First sample of $\rm N_2H^+$ nitrogen isotopic ratio measurements in
  low-mass protostars
  Context. The nitrogen isotopic ratio is considered an important diagnostic
tool of the star formation process, and $N_2H^+$ is particularly important
because it is directly linked to molecular nitrogen $N_2$. However, theoretical
models still lack to provide an exhaustive explanation for the observed
$^{14}N/^{15}N$ values.
  Aims. Recent theoretical works suggest that the $^{14}N/^{15}N$ behaviour is
dominated by two competing reactions that destroy $ N_2H^+$: dissociative
recombination and reaction with CO. When CO is depleted from the gas phase, if
$N_2H^+$ recombination rate is lower with respect to the $N^{15}NH^+$ one, the
rarer isotopologue is destroyed faster. This implies that the $N_2H^+$ isotopic
ratio in protostars should be lower than the one in prestellar cores, and
consistent with the elemental value of ~440. We aim to test this hypothesis,
producing the first sample of $N_2H^+ / N^{15}NH^+$ measurements in low mass
protostars.
  Methods. We observe the $N_2H^+$ and $N^{15}NH^+$ lowest rotational
transition towards six young stellar objects in Perseus and Taurus molecular
clouds. We model the spectra with a custom python code using a constant
$T_{ex}$ approach to fit the observations. We discuss in appendix the validity
of this hypothesis. The derived column densities are used to compute the
nitrogen isotopic ratio.
  Results. Our analysis yields an average of $\rm ^{14}N/^{15}N|_{pro} = 420
\pm 15$ in the protostellar sample. This is consistent with the protosolar
value of 440, and significantly lower than the average value previously
obtained in a sample of prestellar objects. Conclusions. Our results are in
agreement with the hypothesis that, when CO is depleted from the gas-phase,
dissociative recombinations with free electrons destroy $N^{15}NH^+$ faster
than $N_2H^+$, leading to high isotopic ratios in prestellar cores, where CO is
frozen on dust grains.
