CC BY
25
UDK 537.591.15
A. A. Lagutin, A. G. Tyumentsev, N. V. Volkov, A. S. Kuzmin.
Synchrotron radiation of cosmic ray electrons in the anomalous diffusion model
A new study of the cosmic ray electron and synchrotron spectra is presented Anomalous diffusion model, proposed in our recent papers, is used to describe the particles propagation in fractal-like interstellar medium. The parameters defining the anomalous diffusion have been determined from the analysis of nuclear component. We carried out calculation of the synchrotron spectrum in the frequency range ~ 2 MHz - 2 GHz (corresponding to energies of electrons ~ 0.2 — 6 GeV). The computed electron and synchrotron spectra are in a good agreement with the experimental data.
1. Introduction
The studies of the electron component of cosmic rays and their resulting non-thermal synchrotron radiation in galactic magnetic fields allows to obtain information on the origin, acceleration and propagation of particles in the cosmic objects. Calculations of electron spectrum and spectrum of the non-thermal synchrotron radiation were considered in series of papers (see, for example, [1-5]). In this work we consider propagation of cosmic ray electrons in the context of anomalous diffusion model, which recently was discussed in the papers [6-9]. It has been shown that the "knee" in the primary cosmic ray spectrum could be due to large free paths (the so called "Levy flights") of cosmic rays particles between inhomogeneities of magnetic fields - "traps" of the various type. The "Levy flights" are distributed according to inverse power law a Ar~3~a,r —> oo,a < 2, where exponent a is defined by fractal features of the interstellar medium.
In [6, 10, 11] an anomalous diffusion (superdiffu-sion) equation for concentration of the electron in the fractal-like interstellar medium was proposed. This equation has the following form
~ = -D(E, Q)(-A)a/,2yV(f, t, E) +
r\
+—(b(E)N(f,t,E)) + S(f,t,E), (1)
where D(E,a) is the anomalous diffusivity and (-A)"/2 is the fractional Laplacian (called "Riss operator" [12]).
The main goal of this paper is calculation of electron and synchrotron spectra in the framework of anomalous diffusion model. To solve this problem
"This work was presented at the XXXth ICRC (Merida, Mexico, 2007). Also this work is supported by the RFBR grant No. 07-02-01154 and RF president's grant MK - 2873.2007.2.
we resort to two-component model [6, 10, 11]. The sources were separated into two categories: local sources and distant sources, which provides stationary regime of injection.
2. Spectrum of cosmic ray electrons
For a point impulse source with the power-law energy spectrum S{f,t,E) = S0E~PS(r)H{T -t)H(t) that simulates the process of generation of particles in the nearby young sources (so-called local component (L), r < 1 kpc, and injection time T « 104 yr) a flux of electrons is written as [6, 10, 11]
minit,i/62(£+E2)] Ji{f,t,E) — So J drEo(r)-p
max(o,i-7-i \{E,E0(t))-va(l-b2t(E + E2))-2 x
x^a)(rA(£,£o(T))-1/a). (2)
The flux from the distant sources (global component (G), r > 1 kpc) is obtained from the steady-
state model (6, 111
oo
JG(r,E) = dE0EoP x
e
\(E,E0r3'*gM{rX{E,E0)-V°). (3)
In (2) and (3) g{3a){r) is the density of three-dimensional spherically-symmetrical stable distribution with characteristic exponent a < 2 [19, 20]. Equation for E0(t) has been presented in form [1]
p . , __£ + __E
0[T) 1 -&,t(£ + £2)/(£2-£I) 11
A. A. Lagutin, A. G. Tyumentsev, N. V. Volkov, A. S. Kuzmin.
o
100
£
§
10
1
e, GeV
Peterson et al. (1999) ■—•—-Webber etal. (1980)
97 (Barbellini et al. (1997)) *.......*.....<
2000 (Boezio et al. (2000)) --*--HEAT (Barwick et al. (1998)) —a - ■ Golden etal. (1994) >•••» Muller et al. (1997) »■■-<»-< Tang (1984) t-*•■-<
G-component--
L-oomponent --------
ISM spectrum .........
Modulated spectrum .................
Figure 1. L- and Q- components and total energy spectrum of electrons. Experimental data are reported in [4, 5, 13-18].
and
e
where
b{E) = b0 + b,E + b2E2 «
Kb2(E + E1)(E + E2). (4)
Equation (4) gives the energy-loss rate of relativists electrons ¡1], where 60 = 3.06 ■ 10"16n (GeV s_1) is for ionization losses in interstellar medium, byE with b\ — 10_15n (s-1) corresponds to the bremsstrahlung energy losses, and b2E2 with b2 = 1.38- lO-16 (GeV s)-1 represents synchrotron and inverse Compton losses (for 0x ~ 5^G and to % 1 (eV/cm3)), Ei « 60/&:l, E2 « b1/b2.
As a result the flux of electrons from all types of galactic sources may be presented as
Je(r,E)= ]T JUfi,ti,E) + JG{ r,E). (5)
r<ikpc
The nearby sources characteristics, used in our calculations are given in [11].
The parameters defining the anomalous diffusiv-ity and used in our work have been recently derived from the studies of nuclei propagation [21]: a = 0.7, D{E,q) = D0ES where D0 = 2-10-5pc°V~1 and S = 0.27. Our calculations show that the best
agreement with the experimental data may be obtained if electrons are generated by sources with a spectrum S oc E~26 (Fig. 1).
For description of the spectrum in low-energy range near solar system it is necessary to take into account an effect of solar modulation. The solar modulation is calculated as (15]
E2 - mec2
Jmod{r,b) - [E + m]2_77leC2 x
xJe(r,E + $(i)), (6)
where <b(t) = 600 MeV. Results of calculations of cosmic ray electrons spectrum in ISM and the modulated spectrum in low-energy range are demonstrated in Fig. 1.
3. Synchrotron radiation spectrum
The synchrotron radiation emitted by relativistic electrons is the most important diagnostic available for the study of the transport of these particles in the interstellar medium.
Intensity of synchrotron radiation Iu at a given frequency v is defined by average density of electrons pe(E) — J dzJe(r,E) (2 is measured along the line of observation), and equals to [2]
/„ = J P„{E)pe(E)dE,
where PV{E) is the intensity of radiation at a frequency v of single electron with the energy E.
Figure 2. Synchrotron radiation spectrum. Experimental data are taken from [4J.
For E » mec2 the function PU(E) has the peak at a frequency [22]
vm{ MHz) = 16-OBj. (fiG)E2(GeV).
For our purpose (calculation of synchrotron radiation spectrum), it is sufficient to use the delta-approximation in the form PV{E) = Pq5(v - um).
As a result, the intensity of synchrotron radiation is given by the expression [22]
/„ = P0 f dzJe(r,E). (7)
The synchrotron spectrum, calculated by equation (7) with parameters p « 2.6 and a = 0.7 is shown in Fig. 2.
4. Conclusion
We have carried out new calculations of synchrotron radiation spectra using an anomalous diffusion model to describe cosmic ray electrons propagation in the Galaxy. Comparison of results of calculations of the synchrotron spectrum in the frequency range ~ 2 MHz - 2 GHz (corresponding to energy of electrons ~ 0.2 -6 GeV) with the experimental data allowed us to obtain the conclusion on the spectrum power index of the cosmic ray electrons in the low-energy region. Extensive calculations of the cosmic ray electron and synchrotron spectra show that the best fit of experimental data may be obtained for p «s 2.6.
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