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 similar to 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/eta, where eta is the mean fraction of a core's mass that ends up in stars and N-O 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 sigma(O). Observations can be expected to deliver ever more accurate estimates of F, but this still leaves a degeneracy between eta and N-O, and sigma(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 M-alpha, current observations of the binary frequency as a function of primary mass, b(M-1), and the distribution of mass ratios, p(q), strongly favour eta similar to 1.0 +/- 0.3,N-O similar to 4.3 +/- 0.4, sigma(O) similar to 0.3 +/- 0.03 and alpha similar to 0.9 +/- 0.6; eta > 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/dM(1) < 0, in strong disagreement with observation; conversely, if a core typically spawns more than one long-lived binary system, then N-O and eta 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.