Modeling the vertical distribution of the Milky Way's flat subsystem objects

This paper is an initial stage of consideration of the general problem of joint modeling of the vertical structure of a Galactic flat subsystem and the average surface of the disk of the Galaxy, taking into account the natural and measurement dispersions. We approximate the average surface of the Galactic disk with a polynomial model and determine its parameters by minimizing the squared deviations of objects along the normal to the model surface. The developed method allows us to simultaneously identify significant details of the Galactic warping and estimate the offset $z_\odot$ of the Sun relative to the average (non-flat) surface of the Galactic disk and the vertical scale of the object system for an arbitrary area of the disk covered by data. The method is applied to data on classical Cepheids. Significant local extremes of the average disk surface model were found: the minimum in the first Galactic quadrant and the maximum in the second. A well-known warp in the third quadrant has been confirmed. The optimal order of the model was found to be $n_\text{o}=4$. The local (near the Sun, $n_\text{o}=0$) estimate of $z_\odot = 28.1 \pm \left.6.1\right|_{\text{stat.}}\left.{}\pm1.3\right|_{\text{cal.}}$ pc is close to the non-local ($n_\text{o}=4$) $z_\odot = 27.1 \pm \left.8.8\right|_{\text{stat.}}\left.{}^{+1.3}_{-1.2}\right|_{\text{cal.}}$ pc, which suggests that the proposed method eliminates the influence of warping on the $z_\odot$ estimate. However, the non-local estimate of the vertical standard deviation of Cepheids $\sigma_{\rho} = 132.0 \pm \left.3.7\right|_{\text{stat.}}\left.{}^{+6.3}_{-5.9}\right|_{\text{cal.}}$ pc differs significantly from the local $\sigma_{\rho} = \left.76.5 \pm 4.4\right|_{\text{stat.}}\left.{}^{+3.6}_{-3.4}\right|_{\text{cal.}}$ pc, which means the need to introduce more complex models outside the Sun's vicinity.