We use the distant outer halo globular cluster Palomar 14 as a test case for classical versus modified Newtonian dynamics (MOND). Previous theoretical calculations have shown that the line-of-sight velocity dispersion predicted by these theories can differ by up to a factor of 3 for such sparse, remote clusters like Pal 14. We determine the line-of-sight velocity dispersion of Palomar 14 by measuring radial velocities of 17 red giant cluster members obtained using the Very Large Telescope and Keck telescope. The systemic velocity of Palomar 14 is (72.28 ± 0.12) km s −1 . The derived velocity dispersion of (0.38 ± 0.12) km s −1 of the 16 definite member stars is in agreement with the theoretical prediction for the classical Newtonian case according to Baumgardt et al. In order to exclude the possibility that a peculiar mass function might have influenced our measurements, we derived the cluster's main-sequence mass function down to 0.53 M using archival images obtained with the Hubble Space Telescope. We found a mass function slope of α = 1.27 ± 0.44, which is, compared to the canonical mass function, a significantly shallower slope. The derived lower limit on the cluster's mass is higher than the theoretically predicted mass in the case of MOND. Our data are consistent with a central density of ρ 0 = 0.1 M pc −3 . We need no dark matter in Palomar 14. If the cluster is on a circular orbit, our spectroscopic and photometric results argue against MOND, unless the cluster experienced significant mass loss.
We investigate the mean velocity dispersion and the velocity dispersion profile of stellar systems in modified Newtonian dynamics (MOND), using the N‐body code n‐mody, which is a particle‐mesh‐based code with a numerical MOND potential solver developed by Ciotti, Londrillo & Nipoti. We have calculated mean velocity dispersions for stellar systems following Plummer density distributions with masses in the range of 104 to 109 M⊙ and which are either isolated or immersed in an external field. Our integrations reproduce previous analytic estimates for stellar velocities in systems in the deep MOND regime (ai, ae≪a0), where the motion of stars is either dominated by internal accelerations (ai≫ae) or constant external accelerations (ae≫ai). In addition, we derive for the first time analytic formulae for the line‐of‐sight velocity dispersion in the intermediate regime (ai∼ae∼a0). This allows for a much‐improved comparison of MOND with observed velocity dispersions of stellar systems. We finally derive the velocity dispersion of the globular cluster Pal 14 as one of the outer Milky Way halo globular clusters that have recently been proposed as a differentiator between Newtonian and MONDian dynamics.
We present the first ever direct N-body computations of an old Milky Way globular cluster over its entire lifetime on a star-by-star basis. Using recent GPU hardware at Bonn University, we have performed a comprehensive set of N-body calculations to model the distant outer halo globular cluster Palomar 14 (Pal 14). Pal 14 is unusual in that its mean density is about 10 times smaller than that in the solar neighbourhood. Its large radius as well as its low-mass make it possible to simulate Pal 14 on a star-by-star basis. By varying the initial conditions, we aim at finding an initial N-body model which reproduces the observational data best in terms of its basic parameters, i.e. half-light radius, mass and velocity dispersion. We furthermore focus on reproducing the stellar mass function slope of Pal 14 which was found to be significantly shallower than in most globular clusters. While some of our models can reproduce Pal 14's basic parameters reasonably well, we find that dynamical mass segregation alone cannot explain the mass function slope of Pal 14 when starting from the canonical Kroupa initial mass function (IMF). In order to seek an explanation for this discrepancy, we compute additional initial models with varying degrees of primordial mass segregation as well as with a flattened IMF. The necessary degree of primordial mass segregation turns out to be very high, though, such that we prefer the latter hypothesis which we discuss in detail. This modelling has shown that the initial conditions of Pal 14 after gas expulsion must have been a half-mass radius of about 20 pc, a mass of about 50 000 M , and possibly some mass segregation or an already established non-canonical IMF depleted in low-mass stars. Such conditions might be obtained by a violent early gas-expulsion phase from an embedded cluster born with mass segregation. Only at large Galactocentric radii are clusters likely to survive as bound entities the destructive gas-expulsion process we seem to have uncovered for Pal 14. In addition, we compute a model with a 5 per cent primordial binary fraction to test if such a population has an effect on the cluster's evolution. We see no significant effect, though, and moreover find that the binary fraction of Pal 14 stays almost the same and gives the final fraction over its entire lifetime due to the cluster's extremely low density. Low-density, halo globular clusters might therefore be good targets to test primordial binary fractions of globular clusters.
The ultra-diffuse dwarf galaxy NGC 1052-DF2 (DF2) has ten (eleven) measured globular clusters (GCs) with a line-of-sight velocity dispersion of σ = 7.8 +5.2 −2.2 km/s (σ = 10.6 +3.9 −2.3 km/s). Our conventional statistical analysis of the original ten GCs gives σ = 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.
Unlike Newtonian dynamics which is linear and obeys the strong equivalence principle, in any nonlinear gravitation such as Milgromian dynamics (MOND), the strong version of the equivalence principle is violated and the gravitational dynamics of a system is influenced by the external gravitational field in which it is embedded. This so called External Field Effect (EFE) is one of the important implications of MOND and provides a special context to test Milgromian dynamics. Here, we study the rotation curves (RCs) of 18 spiral galaxies and find that their shapes constrain the EFE. We show that the EFE can successfully remedy the overestimation of rotation velocities in 80% of the sample galaxies in Milgromian dynamics fits by decreasing the velocity in the outer part of the RCs. We compare the implied external field with the gravitational field for non-negligible nearby sources of each individual galaxy and find that in many cases it is compatible with the EFE within the uncertainties. We therefore argue that in the framework of Milgromian dynamics, one can constrain the gravitational field induced from the environment of galaxies using their RCs. We finally show that taking into account the EFE yields more realistic values for the stellar mass-to-light ratio in terms of stellar population synthesis than the ones implied without the EFE.
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