation emission.

A.

Synchrotron Intensity

The optically thin synchrotron intensity Isyn. (, t) for an isotropically distributed electron number density is given by Z ∞ Rsource n(γ, t) P (, γ) dγ , (39) Isyn. (, t) = 4π 0 where Rsource is the radius of the spherical emission plasmoid, the function    2 P (, γ) = P0 2 CS γ 3 0 γ 2

(40)

is the pitch-angle-averaged spectral synchrotron power of a single electron in a magnetic field of strength B,  = h ν/(me c2 ) is the normalized photon energy, and P0 = 1.1 × 10−8 erg [9, 14]. The CS-function is discussed in detail in Appendix E. Here, we employ the approximation CS(z) ≈

z 2/3

a0
, 1 + z 1/3 exp (z)

(41)

12 where a0 = 1.15. Substituting the electron number density (10) and the synchrotron power (40) with the CS-function (41) into formula (39), we obtain for the optically thin synchrotron intensity Isyn. (, t) = I0,syn. 1/3

m X i=1

2/3

qi H (t − ti ) Yi (t)   1/3  2 2 2 2/3 1+ Yi (t) exp Yi (t) 3 0 3 0

(42)

2/3

with I0,syn. := 32/3 Rsource P0 a0 0 /(28/3 π) and the abbreviation Yi (t) := G(t) − Gi + xi . For comparisons with observational data or for generic case studies, that is, for fitting or plotting lightcurves, we have to compute the energy-integrated synchrotron intensity I¯syn. (t) with a lower integration limit min. corresponding to the energy of the first data point in the fluence and an upper integration limit max. defined by the last. As (40) is a spectral synchrotron power, i.e., it is normalized with respect to the frequency, we do not integrate the intensity (42) directly over the energy, but over the frequency. This gives rise to an additional factor of me c2 /h. Therefore, the energy-integrated synchrotron intensity becomes Z max. m e c2 ¯ Isyn. (t; min. , max. ) = Isyn. (, t) d . h min. Using (42), this quantity yields me c2 I¯syn. (t; min. , max. ) = h



30 2

4/3 I0,syn.

Z m X τ 1/3 qi H (t − ti ) τmax. dτ , Yi2 (t) 1 + τ 1/3 exp (τ ) τmin. i=1

(43)

where τ := 2  Yi2 /(3 0 ), τmin. := τ (min. ), and τmax. := τ (max. ). In order to solve the integral, we approximate the integrand by τ 1/3 for τ ≤ 1 and exp (−τ ) for τ > 1. This is justified because the approximated CS-function (41) is just adapted to the asymptotics τ  1 and τ  1 of the exact CS-function and extrapolated in between. Moreover, since τmin. and τmax. depend on the time t, we have to consider the three cases where 1 ≤ τmin. , τmin. < 1 ≤ τmax. , and τmax. < 1. Accordingly, the integral results in Z τmax.   τ 1/3 dτ ≈ H(τmin. − 1) exp (−τmin. ) − exp (−τmax. ) 1/3 1+τ exp (τ ) τmin.     3 4/3 (44) + H(τmax. − 1) H(1 − τmin. ) 1 − τmin. + exp (−1) − exp (−τmax. ) 4 h i 3 4/3 4/3 + H(1 − τmax. ) τmax. − τmin. . 4 B.

SSC Intensity

For the computation of the optically thin SSC intensity Z ∞ Rsource ISSC (s , t) = n(γ, t) PSSC (s , γ, t) dγ , 4π 0

(45)

where PSSC (s , γ, t) is the SSC power of a single electron and s = Es /(me c2 ) the normalized scattered photon energy, we have to employ the Thomson limit because the SSC radiative losses in (2) are already restricted to the Thomson regime. In this limit, the SSC power reads [9] PSSC (s , γ, t) =

4 σT c γ 2 E(s , t) 3

with the Thomson cross section σT = 6.65 × 10−25 cm2 , the speed of light in vacuum c, and the total photon energy density E(s , t). For highly relativistic electrons with γ  1 and synchrotron photon energies in the Thomson regime γ   1, the characteristic energy of the SSC-scattered photons is s ≈ 4 γ 2  [17], which corresponds to head-on collisions of the synchrotron photons with the electrons [9, 21, 27]. This allows us to apply a monochromatic approximation in the total photon energy density in form of a Dirac distribution that spikes at this characteristic energy Z ∞
1 E(s , t) =  N (, t) δ s − 4 γ 2  d , 4π 0

13 where N (, t) =

4 π Isyn. (, t) c

is the synchrotron photon number density. We remark that by assuming an isotropic, relativistic electron distribution, the synchrotron photon number density also becomes isotropically distributed. Substituting the latter formulas into (45) and using Fubini’s theorem, we obtain Rsource σT s ISSC (s , t) = 12 π

Z

∞

0

Isyn. (, t) 

Z

1/


n(γ, t) δ s − 4 γ 2  dγ d ,

0

where the upper γ-integration limit arises from the restriction to the Thomson regime. With the electron number density (10) and the synchrotron intensity (42), the optically thin SSC intensity yields, after performing both integrations,   2/3 s m qi qj H (t − ti ) H (t − tj ) H 1 − Yj (t) Yi (t) Yj (t) X 4 ISSC (s , t) = I0,SSC 1/3 (46)   ,  1/3 s   2 s   2/3 s i,j=1 Yi (t) Yj (t) exp Yi (t) Yj (t) 1+ 6 0 6 0 where I0,SSC := Rsource σT I0,syn. /(22/3 12 π). The double sum, in which the index j refers to the jth electron population and the index i to the ith synchrotron photon population, accounts for all combinations of SSC scattering between the different electron and synchrotron photon populations. For the corresponding energyintegrated SSC intensity, we find m X me c2 qi qj H (t − ti ) H (t − tj ) I¯SSC (t; s,min. , s,max. ) = (6 0 )4/3 I0,SSC  2 h Yi (t) Yj (t) i,j=1

(47) Z

min(τmax. ,τ (4/Yj ))

× τmin.

τ 1/3 dτ , 1 + τ 1/3 exp (τ )

where τ := s (Yi Yj )2 /(6 0 ), τmin. := τ (s,min. ), and τ
max. := τ (s,max. ). The integral is given by (44), however, with an upper integration limit of min τmax. , τ (4/Yj ) . IV.

SHORT-TIME VARIABILITY AND BROAD-BAND FLUENCE OF BLAZARS A.

Numerical Implementation

The numerical implementation of G, the energy-integrated synchrotron and SSC intensities, as well as the corresponding total fluences is carried out with Python. Here, we describe the functionality of the code and the specific incorporation of the analytical formulas. The code makes heavy use of the decimal package, which is designed for high floating point precision calculations. This is necessary because the range of possible values between the cooling constants D0 and A0 , the injection strengths qi , and the inverse injection energies xi spans several orders of magnitude, which can lead to a loss of accuracy in expressions where these parameters come (n) up. This problem occurs in its most severe form in the evaluation of the formulas for the transition times tT (f ) and tT , yielding deviations from the expected values by more than 50% with standard floating point precision. But even with higher precision, it is preferable to avoid the evaluation of the transition times altogether. Thus, we compute G on a grid, using fixed time steps. This makes it possible to read out the values of G at the grid (n) (f ) points and directly compare them to the values of GT and GT in order to determine the actual solution branch without referring to the transition times. The code begins with the definitions of the free parameters, namely the injection times ti , the injection strengths qi , the inverse injection energies xi , the synchrotron and SSC cooling constants D0 and A0 , the Lorentz boost D of the plasmoid, the total number of injections ninj. , and the upper time boundary of the grid tend . The value of tend is chosen as one and a half times the injection time of the final injection. In a realistic scenario, however, tend corresponds to the end of the observation time. The time grid is set up homogeneously and linearly, and the number of grid points can be chosen arbitrarily. All computations are performed in the plasmoid rest frame.

14

8

10 10

7

10 8

5 q1 = (1.5e5, 2e5, 5e4, 1e5, 2e4) q2 = (1.5e5, 4e5, 5e4, 1e5, 2e4)

4 3 2

10 6 10 4 10 2

1 00

F [arb. u. ]

I¯ [arb. u. ]

6

q1 q2

10 0 20

40

60

80 100 120 140 160 180 t [s]

10 -18 10 -16 10 -14 10 -12 10 -10 10 -8 10 -6 10 -4 10 -2 10 0 E [GeV]

(a)

(b)

FIG. 1: Lightcurves (a) and total synchrotron (lower curves in (b)) and SSC (upper curves in (b)) fluences for two generic five-injection scenarios with parameter values D0 = 10−9 , A0 = 10−18 , (ti )i=1,...,5 = (0, 23, 50, 80, 120) s, q1 = (qi )i=1,...,5 = (1.5×105 , 2×105 , 5×104 , 105 , 2×104 ) cm−3 , q2 = (qi )i=1,...,5 = (1.5×105 , 4×105 , 5×104 , 105 , 2×104 ) cm−3 , (xi )i=1,...,5 = (5 × 10−4 , 5 × 10−4 , 5 × 10−4 , 5 × 10−4 , 5 × 10−4 ), D = 10, Emin. = 10 MeV, and Emax. = 500 MeV.

For the later evaluation of the lightcurves and fluences, the relevant quantities are transformed into the observer (s) (n) (f ) frame (see Appendix F). Ordered lists of the initial and transition values Gi , GT , GT , GT , and GT , as well as a list keeping track of the elements of the sets S1 and S2 and a list of the various solution branches are implemented. These lists are constantly updated during runtime. The code constructs G incrementally in two loops. The first loop covers G from the time of the first injection t = 0 to the time of the second injection t = t2 using the solutions (35)-(37). The second loop computes G from the time of the second injection to the time tend employing the solutions (15), (22)-(24), and (31)-(33). During each step, the current time tcur. is incremented by a fixed value and Gcur. := G(tcur. ) is determined according to the proper solution branch, which is automatically selected via the above-mentioned lists. Moreover, at each grid point, the analytical expressions for G are glued together continuously. In more detail, after initializing the values of M1 , M2 , N1 , N2 , P1 , and GT at t = 0, the first loop starts its iteration in the first SID solution branch. (s) At each step, it evaluates G at the current time grid point and checks whether Gcur. exceeds or equals GT , i.e., whether G needs to be expressed by the second SID solution branch. If necessary, G is altered accordingly and glued to the previous solution branch continuously. Also, if Gcur. becomes larger than or equal to GT (1), the lists of S1 and S2 are updated and GT (1) is appended to the lists of transition values. The loop ends if either tcur. exceeds or equals the injection time of the second injection t2 , or – in a scenario with only a single injection – tcur. exceeds the upper time boundary tend . In the latter case, the numerical construction of G is completed. The second loop constructs G in a similar way as the first loop, but now more cases, which arise from the more elaborate structure of the analytical multiple injection solution, have to be taken into account. Once the second loop ends, the values of G are known at every point of the grid. This allows us to evaluate the energy-integrated synchrotron and SSC intensities at each grid point by simply substituting these values into the corresponding analytical formulas (43) and (47). The total fluences are approximated by the sums of the areas of rectangles, each of which is defined by the intensity at the left grid point and the size of the time step. Alternatively, one could implement the approximate analytical formulas derived in Appendix G. As the number of grid points can be increased arbitrarily, the precision of these computations is limited only by the machine accuracy and the available CPU time.

B.

Lightcurves and Fluences

In Figure 1, we show plots of lightcurves, i.e., of the energy-integrated synchrotron and SSC intensities (43) and (47), and plots of the corresponding total fluences for two generic five-injection scenarios. The lightcurves in Figure 1(a) illustrate the main features of the model: First and foremost, it can reproduce the short-time variability of blazars down to the minute time scale. Further, we demonstrate the non-linear coupling between injections by comparing two scenarios that differ only in the strengths of the second injection. More precisely,

15

10 7 10 6

ISSCT [arb. u. ]

10 5 10 4 10 3 10 2 10 1

E = 50 MeV E = 10 MeV E = 1 MeV

10 0 10 -10

20

40

60

80 t [s]

100

120

140

FIG. 2: SSC intensity for three different energies. The parameter values are the same as in the q1-scenario of Figure 1.

by doubling the original second injection strength, we find an increase in the peak value of more than 100% indicated by the dotted horizontal line. Also, even after the SSC emission has dropped to virtually the same value in both scenarios (shortly before the third injection occurs), successive injections still have higher emissions due to their coupling with the stronger second injection. We point out that the steep left flanks of the peaks are caused by using Dirac distributions in the injection rate. A convolution of the corresponding electron number density with a suitable source function (modeled by, e.g., a power law) would lead to weaker left slopes and, thus, to Gaussian-like shapes. The total fluences in Figure 1(b) indicate that for realistic parameter values, the SSC component clearly dominates the synchrotron component. Employing the SSC intensity (46) for three different energies, Figure 2 illustrates another feature of blazars, namely that the shortest variability time scales occur for the highest energies. The variations can span several orders of magnitude, whereas towards lower energies they decrease quickly. In this particular scenario, the scale of the intensity variations is reduced from four orders of magnitude to a single order in the energy range from 50 to 1 MeV.

V.

SUMMARY AND CONCLUSIONS

We introduced a leptonic one-zone model that simultaneously explains both the specific short-time variability in the flaring of blazars and the characteristic broad-band fluence behavior. The model assumes that the radiation emission arises out of magnetized plasmoids that propagate relativistically along the general direction of the jet axis, where they successively pick up several electron populations from interstellar and intergalactic clouds. Inside these plasmoids, the electrons undergo synchrotron and SSC radiative losses. This gives rise to a series of strong flares on short time scales. More precisely, we derived an approximate analytical solution of the time-dependent, relativistic transport equation of the volume-averaged differential electron number density for multiple successively and instantaneously injected, mono-energetic, spatially isotropically distributed, interacting electron populations, which are subjected to linear, time-independent synchrotron radiative losses and non-linear, time-dependent SSC radiative losses in the Thomson limit. Using this solution, we computed the optically thin synchrotron and SSC intensities as well as the corresponding total fluences. Finally, we plotted generic lightcurves and fluences for realistic parameter values, which showed that the model can account for the specific short-time variability at all frequencies down to the minute time scale, the typical shape of the broad-band fluence, and the feature that the shortest variability time scales occur for the highest energies. We point out that the SSC radiative loss term considered here is strictly valid only in the Thomson regime, which limits the applicability of the model to GeV blazars. Nonetheless, it can be generalized to describe TeV blazars by using the full KleinNishina cross section in the SSC energy loss rate. This leads to a model for which similar, however, technically more involved, methods apply. We also remark that more sophisticated injection rates Q(γ, t) can be studied by evaluating the convolution of the electron number density (10), as it is the Green’s function of the more general transport equation
∂ ∂n − L n = Q(γ, t) , ∂t ∂γ

16 with a suitable source function. Moreover, the synchrotron loss term can, in principle, be modeled as non-linear and time-dependent, too. This can be achieved by making an equipartition assumption between the magnetic and particle energy densities [32].

Appendix A: Laplace Method

We derive the solution R(x, G) of the PDE (6) by using a composition of two Laplace transformations. First, applying a Laplace transformation with respect to the inverse energy x Z ∞ Lw (x) [ · ] := [ · ] exp (−w x) dx 0

to (6) gives Z

∞

0

∞

Z

∂R exp (−w x) dx + ∂G

0

m

X ∂R exp (−w x) dx = qi δ(G − Gi ) ∂x i=1

∞

Z

δ(x − xi ) exp (−w x) dx . 0

Evaluating the second integral on the left-hand side via integration by parts and employing the Laplace transform of R Z ∞ K(w, G) := R(x, G) exp (−w x) dx , (A1) 0

we obtain m

X   ∂K qi δ(G − Gi ) exp (−w xi ) H(x − xi )|x→∞ − H(x − xi )|x=0 + R exp (−w x)|x→∞ − R(0, G) + w K = ∂G i=1 and, hence, m

X ∂K − R(0, G) + w K = qi δ(G − Gi ) exp (−w xi ) . ∂G i=1

(A2)

As the normalized initial electron energies are finite and bounded from above by γi < 1.94 × 104 b−1/3 for all i : 1 ≤ i ≤ m (due to the restriction to the Thomson regime) and only radiation loss processes are considered, we know that the electron number density has support  supp n(γ, t) = (γ, t) ∈ R2≥0 | γ ≤ γmax. with γmax. := max{γi | 1 ≤ i ≤ m} . Thus, we can write n(γ, t) = H(γmax. − γ) n(γ, t), yielding for the second term   n(x, G) R(0, G) = lim H(x − xmax. ) = 0, x→0 x2 where xmax. := 1/γmax. . Then, (A2) reads m

X ∂K +wK = qi δ(G − Gi ) exp (−w xi ) . ∂G i=1

(A3)

Secondly, applying a Laplace transformation with respect to the function G Z ∞ Ls (G) [ · ] := [ · ] exp (−s G) dG 0

to (A3) results in Z 0

∞

∂K exp (−s G) dG + w ∂G

Z

∞

K exp (−s G) dG = 0

m X i=1

Z

∞

δ(G − Gi ) exp (−s G) dG . (A4)

qi exp (−w xi ) 0

17 This Laplace transformation implies that G is positive. With the Laplace transform of K Z ∞ K(w, G) exp (−s G) dG M (w, s) := 0

and integration by parts as before, Eq.(A4) becomes K exp (−s G)|G→∞ − K(w, 0) + (w + s) M =

m X


qi exp − (w xi + s Gi )

i=1

  × H(G − Gi )|G→∞ − H(G − Gi )|G=0 . Since the Heaviside functions are not well-defined at the jump discontinuities at G = Gi , this equation reduces to − K(w, 0) + (w + s) M =

m X


qi exp − (w xi + s Gi ) − q1 exp (−w x1 ) H(G)|G=0 ,

(A5)

i=1

containing an unspecified contribution in the last term on the right-hand side. At G = 0, no energy losses have yet occurred. Therefore, the electron number density is of the form n(x, 0) = n1 δ(x − x1 ), where n1 = q1 x21 . Substituting this density into (A1) by employing the relation R(x, G) = n(x, G)/x2 , we find Z ∞ exp (−w x) δ(x − x1 ) dx = q1 exp (−w x1 ) . K(w, 0) = n1 x2 0 Choosing H(G)|G=0 = 1, we can use this function in order to compensate the last term on the right-hand side of Eq.(A5), yielding M (w, s) =

m
1 X qi exp − (w xi + s Gi ) . w + s i=1

We point out that both M and the sum are positive. Hence, s > −w, which allows us to apply the inverse Laplace transformations with respect to the variables s and w to M . This gives the solution of (6) −1 −1 R(x, G) = L−1 w (x) Ls (G) M (w, s) = Lw (x)

m X

qi H(G − Gi ) exp − w (G − Gi + xi )




i=1

=

m X

qi H(G − Gi ) δ(x − xi − G + Gi ) .

i=1

Appendix B: NID-FID Transition Time

In the following, we present two different methods for the determination of the transition time tT (j). It is advantageous to start from the integral equation representation of the ODE (11) evaluated at the transition point GT (j) = Gj + xj Z

Gj +xj

tT (j) =

J Gj

−1

e dG e (G)

with

J

−1

e := 1 = (G) e J(G)

D0 + A0

j X i=1

!−1

qi e − Gi + xi G


2

.

(B1)

With J −1 ∈ C ∞ (R≥0 ), the first method applies the first mean value theorem for integration. Thus, there exists a point ξj ∈ [Gj , Gj + xj ] such that the transition time (B1) can be written as tT = xj J −1 (ξj ) . Because the mean value theorem is merely an existence theorem, we approximate the value of ξj by means of an additional input. Since J −1 is strictly increasing, a suitable choice for ξj is the midpoint Gj + xj /2 of the integration interval. The second method employs a trapezoid approximation. Again due to the strictly increasing functional shape of J −1 , the integral in (B1) can be approximated by the area A(j) of a trapezoid, which is

18 computed as the sum of the area Ar (j) of the rectangle defined by the distance between the endpoints Gj and Gj + xj and the height J −1 (Gj ) Ar = xj J −1 (Gj ) and the area At (j) of the right-angled triangle with one cathetus given by the distance between the endpoints and the other one by the height J −1 (Gj + xj ) − J −1 (Gj ) At =


xj −1 J (Gj + xj ) − J −1 (Gj ) . 2

tT =


xj −1 J (Gj ) + J −1 (Gj + xj ) . 2

This leads to the transition time

To obtain a more accurate approximation, one could use additional supporting points in the interval [Gj , Gj +xj ], giving rise to a finer trapezoid decomposition of the integral. Note that here, the trapezoid method yields a better approximation of the transition time.

Appendix C: Constants of Integration – Initial and Transition Conditions and Updating

The constants of integration c4 , ..., c7 of the NID (see (22)-(24)) and d2 , ..., d5 of the FID (see (31)-(33)) are determined. The NID constants of integration c4 , c6 , and c7 are fixed via the initial condition G(t = tj ) = Gj ,
(n) (n) (n) (n) whereas
c5 is fixed via the transition condition G t = tT (j) | tj ≤ t < tT (j) = G t = tT (j) | tT (j) ≤ t < tT (j) . The initial condition provides continuity of G at the transition from the (j − 1)th injection domain to the jth injection domain at t = tj and the transition condition between the non-linear and linear NID (n) solution branches at t = tT (j). The FID constants of integration d2 , d4 , and d5 are specified by the initial

(f ) (f ) condition G t = tT (j) = GT (j) and d3 by the transition condition G t = tT (j) | tj ≤ t < tT (j) = G t =
(f ) (f ) tT (j) | tT (j) ≤ t < tj+1 . These conditions guarantee continuity between the NID and FID solution branches (f ) at t = tT (j) and between the non-linear and linear FID solution branches at t = tT (j), respectively. Further, the constants of integration have to be updated if there are tT (i) ∈ [tj , tj+1 ) for i ∈ {1, ..., j − 1}, as elements (n) of S1 switch over to S2 . Because these updates can in general cause shifts in the transition times tT and (f ) tT towards larger values, we also have to account for updating conditions that cover the changes of ordering
(n) (f ) tT (j) ↔ min tj+1 , tT (j) and tT (j) ↔ tj+1 . Below, the determination of the NID constant of integration c4 is shown in detail. The computations of the remaining constants of integration are accomplished accordingly.
(n) Applying the initial condition G t = tj | tj ≤ t < tT (j) = Gj to the first branch of (22), we obtain c4 (j) = G3j − 3 N1 (j) tj . The conditions and updated constants for an NID-FID transition of the (k < j)th injection at tj ≤ t = tT (k) < tj+1 are: • First branch Sol.(22) → first branch Sol.(22)

(n) (n) G t = tT (k) | tj ≤ t < tT (j; S2 \{k}) = G t = tT (k) | tj ≤ t < tT (j; S2 ∪ {k}) 



c4 j; S2 ∪ {k} = 3 N1 j; S2 \{k} − N1 j; S2 ∪ {k} tT (k) + c4 j; S2 \{k} • First branch Sol.(22) → Sol.(23)

(n) G t = tT (k) | tj ≤ t < tT (j; S2 \{k}) = G t = tT (k) | tj ≤ t < tT (j; S2 ∪ {k}) 



c6 j; S2 ∪ {k} = 3 N1 j; S2 \{k} − N1 j; S2 ∪ {k} tT (k) + c4 j; S2 \{k} .

19 A change from the first branch of (22) to (15) is not possible because elements that are contained in S2 cannot (n) (n) switch back to S1 . Moreover, since tT (j; S2 \{k}) < tT (j; S2 ∪ {k}), a change from the first to the second branch of (22) can also never occur. Finally, as P1 (j; S2 \{k}) > P1 (j; S2 ∪ {k})

and N1 (j; S2 \{k}) < N1 (j; S2 ∪ {k})

and, thus, p p P1 (j; S2 ∪ {k})/N1 (j; S2 ∪ {k})Gj < P1 (j; S2 \{k})/N1 (j; S2 \{k})Gj , a change from the first branch of (22) to (24) is impossible as well. Appendix D: Components of G(t | 0 ≤ t < ∞) and Initial Values Gi

We state the implicit expressions for the SID, NID, and FID components of (38). First, the SID component is given by s !" h

D0 (s) (s) (s)
GSID (t | 0 ≤ t < t2 ) = H 1 − x1 H tT − t G t | 0 ≤ t < tT ; Sol.(35) H t2 − tT A0 q1

+H t−

s +H

(s)
tT G

(s) t | tT

≤ t < t2 ; Sol.(35)


i

# +H

(s) tT




− t2 G t | 0 ≤ t < t2 ; Sol.(36)




!
D0 x1 − 1 G t | 0 ≤ t < t2 ; Sol.(37) . A0 q1

For the NID component, we have to consider the two cases S2 = ∅ and S2 6= ∅. They yield

GNID t | ti ≤ t < tT (i) ; S2 = ∅ = G t | ti ≤ t < tT (i) ; Sol.(15) and s GNID


t | ti ≤ t < tT (i) ; S2 6= ∅ = H 1 −

P1 (i) Gi N1 (i)

!


(n) H tT (i) − tT (i)

h


i
(n) (n) (n) (n) × H tT (i) − t G t | ti ≤ t < tT (i) ; Sol.(22) + H t − tT (i) G t | tT (i) ≤ t < tT (i) ; Sol.(22)

+H

(n) tT (i)

s ! 

P1 (i) Gi − 1 G t | ti ≤ t < tT (i) ; Sol.(24) . − tT (i) G t | ti ≤ t < tT (i) ; Sol.(23) + H N1 (i)


Last, the FID component reads    

2 N1 (i) Gi (f ) GFID t | tT (i) ≤ t < ti+1 = H 1 − β(i) −1 H ti+1 − tT (i) N2 (i) h



i (f ) (f ) (f ) (f ) × H tT (i) − t G t | tT (i) ≤ t < tT (i) ; Sol.(31) + H t − tT (i) G t | tT (i) ≤ t < ti+1 ; Sol.(31)



(f ) + H tT (i) − ti+1 G t | tT (i) ≤ t < ti+1 ; Sol.(32)



   
2 N1 (i) Gi + H β(i) − 1 − 1 G t | tT (i) ≤ t < ti+1 ; Sol.(33) . N2 (i)

20 Further, we obtain the initial values Gi for all i : 3 ≤ i ≤ m by requiring continuity between the solution branches of the (i − 1)th and ith injection domain. If the ith injection enters the system while the (i − 1)th injection is still in the NID and S2 = ∅, the initial values become
Gi = G t = ti | ti−1 ≤ t < tT (i − 1) ; Sol.(15) for ti < tT (i − 1) , whereas for S2 6= ∅, they result in 
(n) G t = ti | ti−1 ≤ t < tT (i − 1) ; Sol.(22)            
 (n)   G t = ti | tT (i − 1) ≤ t < tT (i − 1) ; Sol.(22)    Gi =   
   G t = ti | ti−1 ≤ t < tT (i − 1) ; Sol.(23)           
 G t = ti | ti−1 ≤ t < tT (i − 1) ; Sol.(24)

p P1 (i − 1)/N1 (i − 1) Gi−1 < 1 for (n) and ti < tT (i − 1) < tT (i − 1) p for P1 (i − 1)/N1 (i − 1) Gi−1 < 1 (n) and tT (i − 1) ≤ ti < tT (i − 1) p for P1 (i − 1)/N1 (i − 1) Gi−1 < 1 (n) and ti < tT (i − 1) ≤ tT (i − 1) for

p P1 (i − 1)/N1 (i − 1) Gi−1 ≥ 1 .

If, however, the (i − 1)th injection is already in the FID, we find 

(f ) G t = ti | tT (i − 1) ≤ t < ti ; Sol.(31) for β(i − 1) 2 N1 (i − 1) Gi−1 /N2 (i − 1) − 1 < 1     (f )  and tT (i − 1) < tT (i − 1) ≤ ti      

Gi = G t = ti | tT (i − 1) ≤ t < ti ; Sol.(32) for β(i − 1) 2 N1 (i − 1) Gi−1 /N2 (i − 1) − 1 < 1   (f )  and tT (i − 1) ≤ ti < tT (i − 1)       

 G t = ti | tT (i − 1) ≤ t < ti ; Sol.(33) for β(i − 1) 2 N1 (i − 1) Gi−1 /N2 (i − 1) − 1 ≥ 1 .

Appendix E: The CS-Function

For z ∈ R≥0 , the CS-function is defined by [14] Z π Z ∞ 1 CS(z) := sin (θ) K5/3 (y) dy dθ π 0 z/ sin (θ)

(E1)

= W0, 4/3 (z) W0, 1/3 (z) − W1/2, 5/6 (z) W−1/2, 5/6 (z) , where Ka is the modified Bessel function and Wa, b denotes the Whittaker function [1]. On account of the degree of complexity of (E1), one usually employs an approximate function that is adapted to its asymptotics ( a0 z −2/3 for z  1 CS(z) ' −1 z exp (−z) for z  1 , where a0 = 1.15. Standard approximations are, therefore, given by CS1 (z) :=

a0 exp (−z) , z 2/3

CS2 (z) :=

a0 exp (−z) , z

and CS3 (z) :=

a0
. z 2/3 1 + z 1/3 exp (z)

Figure 3 shows the absolute values of the relative deviations of these approximations with respect to (E1) in percent, that is, CS(z) − CSn (z) for n ∈ {1, 2, 3} . Dev(z) := 100 CS(z)

21

450

CS1 CS2 CS3

400 350 Dev(z) [%]

300 250 200 150 100 50 0 10 -2

10 -1

10 1

10 0 z

10 2

FIG. 3: Absolute values of the relative deviations of the approximate functions CSn for all n ∈ {1, 2, 3} with respect to the exact CS-function in percent.

Since the CS-function is used for the computation of the synchrotron intensity, where the spectrum covers the energy range from radio waves to X-rays, i.e., with a lower energy limit of the order neV and an upper limit of the order keV, and z = 2 /(3 0 γ 2 ) with 0 ∼ 10−14 b, initial electron energies γi ∼ 104 b−1/3 , and normalized magnetic field strength b ∼ 10−3 up to b ∼ 1, we have to consider values z  1 up to z  1. Thus, the only suitable choice for this study is the approximate function CS3 (z), which coincides with both asymptotic ends of (E1) and has the smallest overall deviation. Appendix F: Lorentz Transformation

Under the Lorentz transformation from a comoving frame to an observer frame, where quantities in the observer frame are denoted with an asterisk, the energy ε ∈ {, s } and time transform according to ε? = D ε and t? =

t D

with the boost factor D=

1
, Γ 1 − β cos (θ? )

p the angle θ? between the jet axis and the line of sight of the observer, the Lorentz factor Γ = 1/ 1 − β 2 , and β = v/c. For blazars, one can assume that θ? → 0 and, thus, s 1+β D' . 1−β As the ratio I/ε3 is Lorentz-invariant, i.e., I(ε, t) I ? (ε? , t? ) = , 3 ε (ε? )3 one directly finds that the intensity and the fluence transform as I ? (ε? , t? ) = D3 I(ε, t) and F ? (ε? ) = D2 F (ε) . We remark that since in the comoving frame the maximum normalized SSC energy in the Thomson limit s,max. = 4 0 γ04 is bounded from above by 1.3 × 104 b−1/3 [31], in the observer frame it is restricted by ?s,max. < 1.3 × 104 D b−1/3 , which is the upper SSC energy bound for both the lightcurve and fluence plots.

22 Appendix G: Synchrotron and SSC Fluences

We compute the total fluences corresponding to the synchrotron intensity (42) and the SSC intensity (46). For this purpose, we derive a general expression for the total fluence that, on the one hand, employs the function G and, on the other hand, explicitly displays the various approximate cases of the Jacobian determinant of the integration measure. For simplicity, the updating of constants is suppressed. With (ε, I, F ) ∈ {(, Isyn. , Fsyn. ), (s , ISSC , FSSC )}, the total fluence F is given by Z ∞ m Z Gi+1 X dt I(ε, t) dt = I(ε, G) F (ε) = dG . (G1) dG 0 i=1 Gi By means of (34), (14), (16), and (25), the Jacobian determinant yields  (G + x1 )2   for 0 ≤ G < G2   D0 (G + x1 )2 + A0 q1         1   for Gi ≤ G < GT (i) and S2 = ∅   M1 (i) + M2 (i) G  dt = dG   G2   for Gi ≤ G < GT (i) and S2 = 6 ∅   P (i) G2 + N (i)  1 1         G3   for GT (i) ≤ G < Gi+1 , 3 D0 G + N1 (i) G + N2 (i)

(G2)

where i : 2 ≤ i ≤ m. Substituting (G2) into (G1) and again using G = G + x1 , we obtain

Z min(Gi+1 ,GT (i)) Z G2 +x1 m X I ε, Gi ≤ G < GT (i) I ε, x1 ≤ G < G2 + x1 G 2 F (ε) = dG dG + χ(S2 = ∅) D0 G 2 + A0 q1 M1 (i) + M2 (i) G x1 Gi i=2 Z m X   + 1 − χ(S2 = ∅)

Gi

i=2

+

m Z X i=2

min(Gi+1 ,GT (i))

Gi+1

min(Gi+1 ,GT (i))


I ε, Gi ≤ G < GT (i) G2 dG P1 (i) G2 + N1 (i)


I ε, GT (i) ≤ G < Gi+1 G3 dG . D0 G3 + N1 (i) G + N2 (i)

In the following, only the second integral is calculated in detail. The remaining integrals can be solved accordingly. Note that the first and third integral are special cases of the fourth integral for N2 ≡ 0 and suitably adapted constants and integration limits. 1.

Synchrotron Fluence

With I = Isyn. (, G) according to (42), the second integral reads
Z min(Gi+1 ,GT (i)) Isyn. , Gi ≤ G < GT (i) I2,i,syn. () := dG M1 (i) + M2 (i) G Gi

= I0,syn. 

1/3

i X l=1

Z

min(Gi+1 ,GT (i))

ql Gi

 −1 2/3 M1 (i) + M2 (i) G Yl (G)  1/3   dG , 2 2 2 2/3 1+ Yl (G) exp Y (G) 3 0 3 0 l

where once more Yl (G) = G−Gl +xl . Substituting τ = 2  Yl2 /(3 0 ) and, since M1 (i)  M2 (i) G, approximating the Jacobian determinant by n   ∞  X  −1 1 M2 (i) G 1 M2 (i) G M1 (i) + M2 (i) G = − ≈ 1− M1 (i) n=0 M1 (i) M1 (i) M1 (i)

23 gives
" r # Z τ min(Gi+1 ,GT (i)) i (3 0 )5/6 I0,syn. −1/2 X 1 M2 (i) 3 0 τ I2,i,syn. () =  ql 1− + Gl − xl M1 (i) M1 (i) 2 211/6 τ (Gi ) l=1 (G3) ×

dτ
. τ 1/6 1 + τ 1/3 exp (τ )

Just like in the evaluation of the energy-integrated intensities (cf. the paragraph above formula (44)), we approximate the integrand for τ ≤ 1 by r   M2 (i) M2 (i) 3 0 1/3 −1/6 1− (Gl − xl ) τ − τ M1 (i) M1 (i) 2 and for τ > 1 by ! r   M2 (i) M2 (i) 3 0 −1/2 1− (Gl − xl ) τ − exp (−τ ) . M1 (i) M1 (i) 2 Thus, (G3) can be approximately solved by i
(3 0 )5/6 I0,syn. −1/2 X ql h  H τ (Gi ) − 1 A1 () I2,i,syn. () ≈ M1 (i) 211/6 l=1

   i


 + H τ min(Gi+1 , GT (i)) − 1 H 1 − τ (Gi ) A2 () + H 1 − τ min(Gi+1 , GT (i)) A3 () with    
1 M2 (i) (Gl − xl ) Γ , τ (Gi ), τ min(Gi+1 , GT (i)) A1 () := 1 − M1 (i) 2 M2 (i) + M1 (i)

r



i 3 0 h exp −τ min(Gi+1 , GT (i)) − exp −τ (Gi ) 2

# "   

1 6 M2 (i) 5/6 (Gl − xl ) Γ , 1, τ min(Gi+1 , GT (i)) + 1 − τ (Gi ) A2 () := 1 − M1 (i) 2 5

+

A3 () :=

M2 (i) M1 (i)

r

  

 3 0 3 exp −τ min(Gi+1 , GT (i)) − exp (−1) − 1 − τ 4/3 (Gi ) 2 4

  
 6 M2 (i) 1− (Gl − xl ) τ 5/6 min(Gi+1 , GT (i)) − τ 5/6 (Gi ) 5 M1 (i)

−

33/2 M2 (i) 25/2 M1 (i)

r


 0  4/3 τ min(Gi+1 , GT (i)) − τ 4/3 (Gi ) 

and the generalized incomplete gamma function defined for a ∈ C and 0 by [1] Z z Γ(a, y, z) := ta−1 exp (−t) dt . y

In the special case a = 1/2, the√generalized incomplete gamma function can be expressed in terms of the error   function, namely Γ(1/2, y, z) = π erf(z) − erf(y) .

24 2.

SSC Fluence

With I = ISSC (s , G) given in (46), the second integral becomes
Z min(Gi+1 ,GT (i)) ISSC s , Gi ≤ G < GT (i) I2,i,SSC (s ) := dG M1 (i) + M2 (i) G Gi

= I0,SSC 1/3 s

i X

 qk ql H

k,l=1

Z

4 − Gi + Gl − xl s

min(Gi+1 ,GT (i),4/s +Gl −xl )

× Gi



 −1  2/3 M1 (i) + M2 (i) G Yk (G) Yl (G)  dG . 1/3   2  2/3 s  s exp Yk (G) Yl (G) 1+ Yk (G) Yl (G) 6 0 6 0

An approximate analytical solution of this integral can be found using the same approach as in the synchrotron case. However, one could also employ the methods for the computation of the NID-FID transition time in Appendix B. To this end, we define the function  2/3 Yk (G) Yl (G) Hkl (s , G) := ,  1/3  2  2/3 s s  Yk (G) Yl (G) 1+ Yk (G) Yl (G) exp 6 0 6 0
 −1 and since Hkl (s , G) ∈ C ∞ R2≥0 and M1 (i) + M2 (i) G is positive and integrable, the first mean value 
 theorem for integration can be applied. Thus, there exists a number ξil ∈ Gi , min Gi+1 , GT (i), 4/s + Gl − xl such that   i X 4 1/3 I2,i,SSC (s ) = I0,SSC s qk ql H − Gi + Gl − xl Hkl (s , ξil ) s k,l=1

Z

min(Gi+1 ,GT (i),4/s +Gl −xl )

× Gi

dG . M1 (i) + M2 (i) G

Choosing the midpoint of the integration interval as an approximate value for ξil yields
!   i X Gi + min Gi+1 , GT (i), 4/s + Gl − xl qk ql 4 1/3 I2,i,SSC (s ) ≈ I0,SSC s H − Gi + Gl − xl Hkl s , M2 (i) s 2 k,l=1


min(Gi+1 ,GT (i),4/s +Gl −xl ) × ln M1 (i) + M2 (i) G . Gi

The trapezoid approximation on the other hand results in the expression
  i X min Gi+1 , GT (i), 4/s + Gl − xl − Gi 4 1/3 − Gi + Gl − xl I2,i,SSC (s ) ≈ I0,SSC s qk ql H s 2 k,l=1

"



# Hkl s , min Gi+1 , GT (i), 4/s + Gl − xl Hkl (s , Gi )
. + × M1 (i) + M2 (i) Gi M1 (i) + M2 (i) min Gi+1 , GT (i), 4/s + Gl − xl

Acknowledgments

The authors are grateful to Horst Fichtner and Michael Zacharias for useful discussions and comments.

25

[1] Abramowitz M., Stegun I. A., “Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables,” National Bureau of Standards (1972). [2] Aharonian F. A. et al., “An exceptional very high energy gamma-ray flare of PKS 2155-304,” The Astrophysical Journal 664, L71 (2007). [3] Aharonian F. A. et al., “Simultaneous multiwavelength observations of the second exceptional γ-ray flare of PKS 2155-304 in July 2006,” Astronomy & Astrophysics 502, 749 (2009). [4] Albert J. et al., “Variable very high energy gamma-ray emission from Markarian 501,” The Astrophysical Journal 669, 862 (2007). [5] Biteau J., Giebels B., “The minijets-in-a-jet statistical model and the rms-flux correlation,” Astronomy & Astrophysics 548, A123 (2012). [6] Blandford R. D., Payne D. G., “Hydromagnetic flows from accretion discs and the production of radio jets,” Monthly Notices of the Royal Astronomical Society 199, 883 (1982). [7] Blandford R. D., Znajek R. L., “Electromagnetic extraction of energy from Kerr black holes,” Monthly Notices of the Royal Astronomical Society 179, 433 (1977). [8] Blazejowski M., Sikora M., Moderski R., Madejski G. M., “Comptonization of infrared radiation from hot dust by relativistic jets in quasars,” The Astrophysical Journal 545, 107 (2000). [9] Blumenthal G. R., Gould R. J., “Bremsstrahlung, synchrotron radiation and Compton scattering of high-energy electrons traversing dilute gases,” Reviews of Modern Physics 42, 237 (1970). [10] B¨ ottcher M., “Modeling the emission processes in blazars,” Astrophysics and Space Science 309, 95 (2007). [11] B¨ ottcher M., Reimer A., Sweeney K., Prakash A., “Leptonic and hadronic modeling of Fermi-detected blazars,” The Astrophysical Journal 768, 54 (2013). [12] Cerruti M., Zech A., Boisson C., Inoue S., “Lepto-hadronic modelling of blazar emission,” Proceedings of the Annual meeting of the French Society of Astronomy and Astrophysics, 555 (2011). [13] Cerruti M., Zech A., Boisson C., Inoue S., “A hadronic origin for ultra-high-frequency-peaked BL Lac objects,” Monthly Notices of the Royal Astronomical Society 448, 910 (2015). [14] Crusius A., Schlickeiser R.,“Synchrotron radiation in a thermal plasma with large-scale random magnetic fields,” Astronomy & Astrophysics 196, 327 (1988). [15] Cui W., “X-ray flaring activity of Markarian 421,” The Astrophysical Journal 605, 662 (2004). [16] Dermer, C. D., Schlickeiser, R., “Model for the high-energy emission from blazars,” The Astrophysical Journal 416, 458 (1993). [17] Felten J. E., Morrison P., “Omnidirectional inverse Compton and synchrotron radiation from cosmic distributions of fast electrons and thermal photons,” The Astrophysical Journal 146, 686 (1966). [18] Ghisellini G., Tavecchio F., Bodo G., Celotti A., “TeV variability in blazars: how fast can it be?,” Monthly Notices of the Royal Astronomical Society 393, L16 (2009). [19] Giannios D., Uzdensky D. A., Begelman M. C., “Fast TeV variability in blazars: jets in a jet,” Monthly Notices of the Royal Astronomical Society 395, L29 (2009). [20] Graff P. B., Georganopoulos M., Perlman E. S., Kazanas D., “A multizone model for simulating the high-energy variability of TeV blazars,” The Astrophysical Journal 689, 68 (2008). [21] Jones F. C., “Calculated spectrum of inverse-Compton-scattered photons,” Physical Review 167, 1159 (1968). [22] Kardashev N. S., “Nonstationariness of spectra of young sources of nonthermal radio emission,” Soviet Astronomy 6, 317 (1962). [23] Katarzynski K., Ghisellini G., Mastichiadis A., Tavecchio F., Maraschi L., “Stochastic particle acceleration and synchrotron self-Compton radiation in TeV blazars,” Astronomy & Astrophysics 453, 47 (2006). [24] Li H., Kusunose M., “Temporal and spectral variabilities of high-energy emission from blazars using synchrotron self-Compton models,” The Astrophysical Journal 536, 729 (2000). [25] Marscher A. P., “Turbulent, extreme multi-zone model for simulating flux and polarization variability in blazars,” The Astrophysical Journal 780, 87 (2014). [26] Pohl M., Schlickeiser R., “On the conversion of blast wave energy into radiation in active galactic nuclei and gammaray bursts,” Astronomy & Astrophysics 354, 395 (2000). [27] Reynolds S. P., “Theoretical studies of compact radio sources. II. Inverse-Compton radiation from anisotropic photon and electron distributions: General results and spectra from relativistic flows,” The Astrophysical Journal 256, 38 (1982). [28] R¨ oken C., Schlickeiser R., “Synchrotron self-Compton flaring of TeV blazars II. Linear and nonlinear electron cooling,” Astronomy & Astrophysics 503, 309 (2009). [29] R¨ oken C., Schlickeiser R., “Linear and nonlinear radiative cooling of multiple instantaneously injected monoenergetic relativistic particle populations in flaring blazars,” The Astrophysical Journal 700, 1 (2009). [30] Schlickeiser R., R¨ oken C., “Synchrotron self-Compton flaring of TeV blazars I. Linear electron cooling,” Astronomy & Astrophysics 477, 701 (2008). [31] Schlickeiser R., “Non-linear synchrotron self-Compton cooling of relativistic electrons,” Monthly Notices of the Royal Astronomical Society 398, 1483 (2009). [32] Schlickeiser R., Lerche I., “Nonlinear radiative cooling of relativistic particles under equipartition conditions I. Instantaneous monoenergetic injection,” Astronomy & Astrophysics 476, 1 (2007).

26 [33] Schlickeiser R., B¨ ottcher M., Menzler U.,“Combined synchrotron and nonlinear synchrotron-self-Compton cooling of relativistic electrons,” Astronomy & Astrophysics 519, A9 (2010). [34] Sikora M., Begerlman M. C., Rees M. J ., “Comptonization of diffuse ambient radiation by a relativistic jet: The source of gamma rays from blazars?,” The Astrophysical Journal 421, 153 (1994). [35] Sokolov A., Marscher A. P., McHardy I. M., “Synchrotron self-Compton model for rapid nonthermal flares in blazars with frequency-dependent time lags,” The Astrophysical Journal 613, 725 (2004). [36] Urry, C. M., Padovani, P., “Unified schemes for radio-loud active galactic nuclei,” Publications of the Astronomical Society of the Pacific 107, 803 (1995). [37] Weidinger M., Spanier F., “A self-consistent and time-dependent hybrid blazar emission model,” Astronomy & Astrophysics 573, A7 (2015). [38] Yan D., Zhang L., “Understanding the TeV emission from a distant blazar PKS 1424 + 240 in a lepto-hadronic jet model,” Monthly Notices of the Royal Astronomical Society 447, 2810 (2015). [39] Zacharias M., Schlickeiser R., “Synchrotron lightcurves of blazars in a time-dependent synchrotron-self Compton cooling scenario,” The Astrophysical Journal 777, 109 (2013).