---
abstract: |
    It is common to attribute a flat rotation curve to our Galaxy. However [@Gazinur] in a recent paper have obtained a Keplerian rotation curve for interstellar clouds in outer parts of the Galaxy. They have calculated the distances from equivalent widths of interstellar [Ca[<span style="font-variant:small-caps;">II</span>]{}]{} lines. The radial velocity was also measured on the interstellar [Ca[<span style="font-variant:small-caps;">II</span>]{}]{} absorption line.

    We verify the result by [@Gazinur] basing on observations of old open clusters. We explain the determinations of flat rotation curves assuming elliptical orbits of stars in our Galaxy. The application of formulas derived with the assumption of circular orbits to elliptical ones mimics the flat rotation curve. The interstellar clouds with cross-sections larger than stars may have almost circular orbits. The rotation curve derived from the interstellar clouds, without the assumption of dark matter, will be Keplerian.
author:
- |
    [ P. Gnaciński$^{1}$ and T. Młynik$^{2}$ ]{}\
    $^{1}$Institute of Theoretical Physics and Astrophysics,\
    University of Gdańsk,\
    ul. Wita Stwosza 57,\
    80-952 Gdańsk, Poland\
    email: [*fizpg@univ.gda.pl*]{}\
    $^{2}$Student at the University of Gdańsk.
title: Keplerian rotation of our Galaxy
---

[**Key words:** ]{}[ *Galaxy kinematics and dynamics; dark matter* ]{}

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

Our Galaxy is usually thought to have a flat rotation curve. The flat rotation curves of galaxies are usually explained assuming existence of dark matter. The MOND (MOdified Newtonian Dynamics) models are less popular. However, some galaxies have Keplerian rotation curves which falls as $\sim 1/\sqrt{r}$ in the outer parts of the galaxies. In a sample of 45 galaxies analyzed by [@Honma1997] 11 have Keplerian rotation curve.

A compilation of rotation velocities observed in our Galaxy was made by [@Sofue2009]. They have transformed the rotational velocities from various sources to common parameters $R_{{\odot}}=8$ kpc and $v_{{\odot}}=200$ km/s and these values are also adopted in this paper. Rotation velocities from [@Sofue2009], derived with tangent point method or from radial velocity, are shown on Fig. \[rotVel\](a).

The absence of dark matter in solar neighborhood was postulated by [@Bidin]. Their result is based on stellar kinematics in direction perpendicular to the galactic plane. However, their calculation leads to a flat rotation curve.

The recent paper by [@Gazinur] shows a Keplerian rotation curve of our Galaxy. They have based on distances and radial velocities derived from interstellar [Ca[<span style="font-variant:small-caps;">II</span>]{}]{} absorption lines. The aim of this paper is to reconcile the flat rotation curve from [@Sofue2009] and the Keplerian rotation derived by [@Gazinur].

Old open clusters
=================

![image](Fig1.eps){width="\textwidth"}

In order to verify the result by [@Gazinur] we have analyzed the rotation velocity of old open clusters (age greater than $10^9$ years) located in the outer part of the Galaxy $l\in(90^\circ, 270^\circ)$. All analyzed open clusters were located close to the galactic plane $|b|<20^\circ$.

The rotational velocity was calculated from observed radial velocity. The heliocentric radial velocity $v_h$ was first transformed to the local standard of rest (LSR) $$v_{LSR}=v_h + U_{\odot}\cos b \cos l + V_{\odot}\cos b \sin l + W_{\odot}\sin b               \label{v_LSR}$$ using the Sun velocity $(U_{\odot},V_{\odot},W_{\odot})=(11.1,12.24,7.25)$ km/s from [@Schonrich]. The rotational velocity was derived using the commonly used formula (eg. @Bhattacharjee) $$v(r)=\frac{r}{R_{{\odot}}} \left( \frac{v_\mathrm{LSR}}{\sin l \cos b} + v_{{\odot}} \right).   \label{vr_zb}$$ In this formula $r$ is the projection of galactocentric distance on the galactic plane $$r=\sqrt{R_{\odot}^2 + d^2 \cos^2 b -2R_{\odot}d \cos b \cos l}.             \label{r}$$ The clusters in the direction $l=180^\circ\pm20^\circ$ were excluded from our sample, because the $\sin l$ in the denominator of formula \[vr\_zb\] leads to unphysical (i.e. negative) rotation velocities.

At least some old open clusters have nearly circular orbits. The five old open clusters analyzed by [@Carraro1994] have eccentricities less than 0.14, with two clusters having eccentricities as low as $e=0.03$. We have collected open clusters data from the literature (see table \[tab\_OC\]), and determined the rotation velocity using formulas \[v\_LSR\] and \[vr\_zb\]. The open clusters linear velocity, as well as the angular velocity is presented on figure \[fig\_OC\]. The advantage of the later is, than the angular velocity does not depend from the distance to cluster, which is known with little accuracy. Since the errors of distances to open clusters were not given in the referenced papers, we assumed 20% error of distances to open clusters.

Most open clusters on figure \[fig\_OC\] have rotation velocity close to the Keplerian rotation curve. Only three old open clusters have rotation velocity on the flat rotation curve. The old open clusters are up to 20 kpc from the Galactic centre.

\[tab\_OC\]

  ------------- ------------------- ------- ------ --------- --------- ------- ------- ------ ---------------- ---------------- -------------------
  Cluster           radial velocity   stars   ref.                                       ref.                                   
                                                                                                                                
  Berkeley 20      75.51 $\pm$ 4.85       9      a   203.483   -17.373    8710   9.763      a   16.0 $\pm$ 1.6    5.6 $\pm$ 1.6     89.7 $\pm$ 34.6
  Berkeley 25     134.30 $\pm$ 1.62       4      e   226.612    -9.700   11400   9.699      e   17.7 $\pm$ 2.1    4.6 $\pm$ 0.3     81.8 $\pm$ 14.8
  Berkeley 31      55.80 $\pm$ 1.13       2      o   206.254     5.120    8300   9.301      e   15.8 $\pm$ 1.6   13.3 $\pm$ 0.3    211.1 $\pm$ 26.6
  Berkeley 32     105.00 $\pm$ 1.40       9      m   207.950     4.400    3162   9.720      n   10.9 $\pm$ 0.6    0.9 $\pm$ 0.4     9.9 $\pm$   4.6
  Berkeley 66     -50.65 $\pm$ 0.07       2      a   139.434     0.218    4570   9.580      a   11.9 $\pm$ 0.8   15.2 $\pm$ 0.0    179.9 $\pm$ 12.6
  Berkeley 73      95.70 $\pm$ 0.57       2      e   215.278    -9.424    9800   9.176      e   16.8 $\pm$ 1.9    7.8 $\pm$ 0.1    130.6 $\pm$ 16.5
  Berkeley 75      94.60 $\pm$ 0.35       1      e   234.307   -11.188    9100   9.602      e   15.1 $\pm$ 1.6   12.9 $\pm$ 0.1    194.5 $\pm$ 21.6
  Cr 110           40.00 $\pm$ 1.00              d   209.650    -1.980    1950   9.230      d    9.7 $\pm$ 0.4   18.9 $\pm$ 0.3   184.3 $\pm$   9.2
  King 11         -35.00 $\pm$ 16.0              l   117.160     6.480    2198   9.615      n    9.2 $\pm$ 0.3   21.0 $\pm$ 2.3    193.2 $\pm$ 26.6
  NGC 2243         61.00 $\pm$ 1.00              i   239.480   -18.010    3532   9.681   c, i   10.1 $\pm$ 0.5   18.4 $\pm$ 0.2    186.2 $\pm$ 10.6
  NGC 2506         83.70 $\pm$ 1.40       4      f   230.560     9.940    3311   9.230   j, f   10.4 $\pm$ 0.5   13.7 $\pm$ 0.2   142.4 $\pm$   9.6
  NGC 6939        -18.98 $\pm$ 0.19      26      k    95.900    12.300    1820   9.114      b    8.4 $\pm$ 0.1   24.1 $\pm$ 0.0   202.1 $\pm$   2.9
  Pismis 2         49.20 $\pm$ 7.80       9      h   258.850    -3.340    3467   9.041      g    9.3 $\pm$ 0.4   20.6 $\pm$ 1.0    191.6 $\pm$ 16.9
  Saurer 1        104.60 $\pm$ 0.30       2      p   214.689     7.386   13200   9.699      e   20.2 $\pm$ 2.6    5.2 $\pm$ 0.1    104.2 $\pm$ 14.5
  Tombaugh 2      120.51 $\pm$ 2.19      37      a   232.832    -6.880    7950   9.204      a   14.2 $\pm$ 1.4    8.7 $\pm$ 0.3    123.5 $\pm$ 17.2
  ------------- ------------------- ------- ------ --------- --------- ------- ------- ------ ---------------- ---------------- -------------------

  :  Data of old open clusters and calculated velocities. If the error of radial velocity was not given we assumed 1 km/s. References: a - [@Andreuzzi2011]; b - [@Andreuzzi2004]; c - [@Bonifazi]; d - [@Bragaglia and references therin]; e - [@Carraro2007 and references therin]; f - [@Carretta]; g - [@Di; @Fabrizio]; h - [@Friel]; i - [@Gratton]; j - [@Marconi]; k - [@Milone]; l - [@Scott]; m - [@Sestito]; n - [@Tosi]; o - [@Yong]; p - [@Carraro2004].

Model
=====

We have made a Monte-Carlo simulation of stars on elliptical orbits beyond the Sun – Galactic center ($R_{{\odot}}$) distance. The semi-major axis (a), eccentricity (e), true anomaly ($\nu$) and argument of pericenter ($\omega$) were chosen randomly with uniform distribution. All orbits were located in the Galactic plane. The mass inside the solar orbit was set to $74.4\cdot10^9\ {\mathrm M}_{\odot}$.

The rotation velocities were calculated using formulas, that were derived assuming circular orbits. The rotation velocity was calculated from radial velocity ($v_\mathrm{r}$) of star using $$v(r)=\frac{r}{R_{{\odot}}} \left( \frac{v_\mathrm{r}}{\sin l} + v_{{\odot}} \right),   \label{vr}$$ where $r$ is the distance between star and the Galactic center. The letter $l$ denotes galactic longitude.

The transverse velocities ($v_\mathrm{t}$) were also used to calculate the rotation velocity using the formula $$v(r)=\frac{-r}{d-R_{{\odot}}\cos l} \left( v_\mathrm{t} + v_{{\odot}} \cos l \right),   \label{vt}$$ where $d$ is the star – Sun distance. The equations \[vr\] and \[vt\] are usually used to calculate the outer rotation curve of the Galaxy. We want to test the influence of formulas derived for circular orbits applied to stars on elliptical ones.

The semi-major axes in our simulation were distributed from 5 to 25 kpc to avoid truncation effects at $R_{{\odot}}=8$ kpc. The stars with galactic longitudes less then 15[$^\circ$]{} from 0[$^\circ$]{} or 180[$^\circ$]{}were not shown, because the denominator in eq. \[vr\] is to small. Also stars with rotational velocities calculated from transverse velocity and $d - R_{{\odot}}\cos l <1.5$ kpc (denominator in eq. \[vt\]) were not shown on the plots. A small denominator in equations \[vr\] and \[vt\] leads to unphysically large rotation velocities, up to tens of thousands km per second.

The result of the Monte-Carlo simulation for 200 objects is shown on Fig. \[rotVel\](b)–(d). The rotation velocities derived from transverse velocities (panel (b)) are spread close to the Keplerian curve. However, the rotation velocities derived from radial velocities (panel (c)) have large dispersion and look very similar to the observed rotation velocities from [@Sofue2009]. Only objects with the distance 8–20 kpc from Galactic center are shown on the Monte–Carlo simulations plots.

![image](Fig2.eps){width="\textwidth"}

Rotation of star forming regions
================================

The main argument for the flat rotation curve of our Galaxy, given by @Sofue2009, is the observation of Sharpless 269 star forming region observed by VERA (VLBI Exploration of Radio Astrometry). Both radial and transverse velocities [@VERA] lead to a rotation velocity of $\sim$200 km/s. Also the VLBI measurements of parallaxes and proper motions of star forming regions (SFR) published by @Reid leads to flat rotation curve.

![ The circular velocity component of SFR calculated for interstellar clouds on orbits with eccentricity e$<$0.1. The expelled gas clouds have a circular velocity of 210 km/s and a mass greater than the mass of cloud on nearly circular orbit. The parameter $m_{circ}/(m_{circ}+m_{expelled})$ is uniformly distributed in the range 0–0.5. The lines show the model of flat rotation curve by [@Sofue2009] and the Keplerian rotation curve. []{data-label="SFR"}](Fig3.eps){width="\columnwidth"}

It suggests that SFR rotate differently than interstellar clouds. One of possible explanations comes from the model of @Souza. In this model gas clouds are expelled from inner parts of the Galaxy. Such clouds are postulated to explain the formation of spiral arms. The cloud circular velocity remains constant in this model. The expelled gas clouds collide with interstellar clouds on circular orbits. They form a SFR with velocity that is a weighted average of the expelled cloud velocity and the velocity of ordinary interstellar cloud. If we assume that the mass of cloud on circular orbit is less than the mass of the expelled gas cloud we get a flat rotation for circular velocity component (fig. \[SFR\]).

Discussion
==========

The molecular clouds have the lowest velocity dispersion $\sigma_z=5$ km/s in the direction perpendicular to the galactic plane, as compared to stars or [H[<span style="font-variant:small-caps;">II</span>]{}]{} regions. Because of the large cross section they may be better thermalised than stars. Therefore the orbits of molecular clouds may have lower eccentricities than other objects. Figure \[rotVel\](d) shows rotation velocities for simulated interstellar clouds (e=0–0.07). The rotation velocities were derived from radial velocities. They match very well the Keplerian rotation curve, similar to velocities observed by [@Gazinur].

We have analyzed radial velocities of K and M giants in the Galactic anticenter from the CORAVEL spectrograph given by [@Coravel]. Stars in binary systems have been removed from the analyzed sample. Regardless of the size of square centered on the Galactic anticenter the standard deviations of radial velocities can not be explained assuming circular orbits (Table \[StdDev\]). We obtain standard deviation of radial velocities similar to observed ones, assuming elliptical orbits with eccentricities uniformly distributed in the range 0–0.7 in the outer parts of our Galaxy. So, the assumption of elliptical orbits is consistent with observed radial velocities.

  ---- ------ ----------- -----------
                          
                          
                          
                          
     6   38.0     2.8–4.8   21.7–53.4
     8   43.7     3.4–6.0   21.0–51.6
    10   40.7     6.2–7.9   15.2–48.4
    12   39.6    6.6–10.6   27.7–49.7
    14   40.6    8.6–11.2   34.2–48.2
    16   39.0    9.9–12.2   30.9–52.3
    18   37.6   11.4–13.7   33.5–49.8
    20   37.6   12.8–15.4   26.9–49.0
  ---- ------ ----------- -----------

  :  Observed [@Coravel] and simulated (Monte-Carlo) standard deviations of radial velocity in Galactic anticenter. The square in which we analyze the radial velocities is centered on the Galactic anticenter. The minimum and maximum standard deviation of radial velocity is calculated from 10 Monte-Carlo simulations.[]{data-label="StdDev"}

Eleven directions towards galactic anticenter were observed by [@Gazinur]. The interstellar clouds are located in Galactic longitudes 184[$^\circ$]{}$<$l$<$190[$^\circ$]{}. The radial velocities towards these clouds have a standard deviation of 1.8 km/s. We cannot obtain such low dispersion even with circular orbits. The standard deviation of radial velocity for circular orbits in the mentioned longitude range is 3.5–7.2 km/s.

Conclusions
===========

The rotation curve derived from observations of old open clusters seems to confirm the observations of Keplerian rotation curve for our Galaxy. Our model explains observations of Keplerian rotation of interstellar clouds and the flat rotation curve for stars and star forming regions. The main results are:

-   The observations of flat or Keplerian rotation curve of our Galaxy can be explained assuming Keplerian rotation, elliptical orbits of stars and almost circular orbits of interstellar clouds.

-   The flat circular velocity component of star forming regions can be explained assuming collisions of heavy gas clouds expelled from inner parts of the Galaxy with gas clouds on circular orbits.

-   The Galactic rotation velocity derived from radial velocity in the case of elliptical orbits can not be used to distinguish between flat or Keplerian rotation curve.

-   The Galactic rotation velocity derived from transverse velocity of objects on elliptical orbits is more accurate than rotation velocity derived from radial velocity.

The Keplerian rotation of the Galaxy will have a huge impact on the amount of dark matter in our Galaxy.

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