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 \sim 4 \pm 1 .
This has lead 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 onto the IMF .
If we assume a self-similar mapping , it follows ( i ) that F = { \cal N } _ { { } _ { O } } / \eta , where \eta is the mean fraction of a core ’ s mass that ends up in stars , and { \cal 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 { \cal 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 \sim 1.0 \pm 0.3 , { \cal N } _ { { } _ { O } } \sim 4.3 \pm 0.4 , \sigma _ { { } _ { O } } \sim 0.3 \pm 0.03 and \alpha \sim 0.9 \pm 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 { \cal 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 can not 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 can not be ruled out at this juncture .