Examining the M 67 Classification as an Open Cluster

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 M . We also show that the ratio of the mean kinetic energy of the cluster to its mean gravitational potential energy clust clust 0.3 K U   , while the value predicted by the virial theorem is equal to . So the system is a gravitationally bound. This value of 0.5  clust clust K U 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 t f 70  0 , 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


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][2][3].In the present paper we adopt the value clust Gyr [4,5].It is important to note that open clusters usually get destructed 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 clusters 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

M M 
 and its initial mass was to be approximately past clust [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 (http://ilanramon.bgu.ac.il) (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 particular, M67 and M35 (=NGC 2168).

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 = 27.6 3.6 arcsec.Then, adopting 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 8    taken from the Atlas Image website.Using these photographs (Figures 3 and 4) and this R clust , finally we estimated the total number of stars in the system , where the turn-off point is located, up to .After 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.

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 quasisteady state is negative.Let us show that the assumption of a quasi-steady state for M67 is reasonable. v The average mass of a main-sequence star in the cluster star was evaluated by using the standard massluminosity relation .From the definition abs , one gets the following massabsolute-magnitude ( From the HR diagram (Figure 5) and Equation ( 1), one can estimate the average mass of a star in the cluster 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 3 Taking abs, 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 The typical radial velocity is m•s -1 and the transverse velocity is The average gravitational potential energy clust was calculated using the well-known relation for energy of a self-attracting spherically symmetric mass M. The gravitational potential energy of the mass M of the radius R is given by For the cluster we use , clust and get . This immediately implies that the potential energy of the cluster is Thus, the total energy of M67, 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 clust 0.3 K U   , while the value predicted by the virial theorem for a stable system consisting of N particles, bound by potential forces, is clust clust K U .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 explanation).= 0.5  < 0 E

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 breakup of the cluster.It turned out that the characteristic time for this process of destruction of dense stellar clusters is at most 10 10 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 10 10 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 10 9 yr.We conclude that the time yr might be regarded as an upper limit on the age of clusters in the Galaxy.10 age = 10 t 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][20][21][22].See also Refs.[23][24][25] for a discussion of the problem.Evidently, in this process energy exchange among cluster members will cause individual stars to acquire supercritical velocities (escape velocities) of the order of is the rms velocity) in excess of the parabolic velocity and actually to leave the cluster, whose total mass therefore decreases.The escaping stars carry away a positive energy and a state of statistical equilibrium is impossible for a cluster containing 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  , ,  f t r v of the stars with respect to peculiar (random) velocities to a Maxwellian distribution [22] .)The time of establishment of a Maxwellian distribution of random velocities as a result of encounters between test stars test and field stars field (the "collisional" relaxation time) has the order of magnitude [22,23] , 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 longrange nature of the gravitational force is taken into account, , 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 relaxation 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 relaxation time reached by Chandrasekhar, Spitzer, and others are correct, the discussions were based on a simplified "molecular-kinetic" theory and not on an explicit solution 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 5)), and 3 4πR 9 ~4 10  ~10,0 clust clust , 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 evaporation 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, as a result of mass loss and stellar escapes.

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 . We studied the dynamical features of M67 and argued that  The total energy of the cluster clust clust ust . latter may be considered as evidence of the quasi-stability of the system under consideration. The cluster is a tight group of about 1200 stars which are gravitationally bound, < 0 E .clust  As the cluster is quasi-stable, its characteristic relaxation time is the Chandrasekhar-Spitzer time 7 yr and the half-life time is equal to  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) globular cluster, but got "open cluster" shape due to the dynamical evaporation of the majority of its stars.

Appendix Relaxation of a System of Stars: Numerical Solution of the Fokker-Planck Equation
Let us consider a system of stars interacting through inverse-square-law Coulombian (Newtonian) forces.As is known, the evolution of an isotropic distribution of like particles (stars) through small-angle Coulomb-like collisions is described by the Fokker-Planck equation for an inverse-square force: where   , f v t 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 formalism of the Fokker-Planck equation.The solution of Equation (10), under the same conditions, for a steady state ( Following Refs.[35,36], Equation ( 10) may be put into dimensionless form where 0 = x v v , v 0 is a constant and is a characteristic velocity,  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]).We agree with the second referee of this paper that Equation ( 11) "is neither matching an actual multi-component 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 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 overpopulated.Thus, the higher-and lower-energy parts of the distribution are filled in at a much later time (at times 10   , Figure 7) as one may expect from the Chandrasekhar-Spitzer theory.We conclude that in quantitative terms the collisional relaxation of the high-and low-energy parts of the distribution is only poorly described by the Chandrasekhar-Spitzer simple "molecularkinetic" theory.

Figure 5
shows the observed color-magnitude,   vis .mvs B V , 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 horizontal branch stars.The yellow color indicates the positions 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

Figure 6 Figure 5 .
Figure 5.The color-magnitude diagram for M67, showing the apparent magnitude m vis of stars versus their color index (B -V).
abs, with more stars closer to abs .Also, from Figure5we obtained that the number of 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 -1[7].This value of v  is based on the distance clust pc[7].We corrected = 870 r v  to a new one based on the distance clust = 770 100 r  pc by taking   70 = 0.8 v  1 770 8  km•s -1 .From Equation (4) one easily obtains   3 star = 0.86 0.2 10 m s .v . IJAA has been presented in Ref.[9].It has been shown that at 4 Gyr the total mass has reduced to 2000 M  1200 N 

Figure 7 .
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 Maxwellian of the same density and energy.The time is normalized so that τ = 1 corresponds to a Chandrasekhar-Spitzer relaxation time t chand (Equation (9)).
) numerically, subject to the initial condition that at time = 0  is plotted in Figure7for various values of time  .The final Maxwellian distribution function at    is also shown.In fair agreement with the simple Chandrasekhar-Spitzer theory[22,23], at 1   the distribution function of stars becomes Maxwellian-like.As calculations show, near 0.25 x  and 0.45 x 