The distribution of low-mass stars in the Galactic disc

We quantify the complex interdependence of stellar binarity, the stellar mass-luminosity relation, the mass function, the colour-magnitude relation and Galactic disc structure, all of which must be understood when analysing star-count data and stellar luminosity functions. We derive a mass-Mv relation and a model for the change of stellar luminosity with changes in chemical abundance and age. Combination of this with detailed modelling of all astrophysical and observational contributions to the Malmquist scatter allows us to model star-count data without approximating Malmquist corrections. We show for the first time that a single mass function and normalization explain the stellar distribution towards both Galactic poles, as well as the distribution of stars within a distance of 5.2 pc of the Sun. The initial mass function can be approximated by ξ(m) ∝ m–α with α3 ≈ 2.7 for stars more massive than 1 Mʘ, α 2 ≈ 2.2 in the mass range 0.5 ≤ m ≤ 1 Mʘ and 0.70 < α 1 < 1.85 in the range 0.08 < m ≤ 0.5 Mʘ. If the stars at a distance of about 100 pc from the Galactic mid-plane have a metallicity smaller by about 0.1 dex than do the stars near the plane used to calibrate the mass-Mv and colour-magnitude relations, then both the stars within 5.2 pc of the Sun and the star-count data reaching to a photometric distance of 130 pc lead to a low-mass power-law index of α 1 ≈ 1.5. The change in the power index at 0.5 Mʘ may indicate a characteristic mass-scale in the star formation process. Our model is most consistent with the data if the proportion of binaries among ‘stars’ is larger than 50 per cent and if the component masses are uncorrelated. The possible decline of the proportion of binaries with increasing absolute magnitude, observed in the solar neighbourhood, is in agreement with our models. Two features in the luminosity function for low-mass stars are universal, being solely the result of stellar physics. These features are a flattening at Mv ≈ 7 and a conspicuous maximum at Mv ≈ 12. Binary stars cause the system luminosity function derived from photographic surveys to decrease uniformly with increasing magnitude relative to the single-star luminosity function, but both show the same general features. The solar neighbourhood mass density in main-sequence stars with masses between 0.08 and 100 Mʘ is ρ = 0.05 ± 0.01 Mʘ pc–3, of which main-sequence stars less massive than the Sun contribute about 80 per cent. These estimates explicitly include the effects of binaries. Unresolved binaries cause the apparent disc scaleheight to decrease with decreasing mass to a value of ≈ 150 pc for star-count data in the colour range 4 < V – I < 4.5 if all ‘stars’ are unresolved binaries with uncorrelated component masses. The vertical structure of the Galactic disc is not well described by a single exponential within a few hundred pc of the plane. Neglect of this can lead to a spurious apparent correlation between stellar mass and disc scaleheight.