Deuterium and $^{15}$N fractionation in N$_2$H$^+$ during the formation
  of a Sun-like star
  Although chemical models predict that the deuterium fractionation in
N$_2$H$^+$ is a good evolutionary tracer in the star formation process, the
fractionation of nitrogen is still a poorly understood process. Recent models
have questioned the similar evolutionary trend expected for the two
fractionation mechanisms in N$_2$H$^+$, based on a classical scenario in which
ion-neutral reactions occurring in cold gas should have caused an enhancement
of the abundance of N$_2$D$^+$, $^{15}$NNH$^+$, and N$^{15}$NH$^+$. In the
framework of the ASAI IRAM-30m large program, we have investigated the
fractionation of deuterium and $^{15}$N in N$_2$H$^+$ in the best known
representatives of the different evolutionary stages of the Sun-like star
formation process. The goal is to ultimately confirm (or deny) the classical
"ion-neutral reactions" scenario that predicts a similar trend for D and
$^{15}$N fractionation. We do not find any evolutionary trend of the
$^{14}$N/$^{15}$N ratio from both the $^{15}$NNH$^+$ and N$^{15}$NH$^+$
isotopologues. Therefore, our findings confirm that, during the formation of a
Sun-like star, the core evolution is irrelevant in the fractionation of
$^{15}$N. The independence of the $^{14}$N/$^{15}$N ratio with time, found also
in high-mass star-forming cores, indicates that the enrichment in $^{15}$N
revealed in comets and protoplanetary disks is unlikely to happen at core
scales. Nevertheless, we have firmly confirmed the evolutionary trend expected
for the H/D ratio, with the N$_2$H$^+$/N$_2$D$^+$ ratio decreasing before the
pre-stellar core phase, and increasing monotonically during the protostellar
phase. We have also confirmed clearly that the two fractionation mechanisms are
not related.
