International Journal of Astronomy and Astrophysics, 2012, 2, 167-173 Published Online September 2012 (
Examining the M67 Classification as an Open Cluster
Shim’on Naim1, Evgeny Griv2
1Ilan Ramon Physics Center, Ben-Gurion University of the Negev, Beer-Sheva, Israel
2Department of Physics, Ben-Gurion University of the Negev, Beer-Sheva, Israel
Received May 22, 2012; revised June 28, 2012; accepted July 15, 2012
The cluster M67 (= NGC 2682) in Cancer is a rich stellar cluster, usually classified as an open cluster. Using our own
observations with the 0.4 m telescope, we show that M67 is a tight group of about 1200 stars. The actual radius of the
cluster is about 3.1 pc and the average mass of a star in the system is about 1.33
. We also show that the ratio of the
mean kinetic energy of the cluster to its mean gravitational potential energy clust clust0.3KU , while the value pre-
dicted by the virial theorem is equal to . So the system is a gravitationally bound. This value of
0.5clust clust
KU is
considered as an evidence of quasi-stability of the cluster and allows us to use the Chandrasekhar-Spitzer relaxation
time for M67 chand 30
Myr as a characteristic dynamical relaxation time of the system under study. As the cluster is
almost twice older its half-life time hal chand
, it is argued that M67 was in the past (about 4 Gyr ago, close to its
formation) a relatively small ( stars) globular cluster, but got “open cluster” shape due to the dynamical
evaporation of the majority of its stars.
~ 10,00
Keywords: Photometers; Techniques; Photometric; Open Clusters and Associations; M67; Galaxies; Star Clusters
1. Introduction
Usually classified as an open cluster, the dense stellar
cluster M67 in Cancer is one of the oldest known open
clusters in our Galaxy with an age clust of
Gyr, that is, close to the age of the sun [1-3]. In the
present paper we adopt the value clust Gyr [4,5]. It
is important to note that open clusters usually get de-
structed much faster, over yr timescales. It is
some pc distant the Sun. Following Ref. [6],
we adopt the distance clust pc. The total number
of stars in M67 is estimated at over 700 [7]. M67 is the
nearest old open cluster. The cluster is also anomalous
with respect to the distance of clust pc to the
Galactic equatorial plane, where most open clu-
sters are found. This distance clust is even larger than
scale height pc of the local thick disk of
the Galaxy (see, e.g. Ref. [8]). Other open clusters lie
much closer to the Galactic plane. M67 current mass is
3.5 -4.5
= 770
750- 850
M and its initial mass was to be appro-
astclust [9]. Mass segregation has
been suggested as the cause for the observed structural
properties of M67 in Refs. [10,11]. This process has long
been suspected for globular star clusters (and has been
directly seen in globular cluster 47 Tucanae [12]), but
has never been suggested for open clusters. We conclude
that M67 has properties of globular clusters.
Here, we study M67 with the use of the Meade LX-
200R robotic 0.4 m telescope of the Ilan Ramon Physics
Center Observatory in Israel (
(Figure 1). The telescope operates from a small dome
and is equipped with Meade DSI III pro 1360 1024
CCD camera (Figure 2). In the last four years this
telescope was intensively used in a variety of monitoring
projects, including the study of stellar clusters, in par-
ticular, M67 and M35 (=NGC 2168).
2. Observations
The cluster M67 was photographed in two filters, V and
B, on 7th April 2010. An (f/6.3) local reducer was used,
and therefore the scale is 0.52 arcsec/pixel. Two sets of
exposure times were used in the observations, namely,
long (10 s in both B and V filters) and short (3 s in both B
and V filters).
Figure 3 shows a part of M67. This photograph is a
combination of 13 frames from each exposure, 10 s and 3
s, summed after a registration. The picture was taken
with a blue Johnson filter. First, we estimated the angular
size of the cluster = 27.63.6
= 3.1
arcsec. Then, adopt-
ing the revised distance to the cluster pc, its
actual radius was found clust pc. In order
to cast more light on the structure of M67, in Figure 4
we show a wider field 81 photograph of M67,
clust = 770r
opyright © 2012 SciRes. IJAA
Figure 1. The Meade LX200R robotic 16" telescope of the
Ilan Ramon Physics Center Observatory in Beer-Sheva
Figure 2. The dome of the Meade LX200R robotic 16" tele-
Figure 3. This photograph shows 173 stars of M67 cluster in
our observations.
Figure 4. A wider field 8' × 16' photograph of M67, taken
from the Atlas Image website.
taken from the Atlas Image website. Using these photo-
graphs (Figures 3 and 4) and this Rclust, finally we es-
timated the total number of stars in the system
clust 1200N
Figure 5 shows the observed color-magnitude,
vis . -mvsBV, diagram (or the HR diagram) for M67,
based on our observations. Each dot denotes a star. The
purple color indicates the positions of the main sequence
stars. The red color indicates the positions of the hori-
zontal branch stars. The yellow color indicates the posi-
tions of the giant stars. The positions of blue stragglers
are indicated in the diagram by the blue color. As is seen,
the color-magnitude diagram for M67 shows a short
main sequence extending from , where the
turn-off point is located, up to . After
-0.7BV it starts merging in the field region stars
and is not seen clearly. The giant branch is also clearly
seen. Figure 5 evidently illustrates that M67 is quite old
cluster. There are stars that are brighter and bluer than
the main sequence stars and giant branch stars. These
stars are a part of an exotic class of blue stragglers. A
spectroscopic study of the blue stragglers in M67 has
been done in Ref. [13]. Notice that the HR diagram
shown in Figure 5 is more typical for globular clusters
than for open clusters.
Figure 6 presents our photometric color-magnitude
diagram for the “ordinary” open cluster M35. (The clu-
ster age is about 130 Myr [14].) In sharp contrast to M67,
the color-magnitude diagram for M35 clearly shows a
well-defined long main sequence stretching from
-BV 0.1
-1.BV4, but does not show
horizontal branch stars, the giant branch stars, and blue
stragglers. Such an HR diagram is typical for open clu-
Thus, the following parameters were derived from our
observations: =27.63.6
clust = 770r arcsec, Rclust = 3.1 ± 0.26
pc (adopting pc), and .
clust 1200N
Copyright © 2012 SciRes. IJAA
S. NAIM, E. GRIV 169
Figure 5. The color-magnitude diagram for M67, showing
the apparent magnitude mvis of stars versus their color in-
dex (B - V).
Figure 6. The color-magnitude diagram for M35 (cf. Ref.
3. Total Energy of M67
One of the principal features of stellar clusters is their
quasi-stability: Hundreds of millions of years may elapse
before the cluster will begin to disintegrate by some stars
acquiring velocities exceeding the escape velocity esc .
This is because the total energy of a cluster in a quasi-
steady state is negative. Let us show that the assumption
of a quasi-steady state for M67 is reasonable.
The average mass of a main-sequence star in the
cluster star was evaluated by using the standard mass-
luminosity relation . From the definition
abs , one gets the following mass-
absolute-magnitude (
og c
m) relation:
abs, abs
abs =10 .
Mm M
From the HR diagram (Figure 5) and Equation (1),
one can estimate the average mass of a star in the cluster
star abs,
As is seen, however, this diagram is too noisy for the
dimmest stars (and some of the dimmest stars are not
shown in Figure 5 due to a high noise). Therefore, we
prefer to use a much more simple model. Namely,
examining our and other (e.g. Ref. [1]) HR diagrams, one
can see that the main sequence of this cluster is occupied
mainly by stars with absolute magnitude ranging from
to abs , with more stars closer to
abs . Also, from Figure 5 we obtained that the
number of stars
abs =3m
=3 2=6Nm
Nm abs . Therefore,
in the lowest approximation the cluster consists only of
stars with absolute magnitude abs . In the next
approximation, the average mass of a star is obviously
 
abs abs
2=3 =
Mm Mm
Taking abs, 4.85m
, we get star 1.33
M. Fi-
nally, using g, we have
10= 1.989M
star 2.650.910 g.M
33 (3)
This average mass of a star is in good agreement with
most recent studies of M67.
In order to estimate the kinetic energy, an average
speed of a star was calculated as
star =
The typical radial velocity is
s –1 and the transverse velocity is
=0.48 0.15 10
=0.81 0.1 10v
 m·s–1 [7]. This value of v
based on the distance clust pc [7]. We corrected = 870r
to a new one based on the distance clust = 770100r
pc by taking
70= 0.8v
17708 km·s–1. From Equ-
ation (4) one easily obtains
star=0.860.210 ms.v
 
The kinetic energy of the cluster would then be
star star
clust clust
==1.10.610 J.
KN  (6)
The average gravitational potential energy clust was
calculated using the well-known relation for energy of a
self-attracting spherically symmetric mass M. The gra-
vitational potential energy of the mass M of the radius R
is given by
 (7)
For the cluster we use , clust
and get
clustclust star
clustclust clust
UG . This imme-
diately implies that the potential energy of the cluster is
=35 MNR
Copyright © 2012 SciRes. IJAA
clust =4.3310 JU . (8)
Thus, the total energy of M67,
clustclust clust
=3.2EKU10 J,
is negative but small in its absolute value in a sense that
the kinetic energy and the potential energy are of the
same order of magnitude. Moreover, the value of
clust clust0.3KU , while the value predicted by the virial
theorem for a stable system consisting of N particles,
bound by potential forces, is clust clust
KU . From
the above, one concludes that the assumption of a
quasi-steady state for M67 is indeed reasonable. This is
an important fact from the point of view of stellar
dynamics. One understands, however, that even though
the cluster is dynamically bound, clust, escaping
of stars from the system due to the Maxwellian-like
velocity spread is possible (see Section 4 below for an
4. Evaporation of Stars from M67
Many factors cause stars to dissipate from dynamically
bound clusters. In Ref. [15], it has pointed out that
because of the scattering of stars in clusters on the stars
in the galactic field the total internal energy of a cluster
should rise, ultimately leading to a complete breakup of
the cluster. It turned out that the characteristic time for
this process of destruction of dense stellar clusters is at
most 1010 yr. Spitzer [16,17] has calculated the increase
of energy leading to desintegration of a cluster due to
encounters with interstellar clouds. Spitzer has argued
that the characteristic time for a process of this sort is
also 1010 yr. The importance of the galactic tidal force on
the evolution and stability of clusters has been also
discussed (e.g. Ref. [18]). The characteristic decay time
for this process is more than 109 yr. We conclude that the
time yr might be regarded as an upper limit
on the age of clusters in the Galaxy.
age =10t
Other investigations have studied the most important
mechanism for the destruction of dense stellar clusters,
namely, the dissipation which is caused by the interaction
of stars as they approach one another [19-22]. See also
Refs. [23-25] for a discussion of the problem. Evidently,
in this process energy exchange among cluster members
will cause individual stars to acquire supercritical velo-
cities (escape velocities) of the order of
esc 2vv, where 2
v is the rms velocity) in ex-
cess of the parabolic velocity and actually to leave the
cluster, whose total mass therefore decreases. The escap-
ing stars carry away a positive energy and a state of
statistical equilibrium is impossible for a cluster contain-
ing a finite number of stars if the interactions between
stars is strictly taken into account [26,27]. See also Ref.
[28] for a discussion.
By the time of relaxation of star systems, generally
speaking we mean the characteristic time of approach of
the distribution function
trv of the stars with re-
spect to peculiar (random) velocities to a Maxwellian
distribution [22]. (The distribution function
defined by the condition that is the
number of stars at time t in volume element
and in velocity element .) The time of estab-
lishment of a Maxwellian distribution of random velo-
cities as a result of encounters between test stars test
and field stars field (the “collisional” relaxation time)
has the order of magnitude [22,23]
chand 22
10 log
GM n
where is the relative velocity, which is approximately
the velocity dispersion of the lighter stars, n is the
number density of the field stars, and field are
the masses of test and field stars, is the so-called
Coulomb (Newton) logarithm, by means of which the long-
range nature of the gravitational force is taken into ac-
count, clust
log M
, and field is the total number of
field stars in the system. If we assume that the stellar
distribution in the system tends to a Maxwell-Boltzmann
distribution as a result of stellar encounters, during re-
laxation time the system loses approximately 0.0074 of
its members (e.g. Ref. [28]). Thus we can properly regard
half chand
as a measure of the half-life time of a
cluster. (Although the general conclusions on the relax-
ation time reached by Chandrasekhar, Spitzer, and others
are correct, the discussions were based on a simplified
“molecular-kinetic” theory and not on an explicit solu-
tion of the time-dependent problem. In Appendix 8, we
follow numerically the relaxation of an isotropic system
of like stars through small-angle Coulomb encounters by
using the more accurate approach.)
In M67, assuming fieldteststar =1.33
MM M
(Equation (3)), star
km·s–1 (Equation (5)),
and 3
~4 10
clustclust , one obtains yr
for star-star gravitational encounters, So that the half-life
time of the cluster is relatively short, of the order of
yr only. (We estimate that the dynamical
Chandrasekhar-Spitzer relaxation time for M67 is
yr, which implies that the cluster age is 133
times its relaxation time. In contrast for the “ordinary”
open cluster M35, the cluster age yr is only
1.6 times its relaxation time [14].) We argue therefore
that in the past ( yr ago) the cluster M67 was a
relatively small ( stars) globular cluster, but
got “open cluster” shape due to the dynamical eva-
poration of a substantial portion or even the majority of
its stars. Interestingly, the direct N-body model with
36,000 stars for M67, evolved from zero age to 4 Gyr,
=3nN 7
chand 310
1.3 10t
half =210t
Copyright © 2012 SciRes. IJAA
S. NAIM, E. GRIV 171
has been presented in Ref. [9]. It has been shown that at
4 Gyr the total mass has reduced to 2000
as a
result of mass loss and stellar escapes.
5. Concluding Remark
Summarizing, from our own observations, we built the
HR diagram for the cluster M67. We showed that this
diagram is more typical for relatively old globular
clusters than for open clusters. We also estimated the
total number of stars in the cluster clust , its
actual radius pc, and the average mass of a
star star
clust 3.1R
. We studied the dynamical features
of M67 and argued that
The total energy of the cluster clust clustust
is negative, 39
clust . Moreover
3.210 JE
clust clust0.3KU. The latter may be considered as
evidence of the quasi-stability of the system under
The cluster is a tight group of about 1200 stars which
are gravitationally bound, <0E.
As the cluster is quasi-stable, its characteristic rela-
xation time is the Chandrasekhar-Spitzer time
7 yr and the half-life time is equal to
2 10yr, respectively.
half chand
As the cluster is almost twice older its half-life time,
we conclude that many stars have left the cluster
since its formation. In the past (9
~4 10 yr ago) the
cluster was a relatively small (~ 10,000 stars) glo-
bular cluster, but got “open cluster” shape due to the
dynamical evaporation of the majority of its stars.
6. Acknowledgements
This work was began while the first author (S.N.) was an
undergraduate student at the Ben-Gurion University. S.N.
thanks the Head of Astronomical Programs of the Ilan
Ramon Physics Center, Netzach Farbiash, and the Scien-
tific Director of the Ilan Ramon Physics Center, Moshe
Schechter, for making possible observations with Meade
LX200R telescope. We thank Amir Bernat, Michael Ge-
dalin, and Nimrod Nissim for useful discussions in rela-
tion to this work and both anonymous referees for nu-
merous comments that improved the presentation of the
paper. Atlas Image obtained as part of the Two Micron
All Sky Survey, a joint project of the University of Mas-
sachusetts and the Infrared Processing and Analysis
Center/California Institute of Technology, funded by the
National Aeronautics and Space Administration and the
National Science Foundation. This work was sponsored
in part by the Israel Science Foundation, the Israeli Mi-
nistry of Immigrant Absorption in the framework of the
program “KAMEA,” and the Binational US-Israel Sci-
ence Foundation.
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S. NAIM, E. GRIV 173
Relaxation of a System of Stars:
Numerical Solution of the Fokker-Planck
Let us consider a system of stars interacting through in-
verse-square-law Coulombian (Newtonian) forces. As is
known, the evolution of an isotropic distribution of like
particles (stars) through small-angle Coulomb-like col-
lisions is described by the Fokker-Planck equation for an
inverse-square force:
22 4
=2πlnd d
Gmufu ufu
vvv v
 
 
 
vt is the distribution function of stars,
which is defined by the condition that
is the number density of stars. In this approximation the
distribution function depends only on v, the magnitude of
the velocity, and t, the time. Equation (10) has been
derived by L. D. Landau (e.g. Ref. [29]) and then fully
independent in Ref. [30] for gravitational systems and in
Refs. [31,32] for an ionized gas. See Refs. [33,34] for a
discussion. It can be shown that, in the absence of sources
and losses, the number of particles (stars), as well as total
momentum and energy, are conserved within the for-
malism of the Fokker-Planck equation. The solution of
Equation (10), under the same conditions, for a steady
state (=0ft ) is the Maxwellian distribution,
() exp.
fv v
Following Refs. [35,36], Equation (10) may be put
into dimensionless form
 
where 0
vv , v0 is a constant and is a characteristic
is the dimensionless time, and quantities A,
B, and C are defined in Ref. [36]. We integrate this
equation directly using the difference equation given in
Ref. [36] (see also Ref. [35]).
Figure 7. The relaxation of a system of like stars to the
equilibrium Maxwellian distribution. The dashed line shows
an initial distribution and the dash-dot line shows a Max-
wellian of the same density and energy. The time is nor-
malized so that τ = 1 corresponds to a Chandrasekhar-
Spitzer relaxation time tchand (Equation (9)).
We agree with the second referee of this paper that
Equation (11) “is neither matching an actual multi-com-
ponent system nor a spatially inhomogeneous stellar
system.” This equation can serve only as a convenient
starting point for more realistic computations.
We solve Equation (11) numerically, subject to the
initial condition that at time =0
, the initial distri-
bution is
 
,0exp0.3 0.3fx x
. The result-
ing solution for
is plotted in Figure 7 for
various values of time
. The final Maxwellian distri-
bution function at
 is also shown. In fair agree-
ment with the simple Chandrasekhar-Spitzer theory [22,
23], at 1
the distribution function of stars becomes
Maxwellian-like. As calculations show, near 0.25x
and 0.45x
the distribution function departs slightly
from the equilibrium Maxwellian. However, at =1
the lower energy portion of the spectrum is somewhat
underpopulated, while the higher energy part is over-
populated. Thus, the higher- and lower-energy parts of
the distribution are filled in at a much later time (at times
, Figure 7) as one may expect from the Chan-
drasekhar-Spitzer theory. We conclude that in quanti-
tative terms the collisional relaxation of the high- and
low-energy parts of the distribution is only poorly des-
cribed by the Chandrasekhar-Spitzer simple “molecular-
kinetic” theory.
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