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

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.

Appendix was added to address vigorous objections against using cylindrical coordinates to model the equatorial planes of vertically thin disks

References in corpus (14)

• Filamentary structure of star-forming complexes
• Impact of a global quadratic potential on galactic rotation curves
• Modeling the Resolved Disk Around the Class 0 Protostar L1527
• Testing Fundamental Physics with Distant Star Clusters: Analysis of Observational Data on Palomar 14
• Observations of Infalling and Rotational Motions on a 1,000-AU Scale around 17 Class 0 and 0/I Protostars: Hints of Disk Growth and Magnetic Braking?
• Observation of rotation in star forming regions: clouds, cores, disks, and jets
• The dynamical state of the First Hydrostatic Core Candidate Cha-MMS1
• On local dark matter density
• Dark Matter Deprivation in Field Elliptical Galaxy NGC 7507
• Galaxy rotation curves with log-normal density distribution
• A new equation for the mid-plane potential of power law discs. II. Exact solutions and approximate formulae
• Exact Solutions of the Isothermal Lane--Emden Equation with Rotation and Implications for the Formation of Planets and Satellites
• Extended models of gravity in SNIa cosmological data using genetic algorithms
• Gravitational lensing evidence against extended dark matter halos

Cited by in corpus (4)

• Status, Challenges and Directions in Indirect Dark Matter Searches
• Entropy and Mass Distribution in Disc Galaxies
• Exact Axisymmetric Solutions of the 2-D Lane-Emden Equations with Rotation
• Stability and Damping in the Disks of Massive Galaxies