arX

iv:a

stro

-ph/

0206

276v

1 1

7 Ju

n 20

02

Kinematic study of the disrupting globular cluster Palomar 5

using VLT spectra1

Michael Odenkirchen2, Eva K. Grebel2, Walter Dehnen2, Hans-Walter Rix2,

Kyle M. Cudworth3

ABSTRACT

Wide-field photometric data from the Sloan Digital Sky Survey have recently revealed that theGalactic globular cluster Palomar 5 is in the process of being tidally disrupted (Odenkirchen et al.2001). Here we investigate the kinematics of this sparse remote star cluster using high resolutionspectra from the Very Large Telescope (VLT). Twenty candidate cluster giants located within 6arcmin of the cluster center have been observed with the UV-Visual Echelle Spectrograph (UVES)on VLT-UT2. The spectra provide radial velocities with a typical accuracy of 0.15 km s−1. Wefind that the sample contains 17 certain cluster members with very coherent kinematics, twounrelated field dwarfs, and one giant with a deviant velocity, which is most likely a clusterbinary showing fast orbital motion. From the confirmed members we determine the heliocentricvelocity of the cluster as −58.7 ± 0.2 km s−1. The total line-of-sight velocity dispersion of thecluster stars is 1.1± 0.2 km s−1 (all members) or 0.9± 0.2 km s−1 (stars on the red giant branchonly). This is the lowest velocity dispersion that has so far been measured for a stellar systemclassified as a globular cluster. The shape of the velocity distribution suggests that there isa significant contribution from orbital motions of binaries and that the dynamical part of thevelocity dispersion is therefore still substantially smaller than the total dispersion. Comparingthe observations to the results of Monte Carlo simulations of binaries we find that the frequencyof binaries in Pal 5 is most likely between 24% and 63% and that the dynamical line-of-sightvelocity dispersion of the cluster must be smaller than 0.7 km s−1(90% confidence upper limit).The most probable values of the dynamical dispersion lie in the range 0.12 ≤ σ/kms−1 ≤ 0.41(68% confidence). Pal 5 thus turns out to be a dynamically very cold system. Our results arecompatible with an equilibrium system. We find that the luminosity of the cluster implies atotal mass of only 4.5 to 6.0 × 103M⊙. We further show that a dynamical line-of-sight velocitydispersion of 0.32 to 0.37 km s−1 admits a King model that fits the observed surface densityprofile of Pal 5 (with W0 = 2.9 and rt = 16.′1) and its mass.

Subject headings: globular clusters: individual (Palomar 5) — stars: kinematics

2Max-Planck-Institut fur Astronomie, Konigstuhl17, D-69117 Heidelberg, Germany; odenkirchen@mpia-hd.mpg.de

3Yerkes Observatory, University of Chicago, 373West Geneva Street, Williams Bay, WI 53191;kmc@yerkes.uchicago.edu

1Based on observations obtained at the EuropeanSouthern Observatory, Paranal, Chile (Program 67.D-0298A)

1. Introduction

The globular cluster Palomar 5, an old halocluster located at a distance of about 23 kpc fromthe Sun and 18.5 kpc from the Galactic center (seeHarris 1996), stands out through a number of un-usual and extreme properties. These make it par-ticularly interesting from the viewpoint of dynam-ics, cluster evolution, and galactic structure. Firstof all, Pal 5 is extraordinarily sparse and faint.Its total luminosity of MV ≃ −5.0 (Sandage &

1

Hartwick 1977, hereafter SH77) places it amongthe least luminous globular clusters that we cur-rently know to exist in our Galaxy. Assuming amass-to-light ratio typical of other globular clus-ters this luminosity corresponds to a stellar massof about 1.3 × 104 M⊙ (SH77) or 0.8 × 104 M⊙

(Mandushev, Spassova & Staneva 1991), which liesa factor of 30 below the median mass of Galacticglobulars. Further, Pal 5 has a very extended coreand a low central concentration (see, e.g., Trager,King & Djorgovski 1995). Finally, the faint partof the luminosity function of Pal 5 is unusuallyflat (Smith et al. 1986, Grillmair & Smith 2001),i.e. the fraction of low-mass stars in Pal 5 is muchsmaller than in other galactic globulars. These pe-culiarities suggest that Pal 5 has undergone strongdynamical evolution and mass loss and that it maybe close to complete disruption. The hypothesis ofongoing dissolution was recently confirmed by thediscovery of two massive tails of unbound formercluster members which spread from the cluster inopposite directions over an arc of several degrees(Odenkirchen et al. 2001). The stars observed inthe tails make up at least 35% of the total mass ofcluster members and thus provide direct evidencefor heavy mass loss from a strong tidal perturba-tion. Currently, Pal 5 is therefore the best knownexample of a tidally disrupting globular clusterand is ideally suited for studying this phenomenonin-situ.

In order to reconstruct the mass loss historyof Pal 5 and to understand and interpret the den-sity structures that are visible in its tidal tails oneneeds to simulate the cluster’s dynamics with nu-merical methods. The present-day velocity dis-tribution in the cluster, in particular the velocitydispersion, provides an important boundary con-dition for such simulations and is indispensable forderiving a realistic numerical model of the cluster’sevolution in the tidal field of the Milky Way. How-ever, the internal stellar velocities in Pal 5 havenot been measured to date. The only publishedmeasurements of radial velocities of stars in Pal 5are by Peterson (1985) and by Smith (1985). Thespectral resolution of these measurements is suffi-cient to estimate the radial velocity of the clusteras a whole, but clearly too low to resolve the in-ternal kinematics in the cluster. The results ofSmith (1985) yield an upper limit of 4 km s−1 onthe velocity dispersion in the cluster. Assuming

that the cluster is in virial equilibrium the massof 1.3 × 104 M⊙ estimated by SH77 and a typi-cal radius of about 20 pc imply a velocity disper-sion of the order of 1 km s−1. On the other hand,since Pal 5 has undergone tidal perturbations andis surrounded by a massive population of extrati-dal stars it would be conceivable that the aboveequilibrium model is not valid. Hence, any suchprediction of the velocity dispersion in the clustercan only be a rough guideline. The goal of thispaper is to determine the internal kinematics ofPal 5 directly from observation. We thus set outto obtain high-resolution spectra for a number ofcluster members to derive very precise radial ve-locities. Due to the rather large distance of thecluster, its deficiency in bright giants, and the needfor high spectral resolution, this project requiredan 8-m class telescope.

In Section 2 we provide details on the observa-tions and the reduction and analysis of the spec-tra. In Section 3 we then analyze the observedradial velocities, determine the cluster’s velocityand the overall velocity dispersion along the lineof sight, and compare the observed velocity dis-tribution with a Gaussian model. In Section 4we investigate the influence of binaries and deriveconstraints on the dynamical velocity dispersion ofthe cluster. In Section 5 we rederive the structuralparameters and the total luminosity of the clus-ter from new photometric data and compare theseparameters with the kinematics of the cluster inthe framework of an equilibrium cluster model. InSection 6 we summarize and discuss our results.

2. Observations and data reduction

The stars that are used to probe the kinemat-ics of Pal 5 were selected with the help of multi-band photometry from the Sloan Digital Sky Sur-vey (SDSS; see Stoughton et al. 2002 and Yorket al. 2000 for overviews and Smith et al. 2002,Pier et al. 2002, Hogg et al. 2001, Gunn et al.1998, Fukugita et al. 1996 for different techni-cal aspects of the project). We defined a sampleof 20 program stars located within a radius of 6′

around the center of Pal 5, that have magnitudesin the range 15.0 ≤ i∗ ≤ 17.7, and appear as likelycluster giants according to their magnitude andcolors (see Odenkirchen et al. 2001). The targetsample is presented in Table 1 and its properties

2

are shown in Figures 1 and 2. These stars wereobserved with the echelle spectrograph UVES onESO’s Very Large Telescope (VLT). The observa-tions were carried out in service mode during Mayand June 2001. The spectra were taken in the in-strument’s red arm, using a dichroic beam splitterat 5800 A and a slit width of 1′′. They cover thewavelength ranges 4790 A − 5760 A and 5830 A− 6810 A (recorded on two separate CCD chips)at a spectral resolution of about 40,000. Expo-sure times varied between 7 and 60 minutes andwere chosen such that the spectra reach a signal-to-noise ratio of about 10. To enable precise wave-length calibration each observation of a star wasfollowed by a ThAr lamp exposure taken imme-diately after the sky exposure. In addition to theprogram stars two radial velocity standards of typeK1III (HD 107328 and HD157457) from the list ofUdry et al. (1999) were observed with the sameinstrument set-up in order to provide high qualitytemplate spectra and an absolute velocity refer-ence.

The reduction of the spectra (i.e., bias andbackground subtraction, flatfielding, order extrac-tion, sky subtraction, wavelength calibration, re-binning, and order merging) was carried out withESO’s dedicated UVES reduction pipeline. Thespectra were wavelength calibrated using the at-tached ThAr spectra and were rebinned to a linearwavelength scale of 0.030 A/pix (blue part) and0.035 A/pix (red part). Examples of the reducedand calibrated spectra are shown in Figure 3.

Radial velocities were determined by cross-correlating the spectra of the program stars withthose of the two velocity standards. This wasdone using the routine fxcor in the softwarepackage iraf4. We calculated separate cross-correlation functions in five distinct wavelengthintervals, i.e., at 4910 A – 5210 A, 5210 A –5460 A, 5460 A – 5750 A, 5850 A – 6320 A, and6320 A – 6790 A. Hereby we obtained for eachpair of a program star and a standard star five in-dependent velocity measurements of nearly equalaccuracy. The velocity shift between programstar and template was determined by fitting aLorentzian curve to the cross-correlation functionin a range of ±12.5 km s−1(i.e., 15 data points)

4IRAF is distributed by the National Optical AstronomyObservatory

around its highest peak. The results from thefive wavelength intervals were averaged to definethe relative velocity of each program star with re-spect to the standard star. The rms deviation ofthe individual results from the mean was used toderive an empirical estimate of the random errorof the velocity. Heliocentric corrections for theprogram stars and the templates, depending ontheir position and the time of their observation,were calculated with the routine rvcorrect iniraf. Applying these corrections and adding theknown absolute velocities of the template stars5,the measured velocities were transformed to theheliocentric absolute system.

The results of this procedure are presented inTable 2. The velocities are found to have accu-racies between 0.05 km s−1 and 0.25 km s−1. Inmost cases the random error is below 0.15 km s−1.The results obtained with the two different veloc-ity standards (see column (a) and (b) of Tab. 2)are in good agreeement with each other. Thereis a small mean offset of 0.14 km s−1 between thetwo sets of absolute velocities. When removingthis offset, the remaining rms deviation is lessthan 0.03 km s−1. Cross-correlation of the spec-tra of the two standard stars with one anotheryields velocities that differ from the nominal val-ues by 0.14 km s−1. The difference agrees with themean offset in the results for the program stars andshows that the zero-point of our absolute veloci-ties has an uncertainty of this order. We take foreach program star the mean of the results obtainedwith either of the two standards as the best esti-mate of its absolute velocity. Note, however, thatthe uncertainty of velocity differences is given bythe individual random errors only.

The last two columns of Table 2 give the ve-locities obtained by Smith (1985) and by Peterson(1985) for stars in common with those in our sam-ple. The differences between our results and theprevious ones are of the order of a few km s−1. It iseasily seen that our results for different stars are inmuch closer agreement with each other than theprevious measurements. This suggests that thedifferences between the previous results and oursare largely due to the lower precision of the pre-

5According to Udry et al. (1999) the radial velocitiesof HD107328 and HD157457 are +36.4 km s−1 and+17.8 km s−1, respectively.

3

vious measurements (and perhaps differences inthe absolute calibration) and in general do not re-flect true variations in the radial velocities of thesestars. Variations from binaries are expected to oc-cur mostly with smaller velocity amplitudes (seeSect. 4).

3. Analysis of the kinematics

3.1. Cluster membership

The spectroscopic observations show that mostof the stars in our target sample are indeed gi-ants and that many of them have almost identi-cal velocities. Figure 4 shows the distribution ofthe observed velocities. Seventeen of the twentymeasured stars have radial velocities in the rangebetween −61.2 kms−1 and −56.9 km s−1. Theseare undoubtedly all members of the cluster. Tenstars in this group stand out by having partic-ularly coherent velocities that lie in an intervalof only 1 km s−1. This can be seen by the pro-nounced peak in Figure 4a and by the correspond-ing steep rise of the cumulative number countsshown in Figure 4b. Only three stars appear tobe kinematically distinct from the cluster. Theirvelocities deviate from the rest of the sample byabout 6 km s−1, 14 km s−1, and 35 km s−1, re-spectively. Two of these stars turn out to be fore-ground dwarfs on the basis of a much larger widthof their spectral lines (stars 4 and 12, see Fig. 3).They are definitely not belonging to the clusterand are discarded from the subsequent analysis.The only doubtful case is star 15, which has thespectrum of a giant resembling those of other clus-ter members, but is set off from the cluster byabout 14 km s−1 in radial velocity. Since this staris located at less than 2′ angular distance fromthe center of the cluster it seems unlikely that it isan unrelated field giant at about the same spatialdistance as the cluster and not a member of Pal 5.From the surface density of field stars with magni-tude and color similar to star 15 (i.e., ±0.15 magin magnitude and ±0.05 mag in color) it followsthat the expected number of such field stars within2′ from the cluster center is 0.1 while the expectednumber of cluster stars is 3.5. Since we have ob-served three stars in this range of position, color,and magnitude the probability that (at least) oneof them is a field star is 1−(3.5/3.6)3 = 0.08, only.

The cluster membership of all but stars 4 and 12

of our sample is independently confirmed in propermotion carried out by Cudworth, Schweitzer &Majewski (in preparation). This study determinesproper motions of about 500 stars in the field ofPal 5 using microdensitometer scans of 25 platesfrom large reflectors ranging in epoch from 1949to 1991. From the preliminary results of this workmembership probabilities for our stars were de-rived in the way described in Dinescu et al. (1996).The probabilities are presented in Table 3. Whilestars 4 and 12 again prove to be non-members, theother stars have membership probabilities between72% and 99% and thus qualify as cluster members.In particular, star 15 has a proper motion mem-bership probability of 73%, which is comparable tothat of other stars with radial velocities close tothe cluster mean, thus supporting its membership.

The most plausible explanation then is that star15 is a binary member of Pal 5 and that the ob-served offset of its radial velocity (actually thatof the primary component) is the result of tempo-ral variations due to rapid orbital motion. For abinary in a circular orbit the orbital period T isrelated to the orbital velocity v1 of the primary by

T

yr=

(

30 kms−1

v1

M2

M1 +M2

)3M1 +M2

M⊙

. (1)

Assuming a mass of 0.8M⊙ for the primary onefinds that the orbital period of star 15 can at mostbe 2 yr (M2 = M1) and that it is likely to be ofthe order of a few months because the companionmass is probably smaller than that of the primaryand because the projection on the line of sight hasto be taken into account. Therefore, even only oneadditional precise measurement of the radial veloc-ity of this star should immediately reveal whetherit is indeed a binary member of Pal 5. For thepresent paper we take its occurence as an indica-tion of the likely existence of binaries in Pal 5, butomit it in the analysis of the velocities because ofits large offset from the other cluster stars.

3.2. Mean velocity and dispersion

The median of the velocities of the 17 certainmembers of Pal 5 is −58.8 km s−1, the arithmeticmean −58.9± 0.3 km s−1. By successive omissionof those stars that deviate most strongly from themedian of the sample one obtains mean velocities

4

between −58.6 km s−1 and −58.8 km s−1. Theparticular subgroup of 10 stars whose velocitiescoincide within 1 km s−1 has a mean velocity of−58.7± 0.1 km s−1. Combining these results withthe uncertainty of the zero-point of the absolutevelocity scale we adopt the heliocentric velocityof the cluster vr,cl = −58.7 ± 0.2. The rms dis-persion of the individual velocities of the 17 clus-ter members with respect to this cluster mean is1.14 km s−1. The individual measurement errorsare much smaller, and their contribution to thisdispersion can be neglected.

The colors and magnitudes of the stars revealthat the set of confirmed members consists of 13normal red giants (RGB stars) and 4 asymptoticgiant branch (AGB) stars (see Fig. 1). Compar-ing the velocities of the RGB and AGB stars wefind that three of the four AGB stars have ra-dial velocities that differ from the above clustermean by more than 1 km s−1. Among the RGBstars only three out of 13 have velocities beyondthis limit. This suggests either that the two typesof giants are somehow kinematically different orthat the measurements of the AGB stars are af-fected by some kind of pulsational variations intheir extended atmospheres. Spectroscopic stud-ies of other globular clusters have found evidencefor such ‘atmospheric jitter’ among the most lu-minous red giants, i.e., close to the tip of the redgiant branch (see, e.g., Cote et al. 1996), but thereis so far no information about a similar effect inless luminous AGB stars like those of our sam-ple. The presence of atmospheric jitter in ourAGB stars could in principle be tested and eventu-ally be removed through repeated measurements.However, since further observations of high preci-sion are not available for these stars the questioncannot be clarified at present. If one excludes theAGB stars from the sample and considers only thesubgroup of 13 RGB stars, the velocity dispersiondrops from 1.14 km s−1 to 0.92 km s−1.

3.3. Dependence on magnitude and posi-

tion

In Figure 5 the individual radial velocitiesare plotted versus magnitude, angular distancer from the cluster center (αc = 15h16m04.5s,δc = −00◦07′16′′, J2000.0) and position angle ϕ(measured from north over east). Figure 5a showsthat the mean velocity of the cluster members does

not depend on the magnitude of the stars. Notethat we also find no sign of a dependence of themeasured velocities on the epoch of observation.We thus can eliminate substantial systematic mea-suring errors as a function of brightness or epoch.Figure 5a also demonstrates that while there maybe velocity jitter in the AGB stars there is clearlyno sign for corresponding jitter in the brightestred giants of the sample since the velocities ofthese stars are in extremely close agreement witheach other.

Figure 5b gives the impression that subgroupsof stars at different angular distance from the cen-ter have very different velocity dispersions. Thisis partly, but not entirely due to the AGB stars,which are all located at r > 2.′5 and hence dif-fer from the spatial distribution of the normal gi-ants. An F-test shows that the very low disper-sion of σ = 0.21 km s−1 for the subgroup of the4 innermost stars (r ≤ 1.′5) is indeed significantlysmaller (99% significance) than the dispersion forthe remaining sample, even if the AGB stars areexcluded. On the other hand, it turns out thatthe somewhat lower dispersion of the outermoststars (at r > 4′) as compared to the dispersion atmedium distances from the center is not a statis-tically significant effect.

In Figure 5c one finds weak indications that theobserved velocities may contain azimuthal varia-tions. Such variations could result from of a ro-tation of the cluster. However, there remains al-most no evidence for such an effect if one leavesout the AGB stars. Due to the limited number ofdata points the question of rotation cannot be in-vestigated beyond the simple case of solid bodyrotation. By least-squares adjustment we findthat if the velocities contain a solid body rota-tion component the angular velocity of the rota-tion and the position angle of the rotation axiswould be ω = 0.25 ± 0.12 km s−1 arcmin−1 andPA = +15± 31◦, respectively. When subtractingthis hypothetical rotation from the observed ve-locities the velocity dispersion of the n=17 samplereduces only slightly (from 1.14 to 1.08 km s−1).For the subsample without AGB stars we likewiseobtain ω = 0.17 ± 0.16 km s−1 arcmin−1, i.e., therotation velocity is at the border of statistical sig-nificance, and the rotation model yields no reduc-tion of the velocity dispersion. Therefore we con-clude that rotation is not clearly detectable and

5

that it does not provide an important contribu-tion to the kinematics of the cluster. This conclu-sion also holds for the possibility of an expansionof the cluster along a preferred spatial directionsince from the observational point of view such aneffect is equivalent to a rotation.

3.4. Comparison with an isothermal sys-

tem

If the cluster maintains a state of dynamicalquasi-equilibrium one expects the kinematics inits inner region to be isothermal, which meansthat the line-of-sight velocity distribution of ob-jects with approximately equal masses must (inthe absence of other effects) be Gaussian. Manystellar systems do indeed show velocity distribu-tions that are approximately Gaussian. However,in the case of a cluster with low velocity disper-sion it may happen that contributions from at-mospherically induced variations and from orbitalmotion of binaries are of non-negligible size, andthat the observed line-of-sight velocities are hencenot completely dominated by the dynamics of thecluster. Concerning contributions from binaries,it may also be relevant that a small velocity dis-persion in the cluster is favorable to the survivalof binaries during the cluster’s dynamical evolu-tion. The impact of velocity anomalies in the AGBstars, possibly caused by atmospheric jitter, wasdiscussed in Section 3.3. In order to find out ifbinaries play a significant role in Pal 5 we com-pare the observed velocities to a simple Gaussianrepresenting an isothermal system of single stars.

The Gaussian that best fits the observationscan be found by the method of maximum like-lihood. When observational errors are neglected,the dispersion parameter σ of the best-fit Gaussianis simply given by the rms dispersion of the ob-served velocities. In the general case with observa-tions vri and errors ǫi, one calculates the probabil-ities pi of the observations by convolving the Gaus-sian model Φd with the error distributions Φǫi ,i.e. pi = (Φd ∗ Φǫi)(vri), and maximizes the totalprobability (or likelihood) L =

∏

i pi as a functionof the parameter σ. For the complete sample ofn = 17 stars the solution is σ = 1.14 km s−1 withuncertainties of +0.24 km s−1 and −0.18 km s−1.For the n = 13 subsample of RGB stars we ob-tained σ = 0.91 km s−1 with uncertainties of+0.23 km s−1 and −0.18 km s−1. The given un-

certainties describe the interval around the max-imum of L that contains 68.3% of the likelihoodintegral

∫

Ldσ.

To test the agreement between the maximumlikelihood Gaussian model and the observationswe generated a large number of artificial samplesof n velocities based on the best-fit σ and theobservational errors, and compared the likelihoodof the observed velocities with the likelihood ofthe simulated velocities. The experiment revealsthat the likelihood values of the observed samplesfall close to the median of the likelihood distribu-tion of the corresponding simulated samples, i.e.,47% (n = 17 case) or 45% (n = 13 case) of thesimulated samples have likelihood values that aresmaller than the likelihoods of the observed sam-ples. This means that from the viewpoint of likeli-hood statistics the best-fit Gaussian is an accept-able model for the observed velocities. It wouldhowever be wrong to conclude that a purely Gaus-sian model provides an optimal description of thedata.

In Figure 6 the data and the best-fit Gaussianmodels are compared by plotting their cumulativedistributions. Here it is seen that the agreementbetween the data and the models is in fact lessthan satisfactory. For both samples there are clearsystematic differences between the data and themodel. The observational data are characterizedby a steep rise at small velocities and a flat tailextending from 0.3 to 2.4 km s−1. A Gaussiandistribution is unable to approximate this shapein an appropriate way. In order to maintain theidea of isothermal cluster kinematics we thus needto assume the existence of an additional velocitycomponent.

This suggests that there are indeed signifi-cant contributions from orbital motion of bina-ries. These binaries may either be primordial(as in young open clusters and the field) or mayhave formed by close stellar encounters during theevolution of the cluster. Recent searches for bi-naries in other galactic globulars have revealedthat globular clusters are not generally deficientin binaries as compared to the local populationof field stars and that clusters with low centraldensity and/or indications of strong tidal massloss tend to have enhanced binary frequencies (seeMcMillan, Pryor & Phinney (1998) and referencestherein). Moreover, star 15 provides a direct hint

6

that the existence of binaries in our sample islikely (see Section 3.1). If this object is indeeda binary member of Pal 5 with a relatively shortperiod of a few months as assumed in Section 3.1,then it is natural to expect that the sample alsocontains binaries with longer orbital periods of upto several decades. Such systems will typicallyproduce shifts of the order of 1 km s−1 in the ob-served velocities. This can be seen by consideringthe simple but typical example of a binary with atotal mass of 1 M⊙ and a mass ratio of 1:4 (i.e.,M1 = 0.8M⊙,M2 = 0.2M⊙). Assuming a circularorbit one finds that a radial velocity of the pri-mary of about 1 km s−1 along an average line ofsight is obtained with an orbital period of about30 years.

The presence of significant contributions frombinaries of course makes the determination of thedynamical velocity dispersion of the cluster moredifficult. The above example shows that a di-rect proof of the binary nature of individual starsand the determination of center-of-mass veloci-ties through repeated velocity measurements is notcategorically impossible, but would require a largeamount of observing time over a long time scale(except for star 15). Another way (and currentlythe only practicable one) to eliminate the influ-ence of orbital motion and hence to determine thedynamical velocity dispersion is by estimating thefrequency of binaries and their contribution to theobserved velocity dispersion in a statistical man-ner. This approach works by incorporating theeffects of orbital motion into the model for the dis-tribution of the observed radial velocities, as willbe described in the next section.

4. Simulation of binaries

Since the binarity of the observed objects andtheir orbital motions are not individually knownone may assume that each object has a probabilityxb of being a binary and consider a variety of pos-sible systems with statistically distributed orbitalparameters. If the distribution functions of theorbital parameters of the binary population areknown, then the statistical impact of their orbitalmotions on the radial velocity of the primaries andthe resulting distribution of observable velocitiescan be determined by a Monte Carlo simulation.For binaries in globular clusters one faces the prob-

lem that specific information on the distributionsof their parameters is lacking. Hence one must ei-ther work with plausible general assumptions orrefer to the empirical parameter distributions forfield binaries in the local neighborhood, hopingthat these are not prohibitively wrong for clusterbinaries. We decided to try both ways and thusgenerated different sets of artificial binary popula-tions in the following way:

The mass of the primary component was chosenas M1 = 0.8 M⊙ for normal red giants and M1 =0.5 M⊙ for AGB stars. The mass M2 of the sec-ondary was modelled by a log-normal mass func-tion in the mass range 0.1M⊙ ≤ M2 ≤ M1, i.e. aGaussian for logM , with mean < log(M/M⊙) >=−0.38 and dispersion σlog(M/M⊙) = 0.35. Thismass function closely resembles the empirical massdistribution of secondaries around nearby solar-type stars found by Duquennoy & Mayor (1991).Alternatively, we also tried simpler choices as, forexample, M2 = M1, M2 = 0.4M⊙, or M2 =0.2M⊙. For the semi-major axis a of the bi-nary orbits we adopted a Gaussian in log(a/AU)with mean < log(a/AU) >= 1.5 and dispersionσlog(a/AU) = 1.5. This model is in agreementwith empirical semi-major axis distributions as,e.g., given by Heintz (1969), and with the periodstatistics of local G and K dwarfs given by Mayoret al. (1992). We note that this distribution wasused in truncated form (see below). For the orbitaleccentricity e we used as standard case a flat dis-tribution f(e) = const in the range 0.1 ≤ e ≤ 0.8and e = 0 for periods P < 0.3 yr, taking intoaccount observational results by Duquennoy &Mayor (1991), Mayor et al. (1992) and Lathamet al. (1998). In addition, we also ran simulationswith either e = 0 (circular orbits) or f(e) = 2e(so-called thermal orbits) as extreme alternatives.

In a cluster environment wide binaries are un-likely to survive stellar encounters if their bind-ing energy is less than the typical kinetic energyof relative motion between cluster members. Us-ing a velocity dispersion of 2 to 4 km s−1 (as anestimate for earlier evolutionary stages of Pal 5)the formula given in equation (1) of Pryor et al.(1996) yields upper limits for the semi-major axisof 100 AU to 25 AU, respectively. We thus tenta-tively truncated the distribution of a at differentvalues in this range and after some testing set theupper limit of a to 50 AU. At the lower end the

7

range of distances between the two binary com-ponents also has to be truncated because the pri-maries in our sample are giants and thus have largeradii which can surpass the Roche limit for masstransfer. Roche overflow would increase the sepa-ration between the components and decrease thebrightness of the system (see, e.g., the discussionby Pryor et al. 1988). Therefore we calculatedapproximate radii for our giants using the radius-magnitude relation shown in Figure 6 of Cote etal. (1996), determined the distance of the innerLagrange point L1 from the primary componentat pericenter, and neglected all cases in which thisdistance was smaller than the estimated stellar ra-dius (10 ≤ R/R⊙ ≤ 40). Note that this constraintprecludes the existence of very hard binaries in oursample.

From each simulated set of binaries we deriveda characteristic radial velocity distribution Φb byprojecting the orbital velocities of the primarycomponents onto isotropically distributed line-of-sight directions at randomly chosen fractions ofthe orbital period. Examples of such distributionsare shown in Figure 7. These distributions havelong high-velocity tails which distinguish themfrom a Gaussian. To describe the radial velocitiesof a sample of binaries in a cluster with isother-mal kinematics the binary radial velocity distri-bution Φb needs to be convolved with a GaussianΦd that models the dynamical velocity dispersionas in Section 3.4. Furthermore, for a cluster witha fraction xb of binaries the velocity distributionmust be a composite of Φd ∗ Φb and Φd, namely

Φ = xb · (Φd ∗ Φb) + (1− xb) · Φd . (2)

This yields an extended model Φ with ad-justable parameters σ and xb that should allowa better match with the observed velocities pro-vided that the distributions of the orbital param-eters of the binaries have been set appropriately.The model was fitted to the observations in thesame way as in Section 3.4, namely by convolvingit with the observational errors, calculating theindividual probabilities pi of the observed veloci-ties, and then maximising the probability productL (or actually its logarithm logL =

∑

i log pi) asa function of σ and xb. We calculated maximumlikelihood solutions for a variety of choices of thebinary parameters M2 and e. The results are pre-

sented in Table 4. The upper part of the tablerefers to the entire sample and the lower part tothe subsample of red giants. The quoted uncer-tainties in σ and xb describe the parameter rangethat comprises 68.3% of the integrated likelihood.In addition, we provide the value of the 90% confi-dence upper limit of σ, which is denoted σ90%. Forcomparison, the first line in each part of Table 4repeats the single star solution from Section 3.4.

It turns out that most of the binary models in-deed enable solutions with higher maximum like-lihood than the single star model. The only ex-ception is the extreme case of equal mass binarycomponents (M1 = M2), which fails to producean improved fit to the velocity distribution of then = 17 sample and thus proves to be inadequate.Comparing the values of lnL in Table 4 it is seenthat the improvements achieved with the differentbinary models are generally larger for the n = 13subsample than for the complete n = 17 sample.This supports the assumption that the increasedvelocity dispersion of the AGB stars has a differ-ent origin (see Sect. 3.2) and is not due to binarity.The best fit to the observed velocities is obtainedwith the model that is based on low-mass com-panions (M2 = 0.2M⊙) and preferentially high or-bital eccentricities (f(e) = 2e). This holds forboth samples. Our so-called standard case whichis closest to the properties of the local field bina-ries is found to be less adequate. Other cases withmore massive secondary components or low eccen-tricities are also seen to be less in agreement withthe observed velocities.

Although the solutions are of different qualitywe find that the results for the fit parameters σand xb do not depend much on the details of themodels. For the remainder of this section we focuson the results for the n = 13 subsample of red gi-ants. Table 4 shows that all models require best-fit binary fractions between 0.30 and 0.42, withuncertainties of about ±0.15 to ±0.20. The best-fit values of the dispersion parameter σ fall in allcases close to 0.22 km s−1, and the probable valueslie between 0.12 km s−1 to 0.44 km s−1. Hence itis likely that the dynamical velocity dispersion isby at least a factor of two lower than the directlymeasured total velocity dispersion. Figure 8 showsthe likelihood distribution in the plane of the fitparameters xb and σ for the model that yields thebest fit to the observations (i.e., case (h) of Tab. 4).

8

It is seen that there exists a unique likelihood max-imum at low σ and intermediate xb. However,this maximum is relatively broad and has a tailtowards higher σ, so that the resulting ranges ofprobable values (68.3% confidence) extend from0.24 to 0.63 in xb and from 0.12 to 0.41 kms−1 inσ. Higher values of σ cannot be entirely excludedbut are less likely. To set a reasonable upper limitone can state that with 90% confidence the dy-namical velocity dispersion of the cluster is notlarger than 0.7 km s−1.

In Figure 9, we compare the cumulative distri-bution of observed velocities with the correspond-ing predictions by two of our models with binaries,namely the standard model and the one that yieldsthe best fits. Other than the single Gaussiansshown in the analogous Figure 6, these modelscan (due to the inclusion of binaries) approximateboth the steep inner rise and the extended tail ofthe observed velocity distribution. The standardmodel however is seen to be somewhat less ade-quate than the other model since it predicts toomany stars with |∆vr| > 1.5 km s−1 in comparisonto the observations.

5. Velocity dispersion versus structural

parameters, luminosity, and mass-to-

light ratio

A key question in connection with the deter-mination of the velocity dispersion is: How doesthe result compare to other fundamental param-eters of the cluster? We combine the discussionof this question with a redetermination of Pal 5’ssize, structure, luminosity, and mass using theSDSS data and the HST luminosity function formain-sequence stars of Pal 5 by Grillmair & Smith(2001).

5.1. Surface density profile and best-fit

King model

The wide-field photometry from SDSS providesus with the possibility to derive an improved ra-dial density profile for Pal 5. Using the advan-tage of selective star counts the cluster’s pro-file can be traced down to five times lower sur-face density than in former work by Trager etal. (1995). We separated cluster members fromfield stars through an efficient color-magnitude fil-ter (see Odenkirchen et al. 2001) and determined

the surface density of the members by counts incircular annuli out to a radius of 100 arcminutes.In the outer annuli the member counts are insome places influenced by an overlap with the tidaltails. This was taken into account by splitting thefield into four 90◦ sectors pointing towards north,south, east, and west. The profiles obtained forthe northern and southern sector and for the east-ern an western sector are compared in Figure 10.Since the cluster overlaps with the tails only inthe northern and southern sector, the counts ob-tained in the eastern and western sector yield aprofile that is free of contamination by tidal tailstars and describes the true size and structure ofthe cluster.

Under the assumption of dynamical quasi-equilibrium the density profile of the clustershould match the profile of a so-called King model,i.e., a truncated isothermal sphere (King 1966).We thus took the ’clean’ surface densities mea-sured in the eastern and western sector of thefield, fitted a series of King profiles by means ofweighted least squares, and determined the best-fitting King (1966) model for the cluster. Thebest-fit model has W0 = 2.9, which is equivalentto c = 0.66 for the concentration parameter, anda limiting radius rt = 16.′1 ± 0.′8. The corre-sponding core radius is rc = 3.′6 ± 0.′2. At theassumed distance of d = 23.2 kpc of Pal 5 thismeans a linear core radius of 24.0 pc. Figure 10shows the best-fit King model (plus constant fore-ground density) by a solid curve. The agreementbetween the measurements and the model is sat-isfactory beyond the core radius, but the modelfails to describe the constant surface density inthe central 3′ of Pal 5 and the subsequent abruptdecline. This may be an indication that the dy-namical state of the cluster is in fact somewhatdifferent from a King model. Nevertheless, thebest-fit King model remains a useful approxima-tion because it provides a relation between massand velocity dispersion.

The mass of a truncated isothermal sphere withspatial parameters rc and c and with a line-of-sightvelocity dispersion σlos is given by

M =9

4πGrc µ (β σlos)

2 (3)

(see King 1966). Here, µ denotes a normalized

9

mass parameter that depends on c and that hasto be determined by numerical integration. For amodel with W0 = 2.9 one finds µ = 4.9. The cor-rection factor β depends on the model and on thearea within which the velocity dispersion is sam-pled. If σlos is determined as an average line-of-sight dispersion over a region of one core radius, asit is the case with our observations, the appropri-ate correction factor β for a model with W0 = 2.9is β = 1.5 (see Binney & Tremaine 1987, p.236).Inserting these values into equation (3) we obtainthe relation:

MPal5

M⊙

= 4.4 · 104( σlos

kms−1

)2

(4)

In order to compare this with the observed ve-locity dispersion, one needs an independent esti-mate of the cluster’s mass from its luminosity.

5.2. Total luminosity and mass

To determine the total absolute V band mag-nitude of the cluster we integrated the flux ofall SDSS stars that lie in the region of the gi-ant and subgiant branch, horizontal branch andmain-sequence of Pal 5 and within the limitingradius rt = 16′ of the cluster. The integrationwas carried out down to a magnitude limit ofV = 21.75. Visual magnitudes were derived fromSDSS g∗ and r∗ magnitudes using the transforma-tion V = 0.4g∗+0.6r∗+0.23. This relation was setup by comparing SDSS photometry and CCD pho-tometry in V from Smith et al. (1986) for 18 starsin the color range −0.2 ≤ g∗ − r∗ ≤ 1.4. We cor-rected the integrated flux for the contribution ofresidual field stars by subtracting the statisticallyexpected luminosity of field stars as estimated innearby fields. The resulting integrated magnitudeof Pal 5 from stars down to the limit of V = 21.75is mV = 12.24 ± 0.07. Using a distance modulusof 16.83 ± 0.20 (d = 23.2 ± 2.3 kpc) the abso-lute magnitude of the cluster down to this limit isMV = −4.59± 0.20.

The missing flux of cluster stars fainter thanV = 21.75 was estimated using the deep lumi-nosity function (LF) of Grillmair & Smith (2001).We rescaled this LF to the SDSS star counts inthe range 19.75 ≤ V < 21.75 and then inte-grated the predicted flux from V = 21.75 down

to V = 27, where the LF is flat and the contri-bution to the total flux becomes negligible. Thisyields an additional contribution to the integratedmagnitude of the cluster of 0.18 mag. Our re-sult for the total absolute magnitude of Pal 5 isthus MV = −4.77± 0.20. This is marginally lowerthan the estimate ofMV = −5.0 obtained by SH77and corresponds to a total V band luminosity of(L/L⊙)V = 7.2 × 103. (using MV = 4.87 for theSun).

One way of deriving the mass of the clusterfrom its luminosity is by assuming that the mass-to-light ratio of Pal 5 is not substantially differentfrom those of other Milky Way globular clusters.According to the empirical mass-luminosity rela-tion by Mandushev, Spassova & Stanova (1991)which reads

log(M/M⊙) = −0.456MV + 1.64 (5)

the mean mass-to-light ratio of galactic globu-lars varies between M/LV = 1.1 for faint clusters(MV = −6) and M/LV = 1.8 for very bright clus-ters (MV = −10). The estimates of the massesand mass-to-light ratios of individual clusters de-viate from this relation by at most a factor oftwo. Since the relation is calibrated with clus-ters that have MV ≤ −5.6 its application to Pal 5involves an extrapolation towards fainter magni-tudes. Equation (5) then provides an estimate ofthe cluster’s mass of M/M⊙ = (6.5 ± 1.5) × 103,corresponding to M/LV = 0.90 ± 0.20. For tworeasons this estimate may not appear completelyconvincing by itself. First, the extrapolation toMV < −5.6 is uncertain. Second, it is not a pri-ori clear that Pal 5 fits into the above mean re-lation for other globular cluster because its main-sequence luminosity function is known to be flatterthan that of other clusters.

Therefore we derived the mass of the clusteralso in a more direct way using the luminos-ity function of the cluster and theoretical stellarmasses from a 14 Gyr isochrone by Bergbusch &Vandenberg (1992). Same as for the calculationof the total luminosity the luminosity functionof Grillmair & Smith (2001) was rescaled to theSDSS counts for the entire cluster in the range19.75 ≤ V < 21.75. We then multiplied in eachinterval of width 0.5 mag from V = 14.75 down

10

to V = 28.75 the number of stars with the stel-lar mass for the mean absolute magnitude of theinterval and computed the sum of these masses,i.e., the integral of the mass function. For thefaintest three bins from V = 27.25 to V = 28.75for which no star counts are available from obser-vation we assumed that the luminosity functionat V ≥ 27 is constant. This allowed us to extendthe integral of the mass function down to 0.17M⊙

and hence to be complete down to the transitionfrom stars to brown dwarfs. The mass contribu-tion from the faintest three bins is of the orderof 10%. The total mass of the cluster accordingto this second approach is (5.2 ± 0.7) × 103M⊙,which corresponds to M/LV = 0.73 ± 0.10. Thequoted uncertainty has been estimated by consid-ering the changes that occur when varying the ageof the isochrone, the radius of the field encirclingthe cluster, the transformation between SDSS andstandard magnitudes, and the distance modulus ofthe cluster. The result shows that the extrapola-tion of the mean mass-luminosity relation in equa-tion (5) down to the absolute magnitude of Pal 5 isprincipally correct, but that this may still slightlyoverestimate the mass of the cluster. We concludethat the mass of Pal 5 as revealed by its luminosityis in the range 4.5×103 ≤ MPal5/M⊙ ≤ 6.0×103,which corresponds to 0.63 ≤ M/LV ≤ 0.83 for themass-to-light ratio.

5.3. Implications for the velocity disper-

sion

By putting the empirical mass limits from theend of the previous section into equation (4) weobtain the following result: A King model thatfits the stellar surface density of Pal 5 and has theappropriate mass of 4.5 to 6.0 × 103M⊙ requiresan observable line-of-sight velocity dispersion be-tween σlos = 0.32 kms−1and σlos = 0.37 km s−1.Velocity dispersions at the level of 0.7 km s−1 orhigher are not in agreement with an appropriateKing model because one would need to assumea high mass-to-light ratio of M/LV ≥ 3, whichfrom the results of Section 5.2 is unrealistic. Forσlos = 0.91 km s−1 equation (4) leads to a pre-dicted equilibrium cluster mass of 3.6 × 104M⊙.This shows that if one denies the presence of bina-ries the resulting velocity dispersion is much toohigh to be consistent with the mass that is re-vealed by the cluster’s luminosity. However, when

taking the binaries into account as done in Section4, the estimates of the line-of-sight velocity disper-sion of the cluster decrease in such a way that aconsistent equilibrium model with normal M/L isthen viable and a velocity dispersion in excess ofthe equilibrium value is unlikely. More specifically,the line-of-sight velocity dispersion must lie in theupper third of the 68% confidence interval of prob-able values of σ in order to enable an equilibriummodel that is in agreement with the surface den-sity profile and the mass of the cluster. Dispersionvalues near the point of the highest likelihood (i.e.,at σ ≈ 0.22 km s−1) on the other hand are too lowto be consistent with an equilibrium model of Pal 5since this would require a very small mass-to-lightratio of M/LV ≈ 0.3, which again contradicts theresults of Section 5.2.

6. Summary and discussion

Our study has shown that the radial velocitiesof the giants in the cluster Pal 5 are tightly concen-trated around a mean velocity of−58.7 km s−1 andhave an overall dispersion of at most 1.1 km s−1.Only one out of a total of 18 observed giants ap-pears as a kinematic outlier. Statistical argumentssuggest that this outlier is also a member of thecluster. Its velocity offset of about 14 km s−1 withrespect to the other cluster members leads to theconclusion that it is most likely a binary with anorbital period of a few months. This makes it avery interesting case because the occurrence of abinary with a relatively short period close to thecenter of Pal 5 would fit with the general idea thatsuch binaries, if not primordial, are formed by stel-lar encounters in globular cluster cores during thedynamical evolution the cluster. (see, e.g., Mey-lan & Heggie 1997). Such binaries are believedto be an important energy source that supportsthe evaporation of the cluster. However, the as-sumption that the above object is such a rapidlyorbiting binary still needs to be confirmed.

A peculiar feature in our velocity data is thatthe few AGB stars in the sample show a signif-icantly higher velocity dispersion than the starson the RGB. In principle, such a difference couldresult from the lower mass of the AGB stars. Inpractice, however, the mass ratio of about 0.5:0.8between AGB and RGB stars is by far not smallenough to explain the observed kinematic differ-

11

ences. The possibility that binarity causes the en-hanced velocity dispersion of the AGB stars is alsounlikely because there is no reason to believe thatAGB stars should have a much higher frequency ofbinaries than RGB stars. We thus tend to believethat the enhanced velocity dispersion among theAGB stars is due to so-called atmospheric jitter,i.e., due to pulsations in the extended atmospheresof these evolved stars. This assumption is moti-vated by the fact that very luminous RGB andAGB stars in other globular clusters and in thefield often exhibit photometric and spectroscopicvariations due to pulsating atmospheres. A pos-sible caveat however is that the stars in our sam-ple do not belong to those most luminous typesof giants since they are more than 1 mag fainterthan the tip of the red giant branch. Moreover,the RGB stars in our sample that have the samebrightness as the AGB stars show a very small ve-locity dispersion that leaves no room for any jitter.It will therefore be necessary to check the hypoth-esis of atmospheric pulsation by collecting furtherobservations of these stars. As long as the natureof the enhanced velocity dispersion of the AGBstars is not clear, one must conclude that thesestars are not useful for investigating the dynamicsof the cluster.

Without the AGB stars (and the above outlierof course) the observations yield an overall line-of-sight velocity dispersion of 0.9 km s−1. This valuesets a strict upper limit on the dynamical line-of-sight velocity dispersion of the cluster. How-ever, by further analysis of the data we saw thatthis limit overestimates the dynamical velocity dis-persion substantially because orbital motions of(long-period) binaries have an important influencethat cannot be neglected. The fact that the distri-bution of the binary-induced velocities is differentfrom the Gaussian distribution for an isothermalcluster allowed us to distinguish the two velocitycomponents and to estimate the fraction of bina-ries xb and the dynamical line-of-sight velocity dis-persion σ. Using Monte Carlo simulations we con-structed synthetic velocity distributions for differ-ent types of binary populations, combined themwith the Gaussian model of the cluster kinemat-ics, and fitted these composite models to the ob-served velocity distribution. The results of thisprocedure suggest that the binaries in our samplehave low mass companions of about 0.2M⊙ and

orbits of very high eccentricity. Interestingly, a bi-nary population with the orbital characteristics oflocal field binaries is less in agreement with theobserved velocity distribution. This might be anindication for differences between cluster binariesand field binaries that arise from the special clus-ter environment and the dynamical evolution inthe cluster. On the other hand, the details of thedifferent binary models turned out to be ratherunimportant for the determination of the param-eters xb and σ since all models with a substantialfraction of secondary masses ≤ 0.4M⊙ led to sim-ilar estimates. We thus arrived at the conclusionthat with 68% confidence the frequency of binariesin our sample of giants is in the range from 0.24 to0.63, and that with the same confidence level thedynamical velocity dispersion of the cluster liesbetween 0.12 km s−1 and 0.41 km s−1. The lat-ter means that the dynamical line-of-sight veloc-ity dispersion is by at least a factor of two smallerthan the overall velocity dispersion and that thecontribution from binaries accounts for more than50% of the overall dispersion.

Are these results credible? The binary fre-quency of roughly 40% ± 20% is consistent withresults from systematic searches for spectroscopicbinaries in other galactic globular clusters. Fornormal clusters such studies have reported binaryfrequencies of about 5% to 15% per decade of pe-riod or estimates of overall binary frequencies ofabout 20% to 30% (see the reviews by Meylan &Heggie (1997) and by McMillan, Pryor & Phinney(1998)). Moreover, there is evidence that clus-ters with low density and/or strong tidal massloss can have two to three times higher binaryfrequencies than normal clusters (see, e.g., Pryor,Schommer & Olszewski (1991) and Yan & Cohen(1996)). If we take xb = 0.24 from the low endof our 68% confidence interval as a conservativeestimate, this corresponds to three binaries in thesample of 13 RGB stars. Assuming that at leastone of these binaries, but perhaps two or evenall three of them are responsible for the most de-viant velocities in the sample, we can successivelyomit those three stars that have the highest ab-solute velocities with respect to the cluster meanand calculate the velocity dispersion of the remain-ing subsample. Hereby we obtain velocity disper-sions of 0.64 kms−1, 0.51 km s−1, and 0.32 km s−1

(corrected for measurement errors), respectively.

12

This simple test illustrates that even modest as-sumptions on the frequency and influence of bi-naries result in a substantial reduction of the ve-locity dispersion and easily bring it down to thelevel of 0.5 km s−1. By successive omission of thethree most deviant velocities the velocity distri-bution also becomes progressively more consistentwith an isothermal distribution. The differencesbetween the empirical distribution and the corre-sponding Gaussian model then have Kolmogorov-Smirnov significance levels of 48%, 15% and 10%,respectively, while the significance of the differ-ences is 78% for the original sample.

Another interesting question is: Does the ob-served velocity dispersion admit an equilibriumcluster model that is consistent with other fun-damental parameters of Pal 5? To investigate thispoint we derived the surface density profile and thetotal luminosity of the cluster from recent photo-metric data and determined the King model thatbest fits the density profile. The parameters of thebest-fit King profile are W0 = 2.9, rt = 16.′1. Ac-cording to its absolute magnitude, which we foundto be MV = −4.77 ± 0.20, and by extrapolationwith a general globular cluster mass-luminosity re-lation Pal 5 has a mass in the range 5× 103M⊙ to8× 103M⊙, equivalent to a mass-to-light ratio be-tween 0.7 and 1.1. A direct determination of themass using the cluster’s luminosity function andtheoretical stellar masses along a 14 Gyr isochronegave a similar result, namely a total mass between4.5 × 103M⊙ and 6.0 × 103M⊙. We showed thatthe best-fitting King model comes into agreementwith this mass if the line-of-sight velocity disper-sion is between 0.32 and 0.39 km s−1. Therefore,a line-of-sight velocity dispersion that lies in theupper third of our 68% confidence interval for σdoes indeed admit a consistent equilibrium modelfor Pal 5.

By testing a variety of binary models we sawthat the range of probable values of σ does notcritically depend on the choice of the orbital pa-rameters of the binaries. In order to arrive atprobable velocity dispersions that significantly ex-ceed the equilibrium dispersion of the cluster onewould thus need to assume that the cluster hasonly a small fraction of binaries. This case can-not be strictly ruled out but it appears unlikelyand is not supported by the solutions obtained inSection 4. Based on the more plausible alterna-

tive assumption that binaries are not rare in Pal 5we draw the conclusion that although the clusterhas obviously experienced strong tidal perturba-tion and heavy mass loss, the remaining centralbody still presents a bound stellar system that isclose to a state of dynamical quasi-equilibrium.

This conclusion is in agreement with resultsthat we have recently obtained by simulating thedynamics of Pal 5 in the tidal field of the MilkyWay with an N-body code (Dehnen et al., inpreparation). These simulations suggest that onlyshortly after a tidal shock from a disk crossingthe velocity dispersion in the cluster rises abovethe equilibrium dispersion whereas during otherphases of the orbit the dispersion is settled downat the equilibrium level. Moreover, the numeri-cal models confirm that at the cluster’s presentlocation near the apogalacticon of the orbit theline-of-sight velocity distribution should be nearlyGaussian. The observed departure from a Gaus-sian distribution must therefore indeed be due tobinaries as we assumed above.

The resulting very low dynamical velocity dis-persion inside the cluster suggests that the veloci-ties of the extratidal stars may locally also be verycoherent. The narrow spatial confinement of thetidal tails, which is one of the reasons that led totheir detection, is probably to some extent due tothis kinematical coherence. We are currently con-ducting a similar kinematic study using the sameinstrumental equipment to investigate the veloc-ities in the tidal tails of the cluster. Hereby weaim at measuring the velocity dispersion amongthe extratidal stars and the radial velocity gradi-ent along the tails and hence along the orbit of thecluster.

Acknowledgements. We thank ESO and theVLT staff for providing the observations in servicemode.

The Sloan Digital Sky Survey (SDSS) is a jointproject of The University of Chicago, Fermilab,the Institute for Advanced Study, the Japan Par-ticipation Group, The Johns Hopkins University,the Max-Planck-Institute for Astronomy (MPIA),the Max-Planck-Institute for Astrophysics (MPA),New Mexico State University, Princeton Univer-sity, the United States Naval Observatory, and theUniversity of Washington. Apache Point Observa-tory, site of the SDSS telescopes, is operated by

13

the Astrophysical Research Consortium (ARC).Funding for the project has been provided by theAlfred P. Sloan Foundation, the SDSS memberinstitutions, the National Aeronautics and SpaceAdministration, the National Science Foundation,the U.S. Department of Energy, the JapaneseMonbukagakusho, and the Max Planck Society.The SDSS Web site is http://www.sdss.org/.

The Digitized Sky Surveys were produced at theSpace Telescope Science Institute under U.S. Gov-ernment grant NAG W-2166. The images of thesesurveys are based on photographic data obtainedusing the Oschin Schmidt Telescope on PalomarMountain and the UK Schmidt Telescope. TheSecond Palomar Observatory Sky Survey (POSS-II) was made by the California Institute of Tech-nology with funds from the National Science Foun-dation, the National Geographic Society, the SloanFoundation, the Samuel Oschin Foundation, andthe Eastman Kodak Corporation. The OschinSchmidt Telescope is operated by the CaliforniaInstitute of Technology and Palomar Observatory.

REFERENCES

Armandroff, T.E., Olszewski, E.W., Pryor, C.,1995, AJ, 110, 2131

Bergbusch, P.A., Vandenberg, D.A., 1992, ApJS,81, 163

Binney, J., Tremaine, S., 1987, Galactic Dynam-ics. Princeton University Press, Princeton, NewJersey

Chen, B., Stoughton, C., Smith, J.A., Uomoto, A.,Pier, J.R, et al., 2001, ApJ, 553, 184

Cote, P., Pryor, C., McClure, R., Fletcher, J.M.,Hesser, J.E., 1996, AJ, 112, 574

Dinescu, D., Girard, T.M., van Altena, W.F.,Yang T.-G., Lee, Y.W., 1996, AJ, 111, 1205

Duquennoy, A., Mayor, M., 1991, A&A, 248, 485

Fukugita, M., Ichikawa, T., Gunn, J.E., Doi, M.,Shimasaku, K., Schneider, D.P. 1996, AJ, 111,1748

Grillmair, C.J., Smith, G.H., 2001, AJ, 122, 3231

Gunn, J.E., et al. 1998, AJ, 116, 3040

Hargreaves, J.C., Gilmore, G., Irwin, M.J.,Carter, D., 1996, MNRAS, 282, 305

Harris, W.E., 1996, AJ, 112, 1487

Heintz, W., 1969, JRASC, 63, 275

Hogg, D.W., Finkbeiner, D.P., Schlegel, D.J.,Gunn, J.E. 2001, AJ, 122, 2129

Smith, J.A., et al. 2002, AJ, in press

Pier, J.R., et al. 2002, AJ, submitted

King, I.R., 1962, AJ, 67, 471

King, I.R., 1966, AJ, 71, 64

Latham, D.W., Stefanik, R.P., Mazeh, T., Tor-res, G., Carney, B.W., 1998, in R. Rebolo et al.(eds.), Brown Dwarfs and Extrasolar Planets,ASP Conf. Ser., Vol. 134, p.178

Mandushev, G., Spassova, N., Staneva, A., 1991,A&A, 252, 94

Meylan, G., Heggie D., A&A Rev., 8, 1

Mayor, M., Duquennoy, A., Halbwachs, J.-L.,Mermilliod, J.-C., 1992, in: H.A. McAlis-ter, W.I. Hartkopf (eds.), Complementary Ap-proaches to Double and Multiple Star Research,ASP Conf. Ser., Vol. 32, p.73

McMillan, S., Hut, P., 1994, ApJ, 427, 793

McMillan, S.L.W., Pryor, C., Phinney E.S, 1998,in: J. Andersen (ed.), Highlights in Astronomy,Vol. 11B, p.616

Odenkirchen, M., Grebel E.K., Rockosi C.M., etal., 2001, ApJ, 548, L165

Peterson, R.C., 1985, ApJ, 297, 309

Piotto, G., Cool, A.M., King, I.R., 1997, AJ, 113,1345

Pryor, C., Fletcher, J.M., Hesser, J.E. et al., 1996,IAU Symp. 174, p. 193

Pryor, C., Schommer, R.A., Olszewski, E.W.,1991, in: K. Janes (ed.), The Formation andEvolution of Star Clusters, ASP Conf. Ser., Vol.13, p.439

Sandage, A., Hartwick, F.D.A., 1977, AJ, 82, 459

14

Smith, G.H., 1985, ApJ, 298, 249

Smith, G.H., McClure, R.D., Stetson, P.B.,Hesser, J.E., Bell R.A., 1986, AJ, 91, 842

Stoughton, C., et al., 2002, AJ, 123, 485

Trager, S.C., King, I., Djorgovski, S., 1995, AJ,109, 218

Udry, S., Mayor, M., Maurice, E., Andersen, J.,Imbert, M., Lindgren, H., Mermilliod, J.-C.,Nordstroem, B., Prevot, L., 1999, ASP Conf.Ser., Vol. 185, p.383

Yan, L., Cohen, J.G., 1996, AJ, 112, 1489

York, D.G., et al., 2000, AJ, 120, 1579

This 2-column preprint was prepared with the AAS LATEXmacros v5.0.

15

Table 1

Target identification and observing log

Star SH77 RA (2000) DEC (2000) r Date of Exp. TimeID [h m s] [◦ ′ ′′] [′] Observation [ s ]

1 F 15 15 56.1 −00 06 06 2.4 2001 May 04 4202 G 15 16 08.7 −00 08 03 1.3 2001 May 04 4603 – 15 16 07.1 −00 10 18 3.1 2001 May 04 5104 I 15 16 06.8 −00 10 04 2.8 2001 May 04 6605 H 15 15 52.6 −00 07 40 3.0 2001 May 04 6606 K 15 16 06.5 −00 07 01 0.6 2001 May 10 7307 L 15 16 02.0 −00 08 03 1.0 2001 May 10 8108 – 15 16 09.6 −00 02 40 4.8 2001 May 10 8109 J 15 15 49.7 −00 07 01 3.7 2001 May 10 88010 N 15 15 59.5 −00 08 59 2.1 2001 May 10 90011 U 15 15 54.8 −00 06 55 2.5 2001 May 10 160012 – 15 16 26.5 −00 09 05 5.8 2001 May 10 180013 6 15 15 58.3 −00 09 46 3.0 2001 May 03 180014 19 15 16 08.5 −00 05 10 2.3 2001 May 04 210015 27 15 16 00.3 −00 06 00 1.6 2001 May 04 240016 AB 15 15 48.2 −00 06 07 4.2 2001 May 11 360017 35 15 16 02.6 −00 05 22 2.0 2001 Jun 23 240018 26 15 16 04.8 −00 06 28 0.8 2001 Jun 18 270019 – 15 16 20.7 −00 07 33 4.0 2001 Jun 18 300020 5 15 15 57.1 −00 08 50 2.4 2001 Jun 19 3300

HD107328 12 20 21.0 +03 18 45 – 2001 May 04 1HD157457 17 26 00.0 −50 38 01 – 2001 May 04 10

Note:–SH77 = Sandage & Hartwick (1977)r = angular distance from the center of Pal 5

16

Table 2

Results from spectroscopy

Star Stellar MJD of heliocentric radial velocity in km s−1

type observation vr ǫ vr ǫ vr vr(a) (b) S85 P85

1 RGB 52033.168 −58.51 0.05 −58.66 0.05 −54 –2 RGB 52033.178 −58.31 0.05 −58.47 0.05 −61 –3 AGB 52033.187 −61.16 0.06 −61.30 0.05 – –4 MS 52033.197 −23.44 0.06 −23.62 0.06 −23 –5 AGB 52033.210 −56.92 0.07 −57.06 0.05 −53 −466 RGB 52039.069 −58.72 0.09 −58.87 0.11 −61 –7 RGB 52039.082 −58.79 0.10 −58.93 0.12 −59 –8 AGB 52039.096 −58.98 0.05 −59.14 0.05 – –9 AGB 52039.110 −57.35 0.15 −57.49 0.15 −57 −5510 RGB 52039.124 −60.10 0.14 −60.24 0.14 −52 –11 RGB 52039.138 −58.90 0.09 −59.06 0.07 – −5812 MS 52039.162 −52.97 0.12 −53.13 0.12 – –13 RGB 52032.268 −58.94 0.08 −59.10 0.06 – –14 RGB 52033.222 −61.07 0.17 −61.19 0.19 – –15 RGB 52033.251 −44.67 0.12 −44.73 0.09 – –16 RGB 52040.248 −58.33 0.15 −58.50 0.16 – –17 RGB 52083.033 −58.54 0.06 −58.67 0.04 – –18 RGB 52078.169 −58.42 0.25 −58.55 0.22 – –19 RGB 52078.130 −59.48 0.22 −59.60 0.20 – –20 RGB 52079.064 −57.27 0.06 −57.40 0.05 – –

HD 107328 52033.163 – – +36.26 0.03HD 157457 52032.438 +17.94 0.03 – –

Notes:– (a) by cross-correlation with HD107328(b) by cross-correlation with HD157457

RGB = red giant branch, AGB = asymtotic giant branch, MS = main sequence starǫ = random error of the radial velocity measurement

S85 = Smith (1985), P85 = Peterson (1985)

17

Table 3

Membership probabilities from proper motion

Star SH77 CSM probability(1)

ID ID in %

1 F 782 G 733 – 174 884 I 05 H 926 K 887 L 868 – 32 729 J 8010 N 9811 U 9512 – 593 013 6 9814 19 9915 27 7316 AB 8017 35 8918 26 9719 – 142 9920 5 97

Notes:–SH77 = Sandage & Hartwick (1977)CSM = Cudworth, Schweitzer & Majewski (in preparation)

(1) according to the proper motion study by CSM

18

Table 4

Maximum likelihood solutions for a Gaussian model plus binary component

Model Binary parameters Sample max(lnL) xb σ σ90%

and distributions km s−1 kms−1

– single stars only R+A −26.38 0.0 1.14+0.24−0.19 1.53

(a) standard binary model R+A −25.94 0.45+0.18−0.26 0.24+0.57

−0.10 1.07

(b) e = 0 (*) R+A −26.08 0.44+0.17−0.27 0.24+0.59

−0.10 1.08

(c) f(e) = 2e (*) R+A −25.57 0.50+0.21−0.25 0.24+0.51

−0.11 1.04

(d) M2 = M1 (*) R+A −26.38 0.00+0.24−0.00 1.06+0.38

−0.41 1.39

(e) M2 = 0.4M⊙ (*) R+A −25.98 0.43+0.17−0.26 0.24+0.56

−0.10 1.07

(f) M2 = 0.2M⊙ (*) R+A −25.16 0.53+0.21−0.23 0.24+0.45

−0.12 1.02

(g) M2 = 0.2M⊙, e = 0 (*) R+A −25.32 0.52+0.21−0.23 0.24+0.47

−0.11 1.03

(h) M2 = 0.2M⊙, f(e) = 2e (*) R+A −25.02 0.60+0.22−0.23 0.23+0.46

−0.13 1.01

– single stars only R −17.35 0.0 0.91+0.23−0.18 1.30

(a) standard binary model R −15.30 0.32+0.18−0.16 0.23+0.20

−0.09 0.74

(b) e = 0 (*) R −15.38 0.31+0.17−0.16 0.23+0.21

−0.09 0.75

(c) f(e) = 2e (*) R −15.10 0.35+0.19−0.17 0.22+0.20

−0.09 0.72

(d) M2 = M1 (*) R −16.54 0.23+0.14−0.15 0.24+0.35

−0.09 0.89

(e) M2 = 0.4M⊙ (*) R −15.37 0.30+0.17−0.15 0.23+0.20

−0.09 0.75

(f) M2 = 0.2M⊙ (*) R −14.78 0.37+0.20−0.17 0.22+0.19

−0.09 0.69

(g) M2 = 0.2M⊙, e = 0 (*) R −14.87 0.36+0.19−0.17 0.22+0.19

−0.09 0.70

(h) M2 = 0.2M⊙, f(e) = 2e (*) R −14.67 0.42+0.21−0.18 0.22+0.19

−0.10 0.69

Notes:– (*) = other parameters same as in model (a)R+A = RGB and AGB stars (n = 17), R = RGB stars only (n = 13)xb = best-fit binary fraction, σ = Gaussian dispersion parameter

σ90% = upper limit of σ for 90% confidence

19

Fig. 1.— Color-magnitude diagram of stars from the Sloan Digital Sky Survey in the field of Pal 5. Fat dotsshow stars with angular distance r ≤ 3.′6 from the cluster center, small dots show stars with 3.′6 < r ≤ 6.′0.The stars for which spectra were taken are marked by circles.

20

Fig. 2.— Digitized Sky Survey image of Pal 5 (DSS2) showing the positions of our spectroscopic targets (size12′ × 10′, north up, east to left). The region of the cluster core is marked by the dashed circle (r = 3.′6, coreradius of the best-fitting King model, see Sect. 5.1).

21

Fig. 3.— Examples of the UVES spectra for program stars of different magnitude and for one of the standardstars. The plot shows a 100 A wide section from the blue part of the spectrum, i.e., about 5% of the fullwavelength range covered by the spectra. The left side of the plot contains the Mg I b triplet feature. Notethe broad line profiles of stars 4 and 12, which indicate that they are dwarfs.

22

Fig. 4.— Distribution of the heliocentric radial velocities of the 20 program stars. (a) Histogram showingthe number of stars in 1 km s−1 wide velocity bins. The shaded columns represent stars that have dwarfspectra and hence do not belong to Pal 5. (b) Cumulative distribution, i.e., number of stars with velocities≤ vr, focussing on the range from −62 kms−1 to −56 kms−1.

23

Fig. 5.— Radial velocities of the program stars plotted versus brightness in i∗ (a), angular distance r fromthe cluster center (b), and versus position angle ϕ (c). Filled circles show normal red giants while opencircles show asymptotic giant branch stars. Open triangles mark stars with dwarf spectra. The dashed lineindicates the mean cluster velocity of −58.7 km s−1. The measurement errors are in all cases smaller thanthe sizes of the plot symbols.

24

Fig. 6.— Cumulative distribution of the observed velocities (solid lines) compared with best-fit (error-convolved) Gaussian models (dashed lines). a) Full sample of 17 certain cluster members and best-fit modelwith σ = 1.14 km s−1. In addition, a Gaussian with σ = 0.50 km s−1 is shown (dotted line), which providesa better match to the inner part of the observed distribution but fails in the outer part. b) Subsample of 13RGB stars and best-fit model with σ = 0.91 km s−1.

25

Fig. 7.— Radial velocities induced by the orbital motion of a random-generated population of binaries withparameters as described in the text and in the legend. Here vr is the radial velocity of the primary componentalong isotropically distributed lines of sight at random orbital phase. The so-called standard binary modelassumes a log-normal mass function for the secondary and a constant distribution of ellipticities as observedin local field binaries (for further details see Section 4).

26

Fig. 8.— Likelihood distribution in the plane of the parameters xb (binary frequency) and σ (single starvelocity dispersion) and marginal distributions thereof, calculated with the RGB star sample (n = 13) andthe binary model with M2 = 0.2 M⊙ and f(e) = 2e (see case (h) in the lower part of Tab. 4). The lines inthe xb − σ plane show contours of equal likelihood L that encircle regions with 25%, 50%, 75% and 90% ofthe total integrated likelihood. The small central ellipse marks the location of the maximum of L. Hatchedareas mark the ranges of the probable values of σ and xb, i.e., the intervals around the maxima of σ and xb

that contain 68.3% of the integrated likelihood.

27

Fig. 9.— Cumulative distribution of the observed velocities (step functions, same as in Fig. 6) versusmaximum likelihood predictions for an isothermal system with an adjustable fraction of binaries. For theparameters of the binary populations and the binary frequencies see Tab. 4. (a) Fit to all 17 cluster giants.(b) Fit to the subsample of 13 RGB stars.

28

Fig. 10.— Radial profile of the stellar surface density of Pal 5 derived from SDSS. The datapoints witherrorbars are the results of star counts in circular annuli around the cluster center in two complementarydouble sectors. Filled circles show the surface densities in the eastern and western sector (45◦ ≤ PA ≤ 135◦

and 225◦ ≤ PA ≤ 315◦) while open circles are for the complementary northern and southern sector. Thelatter show an excess at r > 6′ that is due to extratidal stars. The solid line presents the best-fit King(1966) model (plus constant background) for the data shown by filled circles (parameters for that model areW0 = 2.9, rt = 16.1′).

29