  The deuterium fractionation in starless cores gives us a clue to estimate
their lifetime scales, thus allowing us to distinguish between different
dynamical theories of core formation. Cores also seem to be subject to a
differential N2 and CO depletion which was not expected from models. We aim to
make a survey of 10 cores to estimate their lifetime scales and depletion
profiles in detail. After L183, in Serpens, we present the second cloud of the
series, L1512 in Auriga. To constrain the lifetime scale, we perform chemical
modeling of the deuteration profiles across L1512 based on dust extinction
measurements from near-infrared observations and non-local thermal equilibrium
radiative transfer with multiple line observations of N2H+, N2D+, DCO+, C18O,
and 13CO, plus H2D+ (1$_{10}$--1$_{11}$). We find a peak density of
1.1$\times$10$^5$ cm$^{-3}$ and a central temperature of 7.5$\pm$1 K, which are
respectively higher and lower compared with previous dust emission studies. The
depletion factors of N2H+ and N2D+ are 27$^{+17}_{-13}$ and 4$^{+2}_{-1}$ in
L1512, intermediate between the two other more advanced and denser starless
core cases, L183 and L1544. These factors also indicate a similar freeze-out of
N2 in L1512, compared to the two others despite a peak density one to two
orders of magnitude lower. Retrieving CO and N2 abundance profiles with the
chemical model, we find that CO has a depletion factor of $\sim$430-870 and the
N2 profile is similar to that of CO unlike towards L183. Therefore, L1512 has
probably been living long enough so that N2 chemistry has reached steady state.
N2H+ modeling remains compulsory to assess the precise physical conditions in
the center of cold starless cores, rather than dust emission. L1512 is
presumably older than 1.4 Myr. Therefore, the dominating core formation
mechanism should be ambipolar diffusion for this source.
