Is there any mathematical formula or empirical relation that defines the density of stars as a function of distance from the Galaxy centre in a spiral galaxy?
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A very crude, approximate answer would be that the density of stars decreases exponentially with distance $r$ from the center of the galaxy, and also exponentially as a function of distance $z$ above or below the plane of the disk: $$\rho_{\star}(r,z) = \rho_{0} \exp(-r/h) \exp(-z/z_{0})$$ where $h$ and $z_{0}$ are the radial and vertical scale lengths.
This is only approximate because it ignores the presences of "bulges" (excess concentrations of stars in the inner region of some galaxies), local variations due to bars and spiral arms, and the fact that many galaxies have deviations from the radial exponential profile at large radii (e.g., so-called "truncations", where the radial falloff changes to a steeper exponential, along with the presence of low-density, more spherical stellar "halos").
Peter Erwin
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