An Empirical Law to Reproduce the Galaxy Rotation Curves

Abstract

Profiles of galaxy's rotation velocity (rotation curves) remain unexpectedly flat at large distances, where visible matter alone should make the rotation speed decrease with radius. The usual explanation requires large amounts of unseen dark matter. Here, we introduce an empirical law in which spacetime itself appears to store additional energy when distorted by baryonic matter. The baryonic potential $Φ_b$ made by the baryonic energy density $ρ_b$ generates an additional energy density $ρ_s = μ_s Φ_b^2 / c^4$, leading to a Poisson equation for the total potential $Φ_{\rm tot}$, \[ \nabla^2Φ_{\rm tot} = 4πG\,ρ_b /c^2 + 4πG\,μ_s\,Φ_b^2 / c^6. \] When assuming $μ_s = K M_b^{-3/2}$ with a parameter $K$, we applied this equation to 91 galaxies from the SPARC database. The model reproduced quite well both the inner rise and outer flat regions of the observed rotation curves using the observed baryonic mass profiles only. The $K$ values for the 91 galaxies were found to be concentrated within a narrow range. These results suggest that dark matter is directly connected to the baryonic gravitational potential. A theoretical interpretation of the empirical law is also discussed, in which a relation between the additional energy and the cosmological constant is implied.

Figures (2)

• Figure 1: Rotation curves for 12 representative galaxies in the SPARC database spanning a wide range of morphologies and masses. Black points: observed data. Lines: the velocity contributions from gas (green dashed), disk (orange dotted), and bulge (dark-blue dash-dotted line). The total baryonic contribution $V_{b}$ is shown in orange solid, the additional gravity contribution $V_{\mu}$ in purple dashed, and the total velocity $V_{\rm tot}$ in red solid line.
• Figure 2: Fit results for 91 reliable galaxy data in the SPARC database. Left: distribution of the best--fit $\log_{10}K$ values forming a Gaussian distribution with a mean of $\mu_{\log K }= 70.23$ and a width of $\sigma_{\log K}= 0.23$. Right: distribution of the minimum $\chi^2/\rm dof$ values in the fits.