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By solving analytically the various types of Lane-Emden equations withrotation, we have discovered two new coupled fundamental properties ofrotating, self-gravitating, gaseous disks in equilibrium: Isothermal disksmust, on average, exhibit strict power-law density profiles in radius $x$ ontheir equatorial planes of the form $A x^{k-1}$, where $A$ and $k-1$ are theintegration constants, and "flat" rotation curves precisely such as thoseobserved in spiral galaxy disks. Polytropic disks must, on average, exhibitstrict density profiles of the form $\left[\ln(A x^k)\right]^n$, where $n$ isthe polytropic index, and "flat" rotation curves described by square roots ofupper incomplete gamma functions. By "on average," we mean that, irrespectiveof the chosen boundary conditions, the actual profiles must oscillate aroundand remain close to the strict mean profiles of the analytic singularequilibrium solutions. We call such singular solutions the "intrinsic"solutions of the differential equations because they are demanded by thesecond-order equations themselves with no regard to the Cauchy problem. Theresults are directly applicable to gaseous galaxy disks that have long beenknown to be isothermal and to protoplanetary disks during the extendedisothermal and adiabatic phases of their evolution. In galactic gas dynamics,they have the potential to resolve the dark matter--modified gravitycontroversy in a sweeping manner, as they render both of these hypothesesunnecessary. In protoplanetary disk research, they provide observers withpowerful new probing tool, as they predict a clear and simple connectionbetween the radial density profiles and the rotation curves of self-gravitatingdisks in their very early (pre-Class 0 and perhaps the youngest Class YoungStellar Objects) phases of evolution.

journal of modern physics

The Case Against Dark Matter and Modified Gravity: Flat Rotation Curves  Are a Rigorous Requirement in Rotating Self-Gravitating Newtonian Gaseous  Disks