The Deuteration Clock for Massive Starless Cores

To understand massive star formation requires study of its initial conditions. Two massive starless core candidates, C1-N & C1-S, have been detected in IRDC G028.37+00.07 in $\rm N_2D^+$(3-2) with $ALMA$. From their line widths, either the cores are subvirial and are thus young structures on the verge of near free-fall collapse, or they are threaded by $\sim1$ mG $B$-fields that help support them in near virial equilibrium and potentially have older ages. We modeled the deuteration rate of $\rm N_2H^+$ to constrain collapse rates of the cores. First, to measure their current deuterium fraction, $D_{\rm frac}^{\rm N_2H^+}$ $\equiv [\rm N_2D^+]/[N_2H^+]$, we observed multiple transitions of $\rm N_2H^+$ and $\rm N_2D^+$ with $CARMA$, $SMA$, $JCMT$, $NRO~45m$ and $IRAM~30m$, to complement the $ALMA$ data. For both cores we derived $D_{\rm frac}^{\rm N_2H^+}\sim0.3$, several orders of magnitude above the cosmic [D]/[H] ratio. We then carried out chemodynamical modeling, exploring how collapse rate relative to free-fall, $\alpha_{\rm ff}$, affects the level of $D_{\rm frac}^{\rm N_2H^+}$ that is achieved from a given initial condition. To reach the observed $D_{\rm frac}^{\rm N_2H^+}$, most models require slow collapse with $\alpha_{\rm ff}\sim0.1$, i.e., $\sim1/10$th of free-fall. This makes it more likely that the cores have been able to reach a near virial equilibrium state and we predict that strong $B$-fields will eventually be detected. The methods developed here will be useful for measurement of the pre-stellar core mass function.