Distant star clusters of the Milky Way in MOND

0.2 MB·

by  Hosein Haghi

Astronomy & Astrophysics manuscript no.
(will be inserted by hand later)
Distant star clusters of the Milky Way in MOND
Hossein Haghi1,2,,⋆ Holger Baumgardt2,3, ⋆⋆ and Pavel Kroupa2 ⋆⋆⋆
1Institutefor AdvancedStudies in Basic Sciences (IASBS),P.O. Box 11365-9161, Zanjan, Iran
1 2Argelander Institutefor Astronomy (AIfA),Aufdem Hu¨gel 71, D-53121 Bonn, Germany
1 3School of Mathematics and Physics, University of Queensland, Brisbane, QLD 4072, Australia
0
2

n
Accepted
a
J
Abstract. WedeterminethemeanvelocitydispersionofsixGalacticouterhaloglobularclusters,AM1,Eridanus,
2
Pal3,Pal4,Pal15,andArp2intheweakacceleration regimetotestclassical vs.modifiedNewtoniandynamics
1
(MOND).Owing tothenonlinearity of MOND’sPoisson equation,beyondtidal effects,theinternaldynamicsof

] clustersisaffectedbytheexternalfieldinwhichtheyareimmersed.Forthestudiedclusters,particleaccelerations
A aremuchlowerthanthecriticalaccelerationa0ofMOND,butthemotionofstarsisneitherdominatedbyinternal
G accelerations (ai ≫ae) nor external accelerations (ae ≫ai). We use the N-body code N-MODY in our analysis,
which is a particle-mesh-based code with a numerical MOND potential solver developed by Ciotti, Londrillo,
.
h and Nipoti (2006) to derivetheline-of-sight velocity dispersion byadding theexternalfield effect. Weshow that
p Newtoniandynamicspredictsalow-velocitydispersionforeachcluster,whileinmodifiedNewtoniandynamicsthe
- velocitydispersion ismuchhigher.Wecalculatetheminimumnumberofmeasuredstarsnecessary todistinguish
o
r between Newtonian gravity and MOND with theKolmogorov-Smirnov test. Wealso show that for most clusters
t itisnecessarytomeasurethevelocitiesofbetween30to80starstodistinguishbetweenbothcases.Thereforethe
s
a observational measurementoftheline-of-sight velocitydispersion oftheseclusterswill provideatest forMOND.
[

Key words.galaxies: clusters: general- galaxies: dwarf - gravitation - methods: N-body simulations
2
v
2
5 1. Introduction spatial distribution of the subhalos (Klypin et al. 1999,
9 Moore et al. 1999, Metz et al. 2008, Kroupa et al. 2010).
1 Observable matter in galaxies and in clusters of galax-
Even the ability of the dark matter theory to account for
1. ies cannot produce sufficient gravity to explain their dy- the Tully-Fisher and Freeman relations is controversial
0 namics. Cold dark matter (CDM) scenarios or alterna-
(Bosch & Dalcanton 2000, Governato et al. 2010). These
1 tive theories of gravitation are therefore invoked to re-
shortcomings have not led to the rejection of the theory
1
solve the problem. Nowadays,the CDM hypothesis is the
: only because on galactic scales baryons are at least non-
v dominant paradigm. This hypothetical matter does not
negligible contributors to the mass density, consequently
i interact with electromagnetic radiation and only shows
X simulations that include the complex physics of star for-
its presence through its gravitational interaction. Even
r mation are essential for reliable predictions. Currently
a though the dark matter hypothesis has successfully ex- such simulations are still at an experimental stage and
plained the internal dynamics of galaxy clusters, grav-
areusuallysubstitutedby“semi-analytic”argumentsthat
itational lensing, and the standard model of cosmol-
have weak theoretical underpinnings.
ogy within the framework of general relativity (GR)
(Spergel 2003), much experimental effort has failed to One of the alternative theories to CDM is the so-
yield a detection of dark matter particles. Moreover the called modified Newtonian dynamics (MOND) theory,
results of high-resolution simulations of structure forma- which was originally proposed by Milgrom (1983) to ex-
tiondonotreproducesomeobservationsongalacticscales, plain the flat rotation curves of spiral galaxies at large
suchasthecentralstructuresofrotationcurves,thepreva- distances by a modification of Newton’s second law of
lence of low bulge-to-disc ratios, and the numbers and acceleration below a characteristic scale of a0 1.2
10−10ms−2 = 3.6 pc/(Myr)2 without invoking da≃rk ma×t-
Send offprint requests to: H. Haghi ter (Bekenstein & Milgrom 1984). In MOND, Newton’s
⋆ e-mail: [email protected] second law is modified to µ(a/a0)a=aN + H, where
∇×
⋆⋆ e-mail: [email protected] ρ is the mass-density distribution, aN is the Newtonian
⋆⋆⋆ e-mail: [email protected] acceleration vector, a is the MONDian acceleration vec-
2 Haghi et al.: Globular clusters in MOND
tor, a = a is the absolute value of MONDian accel- The mean velocity dispersion of stellar systems for
| |
eration, µ is an interpolating function for the transition the two extreme cases of internal (a a ) or external
i e
≫
from the Newtonian to the MONDian regime,which runs (a a ) field domination have been derived analyti-
e i
≫
smoothly from µ(x) = x at x 1 to µ(x) = 1 at cally by Milgrom (Milgrom 1986, Milgrom 1994), assum-
≪
x 1 (Bekenstein & Milgrom 1984). Different interpo- ing that the systems are everywhere in the deep-MOND
≫
lating functions have been suggested, such as the simple regime (ae,ai a0). Many systems that can be used to
≪
function, µ(x) = x/(1+x) (Famaey & Binney 2005) and test MOND are not completely either internally or exter-
the standard interpolation function, µ(x) = x/√1+x2 nallydominated.Globularclustersordwarfgalaxiesofthe
(Milgrom 1983). Because the simple function fits galac- MilkyWayforexamplehaveinternalandexternalacceler-
tic rotation curves better than the standard function ationsthatareofthesameorder(Baumgardt et al. 2005),
(Gentile, Famaey & de Blok 2010), in this paper we use consequently one has to determine the velocity dispersion
the simple function. The value of the curl field H de- numerically for intermediate cases. Sollima and Nipoti
pends on the boundary conditions and the spatial mass (2009) constructed self-consistent, spherical models for
distribution and vanishes only for some special symme- stellar systems in MOND, neglecting the external field
tries (Bekenstein & Milgrom 1984). The non-linearity of effect and presented a dynamical model for six galactic
theMONDfieldequationleadstodifficultiesforstandard globular clusters. The presence of the external field ef-
N-body codes and makes the use of the usual Newtonian fect breaks the spherical symmetry and validity of their
N-bodysimulationcodesimpossibleintheMONDregime. model. Recently Haghiet al. (2009,hereafter HBK09)in-
It has been shown that on galac- vestigated the dynamics of star clusters by numerically
tic scales MOND can explain many phe- modeling them in MOND, assuming circular orbits.They
nomena better than CDM (Begmann 1989, performed N-body simulations and presented analytical
Begmann et al. 1991, Sanders & McGaugh 2002, formulae for the velocity dispersion of stellar systems in
Sanchez-Salcedo & Hernandez 2007, Haghi et al. 2006, the intermediate MOND regime, which are useful for a
Malekjani et al. 2009, Gentile et al. 2007, Milgrom 1994, comparison with observational data of several GCs and
Brada & Milgrom 2000, Milgrom 1995, Wu et al. 2008, dSph galaxies (for details on the numerical calculations
Zhao & Famaey 2006, Tiret & Combes 2007, see HBK09). In a follow-up paper, Jordi et al. (2009)
Hasani Zonoozi & Haghi 2010). MOND has been gener- determined the velocity dispersion (using 17 stars) and
alized to a general-relativistic version (Bekenstein 2004, mass-function slope of Pal 14 and showed that MOND
Sanders 2005, Zlosnik, Ferreira & Starkman 2007, can hardly explain the low-velocity dispersion of this sys-
Milgrom 2009), making it possible to test its predic- tem. However, Gentile et al. (2010) showed that with
tions for gravitational lensing. Dynamics of galaxies in the currently available data, the Kolmogorov-Smirnov
clusters (Sanders & McGaugh 2002) and the merging (KS) test is still unable to exclude MOND with a suffi-
of galaxy clusters, where the baryonic mass is clearly ciently high confidence level. Moreover, the low density
separatedfromthe gravitationalmass(Clowe et al. 2006) of Pal 14 suggests that binary stars may be an impor-
cannot be completely explained by MOND without tant issue for interpreting its measured velocity disper-
invoking some kind of hot dark matter, perhaps in the sion (Ku¨pper & Kroupa 2010), and the true velocity dis-
form of massive (active or sterile, 2 to 11 eV) neutrinos persion of Pal 14 could be much lower than the value re-
(Angus et al. 2006, Angus et al. 2010). ported by Jordi et al. (2009), thereby possibly posing an
In orderto decide whether MOND is a comprehensive even larger challenge for MOND, but also for Newtonian
theorytoexplainthe dynamicsofthe universe,itis desir- gravityandforanyunderstandingofthe dynamicsofthis
able to study MOND for objects in which no dark matter object as being in equilibrium.
is supposed to exist and where the characteristic acceler- In this paper we calculate the prediction of MOND
ation of the stars is less than the MOND critical acceler- andNewtonian dynamics onthe velocitydispersionofsix
ationparametera0. Globularclusters (GCs) area perfect other distant clusters of the MW (Table 1). In order to
candidate since they are the largest virialized structure see the pure MONDian effects, we concentrate on sys-
that do not contain dark matter (Moore 1996). tems in which the tidal radius is much larger than the
InthedistanthaloofourMilkyWaythereexistseveral gravitationalradius1 andthereforetidaleffectsareunim-
low-mass GCs where both internal and external accelera- portant. In other words, this paper provides the basis for
tions of stars are significantly below the critical accelera- further observationalefforts. The measurements of a low-
tion parameter a0 of MOND. Because GCs are assumed (Newtonian) velocity dispersionwould meanthat MOND
to be dark-matter-free, if MOND is true, the motions of in its present form is in severe trouble and that globular
stars must deviate from the standard Newtonian dynam-
ics.Ithas beenproposedby Baumgardt(2005)thatsome 1 The gravitational radius is a measure of the size of the
ofthesedistantGalacticGCsareperfecttoolstotestgrav-
systemandisrelatedtothemassandpotentialenergyasgiven
itationaltheoriesintheregimeofveryweakaccelerations.
in Eq. 2-132 in Binney and Tremaine (1987). In many stellar
For MOND, the internal velocity dispersion among the systems the gravitational radius can be approximated by the
stars in these clusters would be significantly higher than three-dimensional half-mass radius rh as rg = 1.25rh if the
in Newtonian dynamics. assumption of virial equilibrium is valid.
Haghi et al.: Globular clusters in MOND 3
clusters do not possess dark matter. In contrast, a high- SeveralattemptshavebeenmadeconsideringtheEFE.
velocity dispersion would either favor MOND or could be Wu et al. (2007) simulated an isolated object with a
a hint to the existence of dark matter in globularclusters staticMONDianpotentialsolver,butchangedthebound-
(Baumgardt et al. 2009). ary condition on the outermost grid point to be nonzero.
The paperisorganizedasfollows:InSection2wegive Famaey et al. (2007) estimated the escape velocity of
abriefreviewoftheexternalfieldeffect(EFE)inMOND. galaxies in MOND and assumed µ(ae+ai /a0)ai = aN.
| |
The simulation setup is explained in Section 3. The nu- Asafirstordertesttheyreplaced ae+ai with(ae+ai),or
merical results for six clusters are discussed in Section 4. pa2+a2. This is an approximati|on,sin|ce they neglected
i e
We present our conclusions in Section 5. the direction of ae (i.e. a possible angular difference be-
tween ai and ae).
Inordertostudythe evolutionofthesystemsweneed
2. External field effect in MOND
to calculate the acceleration at each step by adding the
In classicalNewtonian dynamics, a uniform external field constant external gravitationalfield to the internal accel-
does not affect the internal dynamics of a stellar system. erationinsidetheµ-function,consideringthedifferentan-
Inotherwords,intheframeofthesystemrelativemotions glebetweenexternalandinternalaccelerationforallstars
of objects are the same as in an isolated system. throughout the evolution.
In MOND, the situation is entirely different.
Owing to the non-linearity of Poisson’s equation, 3. Simulation setup
the strong equivalence principle (SEP) is violated
(Bekenstein & Milgrom 1984), and consequently the in- We performed a large set of N-body simulations of star
ternal properties and the morphology of a stellar sys- clusters with the N-MODY code, which has been devel-
tem are affected by both the internal and external field. opedbyCiottietal.(2006).N-MODYisaparallel,three-
This so-called external field effect (EFE) significantly af- dimensionalparticle-meshcodeforthetime-integrationof
fectsnon-isolatedsystemsandcanprovideastricttestfor collision-less N-body systems (Londrillo & Nipoti 2009).
MOND. The code numerically solves the non-linear MOND field
The EFE is indeed a phenomenological requirement equations,whichcanbeusedtoperformnumericalexper-
of MOND and was postulated by Milgrom (1983) to ex- iments in either MONDian or Newtonian dynamics. The
plain the dynamical properties of open clusters in the potential solver of N-MODY is based on a grid in spher-
MW that do not show MONDian effects. The EFE al- ical coordinates and is best suited for modeling isolated
lows high-velocity stars to escape from the potential of systems. N-MODY uses the leap-frog method to advance
the Milky Way (Famaey et al. 2007, Wu et al. 2007) and the particles.Thecodeandthe potentialsolverhavebeen
implies that rotation curves of spiral galaxies should fall presented and tested by Ciotti et al. (2006)and Nipoti et
where the internal acceleration becomes equal to the ex- al. (2007).
ternal acceleration (Gentile et al. 2007, Wu et al. 2008). In the present study we used a spherical grid (r,θ,ϕ)
is emFobreaddsetadricnluastheorswtigtahlaaxdyenwsiitthyadidsternibsiuttyiodnisρtrci,bwuhtiiocnh mintaedgeraotfioNnr.×TNheθ×toNtaϕl =nu1m2b8e×r1o2f8p×ar1t2ic8legsridiscNelpls=for1t0h5e.
ρ , the acceleration of stars in the cluster satisfies the The detailsofthe scalingofthe numericalMONDmodels
ext
modified Poisson equation andcode units arediscussedin Nipoti, Londrillo& Ciotti
(2007).
Φ
Our treatment of the external field effect is based on
.[µ(∇ ) Φ]=4πG(ρ +ρ ), (1)
c ext
∇ a0 ∇ the idea that the star clusters we study are much smaller
thantheyaredistantfromtheMilkyWay,sotheexternal
where Φ is the MONDian potential generated by the
∇ field is nearly constant over the cluster area. We carry
total matter density. For star clusters or dwarf galaxies
out the simulations in a rotatingreference frame centered
far out in the halo of the Milky Way the local density
on the cluster. We also assume that the clusters are on
of the Milky Way is negligible ( i.e. ρ << ρ ). One
ext c
a circular orbit, so the external field of the Milky Way is
way to solve Eq. 1 is then to assume that the total ac-
also constant with time in the clusters frame.
celeration is the sum of the internal a and the external
i
We assume that tidal forces arising from a gradient
a acceleration, which both satisfy the modified Poisson
e
of the external field and the Coriolis acceleration caused
equationas(Bekenstein & Milgrom 1984,Wu et al. 2007,
by the rotating reference frame can be neglected. This is
Wu et al. 2008, Angus 2008)
justified as long as the size of the cluster is much smaller
ae+ai than the tidal radius. We then solve Eq. 2 numerically
∇.[µ(| a0 |)(ae+ai)]=4πGρc. (2) with N-MODY, using a constant acceleration ae as the
boundary condition.
In Eq. 2, the direction of the external acceleration is In order to include the EFE for non-isolated systems,
importantforthedynamicsofstarsintheclusters.Forex- we applied several changes to the N-MODY code. The
ample, for two stars the accelerationwould be different if changes were encoded in the source file mond-lib.f90,
a isparalleloranti-paralleltotheirinternalacceleration. which contains all relevant routines implementing the
e
4 Haghi et al.: Globular clusters in MOND
Cluster name Rh[pc] RG[kpc] ae[a0] Mc[103M⊙] σM[kms−1] σN[kms−1] Nmin
AM 1 24 123.2 0.086 12.6 1.50 0.56 25
Eridanus 14.2 95.2 0.113 18.6 1.80 0.90 40
Pal 3 24 95.9 0.112 31.6 1.97 0.87 30
Pal 4 23.2 111.8 0.096 42.6 2.30 1.10 35
Pal 15 21.2 37.9 0.283 26.3 1.42 0.88 80
Arp 2 21.5 21.4 0.504 21.8 1.07 0.80 150
Table 1. Globular clusters modeled in this paper. The half-mass radius and galactocentric distances, R , are taken
G
from Harris (1996). The values of the external acceleration are calculated from ae = √GMa0/RG with M = 1.2
1011M⊙ for all galactocentric distances. Cluster masses, Mc, were calculated from the absolute V-band luminositie×s
by assuming a stellar mass-to-light ratio of M/L = 2, which is close to the measured average mass-to-light ratio of
V
galactic globular clusters (Mieske et al. 2008). σ and σ are the corresponding MONDian and Newtonian values of
M N
the velocity dispersion, respectively. The last column is the minimum number of stars necessary to obtain P 0.05.
≤
MOND potential solver. The Mondian potentials and ac- for several crossing times to reach the equilibrium state.
celerations are assigned to stars in subroutines: ”gmond” TheEFEonthepredictedline-of-sightvelocitydispersion
and”mond-acc”,andweplacedthe vectorofa inthe in- for non-isolated stellar systems with different internal ac-
e
terpolatingfunctionforallstars,i.e.,a isaddedtoa via celerations have been carried out in HBK09.
e i
vector summation and the code solves Eq. 2 by iteration. For the GCs listed in Table 1, observational efforts
At each step of potential solving, we added the constant are underway to determine their velocity dispersion and
external field for all grid points according to Eq. 2. This to constrain their mass (Grebel et al. 2009). Because the
is a first approximation that allows us to focus on the ef- half-mass relaxation times of most clusters in Table 1 are
fects of a constant external field, which in turn allows us on the order of a Hubble time or even larger, dynamical
to deal with the intermediate MOND regime. We would evolution does not play an important role for these clus-
like to stress that our numerical solution agrees very well ters, therefore we can assume that the half-light radius
with analytic ones for the extreme cases ai ae a0, is equal to the half-mass radius. The projected half-mass
≪ ≪
ae ≪ai ≪a0 and ae,ai ≫a0 (HBK09). radii, Rhp, and galactocentric distances, RG, are taken
Our method is a first attempt to take into account from Harris (1996). The projected half-mass radii relate
the constant EFE in N-MODY, in order to avoid solving to the three-dimensional half-mass radii as R = γR
hp h
the modifiedPoissonequationforthelargeareaincluding with γ 0.74. For all clusters in Table 1, the tidal radii
≈
bothgalaxyandcluster,whichisimpossiblewiththecur- are larger than the half-mass radii by a factor of 5 to 10,
rentversionofN-MODY.Thiscanonlybe studiedwitha which means that tidal effects play no significant role for
realistic, high-resolution MOND simulation that includes the internal dynamics of the clusters in our sample. We
the density distribution causing the external field. perform numerical modeling to obtain the mean velocity
In this work we start from Newtonian equilibrium dispersion.
Plummer models and produce MONDian equilibrium ini- Because the cluster masses are not known from ob-
tial systems. In order to have a MONDian equilibrium servations, it is useful to calculate line-of-sight velocity
initial system,we increasedthe initial velocity of the par- dispersions for different values of cluster mass. If the ve-
ticles. This method is useful when the external field is locity dispersion is determined observationally, one can
important. The details of the method are described in constrain the cluster mass. In Fig. 1 the resulting global
HBK09. line-of-sight velocity dispersion as a function of mass is
plottedforthesixstellarclustersandiscomparedwiththe
Newtonian results. The black line shows the Newtonian
4. Modeling the distant globular clusters
prediction for the velocity dispersion, which we calcu-
We calculated the velocity dispersion for the Galactic late using σLOS,N = 0.36pGM/Rh (Haghi et al. 2009).
globularclustersmentioned inTable 1.These clusters are The red line shows the numerical calculation of the ve-
best suited for testing MOND. A large set of dissipation- locity dispersion in the MONDian regime. For each clus-
less N-MODY computations with MONDian equilibrium ter the external acceleration of the Galaxy is given in
initial conditions were performed for stellar systems that Table 1. All clusters are in the intermediate regime, for
are embedded in the outer MW halo and affected by dif- which there is no analytical prediction in MOND. In the
ferent values of the external field. Clusters in Table 1 are low-acceleration region, the numerical solutions show a
generally far out in the Galactic halo so that the external considerable relative difference with Newtonian results.
acceleration caused by the MW is small. In order to pro- As the mass increases, the internal acceleration grows
duce different internal acceleration regimes, we changed and then gets close to the Newtonian results (at about
the cluster mass from 500M⊙ to 107M⊙ and assume the M = 106M⊙). The vertical dashed lines show the ex-
half-mass radius to be constant. The models are evolved pected mass for each cluster calculated from the absolute
Haghi et al.: Globular clusters in MOND 5
AM1 Arp2 Eridanus
3 3.5 4 4.5 5 3 3 3.5 4 4.5 5 3 4 3 3.5 4 4.5 5 4
-1ms] 3 Mondiancase 3 -1ms]2.5 2.5 -1ms]3.5 3.5
dispersion[k2.52 Newtoniancase 22.5 dispersion[k 2 2 dispersion[k2.35 23.5
velocity1.5 1.5 velocity1.5 1.5 velocity1.25 12.5
line-of-sight0.51 01.5 line-of-sight0.51 01.5 line-of-sight0.15 01.5
0 3 3.5 4 4.5 5 0 0 3 3.5 4 4.5 5 0 0 3 3.5 4 4.5 5 0
log(Mass)[sollarmass] log(Mass)[solarmass] log(Mass)[sollarmass]
Pal3 Pal4 Pal15
3.5 3 3.5 4 4.5 5 3.5 3.5 3 3.5 4 4.5 5 3.5 3.5 3 3.5 4 4.5 5 3.5
-1ms] 3 3 -1ms] 3 3 -1ms] 3 3
dispersion[k2.25 22.5 dispersion[k2.25 22.5 dispersion[k2.25 22.5
velocity1.5 1.5 velocity1.5 1.5 velocity1.5 1.5
line-of-sight0.15 01.5 line-of-sight0.15 01.5 line-of-sight0.15 01.5
0 3 3.5 4 4.5 5 0 0 3 3.5 4 4.5 5 0 0 3 3.5 4 4.5 5 0
log(Mass)[sollarmass] log(Mass)[sollarmass] log(Mass)[sollarmass]
Fig.1. Line-of-sight velocity dispersion for clusters listed in Table 1 for various masses as found by N-MODY. In
order to compare with the real cluster, the half-mass radii of all models are fixed. For low masses, which mean low
internalaccelerations,thepredictionofNewtoniandynamicsdiffersfromthenumericalsolution.Asthemassincreases
(at about 106M⊙), the internal acceleration grows and the system enters the Newtonian regime and the numerical
solutions get close to the Newtonian prediction. In order to focus on a reasonable range of mass for these clusters,
we show the results in the range of 500 2 105. Dashed vertical lines show the expected cluster mass, calculated
− ×
from the absolute luminosity by assuming a stellar mass-to-lightratio of M/L =2. This value agrees with observed
V
mass-to-light ratios of galactic globular clusters (Mieske et al. 2008)
luminosities by assuming a stellar mass-to-light ratio of and Nipoti (2009)). This is higher than the value of 1.5
M/L=2 (Mieske et al. 2008). It should be noted that in kms−1 we find, which is to be expected because they did
the simulations we do not assume any M/L, but we vary not include the EFE. If we take the EFE into account,
themassandthereforetheM/Lratiooftheclusters.The the systems tend to be in the quasi-Newotonian regime,
velocity dispersions corresponding to this mass for both where the gravity is weaker than in the deep-Mondian
casesareindicated in Table 1.The M/Lratio of2 is only regime (HBK09).
usedinTable1toillustratethereisaMONDeffect,which In MOND the external field introduces an anisotropy
makesadifferenceforthestudiedclusters.Ourpredictions so that clusters could in principle become elongated. In
fortheMONDiancasearelowerthanthosebyBaumgardt order to test for a possible elongation of the velocity dis-
et al.(2005).The absolute differences between MONDian persioninthedirectionoftheexternalfield(σ ),wedefine
x
and Newtonian results are largest for the clusters Pal 4, the parameter α=1 σ2/(σ2+σ2). Figure 2 shows that
− x y z
Pal3,AM1,andEridanus.Thereforetheseclusterswould the line-of-sightvelocity dispersionin the direction of the
be thebestcasesto testMOND.Thesamequalitativere- externalfielddiffersonlyrandomlycomparedtothatper-
sults were also obtained by Sollima and Nipoti (2009), pendicular to the external field. This small a difference
who have studied some of the clusters of our list (i.e., would be unobservable.
Eridanus, AM1, Pal 3, Pal 4, and Pal 14, which we have In order to show the situation of the isotropy in the
studied in HBK09). They have obtained the cluster’s ve- velocity dispersion, the anisotropy profile, β(r) = 1
locity dispersionprofile.Their predictions forthe velocity (σ2/2σ2) for one cluster is plotted in Fig. 2, where, σ−
t r t
dispersionaregenerallyhigherthanours.ForAM1forex- is the tangential velocity dispersion and σ is the radial
r
ample, the mean value of the velocity dispersion is 1.7 velocity dispersion.
kms−1 sincethepredictedMONDvelocitydispersion∼pro- Figure 3 shows the distribution of radial velocities of
file is flat throughout the cluster area (Fig. 5 of Sollima the N = 105 stars in the velocity range -10 to +10
total
6 Haghi et al.: Globular clusters in MOND
1
Mondmodel
0.9 0.1 Newtonianmodel
α=1-σx2/(σy2+σz2) Gaussianfit
0.8
0.08
0.7
s
0.6 star0.06
f
o
α 0.5 n
o
0.4 cti0.04
a
r
F
0.3
0.02
0.2
0.1 0
0
25 50 75 100 -10 -5 0 5 10
r[pc]
0.02
Ntotol0.01
1
/
0.75 β=1-σ2/2σ2 e)wtonian 0
t r N
N
0.5 -Mond-0.01
N
(
0.25
-10 -5 0 5 10
β 0 Velocity(km/sec)
Fig.3. Upper panel:Radialvelocity distribution ofstars
-0.25
for a Newtonian (red triangles) and MONDian (black
-0.5 dots) system in which both have the same internal ve-
locity dispersion. The smoothed line shows a Gaussian
-0.75 fitted to the distributions. Both distributions follow the
Gaussianfit veryclosely.Lowerpanel:The relativediffer-
-1
25 50 75100 ence between Newtonian and Mondian velocity distribu-
r[pc] tions.Thereisadifferencebetweenthetwomodels,butit
is toosmall(less than1%)to be detected observationally.
Fig.2. Top panel: The ratio of the velocity dispersionin
thedirectionoftheexternalfieldcomparedtotheperpen-
dicular direction for Pal 3 as an example plotted against by this Gaussian. The Newtonian one is slightly different
theradius.Thesystemisisotropicthroughoutthecluster. fromthe Mondiandistribution,butthe relativedifference
Bottompanel:Theanisotropyprofile,β,forPal3asanex- is very small (less than 1%), therefore it is impossible to
ample plotted against the radius. The system is isotropic find a detectable deviation observationally.
throughout the cluster.
5. Distinguishing between MOND and Newtonian
gravity models
km/sec for both Newtonian (red triangles) and Mondian Usually, it is not feasible to measure the velocity disper-
(black dots) cases together with the best Gaussian fit to sion of a large sample of stars in distant and low-mass
this distribution using the least square method. We cre- globularclusters.Ifonlyafewstarsaremeasured,theob-
ated the Newtonian model with a mass of 177000M⊙ and served velocity dispersion has a large error, which makes
theMondianonewithamassof32000M⊙,sothatthe ve- it difficult to distinguish between MOND and Newtonian
locity dispersions are the same. The bottom panel shows dynamics.Recently,Gentile etal.(2010)haveshownthat
therelativedeviation,(N N )/N ,where the small sample size for Pal14 (i.e., radialvelocity of 17
Newtonian Mond total
−
N is the number of stars in a velocity bin. Evidently the starswereusedtodeterminetheobservedvelocitydisper-
observedradialvelocity distribution is well approximated sion) can only reject MOND with a low confidence level.
Haghi et al.: Globular clusters in MOND 7
In order to see how many stars are necessary to reject It should be noted that the clusters could be on ec-
MOND with high statistical confidence if the underlying centric orbits, which means that the MOND predictions
velocitydispersionisNewtonian,wecreatedartificialdata wouldbe differentbecauseofthe variationofthe external
sets of increasing sample size from the Newtonian simu- field (in direction and amplitude) along the orbit.
lations for each of the six clusters. For each sample size Using a KS test, we calculated the minimum number
we produced 20 random realizations of a mock sample, of stars that are sufficient to exclude MOND (under the
and each time applied a KS test with the null hypothesis hypothesis that these globular clusters are on circular or-
being that MOND is correct (thus comparing the mock bits) at the 95% confidence level. We found that between
data set from the Newtonian simulation with the veloc- 30 to 80 stars are necessary for most clusters to distin-
ity dispersion of the MONDian simulation, σ ). We guish between both cases. This number of stars can be
MOND
approximate the cluster mass from the V-band luminos- observedwith current8mclass telescopes.Additional ob-
ity by assuming a stellar mass-to-light ratio of M/L = 2 servationaleffortsto determine the velocitydispersionsof
(vertical dashed line in Fig. 1). Figure 4 shows the mean these clusters and constraining the mass of the clusters
P-value from the KS test as a function of the number of by star counts would be highly important and provide a
stars in the sample. The number of stars changes in the strict test of MOND. On the other hand, if MOND is
range of [10-160]. For any given cluster and a fixed num- the correct theory, these observations could be used to
berofstarsa1-sigmarangeofP-valuesareshownaserror constrain the external field and consequently to put con-
bars. The minimum number of stars, N , which is nec- straints on the potential in which the systems are embed-
min
essary to achieve P < 0.05 (i.e. exclusion confidence of ded. According to the anisotropy profile, the simulated
95%),areobtainedforeachclusterandaregiveninTable systems are isotropic throughout.
1.MeasurementofN starsshouldthereforebeenough
min
toruleoutMONDiftheclustermassissimultaneouslyde- Acknowledgements. WewouldliketothankC.Nipotiforpro-
terminedbyindependentmeasurements,possiblythrough vidinguswiththeN-MODYcodeandhishelpinusingit.H.H
thanks the stellar dynamics group of the Argelander Institute
e.g. star counts.
for Astronomy for giving financial support for this research.
H.B. acknowledges support from the German Science founda-
tionthroughaHeisenbergFellowship andfrom theAustralian
6. Conclusions
ResearchCouncilthroughFutureFellowshipgrantFT0991052.
Wecomparedgloballine-of-sightvelocitydispersionsofsix
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0.5 0.8 0.5
0.7
0.4 0.4
0.6
0.3 0.5 0.3
P-value0.2 P-value00..34 P-value0.2
0.1 0.2 0.1
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Numberofstarsinthesample Numberofstartsinthesample Numberofstarsinthesample
Pal3 Pal4 Pal15
0.5 0.5
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0.4 0.4
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0.3 0.3 0.4
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u u u
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