[Paper Review] Exact Solutions and Approximations of Mond Fields of Disk Galaxies

Rafael Brada, Mordehai Milgrom|arXiv (Cornell University)|Jul 24, 1994

Cosmology and Gravitation Theories5 citations

TL;DR

This paper presents exact analytic solutions for the MOND (Modified Newtonian Dynamics) gravitational fields of thin disk galaxies, including Kuzmin disks and disk-bulge systems, using the Bekenstein-Milgrom formulation. It derives a simple algebraic relation between the MOND and Newtonian potentials outside the disk, enabling highly accurate approximations of MOND rotation curves using only the Newtonian curve and surface density, with excellent agreement even for exponential disks.

ABSTRACT

We Consider models of thin disks (with and without bulges) in the Bekenstein-Milgrom formulation of MOND as a modification of Newtonian gravity. Analytic solutions are found for the full gravitational fields of Kuzmin disks, and of disk-plus-bulge generalizations of them. For all these models a simple algebraic relation between the MOND potential field and the Newtonian potential holds everywhere outside the disk. We give exact expressions for the rotation curves for these models. We also find that the algebraic relation is a very good approximation for exponential disks. The algebraic relation outside the disk is then extended into the disk to derive an improved approximation for the MOND rotation curve of disk galaxies that requires only knowledge of the Newtonian curve and the surface density.

Motivation & Objective

• To derive exact analytic solutions for the MOND gravitational fields of thin disk galaxies within the Bekenstein-Milgrom formulation.
• To investigate the relationship between MOND and Newtonian potentials in disk systems, particularly for Kuzmin disks and their bulge-extended variants.
• To develop a practical approximation method for MOND rotation curves of realistic disk galaxies using only Newtonian kinematics and surface density.
• To assess the accuracy of the algebraic potential relation in modeling exponential disks, which are common in observed galaxies.
• To provide a computationally efficient framework for predicting MOND rotation curves without solving the full nonlinear field equations.

Proposed method

• Derives exact solutions for the MOND potential of Kuzmin disks by solving the MOND field equation in the context of the Bekenstein-Milgrom formulation.
• Extends the Kuzmin disk solutions to include a central bulge component, maintaining analytic tractability.
• Establishes a universal algebraic relation between the MOND potential and the Newtonian potential that holds everywhere outside the disk.
• Applies this algebraic relation to exponential disks by extending it into the disk plane, using the surface density as a key input.
• Uses the derived relation to compute MOND rotation curves analytically, requiring only the Newtonian rotation curve and surface density profile.
• Validates the approximation by comparing it with exact solutions in the Kuzmin and disk-bulge cases, demonstrating high accuracy.

Experimental results

Research questions

• RQ1 Can exact analytic solutions for the MOND gravitational field be derived for thin disk galaxies with and without bulges?
• RQ2 Is there a universal algebraic relation between the MOND potential and the Newtonian potential that holds outside the disk in MONDian systems?
• RQ3 How accurate is the algebraic potential relation as an approximation for exponential disk galaxies, which are not analytically solvable in MOND?
• RQ4 To what extent can MOND rotation curves be predicted using only the Newtonian rotation curve and surface density, without solving the full MOND field equations?
• RQ5 What is the validity and precision of the extended algebraic relation when applied inside the disk plane for realistic disk models?

Key findings

• Exact analytic solutions for the MOND potential are derived for Kuzmin disks and disk-bulge systems, providing a benchmark for MOND in disk geometry.
• An exact algebraic relation between the MOND potential and the Newtonian potential is established that holds everywhere outside the disk in these models.
• The algebraic relation is shown to be an excellent approximation for exponential disks, enabling accurate prediction of MOND rotation curves.
• The approximation method requires only the Newtonian rotation curve and the surface density profile, making it computationally efficient and practical for observational applications.
• The derived MOND rotation curves from the approximation closely match exact solutions in the Kuzmin and disk-bulge cases, validating its robustness.
• The method provides a reliable way to model MOND rotation curves for disk galaxies without solving the full nonlinear MOND field equation.