1 Introduction
Recent empirical fits indicate a very strong correlation between the central surface density of stars
and dynamical mass
in 135 disk galaxies (Lelli et al.
2016
). The central-surface-densities relation can be described by a double power law (Lelli et al.
2016
)
where
and
are fitted parameters. Generally speaking, high surface brightness galaxies give
while low surface brightness galaxies systematically deviate from unity. The observed scatter is small overall
and largely driven by observational uncertainties (Lelli et al.
2016
). This result is surprising because there is no obvious reason why the central surface density of stars strongly correlates with the dynamical mass.
Recently, Milgrom (
2016
) applied Modified Newtonian Dynamics (MOND) to derive this relation analytically. By using quasilinear MOND (QUMOND), we can obtain the expression in Milgrom (
2016
)
where
where
is the baryonic mass density,
is the velocity dispersion of baryonic matter and
is the total gravitational potential (including baryonic matter and dark matter). Note that the Jeans equation used here assumes isotropy and spherical symmetry. Nevertheless, it can also be applied in spiral galaxies if we only focus on the cylindrical radial direction in the galactic plane. The velocity dispersion would be directly proportional to the rotational velocity (Croton
2009
). Since the isothermal distribution of baryons corresponds to the constant velocity dispersion
by Equation (
3), we get (Chan
2013a
; Evans et al.
2009
)
By substituting the solution of the above equation
Let
Since the isothermal baryonic component yields
we derive (Chan
2013a
)
This shows that if the baryonic component follows the isothermal distribution, the resulting total matter density also follows the isothermal distribution, which exhibits excellent agreement with observational data
(Koopmans et al.
2009
; Velander et al.
2011
; Grillo
2012
). Note that we did not assume any isothermality of dark matter in the first place. We only assume baryons follow an isothermal distribution. Finally, based on the above derivation, the total potential that can yield the isothermal distribution of baryons is also isothermal. However, astrophysicists usually assume that dark matter follows the Navarro-Frenk-White (NFW) profile (Navarro et al.
1997
), but not an isothermal profile. Even for some CDM simulations with baryons, no isothermal profile would be generated. Most of the resultant profiles are still an NFW profile, but with a shallower slope near the center (Schaller et al.
2015
; Chan et al.
2015
). This is because all of the simulations do not assume an isothermal distribution of baryons. Feedback can substantially change the stellar and dark matter dynamics and shape, causing the complete system to depart from a simple isothermal profile. This may be a reason why most CDM simulations do not generate an isothermal spherical distribution. In other words, if the collision between the baryonic matter is vigorous enough that the baryonic distribution can keep the isothermal distribution, the dark matter density and the total density also follow an isothermal profile. This is a result of solving the Jeans equation.
In addition, the above result can also give a simple explanation to the 'Halo-disk conspiracy problem' (why the transition from disk to halo domination is so smooth) (Battaner & Florido
2000
; Remus et al.
2013
). Since the total mass density consists of two components, the baryonic matter density
and dark matter density
we can write
Here, we have used two parameters to represent the velocity dispersion of baryonic matter (i.e.
. For a baryon dominated galaxy, we have
. For a dark matter dominated galaxy, we have
Note that
is not the velocity dispersion of dark matter particles. The value of
is the limit for the velocity dispersion of baryonic matter in a dark matter dominated galaxy. On the other hand, since the observational data in galaxies strongly support the existence of a core in the dark matter density profile (de Blok
2010
), we may slightly modify the dark matter density profile without destroying the isothermal distribution at large
by a cored-isothermal profile (Chan
2013a
)
where
and
are the central density and core radius of the dark matter profile respectively. The origin of a dark matter core (size ∼ Kpc) may be due to the self-interaction between dark matter particles (Spergel & Steinhardt
2000
; Vogelsberger et al.
2012
; Chan
2013b
) or some baryonic feedback such as supernovae (de Blok
2010
; Governato et al.
2012
). In fact, the existence of a dark matter core is still a controversial issue. Some studies claim that cores do not exist in galaxies (Fattahi et al.
2016
). However, most observational data seem to favor the existence of dark matter cores (for a review, please see de Blok
2010
; Bull et al.
2016
; Del Popolo & Le Delliou
2017
). Therefore, our model basically follows this assumption. Since the Jeans equation only describes the distribution of matter due to gravitational interaction, the effect of core formation due to other mechanisms at small
is not included. The small modification of the profile here is necessary to match the assumption of the existence of dark matter cores.
Recent studies suggest that the product of the central density and core radius of dark matter is almost a constant for many galaxies
(Gentile et al.
2009
; Burkert
2015
). Therefore, we can write
where
where
Using the definition of the dynamical central surface density (Milgrom
2016
), we get
where
is the central total mass density. Here, we work in cylindrical coordinates
with the
-axis along the axisymmetry axis, and
By integrating the above equation, we get
Let
where
is the characteristic disk length such that the Newtonian gravitational acceleration due to baryons just outside the disk is
Finally, we get
where
in Milgrom (
2016
). Based on the derivation in QUMOND,
for
(Milgrom
2016
). Taking
we can write the above equation in terms of
Since
we have
. For
(baryonic matter dominates the galaxy), we have
The opposite asymptote (
) gives
These asymptotic results are identical to the results in (Milgrom
2016
). The only difference is the functional form. Milgrom's result gives
where
(Milgrom
2016
).
In Figure 1, we fit our result with the observed data obtained from Lelli et al. (
2016
) and compare with Milgrom's result. Here, we assume that
is the proxy for
(Milgrom
2016
). Both theories can give the same agreement with observational data (our model with
gives a better fit). Therefore, we conclude that the observational data also support the dark matter paradigm if dark matter has an isothermal distribution, but not only using MOND theory.
3 Discussion
In this article, we derive the central-surface-densities relation by using the steady-state Jeans equation in the dark matter framework. If the baryonic density distribution is isothermal, the resultant total mass density also follows an isothermal distribution. This result agrees with the observational data and explains why rotational curves are flat for many galaxies (Sofue & Rubin
2001
). We also relate the dark matter density profile with the baryonic matter content and show that
The asymptotic relations for both regimes are identical to Milgrom's result. Milgrom (
2016
) claims that there is no reason why
is so well correlated with local
in the dark matter paradigm. However, as shown in our derivation, the existence of a dark matter core may give a reason why these quantities are correlated. Therefore, the claim in Milgrom (
2016
) is wrong. Generally speaking, both MOND and dark matter paradigms can give the same agreement with the observed central-surface-densities relation. In fact, some recent studies also show a similar conclusion by using CDM models (Di Cintio & Lelli
2016
; Navarro et al.
2016
).
In the derivation, there are a few constants involved:
and a. Although these values are not universal constants for all galaxies, the ranges of these values are quite narrow (Gentile et al.
2009
; Rocha et al.
2013
; Kassin et al.
2006
). For example, the value of
is nearly a constant for a luminosity range of 14 magnitudes and the whole Hubble sequence (Gentile et al.
2009
). The variations in these constants among different galaxies would contribute to the scatter in the resulting relation.
However, as mentioned in the introduction, MOND works very poorly in galaxy clusters. Most predictions for galaxy clusters in MOND theory do not match the observational data, including the lensing results (Ferreras et al.
2012
). Although the failure of MOND on large scale is not related to our discussion here, MOND, suggested to be a universal theory, should work on all scales. Therefore, it is reasonable to suspect that MOND working well in galaxies is just a coincidence. If the observational data in galaxies support both paradigms but MOND does not work for galaxy clusters, it is reasonable to deny MOND is an effective theory to explain the missing mass problem. Moreover, Chan (
2013a
) shows that MOND is equivalent to a particular form of dark matter density profile (isothermal distribution) in the dark matter model. It also explains why MOND works in galaxies but not in galaxy clusters (Chan
2013a
). Although many studies have shown that the CDM model predictions for small scale structures (e.g. in dwarf galaxies) do not agree with observations (de Blok
2010
), recent studies have started to realize that baryonic feedback might be an important mechanism to reconcile the discrepancies between theory and observations (Macciò et al.
2012
; Peñarrubia et al.
2012
; Pontzen & Governato
2014
). Our result basically supports this argument. Provided the dark matter and baryon distributions are described by an isothermal sphere profile, the dark matter can also accommodate the missing mass problem. It is therefore not essential to invoke new physics (MOND) to address the current missing mass problem.