---
abstract: |
    We review the current status of the study of rotation curve (RC) of the Milky Way, and present a unified RC from the Galactic Center to the galacto-centric distance of about 100 kpc. The RC is used to directly calculate the distribution of the surface mass density (SMD). We then propose a method to derive the distribution of dark matter (DM) density in the in the Milky Way using the SMD distribution. The best-fit dark halo profile yielded a local DM density of We also review the estimations of the local DM density in the last decade, and show that the value is converging to a value at $\rho_\odot=0.39\pm 0.09$ .\
    [**Key words**]{} galaxies: DM—galaxies: individual (Milky Way)—galaxies: rotation curve\
    ([*Invited review accepted for Galaxies to appear in special issue on “Debate on the Physics of Galactic Rotation and the Existence of Dark Matter”*]{})
author:
- |
    Yoshiaki Sofue\
    Institute of Astronomy, The University of Tokyo, Mitaka, Tokyo 181-0015, Japan\
    E-mail: sofue@ioa.s.u-tokyo.ac.jp
title: Rotation Curve of the Milky Way and the Dark Matter Density
---

0[ V\_0 ]{}

Introduction
============

The rotation curve (RC) of the Milky Way was obtained by observations of galactic objects in the non-MOND (MOdified Newtonian Dynamics)frame work. The existence of the dark halo (DH) has been confirmed by the analysis of the observed RCs, assuming that  Newtonian dynamics applies evenly to the result of the observations. In this article, current works of RC observations are briefly reviewed, and a new estimation of the local dark matter (DM) density is presented in the framework of Newtonian dynamics.

An RC is defined as the mean circular velocity $\Vrot$ around the nucleus plotted as a function of the galacto-centric radius $R$. Non-circular streaming motion due to the triaxial mass distribution in a bar is crucial for kinematics in the innermost region, though it does not affect the mass determination much in the disk and halo. Spiral arms are another cause for local streaming, which affect the mass determination by several percent, while they do not influence the mass determination of the dark halo much.

There are several reviews on RCs and mass determination of galaxies \[[@SofueRubin2001; @Sofue2017; @Salucci2019]\]. In this review, we revisit recent RC studies and determination of the local DM density in our Milky Way. In Section \[Section2\], we briefly review the current status of the RC determinations along with the methods. We adopt the galactic constants: $(\rzero,\vzero)$=(8.0 kpc, 238 ) \[[@Honma+2012; @Honma+2015]\], where $\rzero$ is the distance of the Sun from the galactic center (GC) and $\vzero$ is the circular velocity of the local standard of rest (LSR) at the Sun \[[@Fich+1991]\].

Rotation Curve of the Milky Way {#Section2}
===============================

Progress in the Last Decades
----------------------------

The galactic RC is dependent on the galactic constants. Accordingly, the uncertainty and error in the RC include uncertainties of the constants. Currently recommended, determined, or measured values are summarized in Table \[tabGal\], where they appear to be converging to around $\sim 8.0-8.3$ and . In this paper, we adopt $R_0= 8.0$ kpc and $V_0 =238$ from the recent measurements with VERA (VLBI Experiments for Radio Astrometry) \[[@Honma+2012; @Honma+2015]\].

  **Authors (Year)**                                               **(kpc)**         **()**
  ---------------------------------------------------------------- ----------------- -------------
  IAU recommended (1982)                                           8.2               220
  Review before 1993 (Reid 1993) \[[@Reid1993]\]                   $8.0 \pm 0.5$     
  Olling and Dehnen 2003 \[[@Olling+2003]\]                        $7.1\pm 0.4$      $184\pm 8$
  VLBI Sgr A$^*$ (Ghez et al. 2008) \[[@Ghez+2008]\]               $8.4 \pm 0.4$     
  ibid (Gillessen et al. 2009) \[[@Gillessen+2009]\]               $8.33 \pm 0.35$   
  Maser astrometry (Reid et al. 2009) \[[@Reid+2009]\]             $8.4\pm 0.6$      $254\pm 16$
  Cepheids (Matsunaga et al. 2009) \[[@Matsunaga+2009]\]           $8.24 \pm 0.42$   
  VERA (Honma et al. 2012, 2015) \[[@Honma+2012; @Honma+2015]\].   $8.05\pm 0.45$    $238\pm 14$
  Adopted in this paper                                            8.0               238

  : Galactic constants ($R_0,V_0$). []{data-label="tabGal"}

The RC of the galaxy has been obtained by various methods as described in the next subsection, and many authors presented their results based on different galactic constants (Table \[tabrcmw\]).

  **Authors (Year)**                                                                                          **Radii (kpc)**   **Method**
  ----------------------------------------------------------------------------------------------------------- ----------------- -------------------------
  Burton and Gordon (1978)\[[@Burton+1978]\]                                                                  0–8               HI tangent
  Blitz et al. (1979) \[[@Blitz+1979]\]                                                                       8–18              OB-CO assoc.
  Clemens (1985)\[[@Clemens1985]\]                                                                            0 -18             CO/compil.
  Dehnen and  Binney (1998)\[[@Dehnen+1998]\]                                                                 8–20              compil. + model
  Genzel et al. (1994–), Ghez et al. (1998–)\[[@Genzel+2010; @Ghez+2008]\]                                    0–0.0001          GC IR spectr.
  Battinelli, et al. (2013)\[[@Battinelli+2013]\]                                                             9–24              C stars
  Bhattacharjee et al.(2014)\[[@Bhattacharjee+2014]\]                                                         0–200             Non-disk objects
  Lopez-Corredoira (2014)\[[@Lopez2014]\]                                                                     5–16              Red-clump giants $\mu$
  Boby et al. (2012)\[[@Bovy+2012b]\]                                                                         4-14              NIR spectroscopy
  Bobylev (2013); — & Bajkova (2015)\[[@Bobylev2013; @Bobylev+2015]\]                                         5–12              Masers/OB stars
  Reid et al. (2014)\[[@Reid+2014]\]                                                                          4-16              Masers SF regions, VLBI
  Honma et al. (2012, 2015)\[[@Honma+2012; @Honma+2015]\]                                                     3–20              Masers,VLBI
  Iocco et al. (2015, 2016); Pato & Iocco (2017a,b)\[[@Iocco+2015; @Iocco+2016; @Pato+2017a; @Pato+2017b]\]   1–25 kpc          CO/HI/opt/maser/compil.
  Huang et al. (2016)\[[@Huang+2016]\]                                                                        4.5–100           HI/opt/red giants
  Kre[ł]{}owski et al (2018)\[[@Krelowski+2018]\]                                                             8–12              GAIA
  Lin and Li (2019)\[[@Lin+2019]\]                                                                            4–100             compil.
  Eilers et al (2019)\[[@Eilers+2019]\]                                                                       5–25              Wise, 2Mass, GAIA
  Mróz et al. (2019)\[[@Mroz+2019]\]                                                                          4–20              Classical cepheids
  Sofue et al. (2009); Sofue (2013, 2015, this work)\[[@Sofue+2009; @Sofue2013; @Sofue2015]\]                 0.01–1000         CO/HI/maser/opt/compil.

  : Rotation curves (RCs) of the Milky Way galaxy.[]{data-label="tabrcmw"}

In the 1970–1980s, the inner RC was extensively measured using the terminal-velocities of HI (neutral hydrogen) and CO (carbon monoxide) gases \[[@Burton+1978; @Clemens1985; @Fich+1989]\]. In the late 1980s to the 2000s, outer rotation velocities were measured by combining optical distances of OB \[[@Blitz+1979; @Demers+2007]\]. The HI thickness method was also useful to measure rotation of the entire disk \[[@Merrifield1992; @Honma+1997]\]. The innermost mass distributions inside the GC have been obtained extensively since the 1990s using the motion of infrared stellar objects \[[@Genzel+2010; @Ghez+2008; @Lindqvist+1992; @Gillessen+2009]\].

Trigonometric determinations of both the 3D positions and velocities have provided the strongest tool to date for measurement of the galactic rotation \[[@Honma+2007; @Honma+2012; @Honma+2015; @Sakai+2015; @Nakanishi+2015]\]. A number of optical parallax measurements of stars such with GAIA have been obtained for RC determination \[[@Lopez2014; @Krelowski+2018]\].

The total mass of the galaxy, including the extended dark halo, has been measured by analyzing the outermost RC and motions of satellite galaxies orbiting the galaxy, and the mass up to kpc has been estimated to be $\sim 3 \times 10^{11}\Msun$ \[[@Sofue2015; @Callingham+2019]\].

Methods to Determine the Galactic RC
------------------------------------

The particular location of the Sun inside the Milky Way makes it difficult to measure the rotation velocity of the galactic objects. Sophisticated methods have been developed to solve this problem, as briefly described below.

### Tangent-Velocity Method 

Inside the solar circle ($-90^\circ \le l \le 90^\circ$), the galactic gas disk has tangential points, at which the rotation velocity is parallel to the line of sight and attains the maximum radial velocity $ {\vr}_{\rm ~max}$ (terminal or tangent-point velocity). The rotation velocity $V(R)$ at galacto-centric distance $R=\Rsun \sin~ l$ is calculated simply correcting for the solar motion.

###  Radial-Velocity + Distance Method 

If the distance $r$ of the object is measured by spectroscopic and/or trigonometric observations, the rotation velocity is obtained by geometric conversion of the radial velocity, distance, and the longitude. The distance has to be measured independently, often using spectroscopic distances of OB stars, and the distances are assumed to be the same as those of associated molecular clouds and HII (ionized hydrogen) regions, whose radial velocities are observed by radio lines. Since the photometric distances have often large errors, obtained RC plots show large scatter.

### Trigonometric Method

If the proper motion and radial velocity along with the distance are measured at the same time, or from different observations, the 3D velocity vector, and therefore the rotation velocity, of any source is uniquely determined without being biased by assumption of circular motion as well as the galactic constants. VLBI (very long baseline interferometer) measurements of maser sources \[[@Honma+2007; @Honma+2012; @Honma+2015; @Nakanishi+2015]\] and optical/IR trigonometry of stars \[[@Roeser+2010; @Lopez2014]\] have given the most accurate RC.

### Disk-Thickness Method

The errors in the above methods are mainly caused by the uncertainty of the distance measurements. This disadvantage is eased by the HI-disk thickness method \[[@Merrifield1992; @Honma+1997]\]. The angular thickness of the HI disk along an annulus ring is related to can be used to determine the rotation velocity by combining with radial velocity distribution along the longitude.

### Pseudo-RC from Non-Disk Objects

Beyond or outside the galactic disk, globular clusters and satellite galaxies are used to estimate the pseudo-circular velocity from their radial velocities based on the Virial theorem, assuming that their motions are at random, or the  rotation velocity is calculated by $\Vrot \sim \sqrt{2} v_g$, where $v_g$ is the galacto-centric radial velocity. [On the other hand, Huang et al. (2016) \[[@Huang+2016]\] have recently employed more sophisticated, probably more reliable, method to solve the Jeans equations for the non-disk stars and clusters.]{}

Unified RC
----------

A RC covering a wide region of the galaxy has been obtained by compiling the existing data by re-scaling the distances and velocities to the common galactic constants \[[@Sofue+2009]\], and later to (8.0 kpc, 238 ) \[[@Sofue2013; @Sofue2017]\]. In these works, the central RC inside the GC has been obtained from analyses of the kinematics of the molecular gas and infrared stellar motions as well as the supposed Keplerian motion representing the central massive black hole. Outer RC beyond $R\sim 30$ kpc has been determined from the radial motions of satellite galaxies and globular clusters.

The RC determination has been improved recently by compiling a large amount of data from a variety of spectroscopic as well as trigonometric measurements from radio to optical wavelengths. An extensive compilation of the data of rotation velocities of the galactic disk has been published recently, and is available as an internet data base \[[@Iocco+2015; @Iocco+2016; @Pato+2015a; @Pato+2015b; @Pato+2017a; @Pato+2017b]\].

Figure \[mwrc\]a shows the presently obtained unified RC using the curves from\[[@Sofue2015; @Sofue2017]\]and RC by Huang et al. (2016)\[[@Huang+2016]\] between $R=4.6$ and $\sim 100$ kpc. Although Huang et al. employed the galactic constants of (8.34 pc, 240 ), we did not apply rescaling to (8.0, 238), because the galacto-centric distances of off-plane objects are less dependent on the solar position compared to the disk objects as used for our RC at $<\sim 20$ kpc where the rotation velocity is rather flat, and also because their $V_0=240$ is close to our 238 .

![ (**a**) Unified RC of the Milky Way used in this paper for the mass distribution obtained by averaging the RCs from references \[[@Sofue2015; @Sofue2017; @Huang+2016]\]. The bars are standard deviations within each Gaussian-averaging bin. The plotted values are listed in the tables in Appendix \[AppendixA\]. (**b**) Logarithmic RC of the Milky Way from \[[@Sofue2015; @Sofue2017]\] (circles), compared with those from the recent literature: Green circles with error bars are from the compilation by \[[@Pato+2017a; @Pato+2017b]\]and blue triangles are their running averages. Red triangles stand for data from \[[@Krelowski+2018]\]based on GAIA data. [These two data are re-scaled to ($R_0, V_0$)=(8.0 kpc, 238 ).]{} Pink rectangles are the RC by \[[@Huang+2016]\].[without re-scaling]{}. (**c**) Same, but in linear scale. (**d**) Same, but close up in the solar vicinity.[]{data-label="mwrc"}](fig1.eps){width="8.5cm"}

The unified RC was obtained by taking Gaussian running averages of rotation velocities from the used RCs in each of newly settled radius bins, where the  statistical weight of each input point was given by the inverse of the squared error.

In Figure \[mwrc\]b,c we compare the unified RC with the recent measurements by \[[@Pato+2017a; @Pato+2017b; @Krelowski+2018]\]re-scaled to the galactic constants of (8.0 kpc, 238 ) following the method described in \[[@Sofue+2009]\].Although individual data points are largely scattered, their averages well coincide with the unified RC. In the figures we also compare the data with the RC by \[[@Huang+2016]\]up to $\sim 100$ kpc without rescaling, which also coincides with the other data within the scatter.

We here comment on the property of the unified RC built by averaging the published data. It must be remembered that the averaging procedure does not satisfy the condition of statistics in the strict meaning, because the data are compiled from different authors using a variety of instruments and analysis methods, which makes it difficult to evaluate common statistical weights for the used data points. So, remembering such a property, in view that the unified RC well approximates the original curves as well as for its convenience for the determination of the mass distribution by the least-squares and/or $\chi^2$ fitting, we shall employ it in our present analysis.

Mass Components
---------------

The rotation velocity is related to the gravitational potential, hence to the mass distribution, as V(R)==, where $\Phi_i$ is the gravitational potential of the $i$-th component and $V_i$ is the corresponding circular velocity. The rotation velocity is often represented by superposition of the central black hole (BH), bulge, disk, and the dark halo as V(R)= .

Here, the subscript BH represents black hole, b stands for bulge, d for disk, and h for the dark halo. The contribution from the black hole can be neglected in sufficiently high accuracy, when the dark halo is concerned. The mass components are usually assumed to have the following functional forms.

### Massive Black Hole

The GC of the Milky Way is known to nest a massive black hole of mass of $M_{\rm BH}\sim 4 \times 10^6\Msun$ \[[@Genzel+2010; @Ghez+2008; @Gillessen+2009]\].The RC is assumed to be expressed by a curve following the Newtonian potential of a point mass at the nucleus.

### De Vaucouleurs Bulge

The commonly used SMD profile to represent the central bulge, which is assumed to be proportional to the empirical optical profile of the surface brightness, is the  law \[[@deV1958]\], \_[b]{}(R) = \_[be]{} [exp]{} , \[eq-smdb\] where $ \Sigma_{\rm be} $ is the value at radius $R_{\rm b}$ enclosing a half of the integrated surface mass \[[@Sofue2017]\].Note that the surface profile, also the exponential disk, has a finite value at the center. The volume mass density $\rho(r)$ at radius $r$ for a spherical bulge is calculated using the SMD by (r) = \_r\^ dx, \[eq-rhob\] and the mass inside $R$ is M(R) =4\_0\^R r\^2(r)dr.

The circular velocity is thus obtained by V\_[b]{}(R) = .

More general form $e^{-(R/r_e)^n}$ called the law is discussed in relation to its dynamical relation to the galactic structure based on the more general profile \[[@Ciotti1991; @Trujillo2002]\].

### Exponential Disk

The galactic disk is generally represented by an exponential disk \[[@Freeman1970]\],where the SMD is expressed as \_[d]{} (R)=\_[d]{} [exp]{}(-R/R\_[d]{}). \[eq-smdd\]

Here, $\Sigma_{\rm d}$ is the central value, $R_d$ is the scale radius. The total mass of the exponential disk is given by $M_{\rm disk}= 2 \pi \Sigma_{dc} R_{\rm d}^2$. The RC for a thin exponential disk is expressed by \[[@Binney+1987]\]V\_[d]{}(R)=, where $y=R/ (2R_{\rm d}) $, and $I_i$ and $K_i$ are the modified Bessel functions.

The dark halo is described in the next section

Dark Halo {#Section3}
=========

The existence of dark halos in spiral galaxies has been firmly evidenced from the well established difference between the galaxy mass predicted by the luminosity and the mass predicted by the rotation velocities \[[@SofueRubin2001; @Sofue2017; @Salucci2019]\].

In the Milky Way, extensive analyses of RC and motions of non-disk objects such as globular clusters and dwarf galaxies in the Local Group have shown flat rotation up to $\sim 30$ kpc, beyond which the RC declines smoothly up to $\sim 300$ kpc \[[@Sofue2013; @Sofue2015]\]. Further analyses of non-disk tracer objects have also shown that the outer RC declines in a similar manner \[[@Bhattacharjee+2014; @Huang+2016; @Li+2017]\]. The fact that the rotation velocity beyond $R\sim 30$ kpc declines monotonically indicates that the isothermal model can be ruled out in representing the Milky Way’s halo.

Dark Halo Models
----------------

There have been various proposed DH models, which may be categorized into two types: The cored halo models \[[@Burkert1995; @Salucci+2000; @Brownstein+2006]\] are a modification of the isothermal model with a steeper decrease of density at large radii. The central cusp models \[[@Navarro+1995; @Navarro+1997; @Moore+1999; @Fukushige+2004]\] are based on extensive $N$-body numerical simulations of the structural evolution in the cold dark matter scenario in the expanding universe, which predict an infinitely increasing central peak. In either type, all the DH models predict decreasing DM density beyond $h$ as $\rho \propto R^{-3}$, or declining rotation velocity as $\Vrot \propto \sqrt{{\rm ln} \ R/R}$.

The cored halo models exhibit a central plateau of finite density with scale radius, or the core radius, $h$, and are often represented by the following functions, where $x=R/h$.

[**Isothermal halo**]{}: \_[Iso]{} (x)=, \[eq-iso\] [**Beta model with $\beta=1$** ]{} \[[@Navarro+1995]\] :\_(x)= . [**Burkert model**]{} \[[@Burkert1995; @Salucci+2000]\] : \_[Bur]{} (x)=, \[eq-bur\] [**Brownstein model**]{} \[[@Brownstein+2006]\] : \_[Bro]{} (x)=. \[eq-bro\]

On the other hand, the central cusp models are often represented by the following functions. [**NFW model**]{} \[[@Navarro+1996; @Navarro+1997]\] : \_[NFW]{} (x)=, \[eq-nfw\] [**Moore model**]{} \[[@Moore+1999; @Fukushige+2004]\] with $\alpha=1.5$: \_[Moo]{} (x) = =. \[eq-nfw\]

Figure \[rhoModels\] shows schematic density profiles for various DH models with $h=10$ kpc combined with the bulge and exponential disk, where the halo density is normalized at $R=20$ kpc..

![ (**Top**) Schematic density profiles of [NFW]{} (Navarro, Frenk, White) (thick solid), Moore (upper long dash), Burkert (long dash), Brownstein (dot), $\beta$ (dash), and isothermal (thin solid) models with $h=10$ kpc normalized at 20 kpc, compared with the disk (straight line) and bulge (inner thick dash). Uppermost thin lines are the sum of bulge, disk and halo. (**Bottom**) Same, but in log–log plot. The NFW cusp and cored halos do not much contribute to the mass density in the GC, whereas the Moore cusp somehow resembles the bulge profile. []{data-label="rhoModels"}](fig2.eps){width="10cm"}

Cusp vs Cored Halo
------------------

The density profiles for the NFW (Navarro, Frenk and White), Moore, Burkert, $\beta$, and Brownstein models are almost identical beyond the core radius $h$, where they tend to $\propto R^{-3}$. Differences among the models appear within the Solar circle. The cusp models (NFW and Moore models) predict steep increase of density toward the center with a singularity. The cored halo models predict a mild and low density plateau in the center with the peak densities not much differing from each other within a factor of two. However, the Burkert model has a singularity with the density gradient being not continuous across the nucleus.

Most of the DH models predict lower density in the innermost galaxy by two to several orders of magnitudes than the bulge’s density. This implies that the DH does not much influence the kinematics in the inner galaxy. Namely, it is practically impossible to detect the DM cusp by analyzing the RC. Only the Moore model predicts cusp density exceeding the bulge’s density in the very center at $R<\sim 0.1$ pc, whereas the applicability of the model to such small sized region is not obvious \[[@Fukushige+2004]\].

Central DM Density
------------------

If we assume that the functional form of the NFW model is valid in the very central region, the SMD at $R\sim 100$ pc could be estimated to be about $\Sigma \sim 2.2\times 10^3\Msun {\rm pc}^{-2}$. This yields an approximate volume density on the order of $\rho \sim \Sigma/R\sim 11\Msun {\rm pc}^{-3} \sim 840$ GeV cm$^{-3}$ for a detector of $\sim 1.4\deg$ resolution.

Such estimations could be a key to the indirect detection experiments of DM in the GC . However, it is stressed that the DM density in the GC is two to several orders of magnitudes smaller than the bulge’s density on the order of $10^4-10^5$ GeV cm$^{-3}$, making the kinematical detection of DM difficult.

Interestingly, the column density of DM, hence brightness (flux/steradian) of self-annihilation emission ($\gamma$-ray) stays almost constant against the radius and is therefore constant regardless the resolution of the detector. On the other hand, the emission measure $\sim \rho^2 R$ varies as $\propto R^{-1}$, hence, the brightness of collision-origin emission ($\gamma$ or microwave haze) increases toward the center \[e.g.,[@Finkbeiner2004]\], so that the detection rate will increase with the detector’s resolution.

Another concern about the DM cusp is the kinetic energy of individual particles. In order for the cusp to be stationary, the particles must be bound to the gravitational potential, so that the particle’s speed must be lower than the escaping velocity $v\sim \sqrt{2}\Vrot \sim 300$ . This will give a constraint on the cross section $\sigma_A$ of the DM annihilation, if the collision rate $\sigma_A v$ is fixed by the detection of DM-origin emissions.

The cored halo models (isothermal, Burkert, Brownstein, and the $\beta$ models) predict a mild and finite-density plateau with scale radius of $h$ ($\sim 10$ kpc). Their central densities are also several orders of magnitude less than the bulge’s density, hence do not contribute to the kinematics of the gas and stars in the GC.

DM Density from Direct SMD {#Section4}
==========================

SMD from RC
-----------

In the decomposition method of the RC, the resulting mass distribution depend on the assumed functional forms of the model profiles. In order to avoid this inconvenience, the RC can be used to directly calculate the surface mass distribution without employing any functional form. Only an assumption has to be made, either if the galaxy’s shape is a sphere or a flat disk.

On the assumption of spherical distribution, the mass inside radius $R$ is given by $$M(R)=\frac{R {V(R)}^{2}}{G}.
\label{masssphere}$$

Then the surface-mass density (SMD) ${\Sigma}_{S}(R)$ at $R$ is calculated by \_[S]{}(R) = 2 \_0\^ (r) dz, \[smdsphere\] where $$\rho(r) =\frac{1}{4 \pi r^2} \frac{dM(r)}{dr}.
\label{rhosphere}$$

If the galaxy is assumed to be a flat thin disk, the SMD ${\Sigma}_{\rm d}(R)$ is calculated by solving Poisson’s equation (Freeman 1970; Binney and  Tremaine 1987) by $${\Sigma}_{\rm d}(R) =\frac{1}{{\pi}^2 G}  
\left[ \frac{1}{R} \int\limits_0^R 
{\left(\frac{dV^2}{dr} \right)}_x K \left(\frac{x}{R}\right)dx 
+ \int\limits_R^{\infty} {\left(\frac{dV^2}{dr} \right)}_x K \left
(\frac{R}{x}\right) \frac{dx}{x} \right].
\label{smdflat}$$

Here, $K$ is the complete elliptic integral, which becomes very large when $x\simeq R$.

The SMD distributions in the galaxy for the sphere and flat-disk cases have been calculated for the recent RCs \[[@Sofue2017]\].In this paper we apply the same method to the here obtained unified RC (Figure \[mwrc\]). Since we aim at studying the dark halo, which is postulated to be rather spherical than a flat disk, we assume spherical mass distribution. The calculated SMD distribution is shown in Figure \[smd\_fit\].

![(**Top**) Direct surface-mass density (SMD) calculated for the unified RC in figure \[mwrc\] in spherical symmetry assumption (dots with error bars) in semi-logarithmic representation. The solid line is the $\chi^2$ fit, and red, blue, and dashed lines represent the NFW halo, disk, and bulge, respectively. (**Bottom**) Same, but in log–log plots. The semi-logarithmic plot makes it easier to discriminate the dark halo from exponential disk, which appears as a straight line. The plotted values are listed in the tables in Appendix \[AppendixA\]. []{data-label="smd_fit"}](fig3.eps){width="10cm"}

The SMD is strongly concentrated toward the center, reaching a value as high as $\sim 10^5 \Msun~{\rm pc}^{-2}$ within $R\sim 10$ pc, representing the core of the central bulge with the extent of several hundred pc. It is followed by a straightly declining profile from $R\sim 2$ to 8 kpc in the semi-logarithmic plot, representing the exponential nature of the galactic disk. In the outer galaxy beyond $\sim 8$ kpc, the SMD profile tends to be displaced from the straight disk profile, and is followed by an extended outskirt with a slowly declining profile, representing a massive halo extending to the end of the RC measurement at .

Fitting by Bulge, Disk, and Dark Halo
-------------------------------------

In order to separate the dark halo from the disk and bulge components, the well established RC decomposition method has been extensively applied to the RCs \[[@Sofue2017; @Salucci2019]\].Besides this traditional method, we here propose to use the SMD distribution. For this, we assume three mass components of bulge, exponential disk, and dark halo. In order to represent the, we employ the NFW profile as a ’tool’ for its popularity and for the dynamics background based on the extensive numerical simulations.

We employ the least $\chi^2$ fitting method, where $\chi^2$ is defined by \^2=\_i\[(SMD\_i\^[direct]{} - SMD\_i\^[calc]{})/\_i\]\^2, with $i$ denoting the value at the $i$-th data point, and $\sigma_i$ is the standard deviation around each data point in the running averaging procedure of the SMD distribution.

Fitting parameters are the scale radius $a_d$ and central SMD $\Sigma_d^0$ for the disk, and the scale (core) radius $h$ and representative DM density $\rho_{\rm model}^0$ for the halo. The bulge SMD is fixed to an assumed profile, which is negligible in the present fitting range at $R\ge 1$ kpc.

The fitting was obtained between $R=1$ and 100 kpc. The fitting result for the NFW halo model is shown in Figure \[smd\_fit\]. The solid line is the $\chi^2$ fit to SMD, and red, blue, and dashed lines represent the halo, disk, and bulge components, respectively. Note that the semi-logarithmic plot makes it visually easier to recognize the dark halo significantly displaced from the exponential disk, which appears as a straight line.

Local DM Density
----------------

We thus obtained the NFW DM halo parameters to be $h=10.94\pm 1.05$ kpc, $\rho_{\rm NFW}^0=0.787\pm 0.037$ , which yields the local DM density $\rho_\odot=0.359\pm 0.017$ . The best-fit parameters for the disk are determined to be $a_d=4.38\pm 0.35 $ kpc and $\Sigma_0=(1.28\pm 0.09)\times 10^3 \Msun {\rm pc}^{-2}$. Table \[tab\_fit\] lists the fitted result along with the minimized $\chi^2$ value.

  **Component**             **Parameter**        **Fitted Value**                                  
  ------------------------- -------------------- ------------------------------------------------- --------
  Expo. disk                $a_d$                $4.38\pm 0.35 $ kpc                               
                            $\Sigma_0$           $(1.28\pm 0.09)\times 10^3 \Msun {\rm pc}^{-2}$   
  NFW dark halo             $h$                  $10.94\pm 1.05$ kpc                               
                            $\rho_{\rm NFW}^0$   $0.787\pm 0.037$                                  
                            $\rho_\odot$         $0.359\pm 0.017$                                  11.9
  Burkert$^\dagger$         $\rho_\odot$         $\sim 0.30\pm 0.02$                               $17.3$
  Brownstein$^\dagger$      $\rho_\odot$         $\sim 0.40\pm 0.02$                               $17.9$
  $\beta$ model$^\dagger$   $\rho_\odot$         $\sim 0.31\pm 0.02$                               $17.3$

  : Best-fit parameters of the direct SMD by NFW halo and exponential disk.[]{data-label="tab_fit"}

$^\dagger$ Rough fitting, not conclusive.

We also obtained $\chi^2$ fitting using the Burkert, Brownstein, and $\beta$ profiles, and listed the local DM density and minimized $\chi^2$ in Table \[tab\_fit\]. In these three models, the $\chi^2 \sim 17-18$ were found to be systematically greater than that for the NFW model ($\chi^2=11.9$). The reason for the difference is due to the systematic difference in the functional behavior between NFW and the other three models: NSF has a cusp steeply increasing toward the center with sharpening scale radius, which results in the possibility of finer fitting to the slightly curved SMD profile at $R<\sim 10$ kpc in the semi-log plot. On the contrary, the other three models predict almost negligible SMD there, so that halo parameters contribute less intensively to the fitting in the innermost region, or the fitting must be done only by the disk’s two parameters there, resulting in worse fitting.

  ----------------------------------------------------------------------- ------------------- ----------- -------- --
  [**Reference**]{}                                                                                                
                                                                                              **(kpc)**   **()**   
  Weber and de Boer (2010)\[[@Weber+2010]\]                               0.2 - 0.4                                
  Catena and Ulio (2010)\[[@Catena+2010]\]                                $0.389 \pm 0.025$                        
  Bovy and Tremaine (2012) \[[@Bovy+2012]\]                               $0.3\pm 0.1$                             
  Piffl et al. (2014) \[[@Piffl+2014]\]                                   0.58                                     
  Pato et al (2015), Pato & Iocco (2015) \[[@Pato+2015a; @Pato+2015b]\]   $0.42\pm 0.25$                  230      
  Huang et al. (2016)\[[@Huang+2016]\]                                    $0.32 \pm 0.02$     8.34        240      
  McMillan (2017)\[[@McMillan2017]\]                                      $0.38\pm 0.04$      8.21        233.1    
  Lin and Li (2019)\[[@Lin+2019]\]                                        $0.51 \pm 0.09$     8.1         240      
  Salucci et al. (2010, 2019) \[[@Salucci+2010; @Salucci2019]\]           $0.43\pm 0.06$      8.29        239      
  Eilers et al (2019) \[[@Eilers+2019]\]                                  $0.3\pm 0.03$       8.1         229      
  de Salas et al. (2019) \[[@deSalas+2019]\]                              $0.3 - 0.4$                              
  Cautun et al (2019) \[[@Cautun+2019]\]                                  $0.34\pm 0.02$      8           229      
  Karukes et al (2019) \[[@Karukes+2019]\]                                $0.43 \pm 0.02$     8.34        240      
  Sofue (2013) \[[@Sofue2013]\]                                           $0.40 \pm 0.04$     8.0         238      
  —– (2020 this paper)                                                    $0.36\pm 0.02$      8.0         238      
  Average$^\ddagger$                                                      $0.387 \pm 0.080$                        
  ----------------------------------------------------------------------- ------------------- ----------- -------- --

  : Current determinations of the local DM density and the literature.[]{data-label="tab_localdm"}

$^\dagger$ =38.2 $\Msun\ {\rm pc}^{-3}$. $^\ddagger$ Simple average of the listed values with equal weighting.\

The local DM density is a key quantity in laboratory experiments by the direct detection of DM, and has been estimated by a number of authors with a variety of methods. In Table \[tab\_localdm\] we list the local DM densities from the literature along with the present value for NFW profile. They are also plotted in Figure \[localDMauthors\] against publication years. The $\rho_\odot$ values seem to be nearly constant in the decade. Averaging all the listed values with an equal weighting yields $\rho_\odot=0.39 \pm 0.09$ , which may be taken as a ’canonical’ value.

![Local dark matter (DM) density from the literature (Table \[tab\_localdm\]) plotted against publication year. The dashed line indicates a simple mean of the plots at $\rho_\odot=0.39 \pm 0.09$ .[]{data-label="localDMauthors"}](fig4.eps){width="9cm"}

Dependence on the Galactic Constants
------------------------------------

We have re-scaled the adopted RC to $(R_0,V_0)=(8.0, 238)$ (kpc, ), which may vary within several %. The resulting local DM density will vary accordingly, depending on the constants. The local mass density of the spherical component is dependent on the constants as $\rho_0 \propto R_0 V_0^2/R_0^3 \sim V_0^2 R_0^{-2}$. For small corrections $\delta V_0$ and $\delta R_0$, the DM density will change as $\delta \rho_0/\rho_0 \sim 2(\delta V_0/ V_0-\delta R_0/R_0)$. For example, for $\delta V_0 \sim \pm $ 10 , the estimated local density varies by $\delta \rho_0/\rho_0 \sim \pm 0.08$, or for $\delta R_0 \sim \pm 0.1$ kpc, $\delta \rho_0/\rho_0 \sim \mp 0.025$.

Summary
=======

We reviewed the current status of determination of the RC of the Milky Way, and presented a unified RC from the GC to outer halo at $R\sim 100$ kpc. The RC was used to directly calculate the SMD without assuming any functional form. The disk appears as a straight line on the semi-logarithmic plot of SMD against $R$, and is visually well discriminated from the DH having an extended outskirt.

The SMD distribution was fitted by a bulge, disk, and NFW dark halo using the $\chi^2$ method. The best-fit DH profile yielded the local DM density of $0.359 \pm 0.017$ . We also reviewed the current estimations from the literature in the last decade, which appear to be converging to a mean value of $\rho_\odot=0.39 \pm 0.09$ .

[[**Acknowledgments**]{}]{} [The data analysis was performed at the Center of Astronomical Data Analysis of the National Astronomical Observatory of Japan. The author is grateful to Professor A. Hofmeister for inviting him to this special issue. ]{}

Tables concerning the RC and SMD of the Milky Way {#AppendixA}
=================================================

Tables \[tabrcA\] and \[tabrcB\] list the running-averaged RC of the Milky Way using the data from \[[@Sofue2015; @Sofue2017; @Huang+2016]\],which is used to calculate the SMD in Figure \[smd\_fit\]. Tables \[tab\_smdsA\] and \[tab\_smdsB\] lists the directly calculated SMD from the RC

\[0.7\]

  ------------ ------- -------------------
   **Radius**           **Standard Dev**.
     (kpc)       ()            ()
     0.100      144.9          3.7
     0.110      147.4          4.2
     0.121      150.4          4.8
     0.133      153.8          6.1
     0.146      158.9         10.3
     0.161      167.4         16.1
     0.177      180.1         22.4
     0.195      196.6         27.1
     0.214      213.6         26.9
     0.236      227.8         22.7
     0.259      237.9         17.0
     0.285      244.4         11.8
     0.314      248.2          7.6
     0.345      250.2          4.7
     0.380      251.0          2.9
     0.418      250.7          2.1
     0.459      249.7          2.3
     0.505      248.0          2.9
     0.556      245.9          3.7
     0.612      243.2          4.6
     0.673      239.8          5.7
     0.740      235.8          6.4
     0.814      231.7          6.5
     0.895      227.8          6.0
     0.985      224.5          5.2
     1.083      221.7          4.5
     1.192      219.1          4.0
     1.311      216.8          3.7
     1.442      214.7          3.4
     1.586      212.7          3.1
     1.745      210.9          2.8
     1.919      209.5          2.3
     2.111      208.5          1.8
     2.323      208.2          1.6
  ------------ ------- -------------------

  : Rotation curve of the Milky Way used in Figure \[mwrc\].[]{data-label="tabrcA"}

\[0.7\]

  ------------ ------- -------------------
   **Radius**           **Standard Dev**.
     (kpc)       ()            ()
     2.555      208.9          2.2
     2.810      210.7          3.6
     3.091      213.4          4.8
     3.400      217.2          5.9
     3.740      222.0          6.6
     4.114      226.6          5.7
     4.526      229.5          4.4
     4.979      231.6          4.3
     5.476      234.1          5.3
     6.024      237.2          5.7
     6.626      239.5          5.0
     7.289      240.1          4.1
     8.018      239.0          4.4
     8.820      236.7          5.4
     9.702      234.5          6.0
     10.672     234.2          7.1
     11.739     237.1          9.8
     12.913     242.8         12.4
     14.204     248.5         13.3
     15.625     249.7         14.8
     17.187     246.2         17.4
     18.906     243.3         18.3
     20.797     243.9         17.5
     22.876     245.6         15.6
     25.164     243.7         15.2
     27.680     237.3         16.1
     30.448     229.6         15.5
     33.493     222.5         14.1
     36.842     215.0         14.0
     40.527     207.1         13.8
     44.579     200.3         12.7
     49.037     194.7         11.9
     53.941     189.8         11.3
     59.335     186.2         10.4
     65.268     184.7          9.6
     71.795     183.9          9.3
     78.975     181.4         11.0
     86.872     175.5         14.6
     95.560     167.7         16.3
  ------------ ------- -------------------

  : Continued from Table \[tabrcA\].[]{data-label="tabrcB"}

\[0.7\]

  ------------ --------- -------------------
   **Radius**             **Standard Dev.**
   **(kpc)**             
     0.100      29933.0         861.3
     0.110      29054.0         654.8
     0.121      28384.0         666.0
     0.133      28160.0         570.8
     0.146      28319.0         637.2
     0.161      28203.0        1406.6
     0.177      27368.0        2481.1
     0.195      25014.0        3514.2
     0.214      21548.0        4357.7
     0.236      17908.0        4806.8
     0.259      14804.0        4733.1
     0.285      12369.0        4231.5
     0.314      10489.0        3549.3
     0.345      8978.9         2929.7
     0.380      7736.5         2384.1
     0.418      6700.8         1959.1
     0.459      5830.8         1636.9
     0.505      5090.6         1374.2
     0.556      4452.0         1158.1
     0.612      3899.9          973.5
     0.673      3464.9          803.4
     0.740      3145.2          644.7
     0.814      2904.3          510.8
     0.895      2701.7          415.1
     0.985      2510.1          354.4
     1.083      2320.9          319.0
     1.192      2144.8          291.9
     1.311      1985.0          266.7
     1.442      1843.6          240.7
     1.586      1718.4          214.5
     1.745      1611.4          188.2
     1.919      1519.3          164.5
     2.111      1440.6          144.4
     2.323      1368.3          130.9
  ------------ --------- -------------------

  : Directly calculated SMD by spherical assumption of the mass distribution.[]{data-label="tab_smdsA"}

\[0.7\]

  ------------ ------------------------- -------------------------
   **Radius**                                  Standard Dev.
     (kpc)      $\Msun \ {\rm pc}^{-2}$   $\Msun \ {\rm pc}^{-2}$
     2.555              1296.4                     125.1
     2.810              1220.5                     126.3
     3.091              1139.8                     134.1
     3.400              1055.2                     146.1
     3.740               944.3                     157.4
     4.114               824.6                     161.5
     4.526               734.9                     150.5
     4.979               668.3                     133.7
     5.476               600.8                     123.8
     6.024               523.5                     119.3
     6.626               446.8                     113.6
     7.289               383.7                     101.6
     8.018               339.1                     83.4
     8.820               314.1                     62.2
     9.702               303.4                     43.9
     10.672              293.2                     39.0
     11.739              272.0                     48.2
     12.913              229.5                     58.5
     14.204              170.5                     65.2
     15.625              127.5                     62.4
     17.187              114.7                     46.4
     18.906              110.0                     35.2
     20.797              91.8                      33.5
     22.876              61.2                      34.5
     25.164              37.2                      33.2
     27.680              28.0                      25.4
     30.448              24.7                      17.1
     33.493              20.3                      11.6
     36.842              17.3                       7.8
     40.527              17.1                       4.7
     44.579              16.7                       3.0
     49.037              15.2                       2.3
     53.941              14.4                       2.3
     59.335              13.4                       3.3
     65.268              10.6                       4.3
     71.795               6.2                       4.9
  ------------ ------------------------- -------------------------

  : Continued from Table \[tab\_smdsA\].[]{data-label="tab_smdsB"}

[**References**]{}

[999]{}

1\. Sofue, Y.; Rubin, V.  Rotation Curves of Spiral Galaxies. *Annu. Rev. Astron. Astrophys.* **2001**, *39*, 137.

2\. Sofue, Y.  Rotation and mass in the Milky Way and spiral galaxies. *Publ. Astron. Soc. Jpn.* **2017**, *69*, R1.

3\. Salucci, P.  The distribution of dark matter in galaxies. *Astron. Astrophys. Rev.* **2019**, *27*, 2.

4\. Honma, M.; Nagayama, T.; Ando, K.; Bushimata, T.; Choi, Y.K.; Handa, T.; Hirota, T.; Imai, H.; Jike, T.; Kim, M.K.; et al.  Fundamental Parameters of the Milky Way Galaxy Based on VLBI astrometry. *Publ. Astron. Soc. Jpn.* **2012**, *64*, 136.

5\. Honma, M.; Nagayama, T.; Sakai, N.  Determining dynamical parameters of the Milky Way Galaxy based on high-accuracy radio astrometry. *Publ. Astron. Soc. Jpn.* **2015**, *67*, 70.

6\. Fich, M.; Tremaine, S.  The mass of the Galaxy. *Annu. Rev. Astron. Astrophys.* **1991**, *29*, 409.

7\. Reid, M.J.  The distance to the center of the Galaxy. *Annu. Rev. Astron. Astrophys.* **1993**, *31*, 345.

8\. Olling, R.P.; Dehnen, W.  The Oort Constants Measured from Proper Motions. *Astrophys. J.* **2003**, *599*, 275.

9\. Ghez, A.M.; Salim, S.; Weinberg, N.N.; Lu, J.R.; Do, T.; Dunn, J.K.; Matthews, K.; Morris, M.R.; Yelda, S.; Becklin, E.E.; et al.  Measuring Distance and Properties of the Milky Way’s Central Supermassive Black Hole with Stellar Orbits. *Astrophys. J.* **2008**, *689*, 1044.

10\. Gillessen, S.; Eisenhauer, F.; Trippe, S.; Alexander, T.; Genzel, R.; Martins, F.; Ott, T.  Monitoring Stellar Orbits Around the Massive Black Hole in the Galactic Center. *Astrophys. J.* **2009**, *692*, 1075.

11\. Reid, M.J., Menten, K.M., Zheng, X.W., Brunthaler, A., Moscadelli, L., Xu, Y., Zhang, B., Sato, M., Honma, M., Hirota, T., Hachisuka, K., et al.  Trigonometric Parallaxes of Massive Star-Forming Regions. VI. Galactic Structure, Fundamental Parameters, and Noncircular Motions. *Astrophys. J.* **2009** *700*, 137.

12\. Matsunaga, N.; Kawadu, T.; Nishiyama, S.; Nagayama, T.; Hatano, H.; Tamura, M.; Glass, I.S.; Nagata, T.  A near-infrared survey of Miras and the distance to the Galactic Centre. *Mon. Not. R. Astron. Soc.* **2009**, *399*, 1709.

13\. Burton, W.B.; Gordon, M.A.  Carbon monoxide in the Galaxy. III. The overall nature of its distribution in the equatorial plane. *Astron. Astrophys.* **1978**, *63*, 7.

14\. Blitz, L.; Lada, C.J. H$_2$O masers near OB associations. *Astrophys. J.* **1979**, *227*, 152–158.

15\. Clemens, D.P.  Massachusetts-Stony Brook Galactic plane CO survey: The galactic disk rotation curve. *Astrophys. J.* **1985**, *295*, 422.

16\. Dehnen, W.; Binney, J.  Mass models of the Milky Way. *Mon. Not. R. Astron. Soc.* **1998**, *294*, 429.

17\. Genzel, R.; Eisenhauer, F.; Gillessen, S.  The Galactic Center massive black hole and nuclear star cluster. *Rev. Mod. Phys.* **2010**, *82*, 3121.

18\. Battinelli, P.; Demers, S.; Rossi, C.; Gigoyan, K.S.  Extension of the C Star Rotation Curve of the Milky Way to 24 kpc. *Astrophysics* **2013**, *56*, 68.

19\. Bhattacharjee, P.; Chaudhury, S.; Kundu, S.  Rotation Curve of the Milky Way out to \~200 kpc. *Astrophys. J.* **2014**, *785*, 63.

20\. L[ó]{}pez-Corredoira, M.  Milky Way rotation curve from proper motions of red clump giants. *Astron. Astrophys.* **2014**, *563*, A128.

21\. Bovy, J.; Prieto, C.A.; Beers, T.; Bizyaev, D.; Da Costa, L.N.; Cunha, K.; Ebelke, G.; Eisenstein, D.J.; Frinchaboy, P.M.; Perez, A.E.G.; et al. The Milky Way’s Circular-velocity Curve between 4 and 14 kpc from APOGEE data. *Astrophys. J.* **2012**, *759*, 131. 22. Bobylev, V.V.; Bajkova, A.T.  Galactic rotation curve and spiral density wave parameters from 73 masers. *Astron. Lett.* **2013**, *39*, 809.

23\. Bobylev, V.V.; Bajkova, A.T.  Determination of the galactic rotation curve from OB stars. *Astron. Lett.* **2015**, *41*, 473.

24\. Reid, M.J., Menten, K.M., Brunthaler, A., Zheng, X.W., Dame, T.M., Xu, Y., Wu, Y., Zhang, B., Sanna, A., Sato, M., et al.  Trigonometric Parallaxes of High Mass Star Forming Regions: The Structure and Kinematics of the Milky Way. *Astrophys. J.* **2014**, *783*, 130.

25\. Iocco, F.; Pato, M.; Bertone, G.  Evidence for dark matter in the inner Milky Way. *Nat. Phys.* **2015**, *11*, 245.

26\. Iocco, F.; Pato, M.  On the dark matter distribution in the Milky Way. *J. Phys. Conf. Ser.* **2016**, *718*, 042031.

27\. Pato, M.; Iocco, F.  *Galkin: Milky Way Rotation Curve Data Handler*; ascl:1711.011; Astrophysics Source Code Library: Houghton, MI, USA, 2017.

28\. Pato, M.; Iocco, F.  Galkin: A new compilation of Milky Way rotation curve data. *SoftwareX* **2017**, *6*, 54.

29\. Huang, Y.; Liu, X.; Yuan, H.-B.; Xiang, M.; Zhang, H.-W.; Chen, B.; Ren, J.; Wang, C.; Zhang, Y.; Hou, Y.-H.; et al. The Milky Way’s rotation curve out to 100 kpc and its constraint on the Galactic mass distribution. *Mon. Not. R. Astron. Soc.* **2016**, *463*, 2623.

30\. Kre[ł]{}owski, J.; Galazutdinov, G.; Strobel, A.  The Milky Way Rotation Curve Revisited. *Publ. Astron. Soc. Pac.* **2018**, *130*, 114302.

31\. Lin, H.-N.; Li, X.  The dark matter profiles in the Milky Way. *Mon. Not. R. Astron. Soc.* **2019**, *487*, 5679.

32\. Eilers, A.-C.; Hogg, D.W.; Rix, H.-W.; Ness, M.K.  The Circular Velocity Curve of the Milky Way from 5 to 25 kpc. *Astrophys. J.* **2019**, *871*, 120.

33\. Mr[ó]{}z, P., Udalski, A., Skowron, D.M., Skowron, J., Soszy[ń]{}ski, I., Pietrukowicz, P., Szyma[ń]{}ski, M.K., Poleski, R., Koz[ł]{}owski, S., and Ulaczyk, K.  Rotation Curve of the Milky Way from Classical Cepheids. *Astrophys. J.* **2012**, *870*, L10.

34\. Sofue, Y.; Honma, M.; Omodaka, T.  Unified Rotation Curve of the Galaxy—Decomposition into de Vaucouleurs Bulge, Disk, Dark Halo, and the 9-kpc Rotation Dip—. *Publ. Astron. Soc. Jpn.* **2009**, *61*, 227.

35\. Sofue, Y.  Rotation Curve and Mass Distribution in the Galactic Center - From Black Hole to Entire Galaxy. *Publ. Astron. Soc. Jpn.* **2013**, *65*, 118.

36\. Sofue, Y.  Dark halos of M 31 and the Milky Way. *Publ. Astron. Soc. Jpn.* **2015**, *67*, 75.

37\. Fich, M.; Blitz, L.; Stark, A.A.  The Rotation Curve of the Milky Way to 2R 0. *Astrophys. J.* **1989**, *342*, 272.

38\. Demers, S.; Battinelli, P.  C stars as kinematic probes of the Milky Way disk from 9 to 15 kpc. *Astron. Astrophys.* **2007**, *473*, 143.

39\. Merrifield, M.R.  The Rotation Curve of the Milky Way to 2.5 R/o From the Thickness of the HI Layer. *Astron. J.* **1992**, *103*, 1552.

40\. Honma, M.; Sofue, Y.  Rotation Curve of the Galaxy. *Publ. Astron. Soc. Jpn.* **1997**, *49*, 453.

41\. Lindqvist, M.; Habing, H.J.; Winnberg, A.  OH/IR stars close to the galactic centre. II. Their spatial and kinematics properties and the mass distribution within 5-100 PC from the galactic centre. *Astron. Astrophys.* **1992**, *259*, 118.

42\. Honma, M.; Bushimata, T.; Choi, Y.K.; Hirota, T.; Imai, H.; Iwadate, K.; Jike, T.; Kameya, O.; Kamohara, R.; Kan-Ya, Y.; et al.  Astrometry of Galactic Star-Forming Region Sharpless 269 with VERA: Parallax Measurements and Constraint on Outer Rotation Curve. *Publ. Astron. Soc. Jpn.* **2007**, *59*, 889.

43\. Sakai, N.; Nakanishi, H.; Matsuo, M.; Koide, N.; Tezuka, D.; Kurayama, T.; Shibata, K.M.; Ueno, Y.; Honma, M. Outer rotation curve of the Galaxy with VERA. III. Astrometry of IRAS 07427-2400 and test of the density-wave theory. *Publ. Astron. Soc. Jpn.* **2015**, *67*, 69.

44\. Nakanishi, H.; Sakai, N.; Kurayama, T.; Matsuo, M.; Imai, H.; Burns, R.A.; Ozawa, T.; Honma, M.; Shibata, K.M.; Kawaguchi, N. Outer rotation curve of the Galaxy with VERA. II. Annual parallax and proper motion of the star-forming region IRAS 21379+5106. *Publ. Astron. Soc. Jpn.* **2015**, *67*, 68.

45\. Callingham, T.M.; Cautun, M.; Deason, A.J.; Frenk, C.S.; Wang, W.; A Gomez, F.; Grand, R.J.J.; Marinacci, F.; Pakmor, R.  The mass of the Milky Way from satellite dynamics. *Mon. Not. R. Astron. Soc.* **2019**, *484*, 5453.

46\. Roeser, S.; Demleitner, M.; Schilbach, E.  The PPMXL Catalog of Positions and Proper Motions on the ICRS. Combining USNO-B1.0 and the Two Micron All Sky Survey (2MASS). *Astron. J.* **2010**, *139*, 2440.

47\. Pato, M.; Iocco, F.; Bertone, G.  Dynamical constraints on the dark matter distribution in the Milky Way. *J. Cosmol. Astropart. Phys.* **2015**, *2015*, 001.

48\. Pato, M.; Iocco, F.  The Dark Matter Profile of the Milky Way: A Non-parametric Reconstruction. *Astrophys. J.* 2015, 803, L3.

49\. de Vaucouleurs, G.  Photoelectric photometry of the Andromeda Nebula in the UBV system. *Astrophys. J.* **1958**, *128*, 465.

50\. Ciotti, L.  Stellar systems following the R1/m luminosity law. *Astron. Astrophys.* **1991**, *249*, 99.

51\. Trujillo, I.; Asensio Ramos, A.; Rubino-Martin, J.A.; Graham, A.W.; Aguerri, J.A.; Cepa, J.; Gutierrez, C.M.  Triaxial stellar systems following the r$^{1/n}$ luminosity law: An analytical mass-density expression, gravitational torques and the bulge/disc interplay. *Mon. Not. R. Astron. Soc.* **2002**, *333*, 510.

52\. Freeman, K.C.  On the Disks of Spiral and S0 Galaxies. *Astrophys. J.* **1970**, *160*, 811.

53\. Binney, J.; Tremaine, S. *Galactic Dynamics*; Princeton University Press: Princeton, NJ, USA, 1987.

54\. Li, Z.-Z.; Jing, Y.P.; Qian, Y.-Z.; Yuan, Z.; Zhao, D.-H.  Determination of Dark Matter Halo Mass from Dynamics of Satellite Galaxies. *Astrophys. J.* **2017**, *850*, 116.

55\. Burkert, A.  The Structure of Dark Matter Halos in Dwarf Galaxies. *Astrophys. J.* 1995, 447, L25.

56\. Salucci, P.; Burkert, A.  Dark Matter Scaling Relations. *Astrophys. J.* **2000**, *537*, L9.

57\. Brownstein, J.R.; Moffat, J.W.  Galaxy Rotation Curves without Nonbaryonic Dark Matter. *Astrophys. J.* **2006**, *636*, 721.

58\. Navarro, J.F.; Frenk, C.S.; White, S.D.M.  Simulations of X-ray clusters. *Mon. Not. R. Astron. Soc.* **1995**, *275*, 720. 59. Navarro, J.F.; Frenk, C.S.; White, S.D.M.  A Universal Density Profile from Hierarchical Clustering. *Astrophys. J.* **1997**, *490*, 493.

60\. Moore, B.; Quinn, T.; Governato, F.; Stadel, J.; Lake, G.  Cold collapse and the core catastrophe. *Mon. Not. R. Astron. Soc.* **1999**, *310*, 1147.

61\. Fukushige, T.; Kawai, A.; Makino, J.  Structure of Dark Matter Halos from Hierarchical Clustering. III. Shallowing of the Inner Cusp. *Astrophys. J.* **2004**, *606*, 625.

62\. Navarro, J.F.; Frenk, C.S.; White, S.D.M.  The Structure of Cold Dark Matter Halos. *Astrophys. J.* **1996**, *462*, 563.

63\. Silk, J.; Bloemen, H.  A Gamma-Ray Constraint on the Nature of Dark Matter. *Astrophys. J.* **1987**, *313*, L47.

64\. Escudero, M.; Hooper, D.; Witte, S.J.  Updated collider and direct detection constraints on Dark Matter models for the Galactic Center gamma-ray excess. *J. Cosmol. Astropart. Phys.* **2017**, *2017*, 38.

65\. Finkbeiner, D.P.  WMAP Microwave Emission Interpreted as Dark Matter Annihilation in the Inner Galaxy. *arXiv* **2004**, arXiv:0409027.

66\. Weber, M.; de Boer, W.  Determination of the local dark matter density in our Galaxy. *Astron. Astrophys.* **2010**, *509*, A25.

67\. Catena, R.; Ullio, P.  A novel determination of the local dark matter density. *J. Cosmol. Astropart. Phys.* **2010**, *2010*, 004.

68\. Bovy, J.; Tremaine, S.  On the Local Dark Matter Density. *Astrophys. J.* **2012**, *756*, 89. 69. Piffl, T.; Binney, J.; McMillan, P.; Steinmetz, M.; Helmi, A.; Wyse, R.F.G.; Bienayme, O.; Bland-Hawthorn, J.; Freeman, K.; Gibson, B.K.; et al.  Constraining the Galaxy’s dark halo with RAVE stars. *Mon. Not. R. Astron. Soc.* **2014**, *445*, 3133.

70\. McMillan, P.J.  The mass distribution and gravitational potential of the Milky Way. *Mon. Not. R. Astron. Soc.* **2017**, *465*, 76.

71\. Salucci, P.; Nesti, F.; Gentile, G.; Frigerio Martins, C.  The dark matter density at the Sun’s location. *Astron. Astrophys.* **2010**, *523*, A83.

72\. de Salas, P.F.; Malhan, K.; Freese, K.; Hattori, K.; Valluri, M.  On the estimation of the local dark matter density using the rotation curve of the Milky Way. *J. Cosmol. Astropart. Phys.* **2019**, *2019*, 37.

73\. Cautun, M.; Benitez-Llambay, A.; Deason, A.J.; Frenk, C.S.; Fattahi, A.; Gomez, F.A.; Grand, R.J.; Oman, K.A.; Navarro, J.F.; Simpson, C.M. The Milky Way total mass profile as inferred from Gaia DR2. *arXiv* **2020**, arXiv:1911.04557. 74. Karukes, E.V.; Benito, M.; Iocco, F.; Trotta, R.; Geringer-Sameth, A.  Bayesian reconstruction of the Milky Way dark matter distribution. *J. Cosmol. Astropart. Phys.* **2019**, *2019*, 46.
