One of the ways to understand the genesis and evolution of the universe is to know how galaxies have formed and evolved. In this regard, the study of star formation history (SFH) plays an important role in the accurate understanding of galaxies. In this paper, we used long-period variable stars (LPVs) for estimating the SFH in the Andromeda galaxy (M31). These cool stars reach their peak luminosity in the final stage of their evolution also their birth mass is directly related to their luminosity. Therefore, using stellar evolution models, we construct the mass function and hence the star formation history.

Some ultra-compact dwarf galaxies have large dynamical mass to light (M / L) ratios and also appear to contain an overabundance of LMXB sources, and some Milky Way globular clusters have a low concentration and appear to have a deficit of low-mass stars. These observations can be explained if the stellar IMF becomes increasingly top-heavy with decreasing metallicity and increasing gas density of the forming object. The thus constrained stellar IMF then accounts for the observed trend of metallicity and M / L ratio found amongst M31 globular star clusters. It also accounts for the overall shift of the observationally deduced galaxy-wide IMF from top-light to top-heavy with increasing star formation rate amongst galaxies. If the IMF varies similarly to deduced here, then extremely young very massive star-burst clusters observed at a high redshift would appear quasar-like (Jerabkova et al. 2017).

The birth of stars and the formation of galaxies are cornerstones of modern astrophysics. While much is known about how galaxies globally and their stars individually form and evolve, one fundamental property that affects both remains elusive. This is problematic because this key property, the birth mass distribution of stars, referred to as the stellar initial mass function, is a key tracer of the physics of star formation that underpins almost all of the unknowns in galaxy and stellar evolution. It is perhaps the greatest source of systematic uncertainty in star and galaxy evolution. The past decade has seen a growing variety of methods for measuring or inferring the initial mass function. This range of approaches and evolving definitions of the quantity being measured has in turn led to conflicting conclusions regarding whether or not the initial mass function is universal. Here I review this growing wealth of approaches, and highlight the importance of considering potential initial mass function variations, reinforcing the need to carefully quantify the scope and uncertainties of measurements. I present a new framework to aid the discussion of the initial mass function and promote clarity in the further development of this fundamental field.

I review what has been learnt so far regarding the origin of stellar properties from numerical simulations of the formation of groups and clusters of stars. In agreement with observations, stellar properties are found to be relatively robust to variations of initial conditions in terms of molecular cloud structure and kinetics, as long as extreme initial conditions (e.g. strong central condensation, weak or no turbulence) and small-scale driving are avoided, but properties may differ between bound and unbound clouds. Radiative feedback appears crucial for setting stellar masses, even for low-mass stars, while magnetic fields can provide low star formation rates.

It is possible to extract, from the observations, distribution functions of the birth dynamical properties of a stellar population, and to also infer that these are quite invariant to the physical conditions of star formation. The most famous example is the stellar IMF, and the initial binary population (IBP) seems to follow suit. A compact mathematical formulation of the IBP can be derived from the data. It has three broad parts: the IBP of the dominant stellar population (0.08–2M⊙), the IBP of the more-massive stars and the IBP of brown dwarfs. These three mass regimes correspond to different physical regimes of star formation but not to structure in the IMF. With this formulation of the IBP it becomes possible to synthesize the stellar-population of whole galaxies.