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

Flaring of Blazars from an Analytical, Time-dependent Model for Combined Synchrotron and Synchrotron Self-Compton Radiative Losses of Multiple Relativistic Electron Populations Christian R¨oken∗ Universit¨ at Regensburg, Fakult¨ at f¨ ur Mathematik, 93040 Regensburg, Germany

Florian Schuppan Ruhr-Universit¨ at Bochum, Institut f¨ ur Theoretische Physik, Lehrstuhl IV: Weltraum- und Astrophysik, 44780 Bochum, Germany

Katharina Proksch

arXiv:1609.00941v2 [astro-ph.HE] 14 Feb 2017

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

Sebastian Sch¨oneberg Ruhr-Universit¨ at Bochum, Institut f¨ ur Theoretische Physik, Lehrstuhl IV: Weltraum- und Astrophysik, 44780 Bochum, Germany (Dated: February 2017) ABSTRACT. A fully analytical, time-dependent leptonic one-zone model that describes a simpli-

fied radiation process of multiple interacting relativistic electron populations and accounts for the flaring of blazars is presented. In this model, several mono-energetic, relativistic electron populations are successively and instantaneously injected into the emission region and subjected to linear, time-independent synchrotron and nonlinear, time-dependent synchrotron self-Compton radiative losses. The corresponding electron number density is computed analytically by solving a transport equation using an approximation scheme that employs specific asymptotics. Moreover, the optically thin synchrotron intensity, the synchrotron self-Compton intensity in the Thomson limit, as well as the associated total fluences are explicitly calculated. In order to mimic injections of finite duration times and radiative transport, flares are modeled by sequences of these instantaneous injections, suitably distributed over the entire emission region. The total synchrotron and synchrotron selfCompton fluence spectral energy distributions are plotted for a generic three-flare scenario with a set of realistic parameter values, reproducing the typical broad-band behavior seen in observational data.

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 Solution 2. FID Solution C. Solution of the Relativistic Transport Equation for {t ∈ R≥0 }

2 3 4 5 6 8 9

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

12 12 13

IV. Synchrotron and SSC Fluences A. Synchrotron Fluence B. SSC Fluence

14 15 16

V. Lightcurves and Fluence SEDs

17

∗

e-mail: [email protected]

2 VI. Summary and Outlook

18

Acknowledgments

19

A. Laplace Method

19

B. NID-FID Transition Time

20

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

21

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

22

E. The CS Function

23

F. Lorentz Transformations

24

G. Numerical Code for Fluence SEDs

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 [37]. Their main jet outflow in form of magnetized plasmoids, which are assumed to arise in the Blandford-Znajek and the Blandford-Payne process [6, 7], constitutes the major radiation zones. These plasmoids pick up – and interact with – particles of interstellar and intergalactic clouds along their trajectories [27], giving rise to the emission of a series of strong flares. A blazar spectral energy distribution (SED) consists of two broad non-thermal radiation components in different domains. The low-energy spectral component, ranging from radio to optical or X-ray energies, is usually attributed to synchrotron radiation of relativistic electrons subjected to ambient magnetic fields. The origin of the high-energy 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 [17], the broad-line region [35], 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 γ-rays from the decay of neutral pions formed in interactions of protons with ambient matter (see, e.g., [12, 14, 38] and references therein). Mixed models including both leptonic as well as hadronic processes are also considered in the literature (e.g., [13, 39]). A main feature of the non-thermal blazar emission is a distinct variability at all frequencies with different 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 [16] in the X-ray domain. So far, only multi-zone models, which feature an internal structure of the emission region with various radiation zones caused by collisions of moving and stationary shock waves, have been proposed to explain the extreme short-time variability of blazars (see, e.g., [5, 20–22, 26, 36]). In the framework of one-zone models, however, extreme short-time variability can, a priori, not be realized as the duration of the injection into an emission knot of finite size with characteristic radius R0 and the light crossing (escape) time in this region are naturally of the order O(2R0 /(cD)), c being the speed of light in vacuum and D the Doppler factor [18]. This leads to a minimum time scale for the observed flare duration, which may exceed the shorttime variability scale in the minute range by several magnitudes. In particular, using the typical parameters R0 = 1015 cm and D = 10, we find 2R0 /(cD) ≈ 1.9 h in the observer frame. Thus, this type of model can only be used to account for variability on larger time scales, that is, from years down to hours. Both one-zone and multi-zone models have been studied extensively over the last decades in order to explain the variability and the SEDs of blazars, incorporating leptonic and hadronic interactions. In these studies, analytical as well as numerical approaches were used, containing, among others, various radiative loss and acceleration processes, details of radiative transport, diverse injection patterns, different cross sections for particle interactions, as well as particle decay and pair production/annihilation (see the review article [11] and the references therein). Thus, the literature on this matter is quite comprehensive. However, models featuring pick-up processes of any kind usually assume only a single injection of particles into the emission knot as the cause of the flaring.

3 This may be unrealistic as blazar jets, which can extend over tens of kiloparsecs, most likely intersect with several clouds, leading to multiple injections. In such an intricate situation, it is of particular interest to have a self-consistent description of the particle’s radiative cooling, the radiative transport, et cetera. Therefore, we propose a simple, but fully analytical, time-dependent leptonic one-zone model featuring multiple uniform injections of nonlinearly interacting relativistic electron populations, which undergo combined synchrotron and SSC radiative losses (for previous works, see [29, 30]). 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 mono-energetic, spatially isotropically distributed 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. This is the first time an analytical model that describes combined synchrotron and SSC radiative losses of several subsequently injected, interacting injections is presented (for a minor, purely numerical study on multiple injections see [25]). 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 emission region, injections of further particle populations into an already cooling system give rise to alterations of the overall cooling behavior [30, 40]. Moreover, as we do employ Dirac distributions for the time profile of the source function of our evolution equation and, further, do not consider any details of radiative transport, we mimic injections of finite duration times and radiative transport by partitioning each flare into a sequence of instantaneous injections, which are appropriately distributed over the entire emission region, using the quantities computed here. 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 intensity, the SSC intensity in the Thomson limit, as well as the corresponding total fluences are calculated in Sections III and IV. We explain how to mimic finite injection durations and radiative transport by multiple use of our results in Section V, and show that the model accounts for the characteristic broad-band fluence SED behavior of blazars obtained from observational data. Section VI concludes with a summary and an outlook. Supplementary material, which is required for the computations of the electron number density and the synchrotron and SSC intensities, is given in Appendices A-F. Moreover, in Appendix G, we briefly describe the numerical code used for the creation of the fluence SED plot. II.

THE RELATIVISTIC TRANSPORT EQUATION

The transport equation describing the temporal evolution and energy dependence of the volume-averaged differential electron number density n = n(γ, t) of m relativistic, mono-energetic, instantaneously injected and 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 [24] m
X ∂n ∂ − Ln = qi δ(γ − γi ) δ(t − ti ) , ∂t ∂γ i=1

(1)

where δ(·) is the Dirac distribution, qi the ith injection strength, γi := Ee,i /(me c2 )  1 the ith normalized initial electron energy, and ti the ith injection time for 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 B) and a nonlinear, time-dependent SSC cooling process in the Thomson limit are considered   Z ∞ L = Lsyn. + LSSC = γ 2 D0 + A0 γ 2 n dγ . (2) 0

The quantities D0 = 1.3 × 10−9 b2 s−1 and A0 = 1.2 × 10−18 b2 cm3 s−1 are the respective cooling magnitudes depending on the magnetic field strength b := kBk/Gauss = const., which is normalized to one Gauss [9, 32]. In the present context, the term linear means that the synchrotron losses do not depend on the electron number density n, thus, resulting in a linear term in the partial differential equation (PDE) (1), whereas the SSC loss term depends on an energy integral containing n, yielding a nonlinearity. The synchrotron loss term can, in principle, be modeled as nonlinear and time-dependent, too. This can be achieved by making an equipartition assumption between the magnetic and particle energy densities [33]. 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 [32]. Therefore, restricting this model to GeV blazars, it is sufficient for the evolution of the electron number density to employ SSC radiative losses in

4 the Thomson limit. As a consequence, the normalized initial electron energies have to be bounded from above by γi < 1.9 × 104 b−1/3 . 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 energy and time points (γi , ti )i∈{1,...,m} . They are to be understood in the distributional sense, that is, they are linear functionals on the space of (smooth) test 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 them 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 qi δ(x − xi ) δ(t − ti ) , +J = ∂t ∂x i=1

(3)

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
1, where Q = Q(γ, t) is a more suitable source function, one cannot simply use Green’s method. Applying successive Laplace transformations with respect to x and G to Eq.(6) yields the solution R(x, G) =

m X

qi H(G − Gi ) δ(x − xi − G + Gi ) ,

i=1

where H(·) is the Heaviside step function ( H(k − k0 ) :=

0 1

for k < k0 for k > k0

(7)

5 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 dG qi H(G − Gi ) = 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 
2   dt  G − G1 + x1      !     dG q q 1 2   = D0 + A0 
2 +
2   dt G − G1 + x1 G − G2 + x2    ..    .           dG    dt = D0 + A0

.. .

for

G1 ≤ G < G2

for

G2 ≤ G < G3

.. .

q1 G − G1 + x1


2 +

.. .

q2 G − G2 + x2


2 + . . . +

!

qm G − Gm + xm

for


2

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. Having a separate analytical solution for each ODE of (9), in Section II C, these are glued together successively requiring continuity at the transition points. Finally, with (7), the electron number density results in n(γ, t) = γ

−2

R(γ, t) = γ

−2

m X






qi H G(t) − Gi δ

i=1

1 1 − − G(t) + Gi γ γi



(10) =

m X

qi H (t − ti ) δ γ −

i=1

B.

γi
1 + γi G(t) − Gi

! .

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

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

(11)

We now introduce the time-dependent sets S1 and S2 that contain injections with G − Gi  xi and G − Gi  xi for i : 1 ≤ i ≤ j, respectively. In order to rewrite Eq.(11) in terms of these sets, we continue their domains of validity 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 represented formally by X dG = D 0 + A0 dt

k∈S1

qk G − Gk + xk


2 + A0

ql

X l∈S2

G − Gl + xl

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

|z| < 1 ,


2 .

(12)

6 we can express Eq.(12) as  −2 −2 X qk  Gl − xl dG A0 X G − Gk ql 1 − + 2 = D0 + A0 1+ dt x2k xk G G l∈S2

k∈S1

= D0 + A0

  n p ∞ ∞ X X G − Gk Gl − xl (n + 1) qk A0 X X − (p + 1) q + . l x2k xk G2 G p=0 n=0 l∈S2

k∈S1

This ODE is suitably approximated by considering only the leading- and next-to-leading-order terms    X qk  Gl − xl dG G − Gk A0 X ql 1 + 2 ≈ D0 + A0 1−2 + 2 dt x2k xk G G l∈S2

k∈S1

(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

k∈S1

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

l∈S2

X

ql (Gl − xl )

l∈S2

depend on the numbers of elements of S1 and S2 , which may change over time. Note that the explicit dependences of these constants on the actual number of injections as well as numbers of elements of S1 and S2 are suppressed in the subsequent calculations for simplicity if possible and given if necessary. An approximate analytical solution of Eq.(13) can be obtained by first computing
asymptotic – and then continued – solutions for the near-injection
domain (NID) Gj ≤ G < min Gj+1 , GT (j) and for the far-injection domain (FID) min Gj+1 , GT (j) ≤ G < Gj+1 , which, in case GT (j) < Gj+1 , are glued together continuously at the transition point GT = GT (j) := Gj + xj that defines the transition between S1 and S2 for the jth injection and corresponds to the transition time tT = tT (j) specified in Appendix B. For Gj+1 ≤ GT (j), we regard solely the NID solution. In the following, the general strategy is to use the leading- and next-to-leading-order contributions in Eq.(13) only if a single 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. 1.

NID Solution


In the NID Gj ≤ G < min Gj+1 , GT (j) , 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 = ∅, Eq.(13) yields
dG = M1 (j) + M2 (j) G for Gj ≤ G < min Gj+1 , GT (j) . dt This ODE can be solved easily using separation of variables. We obtain Z dG 1 = ln (M1 + M2 G) = t + c1 , c1 = const. ∈ R , M1 + M2 G M2

(14)

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 |

7 and, hence, M1 (j) G(t) = − |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, Eq.(13) results in
N1 dG (16) = P1 + 2 for Gj ≤ G < min Gj+1 , GT (j) , 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 1 N1 P1 dG = = G− = t + c3 (17) G − N1 arctan G N 1 + P1 G 2 P1 N1 + P1 G2 P1 P1 N1 with c3 = const. ∈ R. 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 (in case both asymptotic ends exist for G : Gj ≤ G < min(Gj+1 , GT )), 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 .

(19)

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

(20)

This value specifies the upper bound of the domain of validity of (18) and the lower bound of the domain of validity of (19). The corresponding transition time can be directly computed by substituting (20) into (17), in which the constant c3 had to be fixed via the initial condition G(t = tj ) = Gj resulting in r r ! N1 P1 1 c3 = Gj − arctan Gj − tj . P1 P1 N1 This yields (n) tT

1 = tj + P1

r

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

(21) (n)

We point out that in general, one does not have the ordered sequence Gj < GT < GT because GT can in principle be larger than or equal to GT as well as smaller than or equal to Gj . These cases arise when only (n) one asymptotic end of the arctan function exists. Hence, for Gj < GT (j) < GT (j), 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) G(t) = (22)

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

for Gj < GT (j) ≤ GT (j) by
1/3 G(t) = 3 N1 (j) t + c6 (j)


for tj ≤ t < min tj+1 , tT (j) ,

c6 = const. ∈ R ,

(23)

8 (n)

and for GT (j) ≤ Gj 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 continually check during the evolution of G ∈ Gj , min(Gj+1 , GT (j)) whether an injection belongs to the set S1 or the set S2 . Therefore, 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 update at the time of the jth injection verifying 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 accordingly, i.e., the corresponding injection is removed from S1 , while at the same time it is added to S2 . For more details on the updating see Section II C and Appendix C. 2.

FID Solution


In the FID min Gj+1 , GT (j) ≤ G < Gj+1 , the jth injection is, per definition, an element of S2 . Moreover, S1 can only contain injections with comparatively low initial electron energies. However, in the present setting, such injections practically do not occur because the initial electron energies defined in the rest frame of the plasmoid, which primarily result from the relativistic motion of the plasmoid with respect to the ambient medium, are typically of the same order. Consequently, taking into account the leading- and next-to-leading-order terms with respect to S2 , Eq.(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, resulting in an integral equation of the form Z G3 dG = t + d1 , d1 = const. ∈ R . (26) 3 D0 G + N1 G + N2 This integral equation is shown to be approximately equivalent to a specific transcendental equation for which we subsequently derive asymptotic solutions for G that are continued up to an intermediate transition point (if both asymptotic ends exist, else we only consider a continuation of the solution of the present asymptotic end over the entire domain), 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 G3 dG G 1 N1 G + N2 G 1 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 D D 0 0 D0 G N2 1+ 1+ N1 N1 G

Z

dG 1+

D0 N1

 G−

N2 2 N1

2 −

D0 N22 4 N13

≈

Z

dG ∞  2 X

D0 G 1+ N1

G 1 − D0 D0

Z

n=0

N2 − N1 G

n

dG  2 . D0 N2 1+ G− N1 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

9 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 G N1 2 N1 G − 3/2 arctan β −1 = t + d1 . D0 N2 D0

(28)

In order to find an approximate analytical solution of this transcendental equation, we determine G asymptotically for both small and large arguments of the arctan function. Therefore, 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 ,

d0 = const. ∈ R .

Extending the domains of these solutions up to – and connecting them continuously at – the transition point r N1 N2 (f ) GT = + , (29) D0 2 N1 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 tT = tT + + −1 , (30) D0 2 N1 D0 4 N2 (f )

we obtain for GT (j) < GT (j) < Gj+1 the piecewise-defined solution  (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 , (f )

(31) d2 , d3 = const. ∈ R . (f )

In case only one asymptotic end exists, that is for GT (j) < Gj+1 ≤ GT (j) and GT (j) ≤ GT (j), we get G(t) = 3 N1 (j) t + d4


1/3

+

N2 (j) 2 N1 (j)

for tT (j) ≤ t < tj+1 ,

d4 = const. ∈ R ,

(32)

and G(t) = D0 t + d5 (j)

for tT (j) ≤ t < tj+1 ,

d5 = const. ∈ R ,

(33)

respectively. 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 N1 2 N1 (Gj + xj ) d1 = −tT + Gj + xj − arctan β −1 D0 D0 N2 in Eq.(28) by imposing the boundary condition G(t = tT ) = GT , 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}. 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

10 (f )

• Sol.(31)

for tT (1) < tT (1) < t2 ,

• Sol.(32)

for tT (1) < t2 ≤ tT (1) ,

• Sol.(33)

for tT (1) ≤ tT (1)

(f )

(f )

with j = 1. We point out that in this domain, Eq.(8) can also be solved by directly applying separation of variables, resulting in [34] s ! Z Z G2 G G 1 dG (A0 q1 )1/2 D0 = dG = − − arctan G = t + e1 , (34) 3/2 D0 G 2 + A0 q1 D0 D0 D0 A0 q 1 D0 G 2 D 0 1+ A0 q 1 where G := G + x1 and e1 = const. ∈ R. Using the asymptotic-branch-gluing method (cf. Sections II B 1 and II B 2) with the transition time " "s #!# r π A0 q1 D0 1 (s) −x1 + 1 − + arctan x1 tT = D0 D0 4 A0 q 1 (s)

leads for 0 < tT < t2 to the approximate solution ( (3 A0 q1 t + e2 )1/3 − x1 G(t) = D0 t + e3

(s)

for 0 ≤ t < tT (s) for tT ≤ t < t2 ,

(35)

e2 , e3 = const. ∈ R ,

(s)

for 0 < t2 ≤ tT to G(t) = (3 A0 q1 t + e4 )1/3 − x1

for 0 ≤ t < t2 ,

e4 = const. ∈ R ,

(36)

(s)

and otherwise for tT ≤ 0 to G(t) = D0 t + e5

for 0 ≤ t < t2 ,

e5 = const. ∈ R .

(37)

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

e2 = e4 = e5 = 0 ,

x31

(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

and for tT (2) ≤ t < t3 by

• Sol.(15)

for S2 = ∅ ,

• Sol.(22)

for S2 6= ∅ and t2 < tT (2) < tT (2) ,

• Sol.(23)

for S2 6= ∅ and t2 < tT (2) ≤ tT (2) ,

• Sol.(24)

for S2 6= ∅ and tT (2) ≤ t2 ,

(n)

(n)

(n)

11 (f )

• Sol.(31)

for tT (2) < tT (2) < t3 ,

• Sol.(32)

for tT (2) < t3 ≤ tT (2) ,

• Sol.(33)

for tT (2) ≤ tT (2)

(f )

(f )

with j = 2. The initial value G2 is fixed by requiring continuity of time of the second injection, yielding 
(s)  G t = t2 | tT ≤ t < t2 ; Sol.(35)      
G2 = G t = t2 | 0 ≤ t < t2 ; Sol.(36)       G t = t | 0 ≤ t < t ; Sol.(37)
2

2

the SID and DID solution branches at the (s)

for 0 < tT < t2 (s)

for 0 < t2 ≤ tT (s)

for tT ≤ 0 .

In addition to the initial DID 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 DID 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 temporal 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+ , M2 2, S2 = ∅ = −2 A0 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) :

 

2 G2 2 A0 q2 A0 q 2 1 + , , M2 2, S2 = {1} = − M1 2, S2 = {1} = D0 + 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 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 the remaining m − 2 injections results in a formal representation of G for t : 0 ≤ t < ∞ given by 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 i=2



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

12 where (

for S2 = ∅ for S2 = 6 ∅

1 0

χ(S2 = ∅) :=

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 stated explicitly in Appendix D. Note that here 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, we calculate the optically thin synchrotron intensity and the SSC intensity in the Thomson limit. The optically thick component of the synchrotron intensity is not considered because it was shown in [31] that, for all frequencies and times, it provides only a small contribution to the high-energy SSC component. A.

Synchrotron Intensity

The optically thin synchrotron intensity Isyn. (, t) for an isotropically distributed electron number density is given by Z ∞ R0 n(γ, t) P (, γ) dγ , (39) Isyn. (, t) = 4π 0 where the function  P (, γ) = P0 2 CS γ



2 3 0 γ 2

 (40)

is the pitch-angle-averaged spectral synchrotron power of a single electron in a magnetic field of normalized strength b,  := Esyn. /(me c2 ) is the normalized synchrotron photon energy, P0 = 8.5 × 1023 eV s−1 , and 0 := 2.3 × 10−14 b [9]. The CS function is discussed in detail in Appendix E. Here, we employ the approximation CS(z) ≈

a0
, z ∈ R≥0 , z 2/3 1 + z 1/3 exp (z)

(41)

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

 1+

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

(42)

2/3

with I0,syn. := 32/3 R0 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 Z max. I¯syn. (t; min. , max. ) = Isyn. (, t) d min.

with a lower integration limit min. corresponding to the energy of the first data point in the fluence SED and an upper integration limit max. defined by the last. Using (42), this quantity yields I¯syn. (t; min. , max. ) =



30 2

4/3 I0,syn.

Z m X qi H (t − ti ) τi,max. τe1/3 de τ, Yi2 (t) 1 + τe1/3 exp (e τ) τi,min. i=1

where τi,min. := 2 min. Yi2 /(3 0 ) and τi,max. := 2 max. Yi2 /(3 0 ). In order to solve the integral, we approximate the integrand by τe1/3 for τe ≤ 1 and exp (−e τ ) for τe > 1. This is justified because the approximated CS function

13 (41) is only adapted to the asymptotics τe  1 and τe  1 of the exact CS function and extrapolated in between. Moreover, since τi,min. and τi,max. depend on the time t, we have to consider the three cases where 1 ≤ τi,min. , τi,min. < 1 ≤ τi,max. , and τi,max. < 1. Accordingly, the integral results in Z

τi,max.

τi,min.

1+

  τe1/3 de τ ≈ H(τi,min. − 1) exp (−τi,min. ) − exp (−τi,max. ) exp (e τ)     3 4/3 1 − τi,min. − exp (−τi,max. ) + exp (−1) + H(τi,max. − 1) H(1 − τi,min. ) 4 h i 3 4/3 4/3 + H(1 − τi,max. ) τi,max. − τi,min. . 4 (43)

τe1/3

B.

SSC Intensity

In the computation of the SSC intensity R0 ISSC (s , t) = 4π

Z

∞

n(γ, t) PSSC (s , γ, t) dγ ,

(44)

0

where PSSC (s , γ, t) is the SSC power of a single electron and s := Es /(me c2 ) is 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, 19] PSSC (s , γ, t) =

4 σT c γ 2 E(s , t) 3

with the Thomson cross section σT = 6.65 × 10−25 cm2 and the total SSC photon energy density E(s , t). For relativistic electrons with γ  1 and synchrotron photon energies in the Thomson regime for which γ   1, the characteristic energy of the SSC-scattered photons is s ≈ 4 γ 2  [19], corresponding to head-on collisions of the synchrotron photons with the electrons [9, 23, 28]. Thus, it is justified to apply a monochromatic approximation in the total SSC 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 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 becomes inevitably isotropically distributed. Substituting the latter formulas into (44) and using Fubini’s theorem, we obtain ISSC (s , t) =

R0 σT s 12 π

Z 0

∞

Isyn. (, t) 

Z

1/


n(γ, t) δ s − 4 γ 2  dγ d ,

(45)

0

in which 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 SSC intensity (45) yields, after having performed both integrations,   2/3 s m qi qj H (t − ti ) H (t − tj ) H 1 − Yj (t) Yi (t) Yj (t) X 1/3 4 ISSC (s , t) = I0,SSC s (46)  1/3   ,  2/3 2 s s  i,j=1 1+ Yi (t) Yj (t) exp Yi (t) Yj (t) 6 0 6 0 where I0,SSC := R0 σT I0,syn. /(22/3 12 π). The double sum, with the index i referring to the ith synchrotron photon population and the index j to the jth electron population, accounts for all combinations of SSC scattering

14 between the various electron and synchrotron photon populations. For the corresponding energy-integrated SSC intensity, we find I¯SSC (t; s,min. , s,max. ) = (6 0 )4/3 I0,SSC

Z

m X qi qj H (t − ti ) H (t − tj )  2 Yi (t) Yj (t) i,j=1

min(τij,max. ,2 Yi2 Yj /(3 0 ))

× τij,min.

τe1/3 de τ, 1 + τe1/3 exp (e τ)

where τij,min. := s,min. (Yi Yj )2 /(6 0 ) and τij,max. := s,max. (Yi Yj )2 /(6 0 ). The integral is given by (43), however, with adapted integration limits.

IV.

SYNCHROTRON AND SSC FLUENCES

We compute the total fluences associated with 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 once more suppressed. With (ε, I, F ) ∈ {(, Isyn. , Fsyn. ), (s , ISSC , FSSC )}, the total fluence F is given by Z

∞

F (ε) =

I(ε, t) dt = 0

m Z X i=1

Gi+1

I(ε, G)

Gi

dt dG . dG

(47)

By means of Eqs.(34), (14), (16), and (25), the Jacobian determinant yields  (G + x1 )2     D0 (G + x1 )2 + A0 q1         1      M1 (i) + M2 (i) G

dt = dG   G2      P1 (i) G2 + N1 (i)         G3   D0 G3 + N1 (i) G + N2 (i)

for 0 ≤ G < G2


for Gi ≤ G < min Gi+1 , GT (i) and S2 = ∅ (48)
for Gi ≤ G < min Gi+1 , GT (i) and S2 6= ∅


for min Gi+1 , GT (i) ≤ G < Gi+1 ,

where i : 2 ≤ i ≤ m. Substituting (48) into (47), we obtain the expression Z F (ε) = 0

G2

Z min(Gi+1 ,GT (i)) m X I(ε, G) (G + x1 )2 I(ε, G) dG + χ(S = ∅) dG 2 D0 (G + x1 )2 + A0 q1 M (i) + M2 (i) G 1 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, G) G2 dG P1 (i) G2 + N1 (i)

(49)

I(ε, G) G3 dG . D0 G3 + N1 (i) G + N2 (i)

In the following, only the second integral of (49) is calculated in detail for both the synchrotron and the SSC case. The remaining integrals can be solved accordingly. Note that the first and the third integral are special cases of the fourth integral for N2 ≡ 0 and suitably adapted constants and integration limits.

15 A.

Synchrotron Fluence

With I = Isyn. (, G) according to (42), the second integral of (49) reads Z min(Gi+1 ,GT (i)) Isyn. (, G) I2,i,syn. () := dG M (i) + M2 (i) G 1 Gi

= I0,syn. 

1/3

i X

Z

min(Gi+1 ,GT (i))

ql

l=1

Gi

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

Using τl := 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) gives " r # Z τl (min(Gi+1 ,GT (i))) i 1 (3 0 )5/6 I0,syn. −1/2 X M2 (i) 3 0 τe  ql I2,i,syn. () = 1− + Gl − xl M1 (i) M1 (i) 2 211/6 τl (Gi ) l=1 (50) ×

de τ
. τe1/6 1 + τe1/3 exp (e τ)

Similar to the evaluation of the energy-integrated intensities (cf. the paragraph directly above formula (43)), we approximate the integrand for τe ≤ 1 by r   M2 (i) M2 (i) 3 0 1/3 −1/6 1− (Gl − xl ) τe − τe M1 (i) M1 (i) 2 and for τe > 1 by ! r   M2 (i) M2 (i) 3 0 −1/2 1− (Gl − xl ) τe − exp (−e τ) . M1 (i) M1 (i) 2 Thus, (50) is solved approximately by I2,i,syn. () ≈

i
(3 0 )5/6 I0,syn. −1/2 X ql h H τl (Gi ) − 1 Al,1 ()  11/6 M1 (i) 2 l=1

   i


 + H τl min(Gi+1 , GT (i)) − 1 H 1 − τl (Gi ) Al,2 () + H 1 − τl min(Gi+1 , GT (i)) Al,3 () with    
1 M2 (i) Al,1 () := 1 − (Gl − xl ) Γ , τl (Gi ), τl min(Gi+1 , GT (i)) M1 (i) 2 M2 (i) + M1 (i)

r





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

M2 (i) Al,2 () := 1 − (Gl − xl ) M1 (i)

M2 (i) + M1 (i)

r

# "  

1 6 5/6 Γ , 1, τl min(Gi+1 , GT (i)) + 1 − τl (Gi ) 2 5

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

16    5/6
 6 M2 (i) 5/6 Al,3 () := 1− (Gl − xl ) τl min(Gi+1 , GT (i)) − τl (Gi ) 5 M1 (i) 33/2 M2 (i) − 5/2 2 M1 (i)

r


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

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

SSC Fluence

With I = ISSC (s , G) given in (46), the second integral of (49) becomes Z min(Gi+1 ,GT (i)) ISSC (s , G) I2,i,SSC (s ) := dG M 1 (i) + M2 (i) G Gi

= I0,SSC 1/3 s

i X k,l=1

Z

 qk ql H

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)  1/3   dG . 2  2/3 s s  1+ Yk (G) Yl (G) Yk (G) Yl (G) exp 6 0 6 0 (51)

An approximate analytical solution of this integral can be derived with the same method as in the case of the synchrotron fluence. However, one could also employ the methods used for the computation of the NID-FID transition time, which are explained in more detail in Appendix B, as follows. For the mean-value-theorem method, we first define the function  2/3 Yk (G) Yl (G) Hkl (s , G) :=  . 1/3  2  2/3 s s  1+ Yk (G) Yl (G) Yk (G) Yl (G) exp 6 0 6 0
 −1 Since Hkl (s , G) ∈ C ∞ R2≥0 and M1 (i) + M2 (i) G is positive and integrable, we can apply the first mean 
 value theorem for integration in (51). Thus, there exists a number ξil ∈ Gi , min Gi+1 , GT (i), 4/s + Gl − xl such that   i X 4 − G + G − x I2,i,SSC (s ) = I0,SSC 1/3 q q H k l i l l Hkl (s , ξil ) s 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 G + min G , G (i), 4/ + G − x q q 4 i i+1 T s l l k l I2,i,SSC (s ) ≈ I0,SSC 1/3 H − Gi + Gl − xl Hkl s , s M2 (i) s 2 k,l=1


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

17 The trapezoid-approximation method on the other hand results in the expression I2,i,SSC (s ) ≈

I0,SSC 1/3 s

i X k,l=1



4 qk ql H − Gi + Gl − xl s




min Gi+1 , GT (i), 4/s + Gl − xl − Gi 2

"



# 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

V.

LIGHTCURVES AND FLUENCE SEDS

To obtain realistic lightcurves, we have to consider injections of finite duration time and radiative transport inside the emission region. But since, for reasons of computational feasibility, we have only employed Dirac distributions for the time profile of our source function and did not concern ourselves with any details of radiative transport, we have to mimic both by modeling each flare via a sequence of suitably distributed, instantaneous injections. In more detail, we partition the ith flare into ni separate injections labeled by a subscript p ∈ {1, ..., ni }, which are induced equidistantly over the entire emission region given by 2 R0 , i.e., the pth injection of the ith flare occurs at the time ti + κip with κip :=

p − 1 2 R0 , ni − 1 c

and for each of which we use the quantities derived in Section III. Thus, we find for the energy-integrated synchrotron intensity Z

max.

I syn. (t; min. , max. ) =

Z Isyn. (, t) d =

min.

ni m X max. X

min.

qi (p) H(t − ti − κip ) Iip (, t) d ,

(52)

i=1 p=1


where qi (p) p∈{1,...,ni } for all i : 1 ≤ i ≤ m is a specific distribution satisfying the constraint qi =

ni X

qi (p)

(53)

p=1

and the function Iip is, according to the original synchrotron intensity (42), defined by 2/3

I0,syn. 1/3 Yip (t) Iip (, t) :=  1/3   2 2 2 2/3 1+ Yip (t) exp Y (t) 3 0 3 0 ip with Yip (t) := G(t) − G(ti + κip ) + xi . Similarly, we can construct a proper formula for the energy-integrated SSC intensity. We refrain from showing a parameter study for lightcurves because an adequate value for ni , which is at least of the order O(104 ), results in numerical computations that would exceed our available CPU time by far. For the associated total fluence SEDs, it is an entirely different matter as we can illustrate their functional shapes using their suprema given by the special case ni = 1 for all i : 1 ≤ i ≤ m, in which each flare is modeled by a single injection (cf. Section IV). In the following, we show by direct calculation that the supremum of the total fluence of the synchrotron intensity Isyn. (, t) defined in (52) indeed coincides, up to a constant factor, with the total fluence of this special case. To this end, we substitute Isyn. (, t) into the expression for the total fluence and employ the variable G, resulting in Z Fsyn. () =

∞

Isyn. (, t) dt = 0

ni m X X i=1 p=1

Z

∞

qi (p)

Iip (, t) dt = ti +κip

ni m X X

Z

∞

qi (p)

i=1 p=1

Iip (, G) G(ti +κip )

From Eq.(11), it can be deduced that the Jacobian determinant is positive and bounded  D0 + A0

−1 j X qi dt 1 ≤ ≤ , 2 x dG D 0 i i=1

dt dG . dG

(54)

18

10 -4

E · F [arb. u. ]

10 -6 10 -8

10 -10 10 -16 10 -14 10 -12 10 -10 10 -8 10 -6 10 -4 10 -2 10 0 10 2 E [GeV] FIG. 1: Total synchrotron (left curve) and SSC (right curve) fluence SEDs for a generic three-flare scenario with parameter values b = 1, D0 = 1.3 × 10−9 s−1 , A0 = 1.2 × 10−18 cm3 s−1 , (ti )i=1,...,3 = (0, 1.5, 3) 2R0 /c, (qi )i=1,...,3 = (1.5 × 105 , 2 × 105 , 5 × 104 ) cm−3 , (xi )i=1,...,3 = (10−4 , 10−4 , 10−4 ), and D = 10.

and since Iip is non-negative, we can bound the fluence (54) from above by Fsyn. () ≤

Z ∞ m ni 1 XX qi (p) Iip (, G) dG . D0 i=1 p=1 G(ti +κip )

e ip := G − G(ti + κip ) + Gi yields Then, applying the transformation G Z ∞ m ni 1 XX e ip ) dG e ip . qi (p) Iip (, G Fsyn. () ≤ D0 i=1 p=1 Gi Because the integral has the same value for fixed i and arbitrary p, we can always drop the subscript p. Hence, by means of the constraint (53), we obtain Z ∞ Z ∞ m  ni m X 1 X X Fsyn. () e i ) dG ei = 1 Fsyn. () ≤ qi (p) qi Ii (, G Ii (, t) dt = , D0 i=1 p=1 D0 i=1 D0 Gi ti

(55)

proving the claim. In Figure 1, we plot the total synchrotron and SSC fluence SEDs for a generic three-flare, SSC-dominated scenario, in which each flare is modeled by a single injection according to (55), using a set of realistic blazar parameter values. The resulting functional shapes indicate that our model can reproduce the typical fluence SED behavior of blazars visible in observational data.

VI.

SUMMARY AND OUTLOOK

We introduced a fully analytical, time-dependent leptonic one-zone model for the flaring of blazars that employs combined synchrotron and SSC radiative losses of multiple relativistic electron populations. More precisely, we derived an approximate analytical solution of the relativistic transport equation of the volume-averaged differential electron number density for several successively and instantaneously injected, mono-energetic, spatially isotropically distributed, interacting electron populations, which are subjected to linear, time-independent synchrotron radiative losses and nonlinear, time-dependent SSC radiative losses in the Thomson limit. Using this solution, we computed the optically thin synchrotron intensity, the SSC intensity in the Thomson limit, as well as the corresponding total fluences. Moreover, we mimicked finite injection durations and radiative transport by modeling flares in terms of sequences of instantaneous injections. Finally, we plotted generic total synchrotron and SSC fluence SEDs for a set of suitable parameter values, showing that the model can reproduce their characteristic broad-band shapes, which are seen in observational data. We point out that the SSC radiative loss term considered here is strictly valid only in the Thomson regime. Nonetheless, it can be generalized to include the full Klein-Nishina cross section. This leads to a model for which similar yet technically more involved methods apply.

19 Further, in order to make our simple analytical model more realistic, terms accounting for spatial diffusion and for electron escape could be added to the transport equation. Also, more elaborate source functions, e.g., with a power law energy dependence, a time dependence in form of rectangular functions for finite injection durations, and with a proper spatial dependence may be considered. However, judging from the complexity of our more elementary analysis, this is most likely only possible via numerical evaluation.

Acknowledgments

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

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 ∞ [ · ] exp (−w x) dx Lw (x) [ · ] := 0

to Eq.(6) gives Z ∞ Z ∞ Z ∞ m X ∂R ∂R exp (−w x) dx + exp (−w x) dx = qi δ(G − Gi ) δ(x − xi ) exp (−w x) dx . ∂G ∂x 0 0 0 i=1 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 + R exp (−w x)|x→∞ − R(0, G) + w K = qi δ(G − Gi ) exp (−w xi ) H(x − xi )|x→∞ − H(x − xi )|x=0 ∂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 < 1.9 × 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. . Accordingly, Eq.(A2) becomes m

X ∂K +wK = qi δ(G − Gi ) exp (−w xi ) . ∂G i=1 Secondly, applying a Laplace transformation with respect to the function G Z ∞ Ls (G) [ · ] := [ · ] exp (−s G) dG 0

(A3)

20 to Eq.(A3) results in Z 0

∞

∂K exp (−s G) dG + w ∂G

∞

Z

K exp (−s G) dG = 0

m X

Z

∞

δ(G − Gi ) exp (−s G) dG . (A4)

qi exp (−w xi ) 0

i=1

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

and integration by parts as before, Eq.(A4) reads 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 for i : 1 ≤ i ≤ m, 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 Eq.(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 ) .

21 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 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 )
xj −1 At = J (Gj + xj ) − J −1 (Gj ) . 2 This leads to the formula tT =


xj −1 J (Gj ) + J −1 (Gj + xj ) 2

for 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 in the present study, 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 nonlinear and linear NID solution branches
(n) at t = tT (j). The FID constants of integration d2 , d4 , and d5 are specified by the initial condition G t = tT (j) =

(f ) (f ) (f ) (f ) GT (j) and d3 by the transition condition G t = tT (j) | tj ≤ t < tT (j) = G t = tT (j) | tT (j) ≤ t < tj+1 . These conditions guarantee continuity between the NID and FID solution branches at t = tT (j) and between the (f ) nonlinear and linear FID solution branches at t = tT (j), respectively. Further, the NID constants of integration 
have to be updated if there are tT (i) ∈ tj , min(tj+1 , tT (j)) for i ∈ {1, ..., j − 1}, as elements of S1 switch over to (n) S2 . Because these updates cause shifts in the NID transition time tT towards larger values, we have to account
(n) for updating conditions that cover the change of order tT (j) ↔ min tj+1 , tT (j) . Below, the determination of the NID constant of integration c4 is shown in detail. The remaining constants of integration are computed
(n) accordingly. 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 the updated constants – for an NID-FID transition of the (i < j)th injection at tj ≤ t =
(n) tT (i) < min tj+1 , tT (j) are: • first branch Sol.(22) → first branch Sol.(22)

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



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

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



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

22 (n)

(n)

Since S2 ∪ {i} = 6 ∅ and tT (j; S2 \{i}) < tT (j; S2 ∪ {i}), it is obvious that transitions from the first branch of (22) to (15), from the first to the second branch of (22), and from the first branch of (22) to (24) are not possible. 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  h

(s)
(s)
(s) (s) GSID (t | 0 ≤ t < t2 ) = H tT H t2 − tT H tT − t G t | 0 ≤ t < tT ; Sol.(35)

+H t−

(s)
tT G

(s)


+ H −tT

(s) t | tT

≤ t < t2 ; Sol.(35)


i

+H

(s) tT




− t2 G t | 0 ≤ t < t2 ; Sol.(36)







G t | 0 ≤ t < t2 ; Sol.(37) .

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 GNID




(n) (n) t | ti ≤ t < tT (i) ; S2 6= ∅ = H tT (i) − ti 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)




(n) − tT (i) G t | ti ≤ t < tT (i) ; Sol.(23) + H ti − tT (i) G t | ti ≤ t < tT (i) ; Sol.(24) .


Last, the FID component reads 


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

+H

(f ) tT (i)




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








(f ) + H tT (i) − tT (i) G t | tT (i) ≤ t < ti+1 ; Sol.(33) .

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)

(n)

for ti < tT (i − 1) < tT (i − 1) (n)

for ti−1 < tT (i − 1) ≤ ti < tT (i − 1) (n)

for ti < tT (i − 1) ≤ tT (i − 1) (n)

for tT (i − 1) ≤ ti−1 .

23 If, however, the (i − 1)th injection is already in the FID, we find 
(f ) (f )  G t = ti | tT (i − 1) ≤ t < ti ; Sol.(31) for tT (i − 1) < tT (i − 1) ≤ ti      
(f ) Gi = G t = ti | tT (i − 1) ≤ t < ti ; Sol.(32) for tT (i − 1) ≤ ti < tT (i − 1)       (f ) G t = t | t (i − 1) ≤ t < t ; Sol.(33)
for tT (i − 1) ≤ tT (i − 1) . i T i Appendix E: The CS Function

For z ∈ R≥0 , the CS function is defined by [15] Z π Z ∞ 1 sin (θ) K5/3 (y) dy dθ CS(z) := 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) ' z −1 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) :=

z 2/3

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

Figure 2 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) 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 ∼ 10, we have to consider values z  1 up to z  1. Thus, the only suitable choice is the approximate function CS3 (z), which coincides with both asymptotic ends of (E1) and has the smallest overall deviation.

450 400 350

CS1 CS2 CS3

Dev(z) [%]

300 250 200 150 100 50 0 10 -2

10 -1

10 0 z

10 1

10 2

FIG. 2: 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.

24 Appendix F: Lorentz Transformations

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 the 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) . Appendix G: Numerical Code for Fluence SEDs

The numerical implementation of G, the synchrotron and SSC intensities, as well as the corresponding total fluence SEDs 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 up. This problem (n) (f ) occurs in its most severe form in the evaluation of the formulas for the transition times tT 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 points and (n) (f ) 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 , and the inverse injection energies xi for i : 1 ≤ i ≤ m, the synchrotron and SSC cooling magnitudes D0 and A0 , the Lorentz boost D of the plasmoid, 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. 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.

25 (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 list for S1 and S2 is updated. 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 synchrotron and SSC intensities at each grid point by simply substituting these values into the corresponding analytical formulas (42) and (46). The associated total fluences are approximated by 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 Section IV. The total synchrotron and SSC fluence SEDs are then computed as the product of the respective energy and fluence. Since the numbers of grid points and injections can be increased arbitrarily, the precision of these computations is limited only by the machine accuracy and the available CPU time.

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