# A new formulation of the external field effect in MOND and numerical simulations of ultra-diffuse dwarf galaxies $-$ application to NGC 1052-DF2 and NGC 1052-DF4

## TL;DR

This paper introduces a new analytic formulation and self-consistent simulations of the external field effect in MOND, applied to ultra-diffuse dwarf galaxies DF2 and DF4, clarifying their dynamics and distances.

## Contribution

It presents a novel analytic approach and live N-body simulations for the external field effect in MOND, improving understanding of dwarf galaxy dynamics near massive hosts.

## Key findings

- MOND velocity dispersion matches observations if DF2 is within 150 kpc of NGC 1052.
- DF2's globular clusters are brighter and larger unless it is closer than 13 Mpc.
- The new formulation agrees with previous models and supports MOND explanations for dwarf galaxy behavior.

## Abstract

The ultra-diffuse dwarf galaxy NGC 1052-DF2 (DF2) has ten (eleven) measured globular clusters (GCs) with a line-of-sight velocity dispersion of $\sigma=7.8^{+5.2}_{-2.2}\,$km/s ($\sigma=10.6^{+3.9}_{-2.3}\,$km/s). Our conventional statistical analysis of the original ten GCs gives $\sigma=8.0^{+4.3}_{-3.0}\,$km/s. The overall distribution of velocities agrees well with a Gaussian of this width. Due to the non-linear Poisson equation in MOND, a dwarf galaxy has weaker self-gravity when in close proximity to a massive host. This external field effect is investigated using a new analytic formulation and fully self-consistent live $N$-body models in MOND. Our formulation agrees well with that of Famaey and McGaugh (2012). These new simulations confirm our analytic results and suggest that DF2 may be in a deep-freeze state unique to MOND. The correctly calculated MOND velocity dispersion agrees with our inferred dispersion and that of van Dokkum et al. (2018b) if DF2 is within 150 kpc of NGC 1052 and both are 20 Mpc away. The GCs of DF2 are however significantly brighter and larger than normal GCs, a problem which disappears if DF2 is significantly closer to us. A distance of 10-13 Mpc makes DF2 a normal dwarf galaxy even more consistent with MOND and the 13 Mpc distance reported by Trujillo et. al. (2019). We discuss the similar dwarf DF4, finding good agreement with MOND. We also discuss possible massive galaxies near DF2 and DF4 along with their distances and peculiar velocities, noting that NGC 1052 may lie at a distance near 10 Mpc.