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

TL;DR

This paper analytically demonstrates that rotating self-gravitating gaseous disks naturally exhibit flat rotation curves and specific density profiles, challenging dark matter and modified gravity explanations for galactic rotation phenomena.

Contribution

It introduces new fundamental properties of gaseous disks, showing they inherently produce flat rotation curves without dark matter or modified gravity, based on analytic solutions of Lane-Emden equations.

Findings

Isothermal disks follow power-law density profiles with flat rotation curves.

Polytropic disks exhibit density profiles involving logarithmic functions with flat rotation curves.

Results apply to galactic and protoplanetary disks, offering a new perspective on rotation curve origins.

Abstract

By solving analytically the various types of Lane-Emden equations with rotation, we have discovered two new coupled fundamental properties of rotating, self-gravitating, gaseous disks in equilibrium: Isothermal disks must, on average, exhibit strict power-law density profiles in radius $x$ on their equatorial planes of the form $A x^{k-1}$, where $A$ and $k-1$ are the integration constants, and "flat" rotation curves precisely such as those observed in spiral galaxy disks. Polytropic disks must, on average, exhibit strict density profiles of the form $\left[\ln(A x^k)\right]^n$, where $n$ is the polytropic index, and "flat" rotation curves described by square roots of upper incomplete gamma functions. By "on average," we mean that, irrespective of the chosen boundary conditions, the actual profiles must oscillate around and remain close to the strict mean profiles of the analytic singular equilibrium solutions. We call such singular solutions the "intrinsic" solutions of the differential equations because they are demanded by the second-order equations themselves with no regard to the Cauchy problem. The results are directly applicable to gaseous galaxy disks that have long been known to be isothermal and to protoplanetary disks during the extended isothermal and adiabatic phases of their evolution. In galactic gas dynamics, they have the potential to resolve the dark matter--modified gravity controversy in a sweeping manner, as they render both of these hypotheses unnecessary. In protoplanetary disk research, they provide observers with powerful new probing tool, as they predict a clear and simple connection between the radial density profiles and the rotation curves of self-gravitating disks in their very early (pre-Class 0 and perhaps the youngest Class Young Stellar Objects) phases of evolution.