where Qo(p)8(r - ro) is the part of the particle injection source located on the shock front, and the function sign((ro - 0)- u(ro + 0)) reflects that two different possibilities can be realized: when the region 1 (before the shock front) is at r > ro (as in solar or stellar wind), and when the region 1 is at r < ro, as in case of accretion. For simplicity let us consider injection of mono-energetic particles from the undisturbed plasma before the shock front f^p )=(wJ 4npl )i(p - po ), and injection from the front
Let us follow Berezhko et al. (M1988) by considering few important cases in which it is possible to obtain analytical solutions: a standing shock wave in the solar or stellar wind (as terminal shock wave on the boundary of Heliosphere; bow shocks in the magnetospheres of the Earth, Jupiter, Saturn, and so on), a standing shock wave in the case of accretion (as in double star system), a running shock wave (as in interplanetary space from solar flares and from CME - coronal mass ejections; as from supernova explosion).
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