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Cosmic Electrons

Assuming that the initial spectrum for cosmic ray electrons can be represented by a power law:

$\displaystyle Q = k E^{-p} [electrons / cm^2 \; sec \; eV \; sr]$ (1.3)

the spectrum evolution follows the law :

$\displaystyle \frac{dN}{dt} = D \nabla^{2} N + \frac{\partial}{\partial E} \left[ b(E) N(E)\right] + Q(E)$ (1.4)

where D is the diffusion coefficient and :

$\displaystyle b(E) = - \frac{dE}{dt}$ (1.5)

Equation 1.4 can be resolved in first approximation by ignoring the terms due to diffusion (assuming e.g. a uniform sources distribution) and $ \partial N$/ $ \partial t$ we obtain

$\displaystyle \frac{\partial}{\partial E} \left[ b(E) N(E)\right] = - Q (E) = -k E^{-p}$ (1.6)

and integrating :

$\displaystyle N(E)=\frac{kE^{-(p-1)}}{(p-1)b(E)}$ (1.7)

The shape of the initial particle ($ e^-$) spectrum is therefore modified according to the dependence of $ b$ on $ E$. The energy loss can be due to: The radio observation of the synchrotron diffuse emission cover the range between few tens of MHz to a few GHz. These observations show a radiation spectrum with index $ \beta$ between 2.4 and 3, which corresponds to a spectral index of the electron flux $ \alpha$ between 2 and 3. In particular an increase of $ \beta$ from 2.4-2.6 to 2.8-3 is observed for frequency larger than 400 MHz [Strong et al., 2000]. This can be connected to the spectral break due to synchrotron and inverse Compton energy losses, as is foreseen for electron spectrum at energy of about few GeV.
The spectrum of cosmic electrons can also be obtained from the direct measurements performed at the top of the atmosphere, for energies larger than some GeV, where the solar modulation is not important. Recent measurements found that the local electron spectrum can be fitted by a single power-law, with index 3.4, up to 2-3 GeV [Casadei & Bindi, 2004].
However the electron distribution is expected to be very inhomogeneous for energies above some tens of GeV, as more energetic electrons are confined close to the sources [Strong et al., 2000], [Pohl & Esposito, 1998]. This suggests that the spectrum of cosmic electrons can be different from that measured locally.
Figure 1.7: Left: Relation between electron energy and frequency of the emitted synchrotron radiation, (a value of galactic magnetic field of 6 $ \mu G$ has been assumed). Right: Relation between electron energy and energy of the emitted inverse-Compton radiation, for the stellar radiation field (dotted line), the dust radiation field (dashed line) and the cosmic microwave background (solid line)
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next up previous contents
Next: Cosmic Protons Up: Cosmic Rays Previous: Cosmic Rays   Contents
Andrea Giuliani 2005-01-21