External Compton radiation of SSC cooled electrons

# External Compton emission in blazars of non-linear SSC cooled electrons

Michael Zacharias & Reinhard Schlickeiser Institut für Theoretische Physik, Lehrstuhl IV: Weltraum- und Astrophysik, Ruhr-Universität Bochum, 44780 Bochum, Germany
###### Abstract

The origin of the high-energy component in spectral energy distributions (SED) of blazars is still a bit of a mystery. While BL Lac objects can be rather successfully modeled within the one-zone synchrotron self-Compton (SSC) scenario, the SED of low peaked Flat Spectrum Radio Quasars (FSRQ) is more difficult to reproduce. Their high-energy component needs the abundance of strong external photon sources, giving rise to stronger cooling via the inverse Compton channel, and thusly to a powerful component in the SED. Recently, we were able to show that such a powerful inverse Compton component can also be achieved within the SSC framework. This, however, is only possible if the electrons cool by SSC, which results in a non-linear process, since the cooling depends on an energy integral over the electrons. In this paper we aim to compare the non-linear SSC framework with the external Compton (EC) output by calculating analytically the external Compton component with the underlying electron distribution being either linearly or non-linearly cooled. Due to the additional linear cooling of the electrons with the external photons, higher number densities of electrons are required to achieve non-linear cooling, resulting in more powerful inverse Compton components. If the electrons initially cool non-linearly, the resulting SED can exhibit a dominating SSC over the EC component. However, this dominance depends strongly on the input parameters. We conclude that with the correct time-dependent treatment the SSC component should be taken into account to model blazar flares.

radiation mechanisms: non-thermal – BL Lacertae objects: general – gamma-rays: theory

## 1 Introduction

Combined as blazars, flat spectrum radio quasars (FSRQ) and BL Lacertae objects (BL Lacs) are the most violent subgroup of active galactic nuclei (AGN) from the earth’s point of view in the accepted unification scheme (Urry & Padovanni 1995). The broadband spectral energy distribution of blazars is dominated by two broad non-thermal components. The low-energetic one, peaking usually between the infrared and the X-ray parts, is attributed to synchrotron radiation of highly relativistic electrons, while the process behind the high-energy component, peaking in the -rays, is a matter of ongoing discussions.

Albeit the possibility of a hadronic origin (e.g. Mannheim 1993) for the high-energetic component, most authors favor a leptonic origin of the -radiation (for recent reviews see Böttcher 2007, 2012). If highly relativistic electrons (and also positrons) interact with an ambient photon field, the photons can be inverse Compton scattered to very high energies. Such photon fields can be the synchrotron photons of the same population of electrons (the so-called synchrotron self-Compton effect (SSC), Jones, O’Dell & Stein 1974), or photon sources external to the jet (so-called external Compton models (EC)), like photons directly from the accretion disk surrounding the black hole in the center of the active galaxy (Dermer & Schlickeiser 1993), from the broad line regions (Sikora et al. 1994) or the dusty torus (Blazejowski et al. 2000, Arbeiter et al. 2002).

Especially the SEDs of FSRQs are dominated by the inverse Compton component, i.e. most of the luminosity of this type of blazars is emitted in -rays (e.g. Hayashida et al. 2012 for 3C 279, or Vercellone et al. 2011 for 3C 454.3). Since one can measure several thermal emission components in FSRQs, the EC process seems to be a natural choice to model the SED. Being able to detect unbeamed thermal emission, albeit the strongly boosted non-thermal radiation of the jet, proves the abundance of strong external photon fields. This in turn provides lots of seed photons for the electrons, which cool rather strongly by this process, giving rise to a powerful inverse Compton component.

On the other hand, BL Lacs emit most of their power in the synchrotron component having at most comparable inverse Compton fluxes. Especially high-frequency peaked BL Lacs exhibit a much reduces -ray flux compared to the synchrotron flux. This points towards a main cooling by the synchrotron channel, and the SSC process is successfully used to model BL Lacs (e.g. Acciari et al. 2011 for 1ES 2344+514, or Abramowski et al. 2012 for 1RXS J101015.9-311909).

However, such a strict division of the two blazar types has been called into question, recently. Chen et al. (2012) performed a numerical analysis of the multiwavelength variability of the FSRQ PKS 1510-089. Using a time-dependent code they were unable to find a clear preference of the EC over the SSC process, and state that the SSC process might even be preferable, since it matches the X-ray part of the SED far better than the EC process.

The advantage of the time-dependent numerical treatment over the usual steady-state approach is that such codes naturally implement the time-dependent nature of the SSC process, which is normally forgotten in many theoretical investigations (e.g. Moderski et al. 2005, Nakar et al. 2009) and modeling attempts (e.g. Ghisellini et al. 2009, Aleksic et al. 2012). Imagine an electron population that emits synchrotron radiation and then scatters this self-made radiation up to -rays. The electrons, therefore, lose energy, which means that the emitted synchrotron emission will also be less energetic, as will be the SSC emission. This implies that the cooling rate will also become weaker over time, i.e. the cooling rate is time-dependent. Schlickeiser (2009) gave an analytical expression for the time-dependent SSC cooling rate, which is proportional to an energy integral over the electron distribution function itself.

Schlickeiser, Böttcher & Menzler (2010, hereafter referred to as SBM) combined the new SSC cooling term with the well known synchrotron cooling term, in order to give a more realistic treatment. As mentioned before, the SSC cooling becomes weaker over time, which means that after some time it will be weaker than the standard linear cooling terms, such as synchrotron cooling, since they do not depend on time. They also calculated the synchrotron SED with the remarkable result that the synchrotron component exhibits a broken power-law behaviour. Interestingly, this feature is independent of the electron injection function. While SBM used a -like injection, Zacharias & Schlickeiser (2010) performed the same analytical calculation using a power-law injection. Apart from the high-energy end of the synchrotron component the broken power-law behaviour is the same in both cases. This is due to the fact that any extended form of the electron distribution is quickly quenched into a -like structure, as long as no reacceleration is taken into account. Therefore, the -approach is sort of a late time limit for any extended injection.

Following-up on the results described above, Zacharias & Schlickeiser (2012, hereafter referred to as ZS) calculated analytically for the -approach the emerging SSC SED. They obtained the interesting result that the time-dependent SSC cooling leads, in fact, to a dominating inverse Compton (IC) component without the need for rather extreme parameter settings. Additionally, they found that the SSC component also exhibits a broken power-law, which may also be independent of the injection form.111We note that the type of breaks in the SED that we can naturally account for with our model, are usually described with rather complicated electron distributions, requiring sometimes multiple spectral breaks in the electron source energy distribution with practically no theoretical justification (e.g. Abdo et al. 2011a for Mrk 501).

As we already discussed above, the debate whether SSC, EC or both play an important role especially in FSRQs is not yet settled. Since we showed in ZS that a dominating SSC component is easily achievable, it is just straightforward to include the EC scenario in our approach. Therefore, we will extend the loss rate of SBM with the additional contribution of the external Compton losses. This will be done in section 2. In section 3 we will calculate the resulting intensity and fluence spectrum of the EC component, where the fluence is the time-integrated, i.e. averaged, intensity. The lengthy details of these calculations can be found in appendix B and C. Transforming the results into the form of an SED will be done in section 4, where we will also summarise the results of ZS for the sake of completeness. At the end of this section we will give some example SEDs with all contributions of synchrotron, SSC and EC, and will discuss the results. In section 5 we will summarise the results and present our conclusions.

## 2 The extended loss rate

Since we intend to calculate the spectrum due to external Compton emission, we have to include this type of energy transfer between electrons and photons in the energy loss rate of the relativistic electrons. According to Dermer & Schlickeiser (1993) the pitch-angle averaged loss rate of electrons in an external photon field that is isotropically distributed in the lab frame, is

 |˙γec| = 4cσTu′ecγ23mec2Γ2b(43−13Γ2b) (1) ≈ 16cσTu′ecΓ2b9mec2γ2,

with the speed of light , the Thomson cross section , the energy of an electron at rest , the electron Lorentz factor , and the Lorentz factor of the radiating plasma blob .

We assume that the energy density of the external photons is isotropic in the galactic (primed) frame, and is given by

where is the luminosity of the accretion disk surrounding the central supermassive black hole, is the scattering depth of the ambient medium scattering the accretion disk photons, and is the radius up to where the accretion disk photons are scattered by the ambient medium.

Adding the external cooling term to the synchrotron and SSC cooling terms, we obtain the complete electron loss rate:

 |˙γtot| = |˙γsyn|+|˙γec|+|˙γssc| (3) = D0γ2+16cσTu′ecΓ2b9mec2γ2+A0γ2∞∫0dγγ2n(γ,t) = D0(1+lec)γ2+A0γ2∞∫0dγγ2n(γ,t),

with , the magnetic energy density , and the magnetic field . Schlickeiser (2009) gives the constant , which was obtained during the derivation of the time-dependent SSC cooling term. The parameters are given as , , and . We scaled the radius of the spherical emission blob as .

The non-linearity of the SSC cooling manifests itself in the integral over the volume-averaged electron distribution . The factor

 lec = u′ecuB4Γ2b3=8L′adτscΓ2b3B2R′2scc (4) = 0.09L′46τ−2Γ2b,1b2R′2pc

shows the relative strength of external to synchrotron cooling. For the linear cooling is dominated by the external photons, while for the synchrotron process mainly operates. Apart from , which is scaled as , we scaled the quantities in equation (4) as in their respective cgs-units, and .

We can now formulate the differential equation describing the competition between the injection of ultrarelativistic particles with the source function and the energy losses as described by equation (3) inside the spherical emission region (Kardashev 1962):

 ∂n(γ,t)∂t−∂missing∂γ[|˙γtot|n(γ,t)]=S(γ,t). (5)

In order to keep the problem simple, we assume a monoenergetic instantaneous injection . Inserting equation (3) into equation (5) we obtain

 ∂n∂t−∂missing∂γ⎧⎪⎨⎪⎩⎡⎢⎣D0(1+lec)+A0∞∫0dγγ2n(γ,t)⎤⎥⎦γ2n⎫⎪⎬⎪⎭ =q0δ(γ−γ0)δ(t), (6)

which apart from the factor equals the differential equation that was solved by SBM. We can, therefore, use their solution. The only difference is that we have to insert the factor wherever they have a . The solution is given in cases of the injection parameter , which is proportional to the ratio of the nonlinear to linear cooling at time of injection. We will discuss its implications in section 2.1.

For dominating linear cooling , we get the electron distribution

 n(γ,t,α≪1)=q0δ(γ−γ01+D0(1+lec)γ0t). (7)

If initially the non-linear cooling dominates , we obtain

 n(γ,t,α≫1)=q0H[tc−t] ×δ(γ−γ0(1+3D0(1+lec)γ0α2t)1/3), (8)

which is valid for times

 t≤tc=α3−13D0(1+lec)γ0α2. (9)

For late times the linear cooling takes over and the distribution function is described by a modified linear solution:

 n(γ,t,α≫1)=q0H[t−tc] ×δ⎛⎝γ−γ01+2α33α2+D0(1+lec)γ0t⎞⎠. (10)

### 2.1 The injection parameter α

We have written the solutions of the differential equation (6) dependent on the parameter , which we intend to discuss in greater detail, now.

We call it the injection parameter, and it is defined as the square-root of the ratio of the non-linear cooling term to the linear cooling term at time of injection, i.e.

 α2 = |˙γssc(t=0)||˙γsyn|+|˙γec|=A0q0γ20D0(1+lec) (11) = 2.13×103N50γ24R215(1+lec),

where denotes the number of radiating particles, and we applied the same scaling law for the parameters as above.

For we see that the linear cooling dominates, resulting in the solution (7). If , the non-linear cooling at least initially dominates, giving the solutions (8) and (10) for early and late times, respectively. The time marks the transition from non-linear to linear cooling.

Comparing the above given to the injection parameter obtained by SBM (where EC losses were neglected), we find that

 α=αSBM(1+lec)1/2, (12)

where .

This implies that the contribution of the external photons lowers the possibility for non-linear cooling. Solving equation (11) for the electron density we obtain

 q0=D0(1+lec)A0γ20α2>D0A0γ20α2SBM=q0,SBM. (13)

This also demonstrates the afore mentioned fact that with external Compton losses included it is harder to cool the electrons non-linearly. Equation (13) shows that for the same value of , i.e. the relative strength between linear and non-linear cooling, one needs a higher electron density in the blob compared to the case where the external losses are neglected.

## 3 External Compton fluence

The intensity due to inverse Compton collisions of electrons with external photons is given by

 Iec(ϵs,t)=Rjec(ϵs,t), (14)

with the emissivity (see appendix A)

 jec(ϵs,t)=cϵs4π∞∫0dϵu(ϵ)ϵ∞∫1dγn(γ,t)σ(ϵs,ϵ,γ), (15)

where is the normalized target photon energy in units of the electron rest mass , is the normalized scattered photon energy, is the target photon density, and

 σ(ϵs,ϵ,γ)=3σT4ϵγ2G(q,Γ) (16)

being the Klein-Nishina cross-section (Blumenthal & Gould 1970) with

 G(q,Γ) = G0(q)+Γ2q2(1−q)2(1+Γq), (17) G0(q) = 2qlnq+1+q−2q2, (18) Γ = 4ϵγ, (19) q = ϵsΓ(γ−ϵs). (20)

Thus, we obtain for equation (15)

 jec(ϵs,t)=3cσT16πϵs∞∫0dϵu(ϵ)ϵ2∞∫γmindγn(γ,t)γ2G(q,Γ). (21)

Here

 γmin(ϵs,ϵ)=ϵs2[1+√1+1ϵϵs] (22)

denotes the minimum Lorentz factor for the electrons, below which the electrons would gain energy from the photons.

The fluence is the time integrated intensity spectrum, giving an average of the variability in all bands, and also incorporating that observation times can be much longer than the typical flare duration:

 F(ϵs)=∞∫0dtI(ϵs,t). (23)

The intention of this paper is to calculate analytically the complete SED of blazars. We will therefore stick to a rather simple approach and use the simplest approach possible for the external photon density in the comoving frame:

 u(ϵ)=43Γ2bu′ecδ(ϵ−ϵec), (24)

where is the normalized energy of the target photons in the comoving frame. Using the electron densities (7), (8), and (10) we can calculate the intensity, and afterwards the fluence.

### 3.1 Small injection parameter, α≪1

Using equation (7) in equation (14) we find for the case of :

 Iec(ϵs,τ)=I0ϵs(1+τ)2G(q(τ),Γ(τ),ϵ=ϵec,γ=γ01+τ) ×H[τ]H[γ0−γmin(ϵ=ϵec)(1+τ)], (25)

after performing the simple integrations of the -functions, and substituting . Here we defined ,

 G(q(τ),Γ(τ),ϵ=ϵec,γ=γ01+τ) =G0(q)+ϵ2s(1−q)(1+τ)22γ0(γ0−ϵs(1+τ)), (26)
 Γ(τ) = 4ϵecγ01+τ, (27) q(τ) = ϵs(1+τ)24ϵecγ0(γ0−ϵs(1+τ)), (28) γmin(ϵ=ϵec) = ϵs2[1+√1+1ϵecϵs]. (29)

Inserting equation (25) into equation (23) we obtain after some calculations (see appendix B) the final expression for the external Compton fluence in the case of dominating linear cooling:

 F(ϵs)=F0γ30ϵ3/2ecϵ−1/2s(1+ϵecϵs)3/2(1−ϵsϵcut), (30)

where , and .

In the Thomson-limit the spectrum cuts off at . The Thomson-limit also implies that . Hence, there is no break in the spectrum, and the denominator in equation (30) equals unity.

In the Klein-Nishina-limit we find , and the cut-off at . Thus, the spectrum breaks at .

### 3.2 Large injection parameter, α≫1

#### 3.2.1 Early time limit, t≤tc

For we use equation (8) in equation (14), as well as the same substitution for as above, and obtain after solving the simple integrations

 Iec(ϵs,τ)=I0ϵsH[τc−τ](1+3α2τ)2/3 ×G(q(τ),Γ(τ),ϵ=ϵec,γ=γ0(1+3α2τ)1/3) ×H[τ]H[γ0−γmin(ϵ=ϵec)(1+3α2τ)1/3], (31)

with ,

 G(q(τ),Γ(τ),ϵ=ϵec,γ=γ0(1+3α2τ)1/3) =G0(q)+ϵ2s(1−q)(1+3α2τ)2/32γ0(γ0−ϵs(1+3α2τ)1/3), (32)
 Γ(τ) = 4ϵecγ0(1+3α2τ)1/3, (33) q(τ) = ϵs(1+3α2τ)2/34ϵecγ0(γ0−ϵs(1+3α2τ)1/3). (34)

Using equation (23) with equation (31) the fluence for the early time limit can be calculated. The lengthy details can be found in appendix C. The solution depends on the external photon energy , giving three different cases.
For we have

 F(ϵs)=F0α35ϵs(1+ϵsϵγB)(1+ϵs2(√2−1)γB)4 ×(1−ϵsϵcut), (35)

while for we get

 F(ϵs)=F0α35ϵs(1+ϵsϵγB)5/2(1+ϵecϵs)5/2 ×(1−ϵsϵcut). (36)

The last part is for and becomes

 F(ϵs)=F0α35ϵs(1+ϵsϵγB)5/2(1+ϵs2(√2−1)γB)5/2 ×(1−ϵsϵcut). (37)

Here we used , and .

#### 3.2.2 Late time limit, t≥tc

As before, we can find the intensity for , if we insert equation (10) into equation (14). As in the previous cases we exchange , and get

 Iec(ϵs,τ)=I0ϵsH[τ−τc](1+2α33α2+τ)2 ×G⎛⎜⎝q(τ),Γ(τ),ϵ=ϵec,γ=γ01+2α33α2+τ⎞⎟⎠ ×H[γ0−γmin(ϵ=ϵec)(1+2α33α2+τ)], (38)

with

 G(q(τ),Γ(τ),ϵ=ϵec,γ=γ01+τ) =G0(q)+ϵ2s(1−q)(1+2α33α2+τ)22γ0(γ0−ϵs(1+2α33α2+τ)), (39)
 Γ(τ) = 4ϵecγ01+2α33α2+τ, (40) q(τ) = ϵs(1+τ)24ϵecγ0(γ0−ϵs(1+τ)). (41)

Inserting equation (38) into equation (23) one can obtain the fluence in the late time limit. For the details we refer the reader to appendix C, again. The results depend also on the external photon energy, giving two cases this time.
For we find

 F(ϵs<ϵγB)=F0γ30ϵ3/2ecϵ−1/2s(1+ϵecϵs)3/2(1−ϵsϵγB), (42)

while for we obtain a single power-law in the form

 F(ϵs<ϵγB)=163F0ϵ3/2ecγ30ϵ−1/2s(1−ϵsϵγB). (43)

#### 3.2.3 Total fluence in the case α≫1

Combining the results of the previous sections we obtain the total fluence for the inverse Compton component due to interactions with the ambient radiation field in the case that the electrons are at first cooled non-linearly. We have three different cases depending on the value of the normalized external photon energy .
For we find

 F(ϵs)=F0γ30ϵ3/2ecϵ−1/2s(1+ϵecϵs)3/2(1+ϵsϵγB)2 ×(1−ϵsϵcut). (44)

Secondly, if the fluence becomes

 F(ϵs)=163F0γ30ϵ3/2ecϵ−1/2s(1+ϵsϵγB)(1+ϵecϵs)5/2 ×(1−ϵsϵcut). (45)

Lastly, we obtain for

 F(ϵs)=163F0γ30ϵ3/2ecϵ−1/2s(1+ϵsϵγB)(1+ϵs2(√2−1)γ0)5/2 ×(1−ϵsϵcut). (46)

These are all broken power-laws, where at least one break () depends strongly on the value of , and therefore on the non-linear cooling.

We note that the first case corresponds to the extreme Klein-Nishina-limit (), the second one to the mild Klein-Nishina-limit (), and the last one to the Thomson-limit ().

## 4 The complete SED

Using the results of the last section and of ZS222Some of the results of ZS have printing errors, which we correct here. we are now in a position to present the complete spectral energy distribution in a combined picture of SSC end EC radiation. The inverse Compton component will, therefore, be the sum of the SSC and EC contributions. If these contributions are comparable on some scales the resulting spectrum will deviate from pure power-laws.

In order to give the results in a manner that can be easily compared to data, we will give the SEDs depending on the frequency , which is already transformed to the frame of rest of the host galaxy with the Doppler factor , where is the cosine of the angle between the jet and the line of sight, and is the normalized speed of the plasma blob. The SED is then given by the fluence multiplied with the frequency in units of with the transformed fluence .

We should note that we have to adapt the results of ZS to the case discussed here. That is, we have to include the addition of the external Compton cooling to the linear term. This can be done by replacing any ”” of ZS with ””. Interestingly, the synchrotron component will not be affected by this replacement, while the SSC component gains a factor .

Below we will first present the theoretical SEDs and afterwards give a brief discussion of the results.

### 4.1 Synchrotron SED

From equation (ZS-69) we obtain the synchrotron SED for the case , which is

 f′s(ν′)=5.6⋅1038R15δ4α2γ4(ν′νsyn)1/2e−ν′/νsynerg/s, (47)

with .
The maximum value of the synchrotron SED,

 f′s,max=2.4⋅1038R15δ4α2γ4erg/s, (48)

is attained at

 ν′s,max=12νsyn=2.1⋅1014δbγ24Hz. (49)

Using equation (ZS-82) we can write the synchrotron SED for as

 f′s=5.6⋅1038R15δ4α2γ4(ν′νsyn)1/21+ν′νce−ν′/νsynerg/s. (50)

It peaks at

 ν′s,max=νc=2.9⋅1014δbγ24α2Hz (51)

with the maximum value

 f′s,max=2.4⋅1038R15δ4αγ4erg/s. (52)

### 4.2 Synchrotron self-Compton SED

For ZS found two different versions of the SSC SED depending on the Klein-Nishina parameter . In the Thomson limit () we use equation (ZS-72) and get

 f′SSC(K≪1)=2.8⋅1039R15δ4(1+lec)α4γ4 ×(ν′νT)3/4e−ν′/νTerg/s, (53)

while for the Klein-Nishina limit () with equation (ZS-73) we have

 f′SSC(K≫1)=2.1⋅1040R15δ4(1+lec)α4bγ44 ×(ν′νB)3/4(1+ν′νB)7/4[1−(ν′νγ0)7/12]erg/s, (54)

with the constants , the break frequency (Schlickeiser & Röken 2008), and .
The maximum frequencies

 ν′SSC,max(K≪1)=34νT=1.2⋅1023δbγ44Hz, (55)

and

 ν′SSC,max(K≫1)=34νB=1.7⋅1024δb−1/3Hz (56)

imply the maximum values

 f′SSC,max(K≪1)=1.0⋅1039R15δ4(1+lec)α4γ4erg/s, (57)

and

 f′SSC,max(K≫1)=6.3⋅1039R15δ4(1+lec)α4bγ44erg/s, (58)

respectively.

For the SSC SEDs in the case we have to discuss three different cases depending on the Klein-Nishina parameter , and we begin with the Thomson limit () using equation (ZS-85):

 f′SSC(K≪1)=2.8⋅1039R15δ4(1+lec)α4γ4 ×(ν′νT)3/4(1+α4ν′νT)1/2e−ν′/νTerg/s. (59)

The maximum value

 f′SSC,max(K≪1)=1.6⋅1039R15δ4(1+lec)α2γ4erg/s (60)

is attained at

 ν′SSC,max(K≪1)=14νT=4.0⋅1022δbγ44Hz. (61)

For the mild Klein-Nishina limit () equation (ZS-88) yields

 f′SSC(1≪K≪α3)=1.5⋅1039R15δ4(1+lec)α4γ4 ×(ν′νT)3/4(1+α4ν′νT)1/2(1+ν′νB)13/4[1−(ν′νγ0)13/3]erg/s. (62)

This triple power-law peaks at

 ν′SSC,max(1≪K≪α3)≈112νB=1.9⋅1023δb−1/3Hz, (63)

and reaches a maximum value of

 f′SSC,max(1≪K≪α3)≈1.2⋅1039R15δ4(1+lec)α2b1/3γ24erg/s. (64)

The last case is the extreme Klein-Nishina limit (). We obtain the SED from equation (ZS-91), resulting in

 f′SSC(1≪α3≪K)=1.1⋅1040R15δ4(1+lec)α4bγ44 ×(ν′νB)3/4(1+ν′νB)7/4(1+αν′νγ0)2[1−(ν′νγ0)13/3]erg/s. (65)

The SED peaks with a maximum value

 f′SSC,max(1≪α3≪K)≈3.4⋅1039R15δ4(1+lec)α4bγ44erg/s (66)

at a peak frequency

 ν′SSC,max(1≪α3≪K)≈34νB=1.7⋅1024δb−1/3Hz. (67)

### 4.3 External Compton SED

In the case of the SED can be calculated from equation (30), becoming

 f′ec(ν′)=4.0×1034R15lecδ4α2γ24ϵec ×(ν′νbr)1/2(1+ν′νbr)3