From eq. (3.24) and a footnote it seems like we're assuming ballistic propagation across the length $\lambda_{1,2}$. This is not obvious, as these lengths are not obtained as mean free paths, but from a balance between diffusion and advection, yet particles should still cover them in diffusive regime.
I tried to repeat the calculation with this assumption (see proposed edit) and it effectively adds 2 orders of magnitude, since in denominator $c$ is replaced with $u_{1,2}$. This brings the result at 1 GeV from 1 to 100 kyrs.
I'm not sure what to make of it, and if this logic is sound or not.
Another question I had related to this calculation is about downstream diffusion length. It doesn't seem to appear naturally in Fokker-Planck equation solution (at least stationary), as advection and diffusion do not counteract each other. Is there a deep reason to introduce it here, or is it just for convenience of order of magnitude estimation?
(page 54 in the PDF)
This is not (yet) an edit, but a question.
From eq. (3.24) and a footnote it seems like we're assuming ballistic propagation across the length $\lambda_{1,2}$. This is not obvious, as these lengths are not obtained as mean free paths, but from a balance between diffusion and advection, yet particles should still cover them in diffusive regime.
I tried to repeat the calculation with this assumption (see proposed edit) and it effectively adds 2 orders of magnitude, since in denominator $c$ is replaced with $u_{1,2}$. This brings the result at 1 GeV from 1 to 100 kyrs.
I'm not sure what to make of it, and if this logic is sound or not.
Another question I had related to this calculation is about downstream diffusion length. It doesn't seem to appear naturally in Fokker-Planck equation solution (at least stationary), as advection and diffusion do not counteract each other. Is there a deep reason to introduce it here, or is it just for convenience of order of magnitude estimation?