$\beta$--Radioactive Cosmic Rays in a diffusion model: test for a local bubble?

In the present paper, we extend the analysis of Maurin et al. (2001) and Donato et al. (2001) to the $\beta$-radioactive nuclei ${}^{10}$Be, ${}^{26}$Al and ${}^{36}$Cl. These species are be shown to be particularly sensitive to the properties of the local interstellar medium (LISM). As studies of the LISM suggest that we live in an underdense bubble of extent $r_{hole} \sim 50 - 200 \unit{pc}$, this local feature must be taken into account. We present a modified version of our diffusion model which describes the underdensity as a hole in the galactic disc. It is found that the presence of the bubble leads to a decrease in the radioactive fluxes which can be approximated by a simple factor $\exp (-r_{hole}\mathrm{/}{l}_{rad})$ where $l_{rad}=\sqrt{K\gamma {\tau }_0}$ is the typical distance travelled by a radioactive nucleus before it decays. We find that each of the radioactive nuclei independently point towards a bubble radius $\lesssim 100\text{unit}pc$. If these nuclei are considered simultaneously, only models with a bubble radius $r_{hole}\sim 60-100\text{unit}pc$ are marginally consistent with data. In particular, the standard case $r_{hole}=0\text{unit}pc$ is disfavoured. Our main concern is about the consistency of the currently available data, especially ${}^{26}$Al/${}^{27}$Al.