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The Astrophysical Journal, 743:89 (8pp), 2011 December 10 C 2011.

doi:10.1088/0004-637X/743/1/89

The American Astronomical Society. All rights reserved. Printed in the U.S.A.

SYNCHROTRON POLARIZATION AND SYNCHROTRON SELF-ABSORPTION SPECTRA FOR A POWER-LAW PARTICLE DISTRIBUTION WITH FINITE ENERGY RANGE 1

M. Fouka1,2 and S. Ouichaoui2 Research Center in Astronomy, Astrophysics and Geophysics, B.P. 63, Algiers Observatory, Bouzareah, Algiers, Algeria; [email protected] 2 Laboratory of Nuclear Sciences, Faculty of Physics, University of Sciences and Technology H. Boumedi` ene, B.P. 32, 16111 Bab Ezzouar, Algiers, Algeria; [email protected] Received 2011 February 10; accepted 2011 October 24; published 2011 November 23

ABSTRACT We have derived asymptotic forms for the degree of polarization of the optically thin synchrotron and for synchrotron self-absorption (SSA) spectra assuming a power-law particle distribution of the form N (γ ) ∼ γ −p with γ1 < γ < γ2 , especially for a finite high-energy limit, γ2 , in the case of an arbitrary pitch angle. The new results inferred concern more especially the high-frequency range x η2 with parameter η = γ2 /γ1 . The calculated SSA spectra concern instantaneous photon emission where cooling effects are not considered. They have been obtained by also ignoring likely effects such as Comptonization, pair creation and annihilation, as well as magnetic photon splitting. To that aim, in addition to the two usual absorption frequencies, a third possible one has been derived and expressed in terms of the Lambert W function based on the analytical asymptotic form of the absorption coefficient, αν , for the high-frequency range ν ν2 (with ν2 the synchrotron frequency corresponding to γ2 ). We have shown that the latter frequency may not have realistic applications in astrophysics, except in the case of an adequate set of parameters allowing one to neglect Comptonization effects. More detailed calculations and discussions are presented. Key words: gamma-ray burst: general – radiation mechanisms: non-thermal

1. INTRODUCTION Synchrotron radiation is thought to be one of the main emission mechanisms involved in high-energy astrophysical sites such as gamma-ray bursts (GRBs; see, e.g., Katz 1994; Tavani 1996), active galactic nuclei (AGNs; e.g., Biermann & Strittmatter 1987), supernovae remnants (SNRs; Reynolds 1996), and pulsar wind nebulae (PWNs; e.g., Helfand et al. 2001; Komissarov & Lyubarsky 2004). In the astrophysical context, this process is often invoked to be behind relativistic magnetic shocks when a non-thermal ultrarelativistic population of charged particles (mainly electrons and positrons) is generated by Fermi acceleration (Fermi 1949) in highly turbulent regions dominated by intense magnetic fields (Ostrowski & Bednarz 2002). In such sites, as well as in large size media, the emission spectra can be greatly affected by synchrotron self-absorption (SSA) when initially emitted synchrotron photons are absorbed by the same population of relativistic particles before escaping the source region in the presence of strong magnetic fields (Rybicki & Lightman 1979; Granot et al. 1999). High-energy particle densities can also strongly modify the photon emission spectra in two possible ways: (1) by SSA and/or (2) by inverse Compton (IC) radiations. In the latter process, the scattering of synchrotron photons to high energies (in the GeV, TeV ranges) is efficient for large values of the Compton Y parameter (i.e., Y 1, see Rybicki & Lightman 1979). In the current paper, we aim to highlight relevant properties of the preceding emission mechanisms taking place in various astrophysical sites under strong magnetic fields by reporting and discussing new related calculations. It must be noted, however, that likely effects such as Comptonization, pair creation and annihilation, magnetic photon splitting and cooling, were neglected in calculating physical quantities and emission spectra. This assumption leads, indeed, to a simpler picture of a pure optically thin synchrotron (OTS) spectrum in the case of media with small optical depths (τSSA 1) or a pure SSA spectrum for media with large optical depths (τSSA 1). In the following, we first report (in Section 2.1) calculations of the OTS polarization for an arbitrary pitch angle, on the basis of our recent results reported in Fouka & Ouichaoui (2009). Then, we present (in Section 2.2), for the same case, different possible behaviors of SSA spectra depending on the optical depth, τ , before concluding (in Section 3). 2. ARBITRARY PITCH ANGLE As already stated, we make use here of our recent results (Fouka & Ouichaoui 2009) derived for asymptotic forms of the synchrotron spectral power, Pν , the absorption coefficient, αν , and the source function, Sν , for an arbitrary pitch angle to calculate the OTS polarization and SSA spectra. Those results were obtained for highly relativistic electrons described by a power-law distribution of the form N(γ ) ∼ γ −p , with associated universal index p and Lorentz parameter values in the range γ1 < γ < γ2 , especially for finite energies (i.e., finite γ2 ). It must be noted, however, that more realistically, behind relativistic magnetic shocks, one could more appropriately adopt an exponentially decaying power-law particle distribution instead of a power law with sharp energy cutoff. The consideration of such a realistic particle distribution type will be the object of a future treatment. 2.1. Polarization of Optically Thin Synchrotron To calculate the degree of polarization in the case of optically thin media (τ 1), we present calculations of asymptotic forms In of the synchrotron powers for the two emission directions: perpendicular and parallel relative to that of the magnetic field, B. 1

The Astrophysical Journal, 743:89 (8pp), 2011 December 10

Fouka & Ouichaoui

this respect, we first report asymptotic forms of the synchrotron function, F (x), and its complementary function, G(x), for x 1 (Westfold 1959), i.e., aF π 1/2 −x x e F (x) ≈ 1+ 2 x (1) aG π 1/2 −x x e , 1+ G(x) ≈ 2 x where aF = 55/72 and aG = 7/72 are constants. The total spectral power radiated by a power-law particle distribution has been expressed (Fouka & Ouichaoui 2009) as Pν ∝ Zp (x, η) in terms of the dimensionless synchrotron function, Zp (x, η), given by x −(p−1)/2 Zp (x, η) = x x (p−3)/2 F (x )dx . (2) x/η2

In this expression, one has x = ν/(ν1 sin θ ) (with θ depicting the pitch angle and ν1 = (3/4π )γ12 qB/mc), η = γ2 /γ1 , and the function F (x) can be written as (Westfold 1959; Jackson 1962; Rybicki & Lightman 1979) ∞ F (x) = x K5/3 (x )dx , (3) x

in terms of the modified, fractional order, Bessel function K5/3 (x) (Abramowitz & Stegun 1965). Note that Zp (x) ≡ Zp (x, ∞), corresponding to η = ∞ and p > 2. Asymptotic forms of the function Zp (x, η) for finite parameter η and index p > 0 can be written as (Fouka & Ouichaoui 2009) ⎧ κp (η)x 1/3 for x 1 ⎪ ⎪ ⎪ ⎨ for 1 x η2 Cp x −(p−1)/2 Zp (x, η) ≈ (4) ⎪ π −1/2 −p+2 −x/η2 ⎪ −x 2 ⎪ ⎩ x [η e − e ] for x η , 2 where the coefficients κp (η) and Cp are respectively given by (Fouka & Ouichaoui 2009; Rybicki & Lightman 1979) κp (η) =

2F1 (1 − η−p+1/3 ), p − 1/3

p 1 p 19 2(p+1)/2 Γ − Γ + , Cp = p+1 4 12 4 12

(5)

(6)

√ with F1 = π 2 5/3 / 3 Γ(1/3). Similarly to defining the function Zp (x, η) ∼ Pν⊥ + Pν as the total power radiated in the two directions, we define the function Wp (x, η) ∼ Pν⊥ − Pν as the difference between the two powers, with the following expression: x Wp (x, η) = x −(p−1)/2 x (p−3)/2 G(x )dx , (7) x/η2

where G(x) is a complementary synchrotron function given by (Westfold 1959; Jackson 1962; Rybicki & Lightman 1979) G(x) = xK2/3 (x),

(8)

in terms of the modified, fractional order, Bessel function K2/3 (x) (Abramowitz & Stegun 1965). Asymptotic forms of the function Wp (x, η) can be easily derived in a similar way as for the function Zp (x, η). They are expressed by ⎧ 1 ⎪ ⎪ κp (η)x 1/3 for x 1 ⎪ ⎪ ⎨2 −(p−1)/2 for 1 x η2 Wp (x, η) ≈ Dp x (9) ⎪ ⎪ ⎪ π 2 ⎪ ⎩ x −1/2 [η−p+2 e−x/η − e−x ] for x η2 , 2 with the coefficient Dp given by (Rybicki & Lightman 1979)

p p 1 7 Γ . Γ − + 4 12 4 12

Dp = 2

(p−3)/2

2

(10)


Fouka & Ouichaoui

One can remark that for the high-frequency range, x η2 , the functions Zp and Wp have exactly the same asymptotic form. To derive the asymptotic form of Pν ∼ Zp − Wp , and in order to avoid a zero value for the high-frequency range, we need to take into account in Equation (1) the terms in aF /x and aG /x of the functions F (x) and G(x), respectively. We then obtain the following extended asymptotic forms of the functions Zp (x) and Wp (x) for the frequency range x 1, i.e., π −1/2 −x π −3/2 −x x x e − aF e 2 2 π −1/2 −x π −3/2 −x −(p−1)/2 Wp (x) ≈ Dp x − e − aG e . x x 2 2

Zp (x) ≈ Cp x −(p−1)/2 −

(11)

In addition, from the relations (Fouka & Ouichaoui 2009) Zp (x, η) = Zp (x) − η−p+1 Zp (η−2 x) Wp (x, η) = Wp (x) − η−p+1 Wp (η−2 x),

(12)

we also deduce asymptotic forms of the functions Zp (x, η) and Wp (x, η) for the high-frequency range x η2 , i.e., π −1/2 −p+2 −x/η2 π −3/2 −p+4 −x/η2 −x Zp (x, η) ≈ [η e − e ] + aF [η e − e−x ] x x 2 2 π −1/2 −p+2 −x/η2 π −3/2 −p+4 −x/η2 −x x x [η e − e ] + aG [η e − e−x ]. Wp (x, η) ≈ 2 2

(13)

From the latter expressions, one can deduce asymptotic forms for the high-frequency range, x η2 , for the two emission directions, i.e., π −1/2 −p+2 −x/η2 1 ⊥ x [η e − e−x ] Pν ∝ [Zp (x, η) + Wp (x, η)] ≈ 2 2 (14) 1 1 π −3/2 −p+4 −x/η2 −x x [η e − e ]. Pν ∝ [Zp (x, η) − Wp (x, η)] ≈ 2 3 2 Finally, we get

Pν⊥

and

⎧ 3 ⎪ 1/3 ⎪ ⎪ κp (η)x ⎪ 4 ⎪ ⎪ ⎪ ⎨ 1 [Cp + Dp ]x −(p−1)/2 ∝ 2 ⎪ ⎪ ⎪ ⎪ ⎪ π −1/2 −p+2 −x/η2 ⎪ ⎪ ⎩ x [η e − e−x ] 2

for x 1 for 1 x η2

(15)

for x η2

⎧ 1 ⎪ ⎪ κp (η)x 1/3 for x 1 ⎪ ⎪ 4 ⎪ ⎪ ⎪ ⎨1 [Cp − Dp ]x −(p−1)/2 for 1 x η2, Pν ∝ 2 ⎪ ⎪ ⎪ ⎪ ⎪ 1 π −3/2 −p+4 −x/η2 ⎪ ⎪ ⎩ x [η e − e−x ] for x η2 3 2

(16)

for the synchrotron power plotted in Figure 1. One can define the ratio function, Λp (x, η) = Pν /Pν⊥ , admitting the asymptotic forms ⎧ 1 ⎪ ⎪ ⎪ ⎪ 3 ⎪ ⎪ ⎪ ⎪ ⎨ 2 Λp (x, η) ≈ 3p + 5 ⎪ ⎪

⎪ 2 ⎪ ⎪ 1 η−p+4 e−x/η − e−x 1 2 −1 ⎪ ⎪ ⎪ ⎩ 3x η−p+2 e−x/η2 − e−x ≈ 3 η x

for x 1 for 1 x η2

(17)

for x η2 .

This function and corresponding asymptotic forms are plotted in Figure 2 for the three energy ranges, for index p = 2.5 and parameter η = 1000. 3


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Figure 1. Plot of the spectral powers, Pν⊥ and Pν , for the directions perpendicular and parallel to the magnetic field, respectively, for p = 2.5 and η = 1000, in the case of an arbitrary pitch angle.

Figure 2. Plot of the ratio function, Λp (x, η) = Pν /Pν⊥ , for p = 2.5 and η = 1000, in the case of an arbitrary pitch angle. Corresponding asymptotic forms are also plotted.

Finally, asymptotic forms of the degree of polarization can be written as ⎧ 1 ⎪ ⎪ ⎪ ⎪ ⎪2 Wp (x, η) ⎨ p + 1 ≈ Πp (x, η) = ⎪ Zp (x, η) ⎪ p + 7/3 ⎪ ⎪ ⎪ ⎩ 1

for x 1 for 1 x η2 .

(18)

for x η2

Note that the derived asymptotic forms for the middle range, 1 x η2 , are valid for index p > 1/3 corresponding to defined values of the coefficients Cp and Dp (see, e.g., Rybicki & Lightman 1979; Westfold 1959). Furthermore, from the asymptotic form of the function Λp (x, η) for the high-frequency range, x η2 , and given that Π = (1 − Λ)/(1 + Λ), we get Πp (x, η) =

1 − 13 η2 x −1 1 + 13 η2 x −1

for

x η2 .

(19)

In Figure 3 is plotted the degree of polarization, Πp (x, η), together with the corresponding asymptotic forms for the three frequency ranges, for index p = 2.5 and parameter η = 1000. 4


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Figure 3. Plot of the degree of polarization, Πp (x, η) = (Pν⊥ − Pν )/(Pν⊥ + Pν ), for optically thin synchrotron, for index p = 2.5 and η = 1000, in the case of an arbitrary pitch angle.

2.2. Synchrotron Self-absorption Spectra For homogeneous electron densities along the photons’ path inside the medium, strong magnetic fields (but with intensities much lower than the quantum critical limit, B Bcr = e¯h/m2 c3 = 4.413 × 1013 G, such that magnetic photon splitting and pair creation could be avoided) and moderate electron densities (to avoid Comptonization effects), the instantaneous SSA spectrum can be written as (Rybicki & Lightman 1979) Iν ∝ Sν (1 − e−τν ). (20) It is proportional to the dimensionless function Fp (x, η, τp ), defined by Fp (x, η, τp ) = Yp (x, η)[1 − e−τp Hp (x,η) ],

(21)

with the optical depth parameter, τp , given by (Rybicki & Lightman 1979; Fouka & Ouichaoui 2009) q

π −(p+4) τp = αp (γ1 ) = √ . (p + 2)Cγ1 3 3 B sin θ

(22)

The functions Hp (x, η) and Yp (x, η) correspond, respectively, to the absorption coefficient and the source function, and are defined by Fouka & Ouichaoui (2009) Hp (x, η) = x −2 Zp+1 (x, η) (23) Zp (x, η) , Zp (x, η) = x2 Yp (x, η) = Hp (x, η) Zp+1 (x, η) while the constant C can be written in the form (Fouka & Ouichaoui 2009) p−1

C=

(p − 1)γ1 n, 1 − η−p+1

(24)

in terms of the index, p, and the total particle number density, n. Then, one gets the following expression for the parameter τp : π (p2 + p − 2)γ1−5 qn

. τp = √ 1 − η−p+1 B sin θ 3 3

(25)

It is this crucial quantity (the optical depth parameter, τp ), on which the final SSA spectrum is strongly dependent, that determines the SSA absorption frequency. Now we derive, through the function Fp (x, η, τp ), different possible asymptotic forms of the SSA spectra that actually depend on the optical depth parameter values. The first case corresponds to τp Hp (x, η) 1, for which one has τp κp+1 (η)x −5/3 1, leading to x xa(1) , with xa(1) being a self-absorption frequency given by xa(1) = [τp κp+1 (η)]3/5 . 5

(26)


Fouka & Ouichaoui

Figure 4. Plot of the SSA spectrum through the function Fp (x, η, τp ) for p = 2.5, η = 100, τp = 10−5 , τp = 0.737, xa = 1.2 × 10−3 , together with its asymptotic (1) (1) forms for x xa , xa x 1, 1 x η2 , and x η2 , in the case of an arbitrary pitch angle. (1)

(1)

The condition xa(1) 1 corresponds to τp τp(1) with τp(1) given by τp(1) = κp+1 (η)−1 .

(27)

Then, in this case, asymptotic forms of the SSA spectrum can be written as follows: ⎧ Ap (η)x 2 for x xa(1) ⎪ ⎪ ⎪ ⎪ ⎪ 1/3 ⎪ for xa(1) x 1 ⎪ ⎨τp κp (η)x Iν ∝ Fp (x, η, τp ) ≈

−(p−1)/2

τp Cp x ⎪ ⎪ ⎪ ⎪ ⎪ π −1/2 −p+2 −x/η2 ⎪ ⎪ [η e − e−x ] x ⎩τp 2

for 1 x η2

(28)

for x η2

with the coefficient Ap (η) given by (Fouka & Ouichaoui 2009) Ap (η) =

κp (η) . κp+1 (η)

(29)

In Figure 4 is plotted the function Fp (x, η, τp ) together with the corresponding asymptotic forms given by Equation (28). However, if τp τp(1) , i.e., xa(1) 1, one must define another self-absorption frequency based on the middle-range asymptotic form of the function Hp (x, η) (see Fouka & Ouichaoui 2009). Then, one has τp Cp+1 x −(p+4)/2 1, leading to x xa(2) , with xa(2) being another self-absorption frequency given by (30) xa(2) = [τp Cp+1 ]2/(p+4) , with the condition 1 xa(2) η2 corresponding to τp(2a) τp τp(2b) , where −1 τp(2a) = Cp+1

(31)

−1 τp(2b) = ηp+4 Cp+1 .

In the latter case, the asymptotic forms of the SSA spectrum are given by ⎧ Ap (η)x 2 ⎪ ⎪ ⎪ ⎪ ⎪ 5/2 ⎪ ⎪ ⎨Bp x Iν ∝ Fp (x, η, τp ) ≈ τp Cp x −(p−1)/2 ⎪ ⎪ ⎪ ⎪ ⎪ π −1/2 −p+2 −x/η2 ⎪ ⎪ x [η e − e−x ] ⎩τp 2 and are plotted in Figure 5. 6

for x 1 for 1 x xa(2) for xa(2) x η2 for x η2

(32)


Fouka & Ouichaoui

(2a)

(2b)

(2)

Figure 5. Plot of the SSA spectrum through the function Fp (x, η, τp ) for p = 2.5, η = 104 , τp = 1015 with τp = 0.624, τp = 6.24 × 1026 , and xa = 4.77 × 104 , (2) (2) together with its asymptotic forms for x 1, 1 x xa , xa x η2 , and x η2 , in the case of an arbitrary pitch angle.

A third possibility corresponds to the case where xa(2) η2 , for which one can define another self-absorption frequency, xa(3) , based on the third range of the function Hp (x, η), that is a solution for the equation π −5/2 −p+1 −x/η2 x τp [η e − e−x ] = 1. (33) 2 Given that xa(3) η2 , one sets xa(3) = η2 t, with t 1. For η 30, the term in e−x can be dropped relative to the term η−p+1 e−x/η , which leads to the simpler equation t 5/2 et = a (34) π τp η−(p+4) . a= 2 2

The latter admits an exact analytical solution in terms of the Lambert W function3 (see, e.g., Corless et al. 1996), i.e.,

2 2/5 5 . a t= W 2 5 Finally, the frequency xa(3) can be written as xa(3) =

5 2 2 2/5 . η W a 2 5

(35)

(36)

Thus, xa(3) η2 corresponds to τp τp(3) , with τp(3) given by τp(3)

=e

2 p+4 η , π

so that one can write a = e(τp /τp(3) ), where here e denotes the natural logarithm function basis. In the case xa(3) η2 one gets the following asymptotic forms for the SSA spectrum: ⎧ Ap (η)x 2 for x 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ for 1 x η2 Bp x 5/2 ⎪ ⎪ ⎪

⎪ ⎨ −p+2 −x/η2 e − e−x Iν ∝ Fp (x, η, τp ) ≈ x 2 η ≈ ηx 2 for η2 x xa(3) 2 ⎪ ⎪ η−p+1 e−x/η − e−x ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ π −1/2 −p+2 −x/η2 ⎪ ⎪ ⎩τp x [η e − e−x ] for x xa(3) ; 2 these are plotted in Figure 6. 3

The Lambert W function is defined as the inverse of the function y = xex ; then x = W (y).

7

(37)

(38)


Fouka & Ouichaoui

(3)

(3)

Figure 6. Plot of the SSA spectrum through the function Fp (x, η, τp ) for p = 2.5, η = 100, τp = 1020 with τp = 2.17 × 1013 and xa = 1.05 × 105 , together with (3) (3) its asymptotic forms for x 1, 1 x η2 , η2 x xa , and x xa , in the case of an arbitrary pitch angle.

2.2.1. Discussion of the Third SSA Frequency

As already stated, the third absorption frequency can appear in the SSA spectrum if the condition τp τp(3) is fulfilled. In order to test the possible implication of IC effects, we present the two following examples. (1) γ1 = 100, = 1014 cm, B = 1 G (e.g., for GRB sites). In this case, the condition above leads to high particle density values, n > 1019 cm−3 , for which Comptonization effects are very significant and cannot then be neglected. (2) γ1 = 5, = 1014 cm, B = 1 G, η = 10, a case for which the above condition on τp implies particle density values n > 102 cm−3 , and that should be astrophysically possible. However, this case is a limit for the implication of Comptonization effects. On the other hand, for low values of γ1 and the parameter η, Comptonization effects can be avoided. But the relativistic version of synchrotron radiation is no longer valid, and one is therefore forced to jump to the semi-relativistic treatment of this emission process (e.g., Petrosian 1981; McTiernan & Petrosian 1983). Then, one is led to the conclusion that SSA spectra could hardly be expected to be observed in the third absorption frequency. The accuracy of the latter SSA frequency, given by Equation (36) in the case of an arbitrary pitch angle distribution, is shown in Figure 6. Furthermore, by considering the case of an isotropic pitch angle distribution and following the same treatment steps, one can easily show its close similarity to the case of an arbitrary pitch angle distribution. 3. SUMMARY AND CONCLUSION We have derived asymptotic forms for the OTS degree of polarization and for SSA spectra in the case of an arbitrary pitch angle distribution. In addition to the two well-known SSA absorption frequencies, a third possible one has been derived and expressed in terms of the Lambert W function, based on the absorption coefficient asymptotic form described by the function Hp in the highfrequency range. We have shown that this frequency can appear in the SSA spectrum above the particular frequency x = η2 for an adequate set of physical parameters such as, e.g., γ1 = 5, ∼ 1014 cm, B = 1 G, and η = 5, in order to get a pure SSA spectrum avoiding Comptonization effects. One must note that the derived SSA spectra are found to be valid without considering these effects (in the case of very high photon densities), pair creation and annihilation (for very high photon densities), and magnetic photon splitting (for very high magnetic field strengths, B Bcr ). These SSA spectra for instantaneous photon emission are also valid in situations where particles cool much more slowly than the hydrodynamical evolution, as suggested by many authors for GRB prompt emission (see, e.g., Baring & Braby 2004). Note, finally, that by following the same treatment steps, similar results can be derived in the case of an isotropic pitch angle distribution. We thank the referee for precise remarks and constructive suggestions for improving the quality of the manuscript. REFERENCES Jackson, J. D. 1962, Classical Electrodynamics (New York: Wiley) Katz, J. I. 1994, ApJ, 432, L107 Komissarov, S. S., & Lyubarsky, Y. E. 2004, MNRAS, 349, 779 McTiernan, J. M., & Petrosian, V. 1983, Phys. Fluids, 26, 3023 Ostrowski, M., & Bednarz, J. 2002, A&A, 394, 1141 Petrosian, V. 1981, ApJ, 251, 727 Reynolds, S. P. 1996, ApJ, 459, L13 Rybicki, G. B., & Lightman, A. P. 1979, Radiative Processes in Astrophysics (New York: Wiley) Tavani, M. 1996, ApJ, 466, 768 Westfold, K. C. 1959, ApJ, 130, 241

Abramowitz, M., & Stegun, I. A. (ed.) 1965, Handbook of Mathematical Functions (New York: Dover) Baring, M. G., & Braby, M. L. 2004, ApJ, 613, 460 Biermann, P. L., & Strittmatter, P. A. 1987, ApJ, 322, 643 Corless, R. M., Gonnet, G. H., Hare, D. E. G., Jeffrey, D. J., & Knuth, D. E. 1996, Adv. Comput. Math., 5, 329 Fermi, E. 1949, Phys. Rev., 75, 1169 Fouka, M., & Ouichaoui, S. 2009, ApJ, 707, 278 Granot, J., Piran, T., & Sari, R. 1999, ApJ, 527, 236 Helfand, D. J., Gotthelf, E. V., & Halpern, J. P. 2001, ApJ, 556, 380

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