Probabilistic distributions of $M/L$ values for ultra-faint dwarf spheroidal galaxies: stochastic samplings of the IMF

Abstract: As the total mass of stellar populations of fixed IMF, age and metallicity tends to infinity, resulting intrinsic $M/L$ values quickly converge to fixed numbers associated to the particulars of the stars in question. When going to very small stellar populations, the stochastic sampling of the IMF will yield an inherent spread for the $M/L$ values, which will become probabilistic quantities. For the recently discovered ultra-faint dwarf spheroidal galaxies, with total luminosities dropping below 10^3 L_V/L_o, it is important to asses the amplitude of the probabilistic spread in inherent $M/L$ values. The total baryonic masses of these systems are usually estimated from their luminosities, assuming fixed, deterministic $M/L$ values, suitable for the infinite population limit. These total baryonic masses are crucial for the testing of structure formation scenarios, as the local ultra-faint dwarf spheroidals represent the most extreme galactic scales known. Also subject to reliable $M/L$ values, is the use of these systems as possible discriminants between dark matter and modified gravity theories. By simulating large collections of stellar populations, each consisting of a particular collection of individual stars, we compute statistical distributions for the resulting $M/L$ values. For total numbers of stars in the range of what is observed for the ultra-faint dwarf spheroidals, inherent $M/L$ values of stellar populations can be expected to vary by factors of upwards of 3, interestingly, systematically skewed towards higher values than what corresponds to the infinite population limit. This can explain part of the spread in reported baryonic masses for these systems, which also appear as shifted systematically towards high dark to baryonic mass ratios at fixed stellar velocity dispersions.