Integral Gauss's Law of Gravity with Gravitational Field Flux Lines to Interpret Rotation Curves and Tully-Fisher Relation of Disk Galaxies

Objects in disk galaxies can show Non-Keplerian rotational behaviors below a critical acceleration of the order of 10\textasciicircum -10 m/s\textasciicircum 2. MOND (Modified Newtonian Dynamics) theory provided good fits for the rotation curves and the Tully-Fisher relation by postulating that either the Newton's force law or equivalently the Newtonian gravity can be modified. In this report, a physical mechanism of 1/r distance dependent field at non-relativistic limit is proposed within a gravitational field flux conservation picture by generalizing the Integral Gauss's law of gravity. Firstly, 1/r dependence together with a disk thickness dependence of gravitational field and in turn the flat rotation curves are obtained by a Gaussian surface with cylindrical symmetry where most of the gravitational fluxes are distributed eventually along the radial direction of the disk plane. The Gaussian disk thickness as a hidden dynamical variable is discussed with observational evidences. Subsequently, a spherical to cylindrical transition, across a certain critical field, of the Gaussian surface symmetry is shown to give the exact algebraic M$\propto $v\textasciicircum 4 Tully-Fisher relation. The structural-dynamical relations revealed by the radial acceleration relation from SPARC (Spitzer Photometry and Accurate Rotation Curves ) data of Stacy McGaugh's team are mapped to the field flux distribution picture. The bulge to disk structural transition can be directly interpreted by the Newtonian to Non-Newtonian gravitational flux distribution. Extensions of this gravitational lines of force picture to larger scale structures are discussed.