The local vertical density distribution of ultracool dwarfs M7 to L2.5 and their luminosity function

Stephen J Warren,Saad Ahmed,Richard C Laithwaite

doi:10.21105/astro.2010.11093

Abstract

We investigate the form of the local vertical density profile of the stars in the Galactic disk, close to the Galactic plane. We use a homogeneous sample of 34000 ultracool dwarfs M7 to L2.5 that all lie within 350 pc of the plane. We fit a profile of the form sech$^\alpha$, where $\alpha=2$ is the theoretically preferred isothermal profile and $\alpha=0$ is the exponential function. Larger values of $\alpha$ correspond to greater flattening of the profile towards the plane. We employ a likelihood analysis that accounts in a direct way for unresolved binaries in the sample, as well as for the spread in absolute magnitude $M_J$ within each spectral sub-type (Malmquist bias). We measure $\alpha=0.29^{+0.12}_{-0.13}$. The $\alpha=1$ (sech) and flatter profiles are ruled out at high confidence for this sample, while $\alpha=0$ (exponential) is included in the 95% credible interval. Any flattening relative to exponential is modest, and is confined to within 50 pc of the plane. The measured value of $\alpha$ is consistent with the results of the recent analysis by Xiang et al. Our value for $\alpha$ is also similar to that determined for nearby spiral galaxies by de Grijs et al., measured from photometry of galaxies viewed edge on. The measured profile allows an accurate determination of the local space density of ultracool dwarfs M7 to L2.5, and we use this to make a new determination of the luminosity function at the bottom of the main sequence. Our results for the luminosity function are a factor two to three lower than the recent measurement by Bardalez Gagliuffi et al., that uses stars in the local 25 pc radius bubble, but agree well with the older study by Cruz et al.

Highlights

The variation of the space density of stars in the disk of the Milky Way, in the vertical direction, i.e. perpendicular to the plane of the disk, and at the solar radius, approximates to an exponential distribution (Gilmore and Reid 1983) up to heights of 1 kpc
What happens close to the plane? Is there a sharp density peak, or does the exponential soften? We do not have a clear answer to this question for the Milky Way, but the density profile is often modelled by a sech2 distribution (e.g. Gould et al 1996; Siegel et al 2002; Ferguson et al 2017; Bennett and Bovy 2019), which softens by a factor four relative to an exponential
A self-gravitating isothermal sheet has this equilibrium solution (Spitzer 1942; Camm 1950; van der Kruit and Searle 1981), and this may be why the sech2 distribution is popular, even though it is well known that the velocity dispersion of the stars in the disk depends on age

Summary

Introduction

The variation of the space density of stars in the disk of the Milky Way, in the vertical direction, i.e. perpendicular to the plane of the disk, and at the solar radius, approximates to an exponential distribution (Gilmore and Reid 1983) up to heights of 1 kpc. A self-gravitating isothermal sheet has this equilibrium solution (Spitzer 1942; Camm 1950; van der Kruit and Searle 1981), and this may be why the sech distribution is popular, even though it is well known that the velocity dispersion of the stars in the disk depends on age In considering this question a useful flexible functional form for the density distribution as a function of height z from the plane, is the generalised sech distribution proposed by Van der Kruit (1988): ρ(z) = 2−2/nρe sech2/n(nz/2ze) , (1). The exponential, sech, and sech distributions correspond to n = ∞, 2, and 1 respectively For data analysis this representation is unsatisfactory, because we want to constrain the value of the parameter n, but it has an infinite range, causing difficulty in defining the prior.

Methods

Results

Discussion

Conclusion