Exact semianalytical calculation of rotation curves with Bekenstein-Milgrom nonrelativistic MOND

TL;DR

This paper presents a semi-analytical method to accurately compute MOND rotation curves for various density profiles, improving upon the simple algebraic rule by incorporating the AQUAL formalism, with small errors in most cases.

Contribution

It introduces recipes based on Milgrom's proposal to calculate MOND rotation curves analytically for any density distribution, aligning with more sophisticated models like AQUAL.

Findings

Simple algebraic rule differs from AQUAL-MOND by less than 5% in most cases.

Errors exceed 5% when over half the mass is in the MOND regime, reaching over 70% in some models.

The formalism significantly affects the predicted rotation speed slopes and vertical profiles.

Abstract

Astronomers use to derive MOdified Newtonian Dynamics (MOND) rotation curves using the simple algebraic rule of calculating the acceleration as equal to the Newtonian acceleration ($a$) divided by some factor $\mu (a)$. However, there are velocity differences between this simple rule and the calculation derived from more sophisticated MOND versions such as AQUAL or QMOND, created to expand MOND heuristic law and preserve the conservation of momentum, angular momentum, and energy, and follow the weak equivalence principle. Here we provide recipes based on Milgrom's proposal to calculate semianalytically (without numerical simulations) MOND rotation curves for any density distribution based on AQUAL, applying it to different models of thin disks. The application of this formalism is equivalent to the creation of a fictitious phantom mass whose field may be used in a Newtonian way to calculate iteratively the MOND accelerations. In most cases, the differences between the application of the simple algebraic rule and the AQUAL-MOND calculations are small, $\lesssim 5$%. However, the error of the algebraic solution is larger than 5% when more than half of the mass is in the MONDian regime (where Newtonian and MOND rotation speeds differ by more than 10%), reaching in some cases $>70$% discrepance, such as in Maclaurin disks, representative of galaxies for which the rotational velocity rises to the edge of the disk as is seen in irregular galaxies. The slope of the rotation speed in the dependence with the radius or the vertical distance of the plane is also significantly changed.