Numerical solutions of the complete two-body system in QUMOND

TL;DR

This work develops a Green's-function framework to compute exact QUMOND accelerations for $N$-body systems, treating point masses as Dirac sequence limits and recasting the MOND contribution via phantom-dark-matter sources. It systematically tests integrability, implements a two-dimensional numerical integration scheme under axial symmetry, and applies the method to isolated binaries and binaries in an external Galactic field, showing that the effective-one-body approximation generally overestimates internal accelerations in the transition regime. The results quantify boost factors as functions of mass ratio, separation, and external field strength, revealing geometry-dependent differences that challenge commonly used modeling approaches (e.g., Banik 2024a) and highlighting the need for fully consistent MOND $N$-body orbit integrators. The analysis also discusses observational prospects, including potential tests with wide binaries and Solar-system analogs, while outlining computational and methodological challenges for extending to full three-dimensional, non-collinear systems and to AQUAL-type formulations. Overall, the paper provides a rigorous, numerically explicit benchmark for MOND two-body dynamics in QUMOND and clarifies the limitations of simplified effective-one-body treatments in this regime.

Abstract

Due to the non-linearity of the QUMOND field equations, in the modelling of binaries so far the two-body system is replaced by an effective one-body system, where the central particle contains the total mass of both binary components and is orbited by a massless test particle. In this work, the discrepancy between the effective one-body treatment and the complete two-body solution in QUMOND is quantified. Particles are treated as limits of Dirac sequences. Then, the QUMOND contribution to the total kinematical acceleration of a particle is expressed as a Green's integral which is calculated numerically. In the non-linear transition regime the kinematical acceleration of the effective one-body system with a total mass of 2 Msun is up to a factor of 1.44 higher than the Newtonian acceleration, whereas the acceleration is only boosted by a factor of 1.2--1.3 in the two-body system in the case of the simple transition function.

Figures (21)

• Figure 1: Set up of particle positions.
• Figure 2: Phantom dark matter density distribution of an isolated two-body system.
• Figure 3: Speed of the convergence of the numerical integration of Eq. (\ref{['eq_alpha']}) for an equal-mass isolated binary and the standard transition function.
• Figure 4: Speed of the (non)-convergence of the numerical integration of Eq. (\ref{['eq_alpha']}) for an equal-mass isolated binary and the simple transition function (symmetric evaluation for particle one, non-symmetric evaluation for particle two).
• Figure 5: Forces of both components of a non-equal-mass isolated binary (simple transition function, symmetric evaluation).