Spiral galaxies do not need dark matter to describe their rotational velocities

We show [1] using integral techniques from fluid mechanics that there is no need for `Dark Matter’ or `Alternative Gravity' to explain `anomalous' galactic rotational velocities.. The integral momentum balance equation reduces to a simple balance between the product of momentum thickness, say z<sub>m</sub>, times (V<sub>φ</sub><sup>2</sup> - V<sub>r</sub> dV<sub>r</sub>/dr) and 4 π G times the integral of the `two-dimensional' baryonic density times r of stars and/or dust. The coupling between stars and dust has been included by using the mass conservation equation. The integral on the right-hand side is asymptotic to the total mass in the galaxy; and the momentum thickness times velocity-squared term is proportional to 2 G M_tot where M_tot is the total mass in the galaxy.

We also carry out an equilibrium similarity analysis and find two similarity regimes: one in which both z_m and the velocities are asymptotically constant; and the other in which z_m increases linearly while the velocities are power laws in r. The Triangulum Galaxy (M33) and the Milky Way show both similarity regimes. The integral equation results describe well both the shape and magnitude of the velocity measurements, even at large radius. `Velocity Mean Streamlines' computed for the Milky Way are log spirals, consistent with the equilibrium similarity analysis.

[1] George, W.K. and T.G. Johansson (2023) ``Fluid Mechanics of Isolated Spiral Galaxies'', http://www.turbulence-online.com/Publications/A_Fluid_Mechanics_Solution_Galaxy_Rotation_Problem_Oct_2023_final_Version_3LP.pdf