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

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.