Does the galaxy NGC1052-DF2 falsify Milgromian dynamics?
Pavel Kroupa Hosein Haghi Behnam Javanmardi Akram Hasani Zonoozi Oliver Mueller Indranil Banik Xufen Wu Hongsheng Zhao Joerg Dabringhausen


Branes with Brains: Exploring String Vacua 
 with Deep Reinforcement Learning
    [
    December 30, 2023
=============================================================================






  * Helmholtz-Institut für Strahlen- und Kernphysik, University
  of Bonn, Nussallee 14-16, D-53115 Bonn, Germany


  * Charles University in Prague, Faculty of Mathematics and
  Physics, Astronomical Institute, V Holešovičkách 2, CZ-180
  00 Praha 8, Czech Republic


  * Institute for Advanced Studies in Basic Sciences, Physics
  department, Gavazang road, Zanjan, 4513766731 Zanjan, Iran


  * School of Astronomy,
Institute for Research in Fundamental Sciences (IPM)
P. O. Box 19395-5531, Tehran, Iran


  * Departement Physik, Universität Basel, Klingelbergstrasse 82,
 CH-4056 Basel, Switzerland


  * Scottish Universities Physics Alliance, University of St
  Andrews, North Haugh, St Andrews, Fife KY16 9SS, UK 


  * CAS Key Laboratory for Research in Galaxies and Cosmology, Department of 
Astronomy, University of Science and Technology of China, Hefei, 230026, P.R. China


  * School of Astronomy and Space Science, University of Science and Technology 
of China, Hefei 230026, China


  * Department of Physics and Astronomy, LaserLaB, Vrije Universiteit, De Boelelaan
  1081, NL-1081 HV Amsterdam, the Netherlands





A great challenge in present-day physics is to understand
  whether the observed internal dynamics of galaxies is due to dark
  matter matter or due to a modification of the law of
  gravity. Recently, van Dokkum et al.<cit.> reported
  that the ultra-diffuse dwarf galaxy NGC1052-DF2 lacks dark matter,
  and they claimed that this would – paradoxically – be problematic
  for modified gravity theories like Milgromian dynamics
  (MOND<cit.>). However, NGC1052-DF2 is not isolated,
  so that a valid prediction of its internal dynamics in MOND cannot
  be made without properly accounting for the external gravitational
  fields from neighbouring galaxies. Including this external field
  effect following Haghi et al.<cit.> shows that NGC1052-DF2
  is consistent with MOND.  

In any viable cosmological model, both primordial and tidal dwarf
galaxies, which form in gas-rich tidal debris when galaxies interact,
should exist. Within the standard dark-matter based cosmological
model, primordial dwarfs are dark-matter dominated, whereas tidal
dwarf galaxies contain very little (if any) dark
matter<cit.>. In MOND – a classical potential theory
of gravity derivable from a Lagrangian with conserved energy, total
momentum and angular momentum <cit.> – the two types
of dwarf galaxies cannot be distinguished.  Until now, all known dwarf
galaxies have shown similar dynamical behaviour, possibly implying
falsification of the dark-matter models<cit.>. The discovery
by van Dokkum et al.<cit.> of a galaxy lacking dark
matter would thus constitute an important verification of the standard
dark-matter cosmological model.

Given the density distribution of baryonic matter, ρ, the
gravitational potential of Milgromian gravitation, ϕ, is
determined by the generalised nonlinear Poisson equation<cit.>
∇⃗·[ μ( | ∇⃗ϕ |/a_o )
  ∇⃗ϕ] = 4 π G ρ, where the function
μ( | ∇⃗ϕ |/a_o ) describes the transition
from the Newtonian (| ∇⃗ϕ |/a_o ≫ 1) to the
Milgromian (| ∇⃗ϕ |/a_o ≪ 1)
regime<cit.>, a_0 = 3600km^2s^-2kpc^-1 is
Milgrom's constant<cit.> and G is Newton's universal
constant of gravitation.  An equation of this form has been used in
investigations of classical models of quark dynamics and is therefore
not without precedent in physics<cit.>. One implication of this
equation is that the coupling strength of an object described by
ρ depends on the position of another object, whereas its inertial
mass is given only by its baryonic component. This breaking of the
equivalence of the active gravitating and inertial mass of an object
constitutes a prediction of a new physical phenomenon, which does not
exist in standard gravitation models and may be visualised as the
phantom dark-matter halo (incorporated in ϕ) being reduced in the
presence of a sufficiently strong constant external
field<cit.>. This may be related to the quantum
vacuum<cit.>. A dwarf galaxy in the vicinity of a major
galaxy may thus lose its phantom dark-matter halo appearing as a
purely Newtonian system<cit.>.

Following van Dokkum et al.<cit.> we assume
NGC1052-DF2 is located at a distance of D=20Mpc from the
Earth in the NGC1052 group, has a globular cluster population
with an effective radius of r_e = 3.1kpc (the stellar body has
r_e≈ 2.2kpc), an absolute V-band magnitde M_V = -15.4 and
a mass-to-light ratio of the stellar population of
M_ DF2/L_V=2, where M_ DF2 is the baryonic mass of
NGC1052-DF2 and L_V is its absolute luminosity in the visual
photometric band.




If NGC1052-DF2 were isolated, then its MOND internal
acceleration would be
a_i=570  km^2  s^-2 kpc^-1 = 0.15 a_0.  This
weak gravity would imply a MONDian velocity dispersion of
σ_ M≈( (4/81)  G M_
    DF2 a_o)^1/4≈ 20km/s.  The 2σ
(3σ) confidence range for the observed velocity dispersion of
the globular clusters, calculated using the robust biweight
estimator<cit.>, is 0.0 - 14.2km/s (0.0 - 19.7km/s),
falsifying MOND with high significance<cit.>.

This inference is only correct if NGC1052-DF2 is sufficiently
isolated and at the distance of D≈ 20Mpc.  The bright
elliptical host galaxy NGC1052 has a baryonic mass<cit.>
M_ NGC1052≈ 10^11  M_⊙.  The projected
separation between NGC1052-DF2 and NGC1052 is
D_ sep = 80kpc if both are at the same distance of
D = 20Mpc. At this separation, the acceleration induced by the
MOND external-field (EF) of the host,
a_e = √(a_0× G × M_ NGC1052) / D_ sep, is
a_e ≈ 0.14 a_0, such that a_i ≈ a_e ≪ a_0. Because
the accelaration by the external-field is as important as internal
galactic acceleration, it needs to be properly taken into account in
calculating the velocity dispersion in MOND, σ_ M.  This
is true for any object of mass m whenever its physical size
r_e > (m/M)^1/2  D_ sep and the external field is caused by
another object of mass M at a distance of D_ sep.

Using a MOND N-body integrator<cit.>, Haghi et al.<cit.>
quantified the global 1-D line-of-sight velocity dispersion
(σ_ M, EF) of a non-isolated stellar system lying in the
intermediate external-field regime (see Supplementary Information for
more details).  The dependence of the velocity dispersion of
NGC1052-DF2, σ_ M, EF, on D_ sep and
M_ NGC1052 is shown in Fig. <ref>.  The velocity
dispersion declines from the isolated, asymptotic value
σ_ M = 20km/s with decreasing D_ sep and
increasing mass because the external field of NGC1052
suppresses the phantom dark-matter halo, leading to Newtonian
behaviour in the case of a_e ≫ a_i.  For the nominal host
mass<cit.>, M_ NGC1052=10^11 M_⊙ and for
D_ sep=113kpc, the MOND velocity dispersion is in agreement
within the 2σ confidence range of the observational value of
van Dokkum et al.<cit.>, as shown by the star in
Fig. <ref>.  Thus, the estimate of the internal velocity
dispersion of the globular cluster system of NGC1052-DF2
according to MOND is σ_ M, EF = 14km/s for
M_ NGC1052=10^11 M_⊙ and D_ sep=113kpc
(Fig. <ref>), in agreement with the observed value within
the 2σ confidence level.  NGC1052-DF2 is thus consistent
with an important MOND prediction.

For the external-field to suppress the phantom dark-matter halo of
NGC1052-DF2, this galaxy needs to be close to
NGC1052. In ref.<cit.> the distance of
NGC1052-DF2 is estimated using the surface-brightness
fluctuation method.





The calculations performed here show that NGC1052-DF2 falsifies
MOND if its distance from Earth is D > 18Mpc and its physical
separation from NGC 1052 or any other massive galaxy is
D_ sep > 300kpc. In this case, the external field becomes
negligible and the galaxy has too small a velocity dispersion for its
stellar mass in MOND. On the other hand, the observed apparent
luminosity of the galaxy is l_V = L_V / D^2. In Milgromian dynamics,
σ_ M∝ M_ DF2^1/4∝ L_V^1/4∝
D^1/2, so NGC1052-DF2 would be consistent with MOND at the
2σ confidence level if D<13Mpc even if it were isolated.
We note that the velocity dispersion predicted by Newtonian dynamics
is also proportional to D^1/2 for a galaxy without dark matter.


Future high-precision observations will ascertain the distance of
NGC1052-DF2 and will constrain its gravitating mass using
stellar velocities within NGC1052-DF2, instead of globular
cluster velocities.  As a word of caution, if this galaxy is not in
virial equilibrium then using the globular clusters as tracers for the
mass may be unreliable, independently of the underlying cosmological
model.


Online Supplementary Information contains the external
field effect calculation in MOND.




References
naturemag






Acknowledgements

We acknowledge support from the Alexander von Humboldt Foundation, the
German Academic Exchange Service (DAAD), the Swiss National Science
Foundation, the Natural Science Foundation of China (grant numbers
11503025 and 11421303), the Anhui Natural Science Foundation (grant
number 1708085MA20), the 'Hundred Talents Project of Anhui Province',
the 'Fundamental Research Funds for the Central Universities', the
'DAAD- Ostpartnerschaftsprogramm für 2018' and the Iran Science Elites
Federation (ISEF).


Author contributions:

P.K. conceived the work and wrote the mansucript; H.H. and
A.H.Z. performed the external field calculations; O.M.  and
B.J. contributed observational expertise on galaxies; I.B., X.W.  and
H.Z. contributed MOND expertise; J.D.  contributed dwarf galaxy
expertise. All co-authors contributed to the manuscript with comments
and passages.

Statement on competing interests:

Declared none.










































Supplementary Information




The external field effect in MOND:
In a self-bound gravitational system with baryonic mass M and
half-mass radius r_h, its components experience a Newtonian internal
acceleration, a_N ≡ G M/( 2 r_h^2), which is in
Newtonian gravitation linearly additive to any external acceleration
a_e. If this external acceleration is constant across the system
then it does not influence the internal eigengravitation.  However, in
MOND, the internal gravitation is suppressed by the external field if
it is sufficiently strong.  This is known since the original work by
Milgrom (1983, ApJ 270, 365)<cit.> and follows from the
Lagrangian formulation of MOND which implies the generalised
non-linear Poisson equation (Bekenstein & Milgrom 1984, ApJ 286,
7)<cit.>.  The details of computing the acceleration on the
systems's constituents are simple only in the limits where the
internal acceleration is much larger than Milgrom's constant a_0 (in
which case the system is Newtonian), or when the external field is
negligible and the internal acceleration is much smaller than
a_0. In the latter case the system is in the deep MOND limit such
that its MOND internal acceleration is
a_i = √((G  M  a_0))/r_e for a system with effective radius
r_e, and effectively behaves as if it is embedded in a dark-matter
halo which can be described mathematically as a Newtonian isothermal
logarithmic potential. This Milgromian potential is referred to as a
phantom dark-matter halo as it is not made of particles, and this
limiting case was used by van Dokkum et al.<cit.> to
calculate the value of the velocity dispersion predicted by MOND.  In
the case when the external field is significant compared to the
internal acceleration and when both are smaller than or comparable to
Milgrom's constant a_0 then the calculations become less
straightforward, and again, only in the limiting cases can
calculations be performed.  For example, when a_e > a_0 and
a_i ≪ a_ e forces from different masses can be superposed (Banik
& Zhao 2015, arXiv150908457B), though this is generally not correct
as MOND is non-linear.  Effectively an external field truncates (or
reduces or “melts”) the phantom dark-matter halo of the system.  The
external field effect thereby allows observational tests for MOND,
because it has no counterpart in Newtonian dynamics.  Simplified
descriptions of the external-field effect are available in Wu &
Kroupa (2015, MNRAS 446, 330) and an in-depth discussion is found in
the major review by Famaey & McGaugh (2012)<cit.>.

Observational evidence for the MOND external-field effect has been
found in the form of falling rotation curves of disk galaxies (Wu &
Kroupa 2015, MNRAS, 446, 330; Hees et al. 2016, MNRAS, 455, 449; Haghi
et al. 2016, MNRAS 458, 4172), and in the asymmetric tidal tail of the
globular star cluster Pal 5 (Thomas et al. 2018<cit.>).  The
successful prediction of the very low velocity dispersion of the
ultra-diffuse dwarf satellite galaxy Crater II<cit.> and the
velocity dispersions of Andromeda satellite galaxies (McGaugh &
Milgrom 2013, ApJ 775, 139) are highly relevant for the problem at
hand.

Haghi et al.<cit.> quantified the global 1-D line-of-sight
velocity dispersion (σ_ M, EF ) of a non-isolated stellar
system lying in the intermediate external-field regime, using the code
N-MODY, which has a N-body MOND mean-field potential solver
developed by Ciotti, Londrillo & Nipoti (2006, ApJ 640,
741)<cit.>.  Haghi et al.<cit.> fit their results for
σ_ M, EF in the intermediate regime
(a_i ≈ a_e < a_0) for different values of the external
field. They quantified the different asymptotic behavior in fig. 5 of
Haghi et al.<cit.> (i.e. in the Newtonian, the deep-MOND and
and the external-field-dominated regime). The formulation was
presented as their eq. 16 and 17 with coefficients provided for
different values of a_e in their table 1.

Thus, for example, the M=10^11 M_⊙,
D_ sep =√(2)× 80 = 113kpc case corresponds to
a_e = 0.1 a_0.  Adopting the three coefficients tabulated in the
row a_e=0.1 a_0  in table 1 of Haghi et al. (2009)<cit.>,
we use their eqs. 13, 15 and 17, to obtain σ = 14.8km/s.  This
is indicated by the star symbol in Fig. <ref>.  Likewise for
other values of D_ sep and M combinations, we can work out
a_e, and the corresponding σ_M, EF, and these are plotted as curves
in Fig. <ref>.  

The interpolation to other-than-the tabulated values of a_e in
table 1 in Haghi et al. (2009)<cit.> is not unique and for the
purpose of documentation we provide in the following one possible
analytical formulation (eq. <ref>) for the data in their table 1
allowing σ_ M, EF to be calculated as a function of the
internal field in the system with mass M exposed to an external
field, a_e.  The one-dimensional (line of sight) velocity
dispersion, σ_ M, EF (in km/s),

    log_10σ_ M, EF =  log_10σ_ M + F(a_e),

where the velocity dispersion of an isolated system

    σ_ M =
    ( 4 81  G  M  a_0 )^14×(1 +
    0.56  exp( 3.02   x )
    )^0.184,

where x ≡ log_10( a_N / a_0 ) and

    F(a_e) = -  A(a_e)  4 (  ln[  exp( -x A(a_e)) + B(a_e) ] + C(a_e) ),

with G being Newton's gravitational constant. Note that
Eq. <ref> supersedes the equation used by van Dokkum et
al.<cit.> by a correction factor which ensures the
correct behaviour as the internal acceleration approaches the
Newtonian value.

The functions A, B, C are fits to the data in table 1 in Haghi et al.<cit.>:

    A(a_e)    = 5.3 (
    10.56 + ( y +2 )^3.22)  , 
    
    B(a_e)    =
    0.44   exp(-4.26  y ),
    
    C(a_e)    = -12.83  ln(
     exp(  -( y + 2 )   1.28)
     + 0.85
    ),

where y ≡ log_10( a_e/a_0 ). The important
issue is to obtain an accurate fit to the combination of A, B and
C given in eq. 15 of Haghi et al. (2009)<cit.> - fitting
the coefficients individually is less important.

These formulae reproduce well the previous analytical velocity
dispersion estimators in systems 



in the isolated deep-MOND regime [that is,
σ_ M, EF≈σ_M = ( 4 G M   a_0 / 81
)^1/4] (Milgrom, M. 1994, ApJ 429, 540; McGaugh, S. &
Milgrom, M., 2013, ApJ 775, 139) and for the deep-MOND
external-field-dominated regime
(σ_ M, EF∝ 1/√((a_e)), eq. 2 of McGaugh
2016<cit.>).

We note that the three orange points in Fig. <ref> can be obtained
directly from the a_e = 0.1   a_0 line in table 1 of Haghi et
al. (2009)<cit.> without interpolation.




