Certain types of globular clusters have the very important property that the
predictions for their kinematics in the Newtonian and modified Newtonian
dynamics (MOND) contexts are divergent. Here, we caution the recent claim that
the stellar kinematics data (using 17 stars) of the globular cluster Palomar 14
are inconsistent with MOND. We compare the observations to the theoretical
predictions using a Kolmogorov-Smirnov test, which is appropriate for small
samples. We find that, with the currently available data, the MOND prediction
for the velocity distribution can only be excluded with a very low confidence
level, clearly insufficient to claim that MOND is falsified.Comment: Research note accepted for publication in A&

Angus, Garry W.

Famaey, Benoit

Gentile, Gianfranco

Kroupa, Pavel

English

arXiv

Gentile, G.

Angus, G.

Kroupa, P.

HAL-INSU

Current data on the globular cluster Palomar 14 are consistent with MOND

G. Gentile

B. Famaey

G. Angus

P. Kroupa

Crossref

Context. Certain types of globular clusters have the very important property that the predictions for their kinematics in the Newtonian and modified Newtonian dynamics (MOND) contexts are divergent.
Aims. Here, we advice regarding the recent claim that the stellar kinematics data (using 17 stars) of the globular cluster Palomar 14 are inconsistent with MOND.
Methods. We compare the observations to the theoretical predictions using a Kolmogorov-Smirnov test, which is appropriate for small samples.
Results. We find that with the currently available data the MOND prediction for the velocity distribution can only be excluded with a very low confidence level, clearly insufficient to claim that MOND is falsified

Famaey, B

Angus, G

Kroupa, P

Ghent University Academic Bibliography

A&A 509, A97 (2010)DOI: 10.1051/0004-6361/200913296c© ESO 2010Astronomy&AstrophysicsCurrent data on the globular cluster Palomar 14are consistent with MOND(Research Note)G. Gentile1 ,2, B. Famaey3,4, G. Angus5 ,6, and P. Kroupa41 Institut d’Astronomie et d’Astrophysique, Université Libre de Bruxelles, CP 226, Bvd du Triomphe, 1050, Bruxelles, Belgium2 Sterrenkundig Observatorium, Universiteit Gent, Krijgslaan 281, 9000 Gent, Belgiume-mail: gianfranco.gentile@ugent.be3 Observatoire Astronomique, Université de Strasbourg, CNRS UMR 7550, 67000 Strasbourg, Francee-mail: benoit.famaey@astro.unistra.fr4 AIfA, Universität Bonn, 53121 Bonn, Germanye-mail: pavel@astro.uni-bonn.de5 Dipartimento di Fisica Generale “Amedeo Avogadro”, Università degli studi di Torino, via P. Giuria 1, 10125, Torino, Italye-mail: angus@ph.unito.it6 Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Torino, Torino, ItalyReceived 15 September 2009 / Accepted 15 November 2009ABSTRACTContext. Certain types of globular clusters have the very important property that the predictions for their kinematics in the Newtonianand modified Newtonian dynamics (MOND) contexts are divergent.Aims. Here, we advice regarding the recent claim that the stellar kinematics data (using 17 stars) of the globular cluster Palomar 14are inconsistent with MOND.Methods. We compare the observations to the theoretical predictions using a Kolmogorov-Smirnov test, which is appropriate forsmall samples.Results. We find that with the currently available data the MOND prediction for the velocity distribution can only be excluded with avery low confidence level, clearly insuﬃcient to claim that MOND is falsified.Key words. globular clusters: individual: Palomar 14 – gravitation – methods: statistical1. IntroductionA plethora of observational data on various astronomical sca-lesseem to support the idea that the amount of visible matterin the Universe is several times smaller than the total amountof matter (e.g. Hinshaw et al. 2009). The current paradigm ofstructure formation and evolution in the Universe is known asthe Λ Cold Dark Matter (ΛCDM) model. However, as long asthe dark matter particle has not been discovered (and its cosmo-logical abundance confirmed), it is worth considering alternativetheories to explain the current data.For instance, MOND (modified Newtonian dynamics) wasproposed by Milgrom (1983) as an alternative to galactic darkmatter. MOND stipulates that below a certain gravitational ac-celeration a0 ∼ 1.2 × 10−8 cm s−2 the actual gravitational ac-celeration g is stronger than expected in Newtonian gravity(gN). Asymptotically, in MOND it reaches the value √gNa0 forgN  a0. This allows it to naturally explain various galaxy scal-ing relations (e.g., Faber & Jackson 1976; Tully & Fisher 1977;McGaugh et al. 2000; McGaugh 2004; Gentile 2008; Donatoet al. 2009; Gentile et al. 2009).The eﬀects of MOND and dark matter are however oftenrather degenerate and model-dependent since the gravitationalpotential predicted by MOND can almost always be attributedby a Newtonist to an ad hoc dark matter distribution. Objectsfor which the predictions of the two theories are unambiguouslydiﬀerent are unfortunately rare. We note that galaxy clusters arenot good discriminant tests: indeed, in galaxy clusters, the accel-eration predicted by MOND is not large enough (e.g., Sanders1999), but some form of hot dark matter can be added within theMOND context to make the data consistent with the predictions(e.g., Angus et al. 2009). If there were systems where MONDpredicted more gravity than observed, they would make a strongcase against MOND.One example of such objects that should be (almost) devoidof cold dark matter in the ΛCDM cosmological model, and forwhich MOND predicts a significantly stronger gravity than thatattributable to visible matter, are tidal dwarf galaxies (TDGs).However, from the observations of three young TDGs aroundNGC 5291, Gentile et al. (2007) found that MOND fits the dataremarkably well with zero free parameters, whereas CDM failsto explain them.Apart from TDGs, another type of object that has recentlybeen put forward as a distinguishing test for CDM and MONDare the globular clusters of our own Galaxy (Baumgardt et al.2005). Of particular interest are those which are diﬀuse and dis-tant from the Milky Way, to ensure that the internal accelerationprobes the deep MOND regime (which is not always necessar-ily the case: see NGC 2419 in Baumgardt et al. 2009) and that,simultaneously, the gravitational acceleration due to the MilkyWay (external field) is weak enough. In MOND the deviationfrom a Newtonian behaviour should start appearing around a0,Article published by EDP Sciences Page 1 of 4A&A 509, A97 (2010)whereas in the “Newtonian gravity plus CDM” picture no dis-crepancy is expected since globular clusters should contain (al-most) no cold dark matter.In a recent paper, Jordi et al. (2009) analysed the data of17 stars from the globular cluster Palomar 14 and claimed that(within the assumption of Palomar 14 being on a circular or-bit) MOND is inconsistent with the observed velocity disper-sion: the MOND prediction is too high, whereas the data areconsistent with the Newtonian prediction (with no dark matter).They discuss two separate cases, depending on the inclusion orexclusion of a star (Star 15), which based on its line-of-sight ve-locity is not a definite member of the cluster. However, givensuch a small sample size, an appropriate test should be used todiscern between the hypotheses. An example of such a test isthe Kolmogorov-Smirnov (KS) test (e.g. Soong 2004). An illus-trative example of the uncertain value of the velocity dispersionmeasured with a small number of stars is NGC 2419: the orig-inal analysis by Olszewski et al. (1993) yielded a velocity dis-persion for 12 stars of 2.7 ± 0.8 km s−1, which was re-evaluatedby Baumgardt et al. (2009) with 40 stars to be 4.1 ± 0.5 km s−1,i.e. about two sigma greater than the original. This gives a per-fectly reasonable M/L in MOND, although the velocity dis-persion profile, if confirmed, could be a problem for MOND(Sollima & Nipoti 2009). This will allow for a very interestingtest of MOND when data points at larger radii will be obtainedin NGC 2419. In any case, this is an excellent example of anunderestimation of the true velocity dispersion due to a samplesize originally too limited.Therefore, the question we ask in the present research noteis the following: do the Palomar 14 data really exclude MOND,given the small number of stars used in the analysis of Jordiet al. (2009)? We then also investigate the minimum number ofstars needed to exclude MOND in a globular cluster similar toPalomar 14. Or, equivalently, we ask how many globular clusterslike Palomar 14 would be needed to exclude MOND.2. MethodGiven the small sample size, the formal error on the veloc-ity dispersion is not suﬃcient to distinguish between variousmodels, so we use the KS-test to redo the analysis of Jordiet al. (2009). The KS-test compares two cumulative distributionfunctions (cdfs), then the maximum diﬀerence D between thesetwo cdfs yields a P-value.Here we compare the cdf of the data (separately for the sam-ples with and without Star 15) to the cdf of a Gaussian witha dispersion equal to 1.27 km s−1 (the MOND prediction withM/L = 2, Baumgardt et al. 2005; Jordi et al. 2009). We note thatthere is no evidence, apart from the dynamics itself, about thecentre of the Gaussian, therefore we chose the centres that min-imise D (in other words, the centres that maximise the P-value).The MOND prediction for the velocity dispersion was de-rived by Baumgardt et al. (2005). First, they calculated the in-ternal and external field gravitational accelerations (aint and aext,respectively) of a number of globular clusters, including Pal 14.Then, based on results obtained by Milgrom (1986), they foundthat the MOND prediction is simply given by the Newtonian onemultiplied by√a0/aext, because Pal 14 is in the deep MONDregime and aext is larger than aint. Baumgardt et al. (2005) as-sume M/L = 2 from the observed M/L ratios of globular clus-ters: Mandushev et al. (1991) find M/L = 1.21 from 32 clusters,whereas Pryor & Meylan (1993) find M/L = 2.3 from 56 clus-ters. Baumgardt et al. (2005) state that the latter value is moreplausible because the modelling used in Pryor & Meylan (1993)Table 1. Velocities and cumulative distribution functions (observed andGaussian, with and without Star 15).Name velocity obs cdf Gauss cdf obs cdf Gauss cdf(km s−1) with Star 15 with Star 15 w/o Star 15 w/o Star 1515 69.99 0.059 0.034 - -8 71.38 0.118 0.234 0.063 0.2093 71.75 0.177 0.332 0.125 0.30214 71.80 0.235 0.347 0.188 0.31612 71.83 0.294 0.356 0.250 0.324HH042 71.94 0.353 0.388 0.313 0.35616 72.14 0.412 0.450 0.375 0.4165 72.21 0.471 0.472 0.438 0.43713 72.33 0.529 0.509 0.500 0.47517 72.39 0.588 0.528 0.563 0.4942 72.47 0.647 0.553 0.625 0.5191 72.53 0.706 0.572 0.688 0.5387 72.64 0.765 0.606 0.750 0.5726 72.65 0.824 0.609 0.813 0.575HV004 73.23 0.882 0.768 0.875 0.7409 73.50 0.941 0.828 0.938 0.805HV055 73.62 1.000 0.851 1.000 0.830The star names are taken from Hilker (2006) when present, otherwisefrom Harris & van den Bergh (1984) and Holland & Harris (1992).Fig. 1. Cumulative distribution function (cdf) vs. radial velocity forthe sample with Star 15. The solid step-like line represents the data,whereas the dotted line is the cdf of a Gaussian with a standard deviationof 1.27 km s−1 (the MOND prediction) and the centre at 72.30 km s−1.The red segment shows D, the maximum diﬀerence between the twocdfs.takes mass segregation into account. To get an estimate of theuncertainty on the M/L, the data compilation by Dabringhausenet al. (2008) suggest that M/L ratios of globular clusters have aspread of roughly 0.5 dex.3. ResultsIn Table 1 we list the observed and predicted cdfs (Gaussianswith a standard deviation of 1.27 km s−1) separately for the sam-ple with and without Star 15. The two predicted cdfs are dif-ferent because they have diﬀerent centres (72.30 km s−1 for thesample with Star 15 and 72.41 km s−1 for the sample withoutStar 15). We also show them in Figs. 1 and 2. In both cases,there are two regions where the diﬀerence almost reaches themaximum: around 71.75 km s−1 and around 72.65 km s−1, re-spectively. And in both samples, the diﬀerence between the twoobserved and predicted cdf are very similar at these two veloc-ities. With Star 15, the maximum diﬀerence is 0.215, and with-out Star 15 the maximum diﬀerence is 0.239. These values ofD correspond to P-values of 0.360 and 0.273, respectively. Thismeans that using the KS-test, the data presented in Jordi et al.(2009) can exclude the MOND with M/L = 2 hypothesis onlywith 64% and 73% confidence, depending on the inclusion ofPage 2 of 4G. Gentile et al.: Current data on the globular cluster Palomar 14 are consistent with MOND (RN)Fig. 2. Cumulative distribution function (cdf) vs. radial velocity for thesample without Star 15. The lines are the same as in Fig. 1, but theGaussian’s centre is at 72.41 km s−1.Fig. 3. Mean P-value vs. number of stars in the sample for a number ofMonte-Carlo realisations (10 per sample size) of distributions of starsfollowing a Gaussian with a standard deviation of 0.85 km s−1 (equiva-lent to the sample with Star 15).Star 15. These confidence levels are clearly not suﬃcient to ex-clude MOND. We note that if the M/L is not 2, but rather 1.1(a possibility considered by Jordi et al. 2009), then MOND isperfectly consistent with a velocity dispersion of 0.85 km s−1.Now we can ask ourselves: if one wanted a P-value of 0.1(i.e. an exclusion confidence of 90%), how many stars wouldone need (with the hypothesis that they follow a Gaussian witha standard deviation equal to the measured one)? We measure adispersion of 0.85 km s−1 with Star 15 and 0.63 km s−1 withoutStar 15. Hence, we create a series of Monte-Carlo realisations,using the software package R (Ihaka & Gentleman 1996), ofan increasing sample size. We create 10 realisations per samplesize. Then, we compare these mock data sets to the MOND pre-diction using the same method as above, take the mean P-valuefor each sample size, and look for the minimum number of starsnecessary to obtain P ≤ 0.1. Our results (shown in Figs. 3 and 4)are that for the mock data sets equivalent to the sample withStar 15 (i.e., the mock data sets that follow a Gaussian with astandard deviation = 0.85 km s−1) a minimum of about 80 stars(or alternatively 80/17 ∼ 5 clusters) would have been needed toexclude the MOND hypothesis with a confidence level of 90%,whereas without Star 15 this minimum number of stars decreasesto about 30.4. DiscussionIn this research note, we showed that current observational dataon Pal 14 are not significantly discrepant with the theoreticalFig. 4. Same as in Fig. 3, but for a Gaussian with a standard deviationof 0.63 km s−1 (equivalent to the sample without Star 15).prediction of MOND. Let us however also note that this the-oretical prediction is not really unique and depends on manyfactors, such as stellar M/L, rotation and anisotropy. For in-stance, in Angus (2008, thesis), the velocity dispersion of Pal 14was discussed before any velocity dispersion was published: theline of sight velocity dispersion for various M/Ls and veloc-ity anisotropies were computed (radial, isotropic and centrallyisotropic with increasing tangentially biased orbits). It was foundthat lower velocity dispersions (consistent with 0.85 km s−1)were attainable with very radial orbits, and in that case the lineof sight velocity dispersion profile would moreover not be flatin the outskirts. Rotation could also play a role although theround appearance of Pal 14 justifies the no-rotation assumption.Of course, the most important uncertainty comes from the M/Lratio itself.Then, another possible oversight may be the mass segrega-tion of stars within the globular cluster: if significant energy par-titioning has occured due to the short MOND relaxation time(Ciotti & Binney 2004), this would mean that the low-mass, un-observed stars could have a larger spread in configuration spaceand a higher velocity dispersion. Also, as already noted in Jordiet al. (2009), the MOND theoretical prediction was based on as-suming a purely circular orbit for Pal 14 around the Milky Way,while if it is on an eccentric orbit, it could (i) have lost many lowmass stars at perigalacticon leading to a decrease of the theoreti-cal stellar M/L, and (ii) be “frozen" in the Newtonian regime dueto the period of recovery while transiting from the large externalfield endured at perigalacticon. Finally, we note that the fact thatMOND cannot be excluded also implies, in the Newtonian plusdark matter cosmological framework, that the current data of Pal14 cannot exclude a certain amount of unseen mass in the cluster.With the assumption of a stellar M/L of 2, a velocity dispersionof 1.27 km s−1 would imply a total-to-luminous mass ratio of 2to 4.5. ConclusionEven assuming an isotropic MOND model with no rotation fora globular cluster on a circular orbit with M/L = 2, we showedthat, based on a KS test, which is the relevant statistical test forsmall samples, the currently available data are insuﬃcient to dis-tinguish between Newtonian gravity and MOND in Palomar 14,contrary to the claim of Jordi et al. (2009). While the objectsproposed by Baumgardt et al. (2005) provide one of the bestdistinguishing tests between MOND and cold dark matter plusNewtonian dynamics, more observations would be needed to ex-clude the aforementioned MOND model if the observed veloc-ity dispersion is representative of the true one: about 80 stars inPage 3 of 4A&A 509, A97 (2010)Pal 14, or about five similarly problematic globular clusters forthis model. In this respect, even though it might still not be con-clusive, velocity data of 21 stars in Pal 3 and 24 stars in Pal 4will be of prime interest for testing fundamental physics in ourGalactic backyard.Acknowledgements. G.G. is a postdoctoral researcher of the FWO-Vlaanderen(Belgium). B.F. acknowledges support from the Hulmboldt foundation and theCNRS. G.W.A. is supported by the Univsersity of Torino and Regione Piemonte,and by the INFN grant PD51.ReferencesAngus, G., Famaey, B., & Diaferio, A. 2009, MNRAS, in press[arXiv:0906.3322]Baumgardt, H., Grebel, E. K., & Kroupa, P. 2005, MNRAS, 359, L1Baumgardt, H., Côté, P., & Hilker, M. 2009, MNRAS, 396, 2051Ciotti, L., & Binney, J. 2004, MNRAS, 351, 285Dabringhausen, J., Hilker, M., & Kroupa, P. 2008, MNRAS, 386, 864Donato, F., Gentile, G., & Salucci, P. 2009, MNRAS, 397, 1169Faber, S., & Jackson, R. 1976, ApJ, 204, 668Gentile, G. 2008, ApJ, 684, 1018Gentile, G., Famaey, B., & Combes, F. 2007, A&A, 472, L25Gentile, G., Famaey, B., Zhao, H. S., & Salucci, P. 2009, Nature, 461, 627Harris, W. E., & van den Bergh, S. 1984, AJ, 89, 1816Hilker, M. 2006, A&A, 448, 171Hinshaw, G., Weiland, J. L., & Hill, R. S. 2009, ApJS, 180, 225Holland, S., & Harris, W. E. 1992, AJ, 103, 131Ihaka, R., & Gentleman, R. 1996, Journal of Computational and GraphicalStatistics, 5(3), 299, URL: http://www.r-project.org/Jordi, K., Grebel, E. K., & Hilker, M. 2009, AJ, 137, 4586Mandushev, G., Staneva, A., & Spasova, N. 1991, A&A, 252, 94McGaugh, S. S. 2004, ApJ, 609, 652McGaugh, S. S., Schombert, J. M., Bothun, G. D., & de Blok, W. J. G. 2000,ApJ, L99Milgrom, M. 1983, ApJ, 270, 365Milgrom, M. 1986, ApJ, 302, 61Olszewski, E. W., Pryor, C., & Shommer, R. B., ASP Conf. Ser., 48, 99Pryor, C., & Meylan, G. 1993, in Structure and Dynamics of Globular Clusters,ASP Conf., 50, 357Sanders, R. H. 1999, ApJ, 512, L23Sollima, A., & Nipoti, C. 2009, MNRAS, 401, 131Soong, T. T. 2004, Fundamentals of Probability and Statistics for Engineers(Publisher: John Wiley & Sons)Tully, R. B. & Fisher, J. R. 1977, A&A, 54, 661Page 4 of 4

Current data on the globular cluster Palomar 14 are consistent with MOND

Context. Certain types of globular clusters have the very important property that the 
predictions for their kinematics in the Newtonian and modified Newtonian dynamics (MOND) contexts are 
divergent.
        Aims. Here, we advice regarding the recent claim
that the stellar kinematics data (using 17 stars) of the globular cluster Palomar 14 are inconsistent with MOND.
        Methods. We compare the observations to the theoretical predictions using a Kolmogorov-Smirnov
test, which is appropriate for small samples. 
        Results. We find that with the currently available data the MOND prediction for the velocity distribution can only be
excluded with a very low confidence level, clearly
insufficient to claim that MOND is falsified

EDP Sciences OAI-PMH repository (1.2.0)

Current data on the globular cluster Palomar 14  are consistent with MOND

HAL: Hyper Article en Ligne

Context: Certain types of globular clusters have the very important property that the predictions for their kinematics in the Newtonian and modified Newtonian dynamics (MOND) contexts are divergent. Aims: Here, we advice regarding the recent claim that the stellar kinematics data (using 17 stars) of the globular cluster Palomar 14 are inconsistent with MOND. Methods: We compare the observations to the theoretical predictions using a Kolmogorov-Smirnov test, which is appropriate for small samples. Results: We find that with the currently available data the MOND prediction for the velocity distribution can only be excluded with a very low confidence level, clearly insufficient to claim that MOND is falsified. © 2010 ESO.SCOPUS: ar.jinfo:eu-repo/semantics/publishe

Famaey, Benoît

Angus, Garry

DI-fusion

Current data on the globular cluster Palomar 14 are consistent with MOND (Research Note)