Mapping the core mass function on to the stellar initial mass function: multiplicity matters

Observations indicate that the central portions of the present-day prestellar core mass function (hereafter CMF) and the stellar initial mass function (hereafter IMF) both have approximately log-normal shapes, but that the CMF is displaced to higher mass than the IMF by a factor F ∼ 4 ± 1. This has led to suggestions that the shape of the IMF is directly inherited from the shape of the CMF – and therefore, by implication, that there is a self-similar mapping from the CMF on to the IMF. If we assume a self-similar mapping, it follows (i) that F = N O/η, where η is the mean fraction of a core’s mass that ends up in stars and NO is the mean number of stars spawned by a single core; and (ii) that the stars spawned by a single core must have an approximately log-normal distribution of relative masses, with universal standard deviation σ O. Observations can be expected to deliver ever more accurate estimates of F, but this still leaves a degeneracy between η and NO, and σ O is also unconstrained by observation. Here we show that these parameters can be estimated by invoking binary statistics. Specifically, if (a) each core spawns one long-lived binary system, and (b) the probability that a star of mass M is part of this long-lived binary is proportional to Mα, current observations of the binary frequency as a function of primary mass, b(M1), and the distribution of mass ratios, pq, strongly favour η ∼ 1.0 ± 0.3, NO ∼ 4.3 ± 0.4, σ O ∼ 0.3 ± 0.03 and α ∼ 0.9 ± 0.6; η > 1 just means that, between when its mass is measured and when it finishes spawning stars, a core accretes additional mass, for example from the filament in which it is embedded. If not all cores spawn a long-lived binary system, db/dM1 < 0, in strong disagreement with observation; conversely, if a core typically spawns more than one long-lived binary system, then NO and η have to be increased further. The mapping from CMF to IMF is not necessarily self-similar – there are many possible motivations for a non-self-similar mapping – but if it is not, then the shape of the IMF cannot be inherited from the CMF. Given the limited observational constraints currently available and the ability of a self-similar mapping to satisfy them, the possibility that the shape of the IMF is inherited from the CMF cannot be ruled out at this juncture.