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Field Equation of Correlation Function of Mass Density Fluctuation for Self-Gravitating Systems

We study the mass density distribution of the Newtonian self-gravitating system. Modeling the system either as a gas in thermal equilibrium, or as a fluid in hydrostatical equilibrium, we obtain the field equation of correlation function $ξ(r)$ of the mass density fluctuation itself. It can apply to the study of galaxy clustering on Universe large scales. The observed $ξ(r)\simeq (r_0/r)^{1.7}$ follows from first principle. The equation tells that $ξ(r)$ depends on the point mass $m$ and Jeans wavelength scale $λ_{0}$, which are different for galaxies and clusters. It explains several longstanding, prominent features of the observed clustering: the profile of $ξ_{cc}(r)$ of clusters is similar to $ξ_{gg}(r)$ of galaxies but with a higher amplitude and a longer correlation length, the correlation length increases with the mean separation between clusters $r_0\simeq 0.4d$ as the observed scaling, and on very large scales $ξ_{cc}(r)$ exhibits periodic oscillations with a characteristic wavelength $\sim 120$Mpc. With a set of fixed model parameters, the solution $ξ(r)$ for galaxies and for clusters, the power spectrum, the projected, and the angular correlation function, simultaneously agree with the observational data from the surveys, such as Automatic Plate Measuring (APM), Two-degree-Field Galaxy Redshift Survey (2dFGRS), and Sloan Digital Sky Survey (SDSS), etc.