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
  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.
