  We determine an asymptotic expression of the blow-up time t_coll for
self-gravitating Brownian particles or bacterial populations (chemotaxis) close
to the critical point. We show that t_coll=t_{*}(eta-eta_c)^{-1/2} with
t_{*}=0.91767702..., where eta represents the inverse temperature (for Brownian
particles) or the mass (for bacterial colonies), and eta_c is the critical
value of eta above which the system blows up. This result is in perfect
agreement with the numerical solution of the Smoluchowski-Poisson system. We
also determine the asymptotic expression of the relaxation time close but above
the critical temperature and derive a large time asymptotic expansion for the
density profile exactly at the critical point.
