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We investigate the form of the local vertical density profile of the stars inthe Galactic disk, close to the Galactic plane. We use a homogeneous sample of34000 ultracool dwarfs M7 to L2.5 that all lie within 350 pc of the plane. Wefit a profile of the form sech$^\alpha$, where $\alpha=2$ is the theoreticallypreferred isothermal profile and $\alpha=0$ is the exponential function. Largervalues of $\alpha$ correspond to greater flattening of the profile towards theplane. We employ a likelihood analysis that accounts in a direct way forunresolved binaries in the sample, as well as for the spread in absolutemagnitude $M_J$ within each spectral sub-type (Malmquist bias). We measure$\alpha=0.29^{+0.12}_{-0.13}$. The $\alpha=1$ (sech) and flatter profiles areruled out at high confidence for this sample, while $\alpha=0$ (exponential) isincluded in the 95% credible interval. Any flattening relative to exponentialis 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 spiralgalaxies by de Grijs et al., measured from photometry of galaxies viewed edgeon. The measured profile allows an accurate determination of the local spacedensity of ultracool dwarfs M7 to L2.5, and we use this to make a newdetermination of the luminosity function at the bottom of the main sequence.Our results for the luminosity function are a factor two to three lower thanthe recent measurement by Bardalez Gagliuffi et al., that uses stars in thelocal 25 pc radius bubble, but agree well with the older study by Cruz et al.

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